the semiclassical approach to mesoscopic physicsthe semiclassical approach to mesoscopic physics...
TRANSCRIPT
The Semiclassical Approach to Mesoscopic Physics
Philippe Jacquod1, ∗
1Department of Physics, University of Arizona,
1118 E. Fourth Street, Tucson, AZ 85721
(Dated: January 19, 2009)
Abstract
These lecture notes are the result of a class I taught at the EPFL in the framework of the
”troisieme cycle de la physique en Suisse romande” in the fall of 2008. They are largely based on
several already published works, and therefore are not intended to be published, at least not in
their present form.
∗Electronic address: [email protected]
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Contents
I. Preamble : What is Mesoscopic Physics ? 4
A. Background 4
B. Length scales 6
C. What is new in mesoscopic physics ? 9
1. Sample specificity 9
2. Nonlocality 10
3. Transport properties depend on the sample geometry 11
4. Macrosopic symmetries appear to be violated 13
D. The phase coherence length 14
E. Alternative to semiclassics: random matrix theory 18
II. Old Semiclassical Methods 21
A. The WKB method 22
1. Method of stationary phases 25
2. WKB quantization 25
B. Failure of eikonal methods for chaotic systems 26
1. Few words on chaotic vs. integrable systems 27
2. The eikonal method and its failure 29
III. From Path Integrals to the Semiclassical Green’s Function 36
A. The Green’s function 36
B. The Feynman path integral 37
C. The Van Vleck–Gutzwiller propagator and the semiclassical Green’s function 39
1. Generalization of the stationary phase method 39
2. Semiclassical approximation to the path integral propagator 39
3. The semiclassical Green’s function 40
D. The trace formula 42
E. Beyond the trace formula: two-point correlations 43
1. Time scales 43
2. The diagonal approximation 45
3. Semiclassical sum rules 46
2
IV. Persistent Currents 49
A. Generalities 50
B. Persistent currents in a clean one-dimensional ring 51
C. Even-odd effect, finite temperature, disorder and all that 52
D. Average and typical current in the canonical ensemble 54
E. Semiclassical approach to persistent currents in normal metallic rings 55
V. Semiclassical Theory of Transport: Ray Optics for the XXIst Century 58
A. Diagonal approximation: the Drude conductance 58
B. Beyond the diagonal approximation I : weak localization for transmission 61
C. Beyond the diagonal approximation II : quantum corrections to reflection 67
1. Weak localization 68
2. Coherent backscattering 68
3. The off-diagonal nature of coherent backscattering 69
D. Magnetoconductance 70
E. Spin-orbit interaction and weak antilocalization 72
F. Universal conductance fluctuations 75
G. Shot-noise 78
VI. Conclusion: Regime of Applicability of These Semiclassics 83
Acknowledgments 84
References 84
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I. PREAMBLE : WHAT IS MESOSCOPIC PHYSICS ?
A. Background
The term “mesoscopic physics” was first coined by the Dutch physicist Nico van Kampen
[1]. It obviously broadly refers to systems which are somewhere in between macroscopic
(i.e. classical) and microscopic (i.e. quantal, atomic or subatomic) systems. Accordingly,
the realm of mesoscopic physics is defined by systems which are coherent, i.e. whose (dy-
namical, transport or equilibrium and so forth) properties derive from and can be described
by quantum mechanics. Thus, the physics of those systems is in principle included in the
Schrodinger equation. However, those systems are big and complex enough that any at-
tempt to solve exactly the latter equation is doomed to fail. One therefore has to rely
on approximate and/or statistical methods. In particular, because mesoscopic sizes exceed
atomic sizes by far, the spatial extension of the particles involved (their Fermi or de Broglie
wavelength λF ) is by far the smallest length scale in the system. It makes then perfect sense
to use semiclassical methods to get approximate solutions to the Schrodinger equation in a
controlled expansion in λF/ζ , where ζ is a (macroscopic) classical length scale such as the
elastic mean free path ℓ, or the system size L. The theory of weak localization is one of the
most spectacular illustration of the power of such methods.
Weak localization [2] is arguably the place where mesoscopic physics was born [3]. The
quasiclassical, Drude-Boltzmann theory of electronic transport in macroscopic metals al-
lows to calculate the spatially-averaged current through a sample in presence of impurities,
phonons, externally applied electromagnetic fields, and so forth. From there, the longitudi-
nal and transversal (Hall) conductivities can be calculated. They turn out to be intensive
quantities, and in particular, they do not depend on the geometry (size and shape) of the
sample. One may however ask oneself how small a sample has to be made in order to exhibit
quantum mechanical corrections to the quasiclassical theory of transport. The latter theory
is indeed based on at least two, non-quantal assumptions : that the current carriers, the
electrons, are point-like objects, and that successive scattering at impurities are uncorre-
lated. The Boltzmann theory thus sets the de Broglie wavelength of the electrons to zero
and neglects quantum mechanical phase interferences. Mesoscopic physics essentially sets
itself the task of dealing with systems which do not satisfy these assumptions. With the
4
theoretical prediction of weak localization, shortly followed by its experimental confirmation,
it became clear that the assumptions of the Drude-Boltzmann theory breaks down even for
macroscopic conductors, at low enough temperature. Moreover, Aharonov-Bohm transport
experiments, and the discovery of reproducible mesoscopic conductance fluctutations made
clear that low-temperature transport on small scales definitely differs from the quasiclassical,
macroscopic transport.
The very existence of mesoscopic systems is not at all trivial. After all, and despite
the fact that most natural systems are deep down quantum mechanical in essence, almost
everything we see and experience in our daily life is classical, i.e. can be explained by
classical mechanics. The manifestations of, say, the Heisenberg uncertainty principle or of
the Pauli exclusion principle are totally absent from our lives (even though our very existence
and the stability of matter as we know it eventually relies on them). Indeed, mesoscopic
physics is a relatively recent branch of solid-state physics. The ability to devise cleaner and
cleaner semiconductor samples (where the elastic mean free path becomes larger and larger,
allowing for a good metallic behavior), as well as more and more precise and sophisticated
constrictions, together with the advent of powerful, subkelvin cryogenic apparatus resulted
in a booming expansion of mesoscopics in the late 80’s and the 90’s. Due to these advances,
mesoscopic physics truly became one of the most important subfields of solid-state physics.
Mesoscopic systems are defined by an ordered sequence of length scales
lϕ ≫ L & ℓ≫ λF , (1.1)
where λF is the Fermi wavelength, ℓ is the elastic mean free path, and L is the system size.
The fourth of these length scales is the coherence length lϕ, which measures how long the
system keeps memory of its phase and remains quantum mechanical. A precise definition
of lϕ will be given shortly. For the time being, it is important to keep in mind that, unlike
the three other length scales, lϕ is not fixed, rather, it depends on physical conditions,
most notably on the temperature. Thus, mesoscopics has to be assimilated to a regime
— mesoscopic samples become classical samples at room temperature. Conversely, room-
temperature classical metallic samples may become mesoscopic at low temperature – though
for very large samples, the temperature below which mesoscopic effects occur turns out to
be inaccessible. Most of this course will be devoted to purely coherent systems, for which
lϕ → ∞ (which requires at least T → 0). We will thus assume that quantum mechanics
5
applies without restriction.
Before we get into the heat of the action, we devote the rest of this introduction first to a
discussion of the four length scales mentioned above, second to a description of mesoscopic
systems via a list of perhaps the most strikingly peculiar behaviors and properties they
exhibit, and which make them fundamentally different from classical samples. We will
finally close this introduction with a brief formal discussion of decoherence to give a better
qualitative idea of the meaning of lϕ.
B. Length scales
In atomic systems, electrons have a spatial extension given by their Bohr radius aB =
~2/me2 ≃ 0.5 × 10−10m = 0.5A, which follows from solving the Schrodinger equation for
the Hydrogen atom. Solid-state physics deals however with electrons in more complicated
structures, such as disordered or clean lattices, where aB loses its relevance. Let us model
a solid-state system by a square d-dimensional box of linear sizes L, where we impose
periodic boundary conditions. The solutions to the time-independent Schrodinger equation
are plane-waves ψα(x) = L−d/2 exp[ikα ·x] with eigenergies ǫα = ~2
2mk2
α. The wavevectors are
determined by the periodic boundary conditions as kα = 2πα/L, with α = (α1, . . . αd) and
αi ∈ [−L,L] are integers (one of them being nonzero at least). We populate this spectrum
with N electrons which, being fermions, must obey the Pauli exclusion principle. It follows
that the Fermi wavelength λF = 2π/kF is given by
λF =
[
2Ωd
n
]1/d
, Ω1 = 2, Ω2 = π, Ω3 = 4π/3, (1.2)
in terms of the density of electrons n = N/Ld. Typical two-dimensional semiconduc-
tor heterostructures have electronic densities n ≈ 1011cm−2 and thus a Fermi wavelength
λF ≈ 40nm. Higher densities are achieved in bulk, ordinary three-dimensional metals,
n & 1022cm−3, leading to a resulting Fermi wavelength which is two orders of magnitude
smaller, λF ≈ 0.5nm.
The second length scale is the elastic mean free path ℓ, which is in principle determined
by the size and density of impurities. The latter may be lattice site vacancies (an atom is
missing) or anomalies (an atom of the expected species is replaced by another kind of atom)
in a material’s crystal structure, a lattice dislocation, doping (where few atoms are replaced
6
by atoms with a different valence in order to change the charge carrier concentration) or
surface roughness as in two-dimensional electron/hole gases. It is important to realize that
ℓ is determined by scattering at static impurities, in particular, the momentum and energy
are conserved by this kind of scattering. From a mathematical point of view, ℓ is defined
as the characteristic length over which the direction of the momentum gets randomized, i.e.
by the decay of the momentum-momentum correlator. Here we will limit ourselves to either
spatially small, but strong impurities, in which case the elastic mean free path is given by
the distance between impurities ℓ = n−1/dimp (nimp is the density of impurities), or to ballistic
systems with ℓ → ∞. We only note that in the weak scattering limit, many impurity
collisions are necessary to significantly bend a trajectory, and that different trajectories
stay together for a longer time for extended impurities. In these two instances, ℓ is thus
larger than the distance between impurities. In standard mesoscopic systems (e.g. the
2DEG formed in GaAs heterostructures), ℓ roughly lies in the range 1−10µm, thus one has
ℓ/λF ≃ 104 ≫ 1. The elastic mean free path is smaller by one or two orders of magnitude
in commercial devices (e.g. computer microprocessors).
The third length scale is the system size itself. The smallest systems we will discuss
are semiconductor quantum dots, or metallic nanograins which lie in the submicronic range,
L ≤ 1µm. Mesoscopic systems can be divided into roughly two classes : diffusive systems
for which ℓ < L and ballistic systems when ℓ & L. Except in our discussion of persistent
currents, we will focus on balllistic systems. Table 1 gives orders of magnitude of typical
length scales.
The fourth length scale is the coherence length or phase relaxation length, defined as
the typical length scale over which the phase-phase correlator of the electronic wavefunc-
tion decays. In an isolated, quantum mechanical system, this correlator does not decay, —
quantum mechanics is by definition a coherent theory. One of the important steps toward
the understanding of mesoscopic physics is actually that static impurities do not lead to de-
coherence. Because they can be incorporated as an additional, time-independent potential
Wimp(r) in the Schrodinger equation, the time-evolution of the wavefunction is still deter-
ministic [4], and in particular, its phase is uniquely defined at any later time, for a given
initial condition.
The situation is however different when the system is in contact with an environment.
The latter is so big and complicated that it is hopeless to treat the full problem exactly.
7
Figure 1: Some relevant length scales and their order of magnitude. Table taken from Ref. [5].
Rather, one attempts to integrate the environment out analytically, within some approxi-
mate scheme, and try to map the total problem (system + environment) onto an effective
(system∗). This procedure usually leads to a loss of memory of the phase of the wavefunction,
i.e. to decoherence, as well as energy and momentum relaxation.
If the coupling between the system and the bath is sufficiently weak, decoherence dom-
inates and relaxation can be neglected altogether. The coupling to the bath then induces
random jumps in the electron’s phase, which add up with time, until the phase has been
completely randomized after a time τϕ. The coherence length corresponds to the average
distance an electron travels during τϕ, and is given by lϕ = vF τϕ in a ballistic system, or
lϕ = (Dτϕ)1/2 in a diffusive system, where the electronic motion is described by a diffusion
equation with diffusion constant D. In a purely classical system, lϕ = 0, while in a strictly
quantum mechanical system, lϕ → ∞. Mesoscopic systems are somewhere in between,
which, in principle, necessitates to look at the problem in a bit more details. This is not our
purpose here, and while semiclassical theories of dephasing exist, we will restrict ourselves
in this manuscript to a short, qualitative discussion of the dephasing length at the end of
this introductory chapter.
8
Let us finally mention a fifth length scale, the localization length ξ, which emerges once
the impurity concentration becomes so strong that quantum mechanical wavefunctions are
spatially trapped inside volumes smaller than the system size as ψ(x) ∝ exp[−|x − x0|/ξ.In the reminder of this manuscript we assume that ξ ≫ L, so that strong localization effects
can be neglected.
C. What is new in mesoscopic physics ?
We will see below that cold and/or small enough samples are characterized by nonlocal
corrections to the conductivity σ – the latter in particular losing its nature of an intensive
quantity, and its relation to the conductance as G = σLxLy/Lz breaking down. This has a
number of consequences, and in this section, we list some of the most striking consequences
of nonlocality of the conductivity tensor [6].
1. Sample specificity
Two “identical” mesoscopic samples, with a similar geometry, made out of the same ma-
terial, and following the same fabrication procedure will in general exhibit different transport
properties. Worse, a given mesoscopic sample typically exhibits fluctuations of its conduc-
tance, when parameters such as the chemical potential or an external magnetic field are
varied. This is due to different microscopic conditions, like different impurity potential con-
figuration in two otherwise similar samples (even though the elastic mean free path is kept
the same). This is illustrated in Fig. 2 which shows the fluctuations of conductance ex-
hibited by one sample as an external magnetic field is varied — corresponding to changing
microscopic conditions. A similar pattern is observed when the electrochemical potential is
varied.
The existence of these conductance fluctuations forces us to deal not only with average
quantities, but also with their variance, and higher moments of their distribution.
9
Figure 2: Mesoscopic conductance fluctuations as a function of an externally applied magnetic
field. The conductance fluctuations are reproducible (i.e. this is not experimental noise) and have
a universal character, in that their variance is ∝ O(e4/h2), independently on the sample size and
its average conductance (note the three orders of magnitude difference in conductance between the
three figures). From left to right: a) magnetoconductance for a gold ring (after Ref. [7]), b) for
a silicon mosfet (after Ref. [8]), and c) numerical results for an Anderson model (after Ref. [9]).
Figure taken from Ref. [10].
2. Nonlocality
Fig. 3 sketches the result of a non-local Aharonov-Bohm experiment. The conductance
through a microstructure is measured via a four-terminal measurement, by which we mean
that the sample is connected to four external leads, and a voltage difference V is applied
on two of those leads, while the current I through the two other leads is measured. The
four-terminal conductance is then defined as the ratio G = I/V .
The data shown on the right-hand side of Fig. 3 differs from those on its left-hand side
by additional periodic conductance oscillations. Indeed, the right side corresponds to the
same sample as on the left side, except a small mesoscopic ring was attached to the side of
the sample outside the main current path! These periodic oscillations arise of course due to
the added ring and the resulting Aharonov-Bohm effect, but the striking feature of Fig. 3 is
that those interferences occur, even though the ring is placed well outside the main current
path.
10
0 1 2 3B[T]
10
11
12
G[e
2 /h]
0 1 2 310
11
12
0 1 2 3B[T]
0 1 2 3
I I I
VVV
Φ
Figure 3: Measurement of non-local Aharonov-Bohm oscillations. The sample with an additional
ring outside the main current path (right) exhibits additional periodic oscillations superimposed
on the aperiodic universal conductance fluctuations. Figure inspired by Ref. [11].
3. Transport properties depend on the sample geometry
The conductivity of macroscopic conductors is an intensive quantity which does not
depend on the geometry of the sample under consideration - a cubic piece of aluminium
has the same conductivity as a sphere, and this holds for both longitudinal and transverse
(i.e. Hall) conductivities. We have already seen, however, that because of nonlocal effects,
the conductivity is no longer intensive in the mesoscopic regime, where both longitudinal
and Hall resistances become geometry-dependent. This has been best illustrated by Ford
et al. [12], whose experimental data are shown on Fig. 4. They showed that the presence
of special trajectories leads to the wrong Hall response, i.e. the Hall resistance of some
samples acquires the wrong sign. This is (of course!) not due to an inversion of the sign of
the Lorentz force, instead, some trajectories undergo reflection at well positioned obstacles
in the sample that reflects them to the ”wrong” Hall probe. These trajectories are depicted
in the left central panel of Fig. 4. Their effect can be removed by placing a central island
on the sample (top right panel) in which case the Hall resistance regains its normal sign.
11
Figure 4: Left: Panel a): Hall resistance for the widened cross (top left, solid line) and normal
cross (top right, dashed line) hall samples. At low magnetic field, the specific geometry of the
widened cross allows for special trajectories (see top left inset on panel a) that reverse the sign of
the Hall resistance. The bottom right inset shows that the sign of the Hall resistance is wrong for
all gate voltages defining the samples (applied on the black regions on the sample’s micrographs).
Panel b) shows the the longitudinal resistance. Right: Hall resistance as a function of magnetic
field for samples with different geometries. The central island in the top left micrograph removes
the electronic orbits giving rise to the sign inversion of the Hall resistance in the widened cross.
Figures taken from [12].
For our purpose in this manuscript, these experimental data present the additional advan-
tage that they obviously motivate to construct a theory of transport that resolves classical
trajectories.
12
Figure 5: Four-terminal magnetoconductance traces for an Aharonov-Bohm loop as sketched on
the right-hand side. The top curve corresponds to G14;23 and is the H-symmetric of the bottom
curve, corresponding to interchanged current and voltage leads, G23;14. Figure taken from Ref. [14].
4. Macrosopic symmetries appear to be violated
Magnetoconductance experiments on mesoscopic systems have revealed an asymmetric
behavior of the conductance under reversal of the magnetic field. The question arose as
to whether this signals a violation of the macroscopic Onsager relation G(B) = G(−B).
Instead, Buttiker argued that the standard four-terminal set-ups used in these experiments
do not actually measure diagonal Onsager coefficients. Instead, they have to be compared
to specific multiterminal formulae for the conductance, which keep track of whether a given
terminal measures a voltage or a current [13]. In a four-terminal set-up, one ends up with
conductances Gij;kl = Ii,j/(Vk − Vl) with i and j referring to the current flow (Ii = −Ij)and k and l referring to voltage measuring leads. The reciprocity relations then translate
into Gij;kl(B) = Gkl;ij(−B), i.e the role of current and voltage leads has to be interchanged
when the direction of the magnetic field is reversed. This symmetry has been beautifully
illustrated experimentally by Benoit et al. [14], see Fig. 5.
13
D. The phase coherence length
”Mesoscopic” refers to a regime rather than to specific samples. To be mesoscopic, a
sample needs to show some degree of coherence/quantumness, which requires that the phase
coherence length lϕ is at least comparable to the system size. This is achieved by cooling
down the sample to a low enough temperature. In this paragraph, we present a short
qualitative discussion of lϕ. Our purpose here is not to give a full account of decoherence
and of all the possible processes leading to it, rather to illustrate with the help of a toy model
how decoherence comes about, and how lϕ can be defined and determined. The discussion
follows some of the lines of the Ref. [15].
We consider the set-up shown on Fig. 6, where the Aharonov-Bohm interference pattern
can be measured on one point, denoted B, on the right-hand side of the set-up, as the
magnetic flux through the ring is varied between Φ ∈ [0,Φ0]. To simplify the discussion,
we assume that the apparatus is made solely of one-dimensional clean (ballistic) segments.
In addition, electrons are transmitted with equal probability through path 1 and 2. Fur-
thermore, we neglect backreflection at the two “forks” (beamspllitters). This is a rather
idealized situation – unitarity of the scattering matrix at the forks requires that backreflec-
tion be present – but for the sake of our short, qualitative discussion of decoherence it is
sufficient.
Quantum mechanics tells us that, on a given trajectory γ, ballistic electrons accumulate
a phase ϕ given by the action integral
ϕ =1
~
∫
γ
p · dq. (1.3)
Including a magnetic field by means of a vector potential satisfying curlA = H, and calcu-
lating the action integral in Eq. (1.3) using Stokes’ theorem and under the assumption that
the loop is perfectly up-down symmetric gives
δϕ =2πφ
φ0
, (1.4)
for the phase difference accumulated between the top and bottom paths. We introduced the
quantum of flux φ0 = h/e, and the flux φ = HA with the area A of the loop.
To discuss decoherence, we consider that one of the two arms of the ring, say arm 2, is
not isolated from the outside world. Generally, the Hamiltonian of the problem is given by
H = Hsys ⊗ Ienv + Isys ⊗Henv + εU, (1.5)
14
1
2A B
Figure 6: Geometry of a mesoscopic Aharonov-Bohm experiment. Electrons are injected at point
A, on the left hand side. The top (1) and bottom (2) electronic paths enclose a magnetic flux Φ.
The resulting interference pattern is measured on the right hand side, at point B.
where Iα indicates the identity in the α subspace (system or environment) and the initial
wavefunction is a product state
Ψ(r; η; t = 0) = ψ(r; 0) ⊗ χ(η; 0), (1.6)
where η gives the set of coordinates of the environment state. As the experiment is performed
and the coupling between system and environment starts to play a role, Ψ is no longer a
product state — this occurence is closely related to the phenomenon of decoherence that
we want to describe here. Another way to look at things is to realize that, as the strength
ε of the coupling εU between the system (i.e. the ring) and the environment (which, at
this point is not yet specified) is increased, the amplitude of the Φ-dependent interference
pattern of the electronic intensity at point B will decrease — as will be shown below, this is
related to the above stated fact that Ψ is no longer a product state. Before we discuss this
occurence for a specific situation, let us make some general statements about the problem.
Assuming that one can solve the Schrodinger equation, the electronic intensity at point
B is obtained via the total wavefunction (subindices refer to the followed branch of the
interferometer)
Ψ(r = B; η; t) = ψ1(B; t) ⊗ χ1(η; t) + ψ2(B; t) ⊗ χ2(η; t), (1.7)
15
as
I(t) = |Ψ(B; η; t)|2 = |ψ1(B; t) ⊗ χ1(η; t)|2 + |ψ2(B; t) ⊗ χ2(η; t)|2
+2Re[ψ1(B; t) ⊗ χ1(η; t)ψ2(B; t) ⊗ χ2(η; t)∗]. (1.8)
Obviously, the last term on the right-hand side of this latter expression is the interference
term we’re interested in. The question now is “what do we do with the degrees of freedom
of the environment?” As mentioned above, we are aiming at a statistical/probabilistic
description of the effect of the environment — in most instances we cannot do any better
anyway, and are actually quite happy if we can reach such a statistical description at all!
Therefore, as in the case of joint probabilities, we integrate over the degrees of freedom of
the environment (with an appropriate measure) to get the electronic amplitude. Physically
this means that we average over all possible configurations of the environment, which makes
sense considering its much larger phase space, which ensures both its ergodic behavior and
variations of its configuration on a much shorter time scale than the system itself. In
addition, the large size of the environment’s Hilbert space allows it to change its state
without affecting more than only the phase of the system itself. Thus we get the electronic
intensity at point B as
Iel(t) =
∫
dη|Ψ(B; η; t)|2 = |ψ1(B; t)|2+|ψ2(B; t)|2+2Re[ψ1(B; t)ψ∗2(B; t)
∫
dη χ1(η; t)χ∗2(η; t)].
(1.9)
We see that (i) the classical terms are not affected by the coupling to the environment,
and (ii) the interference term gets reduced by the integral∫
dηχ1(η; t)χ∗2(η; t) ≤ 1, which is
equal to one only if χ1(η; t) = χ2(η; t) i.e. in the trivial cases when there is no coupling with
the environment, or for identical coupling between environment and each arms of the AB
interferometer. Decoherence is complete, i.e. the interference term gets totally erased once
the environmental wavefunction gets orthogonalized by its coupling to the system.
Since it may seem weird to discuss a system’s decoherence by looking at how it affects the
environment to which it is coupled, we discuss a second point of view, where the environment
induces a potential V (x, t) which modifies the phase of the system’s wavefunction as
δϕ = −1
~
∫ t0+τ
t0
V (x(t); t)dt, (1.10)
in term of the duration τ of the interaction between system and environment. In the case
when only ψ2 feels the effect of the environment, then the interference term is multiplied
16
by exp(iδϕ). Since V is generated by an environment with a dynamics of its own, it will
fluctuate, therefore we assume that it becomes a random variable, so that
exp(iδϕ) → 〈exp(iδϕ)〉 =
∫
dϕP (ϕ) exp(iδϕ) ≤ 1. (1.11)
It turns out — though we will not show it here — that the two above formulations are
equivalent, and that under quite general assumptions, one has 〈exp(iδϕ)〉 =∫
dηχ1(η)χ∗2(η).
After these conceptual considerations, we close this chapter with a simple practical illus-
tration. Let us consider that electrons travelling on the lower arm of the AB interferometer
are coupled to a (too ?) simple environment in the form of a two-level system. We write
the coupling Hamiltonian as
HC = V ⊗ σz , (1.12)
where σz is a Pauli matrix, and V vanishes everywhere but on a segment of length l (smaller
than the length of the arm) where we have V = V0 Isys. For this very simple model, there
is no intrinsic “environment” dynamics (i.e. the two-level system has no Henv). Two cases
now should be differentiated: (i) initially, the two-level system is in either of the eigenstates
of σz , i.e. | ↑〉 or | ↓〉. Then nothing happens, since the coupling is not altering the state of
the “environment”. This is a nongeneric situation which one could cure by taking a more
complicated coupling Hamiltonian containing other Pauli matrices σx,y; (ii) initially, the
two-level system is in a superposition of | ↑〉 and | ↓〉. In this case, decoherence will take
place. To see how this happens, let us take | →〉 = (| ↑〉+ | ↓〉)/√
2 as initial two-level state.
At t = 0 one has
Ψ(r; η; 0) = [ψ1(r, 0) + ψ2(r, 0)] ⊗ | ↑〉 + | ↓〉√2
. (1.13)
Then the time-evolution delivers
Ψ(r; η; t) = ψ1(r, t)⊗| ↑〉 + | ↓〉√
2+
1√2ψ2(r, t)[| ↑〉 exp(−i
∫ t
0
V dt/~) + | ↓〉 exp(i
∫ t
0
V dt/~)],
(1.14)
so that at point B and time t one has
Ψ(B; η; t) = ψ1(B, t) ⊗| ↑〉 + | ↓〉√
2+
1√2ψ2(B, t)[| ↑〉 exp(−iV0ml/~p) + | ↓〉 exp(iV0ml/~p)].
(1.15)
The time spent by the electron in the interaction zone is indeed given by ml/p. One finally
has
Iel(t) = |ψ1(B; t)|2 + |ψ2(B; t)|2 + 2Re[ψ1(B; t)ψ∗2(B; t)] cos
(
V0ml
p
)
. (1.16)
17
In this simple example, the interference term is reduced by a prefactor cos(V0ml/p), which
decays on the typical length scale
lϕ =π
2
p
mV −1
0 . (1.17)
From this simple toy model, we have learned that the suppression/reduction of quantum
interference effects does not need a huge amount of degrees of freedom to occur, nor does
it require energy exchange — the above two-level system was degenerate. True decoherence
sets in when generalizing Eq.(1.16) to the case where the environment contains more than
one two-level system – say with a broad enough distribution of P (V0). The periodicity of
the damping factor is then lost, and one typically gets a monotonously decaying damping
factor – it is often exponential, but this depends on the exact form of P (V0).
As a final remark in this introductory chapter, let us emphasize that, in quantum physics,
inelastic collisions affecting the phase of the wavefunction may be energy-conserving. In our
semiclassical treatment of decoherence, to be presented below, we will restrict ourselves
to this regime of ”pure dephasing” when the environment’s only effect is to induce phase
randomization.
E. Alternative to semiclassics: random matrix theory
We provide a quick summary of the random matrix theory of transport for metallic
cavities. For more details of the method, the reader is referred to Ref. [16, 17].
The starting point is the scattering approach to transport according to which transport
quantities such as the conductance and shot-noise can be computed from the S-matrix of
the system. The latter connects outgoing and incoming fluxes. For a M-terminal set-up one
has
S =
r11 t12 ... t1M
t21 r22 ... t2M
... ... ... ...
tM1 tM2 ... rMM
. (1.18)
Here, rii is a Ni × Ni matrix, containing the reflection amplitudes from the Ni channels of
the ith lead back to the same set of channels, and tij is a Ni × Nj matrix of transmission
amplitudes from the Nj channels of the jth lead to the Ni channels of the ith lead. The
S-matrix is a square N × N matrix with N =∑M
i=1Ni. For a two-terminal set-up, which
18
will occur exclusively in this manuscript one has
S =
rLL tLR
tRL rRR
. (1.19)
Flux/particle current conservation implies the unitarity of the S-matrix, i.e.
SS† = S
†S = I. (1.20)
In the absence of other symmetries, this is the only requirement that the S-matrix has to
satisfy. If the system is time-reversal symmetric, the scattering matrix has to be symmetric,
i.e.
S = ST. (1.21)
Finally, in presence of spin-rotational symmetry, the S matrix is self-dual
σySTσy = S, (1.22)
with the Pauli matrix σy. Random matrix theory assumes that the matrix elements of S
are randomly distributed, up to these three requirements. Accordingly, one defines three
ensembles, labeled β = 1, 2, 4, where β = 1 corresponds to time-reversal and spin-rotational
symmetric systems (the orthogonal ensemble), β = 2 corresponds to systems without spin
rotational symmetry (the unitary ensemble) and β = 4 corresponds to time-reversal sym-
metric systems with broken spin rotational symmetry (the symplectic ensemble). In our
discussion of transport in noninteracting electronic systems, β = 4 corresponds to systems
with strong enough spin-orbit interaction but no magnetic field (or a weak one), β = 2 to
systems with strong enough magnetic field and β = 1 has neither spin-orbit interaction nor
magnetic field.
We are interested in the electronic conductance of a microstructure in the shape of a
cavity/quantum dot. If we assume that the latter is placed between two reservoirs at different
chemical potentials with negligible reflection back to the microstructure, then the scattering
approach to transport expresses the conductance G in terms of the scattering properties of
the microstructure itself as
G = G0Tr[t†t], (1.23)
and Random matrix theory gives
〈G/G0〉 =NLNR
NL +NR + 2/β − 1≃ NLNR
NL +NR+ (1 − 2/β)
NLNR
(NL +NR)2, (1.24)
19
with G0 = 2e2/h. The first term on the right-hand side of Eq.(1.24) gives the classical
Drude conductance. It corresponds to the series conductance of two contact resistances,
Gser = G0(N−1L +N−1
R ). The second term is the leading quantum, weak localization correction
to the classical conductance in the sense of a largeN expansion. This contribution is negative
for β = 1, zero for β = 2 and positive for β = 4 (in which case one sometimes talk of weak
antilocalization).
One can go beyond the ensemble-averaged conductance and calculate the conductance
variance. One obtains to leading order in N
var(G/G0) =2
β
(NLNR)2
(NL +NR)4. (1.25)
Random matrix theory can interpolate between ensembles. Applying a magnetic field
breaks time-reversal symmetry. For the β = 1 → 2 transition, one gets that the weak
localization disappears as
δG/G0 = − NLNR
(NL +NR)2
1
1 + (φ/φc)2, (1.26)
where the typical flux is φc = αφ0(τf/τD)1/2, with α a system-dependent constant of
order one, τf = L/vF is the time of flight across the cavity (with linear size L) and
τD = mA/[~(NL + NR)] is the average dwell time spent by an electron inside the cavity
(with area A = L2). Likewise, the conductance fluctuations are reduced as
var(G/G0) =(NLNR)2
(NL +NR)4
(
1 +
[
1
1 + (φ/φc)2
]2)
. (1.27)
Another transport property of interest is shot-noise. This is the non-thermal component
of the electronic current fluctuations [18]
P (ω) = 2
∫ ∞
−∞
〈δI(t)δI(0)〉 exp(iωt) dt, (1.28)
where δI(t) = I(t) − I is the deviation of the current from the mean current I, and 〈. . .〉denotes the expectation value. This noise arises because of stochastic uncertainties in the
charge carrier dynamics, which can be caused by a random injection process, or may develop
during the transport through the system. For completely uncorrelated charge carriers, the
noise attains the Poissonian value P0 = 2e2GV . Deviations from this value are a valuable
indicator of correlations between the charge carriers.
20
Phase coherence requires sufficiently low temperatures, at which the Pauli blocking results
in a regular injection and collection of the charge carriers from the bulk electrodes. The only
source of shot noise is then the quantum uncertainty with which an electron is transmitted
or reflected. This is expressed by the quantum probabilities 0 ≤ Tn ≤ 1. In terms of these
probabilities, the zero-frequency component of the shot noise is given by [18]
P (ω = 0) = 2G0eV∑
n
Tn(1 − Tn). (1.29)
This expression can alternatively be rewritten in terms of the transmission matrix t as
P (ω = 0) = 2G0eV Tr[t†t(1 − t†t)]. (1.30)
In the limit of small transmissions, Tn ≪ 1, shot-noise goes to its Poissonian value, otherwise,
it is always smaller than P0, which can be attributed to Pauli blocking.
In random matrix theory, the universal value of shot noise in cavities follows from the
distribution P(T ) of transmission eigenvalues, which gives the Fano factor
F ≡ P (ω = 0)
P0=
NLNR
(NL +NR)2(1.31)
Below, we reproduce these results using semiclassics, and precise their range of validity.
II. OLD SEMICLASSICAL METHODS
The old quantum theory was a collection of results from the years 1900-1925 which predate
modern quantum mechanics. The theory was never complete or self-consistent, but was a
collection of heuristic prescriptions which are now understood to be semiclassical in nature
in that they derive the first quantum corrections to classical mechanics. The Bohr model
was the focus of study, and Arnold Sommerfeld made a crucial contribution by quantizing
the z-component of the angular momentum.
The old quantum theory lives on as an approximation technique in quantum mechanics,
called the WKB (after Wentzel, Kramers and Brillouin) method. It is, even today, quite un-
derestimated that the reason why the old quantum theory was abandoned is classical chaos.
Semiclassical methods are built on classical trajectories, and if the latter are complex, the
semiclassical approach becomes inaccurate or even breaks down. It is chaos that kept Bohr
and Co from quantizing the helium atom despite their success in quantizing the hydrogen
21
atom – while the two-body problem is integrable, the three-body problem is not. In this
chapter we review some of the old semiclassical approaches and discuss their breakdown.
A. The WKB method
In this paragraph we take our inspiration from Ref. [19].
The wave function for a particle of energy E moving in a constant potential V is
ψ = A exp[i pq/~] (2.1)
with a constant amplitude A, and constant wavelength λ = 2π/k, k = p/~, and p =
±√
2m(E − V ) is the momentum. WKB quantization generalizes this solution to cases
where V (q) varies slowly over many wavelengths. This semiclassical (in a sense to be made
clear below) approximate solution of the Schrodinger equation however breaks down at
classical turning points, which are configuration space points where the particle momentum
vanishes. This can be cured, however, as long as these points are isolated (which they are).
In their neighborhoods, one needs to solve locally the exact quantum problem, in order to
compute connection coefficients which patch up semiclassical segments into an approximate
global wave function.
Consider a Schrodinger equation in one spatial dimension
− ~2
2mψ
′′
(q) + V (q)ψ(q) = Eψ(q), (2.2)
and assume that V (q) grows sufficiently rapidly at |q| → ∞, so that the classical motion is
confined for any energy. This alternatively means that there are two turning point, x1 and
x2 for any (obviously periodic) classical trajectory. The situation is depicted in Fig. 7.
We define local momenta and wavenumbers
p(q) = ±√
2m[E − V (q)] , p(q) = ~k(q), (2.3)
and rewrite Eq. (2.2) as
ψ′′
(q) + k2(q)ψ(q) = 0. (2.4)
This suggests to formally write solutions as ψ(q) = A(q) exp[iS(q)/~] in terms of real-valued
22
Figure 7: A 1-dimensional potential V (q), and the location of the two turning points at energy E.
functions A(q) and S(q). This gives a set of two equations
(S′
)2 = p2 + ~2A
′′
A, (2.5)
1
A
d
dq(S
′
A2) = 0. (2.6)
The WKB approximation consists in dropping the ~2-term in the first of these two equations.
This is justified if k2 ≫ A′′
/A, i.e. when the phase of the wavefunction varies much faster
than its overall envelope. It is a generic assumption whenever semiclassical approximations
are made. It also makes it somehow more understandable why the semiclassical limit is
sometimes referred to as the ~ → 0 limit.
The WKB solution then reads
ψ(q, q′;E) =C
|p(q)|1/2exp[iS(q, q′;E)/~], (2.7)
S(q, q′;E) = ±∫ q
q′dq”p(q”), (2.8)
C = |p(q′)|1/2ψ(q′). (2.9)
The WKB generalization of Eq. (2.1) replaces pq/~ by the action integral S(q, q′;E) along
a classical path, starting from an arbitrary point q′. It makes full sense that the probability
|A(q)|2 = |C|2/|p(q)| to find the particle at q is inversely proportional to its momentum
there. Simultaneously, however, this means that the condition k2 ≫ A′′
/A on which the
whole approach is based breaks down at turning points, where p(q) = 0. This is not so
dramatic, as these points are isolated. The standard textbook treatment of WKB relies on
23
q’
x21x
qa
qb
Figure 8: Phase space trajectory of a particle moving in a bound potential of a 1 degree of freedom
system.
connecting solutions in different regions separated by turning points in such a way that ψ
and its derivative are continuous [see for instance Ref. [20]]. We here present an alternative,
qualitative approach which has the advantage that it allows us to develop some useful
semiclassical concepts along the way.
Our strategy is to Fourier-transform the wavefunction in the vicinity of the turning points
as there is no problem there in p−representation. There are two key points that help us
with this strategy in the semiclassical limit (they are actually related). First, the Fourier
transform cannot be performed exactly, but it can easily be evaluated with semiclassical
accuracy by the method of stationary phases. Second, the resulting Fresnel integral giving
ψ(p) in the vicinity of p = 0 is dominated by a semiclassically small range of q-components
ψ(q) – we do not need the full knowledge of ψ(q) to get ψ(p) close to turning points. The
strategy is thus to perform the Fourier transform close enough to p = 0 that the Fresnel
integration gives an accurate estimate of ψ, but far enough from p = 0 that ψ(q) still has an
acceptable magnitude satisfying k2 ≫ A′′
/A. This is indicated by qa on Fig. 8. Accordingly,
we consider the wavefunction in p-representation between qa and qb, Before we proceed, let
us first look into the method of stationary phases and Fresnel integrals.
24
1. Method of stationary phases
All semiclassical approximations to be used in this manuscript are based on saddle point
/ stationary phase approximation of integrals of the form
F =
∫
dxA(x) exp[isΦ(x)], (2.10)
where eventually we have s = ~−1 ≫ 1, in the sense that for most x, sΦ(x) ≫ 1. Since
Φ(x) is in our case a classical action, this condition is almost always satisfied – and we are
in business. Under this condition, the complex exponential in the integral above oscillates
very fast with x, thus contributions from most x average to zero. This is the case except
close to points with Φ′(x0) = 0. One therefore approximates
F =
∫
dxA(x) exp[isΦ(x0) +s
2Φ
′′
(x0)(x− x0)2], (2.11)
which is fine as long as Φ′′
(x0) 6= 0. If this is not the case, one needs to take the next term
in the Taylor expansion of Φ(x) close to x0. While this can be done, we restrict ourselves
to Φ′′
(x0) 6= 0 here. One further assumes that A(x) varies more slowly than Φ(x) to obtain
F = A(x0) exp[isΦ(x0)]
∫
dx exp[is
2Φ
′′
(x0)(x− x0)2]. (2.12)
This is a Fresnel integral, which, using Fresnel’s integral formula, gives
F ≃ A(x0)
∣
∣
∣
∣
2π
sΦ′′(x0)
∣
∣
∣
∣
1/2
exp[isΦ(x0) + iπ
4sgnΦ
′′
(x0)] (2.13)
Note that the Fresnel integral is dominated by contributions from the range (x − x0)2 .
1/sΦ′′
(x0).
2. WKB quantization
A stationary phase approximation gives, for the Fourier transform of the WKB wave-
function
ψsc(p) =C√2π~
exp[i(S(q∗) − q∗p)/~]
|p(q∗)|1/2
∫
dq exp[ i
2~S
′′
(q∗)(q − q∗)2]
, (2.14)
where q∗ is determined by S ′(q∗)− p = p(q∗)− p = 0 (from now on we use p to differentiate
the argument of ψ from the q-dependent momentum p(q). Fresnel integration gives
ψsc(p) =C
√
|p(q∗) p′(q∗)|exp[i(S(q∗) − q∗p)/~] exp
[
iπ
4sgnS
′′
(q∗)]
. (2.15)
25
Note that p(q∗) p′(q∗) is finite, even at turning points, since
p(q) p′(q) =1
2
dp2(q)
dq= −mV ′(q). (2.16)
In the lower left quadrant, one has S′′
= p′ ≤ 0 and Eq. (2.15) gives a well-behaved ψ, with
a phase-shift of −π/4 occuring at the turning point. One then goes back to q-representation
via the inverse Fourier transform. This has to be done some distance away from the turning
point – say at qb as indicated on Fig. 8. The inverse Fourier transform delivers an additional
phase shift of −π/4. The message is that each turning point gives a −π/2 phase shift, to
be added to the action integral – for m turning points, one gets −mπ/2.
The procedure can be iterated, and the wavefunction must be single-valued once one gets
back to the initial point q′ – this gives a quantization condition which now reads
∮
p(q)dq − π~l
2= 2π~n, (2.17)
with n = 0, 1, 2, ... and l the number of turning points on the (periodic) orbit. For a harmonic
oscillator with H = (p2 + (mωq)2)/2m, for instance, one has l = 2, the action integral gives
S = 2πE/ω, and the quantization condition is
En = ~ω(n+1
2), (2.18)
the origin of the 1/2 being the topological phases accumulated at the two turning points.
There is apparently no such phase for the hydrogen atom, where Bohr-Sommerfeld quan-
tization correctly gives En = −1/2n2. This is however not totally correct – in this two-
dimensional problem, there are four turning points per orbit (points for which pi = 0,
i = x, y), and so the shift is l/4 = 1. This is the reason why the energy quantum number
is n = 1, 2, 3, ..., not including n = 0 – including the topological index, the quantization
condition is, in other words, En = −1/2(n+ 1)2, n = 0, 1, 2, ... For the helium atom, l = 6,
which is one of the reasons why Bohr and Co could not quantize it.
B. Failure of eikonal methods for chaotic systems
The WKB method can be generalized to systems with higher dimensionality. This gen-
eralization is often called the eikonal method. As the WKB approach, it makes a bridge be-
tween point-particles traveling on classical trajectories and waves. The additional difficulty,
26
however, is that more complicated dynamics arise in higher dimensions, where autonomous
systems (whose dynamics is determined by a time-independent Hamiltonian) can exhibit
chaos. This is forbidden in one dimension, where the conservation of energy is sufficient to
guarantee integrability. As we are going to see, chaos results in a proliferation of different
return momenta to a given initial spatial position, which requires an expansion in an in-
finitely large number of eikonals. As but one consequence of this fact, the eikonal solution
can no longer be normalized, it breaks down. The qualitative discussion we are about to
present is taken from Ref. [21]. Before that, however, we briefly introduce some concepts of
dynamical systems we will be using in discussions to come.
1. Few words on chaotic vs. integrable systems
We quickly discuss some concepts related to dynamical systems that are required to make
this manuscript self-contained. This discussion is qualitative. For a more rigorous discussion
of classical dynamical systems, see for instance Refs. [19, 22].
We consider a d-dimensional Hamiltonian system. The system is said to be autonomous
if the Hamiltonian does not depend explicitly on time. In this manuscript we restrict our-
selves to autonomous systems which are energy-conserving. The system’s phase space is
2d-dimensional and the Hamiltonian flow ensures that phase-space volumes are conserved
– this is the Liouville theorem. The dynamics in phase space is restricted by the number
of integrals of motions, i.e. observables K such that K,H = 0 with the Poisson bracket
..., ... [23]. For d = 1, energy conservation is sufficient to ensure that the phase-space mo-
tion is restricted to one-dimensional curves. Systems for which the dynamics is restricted in
this way are generically called integrable systems. They are characterized by predictability
of motion over very large times. Integrability requires further constant of motions for d > 1.
As a matter of fact, for a system to be integrable, it needs to have d integrals of motion in
involution (the latter term meaning, qualitatively speaking, that one should not count trivial
constants of motion such as twice, three times... the energy, or linear combination of inte-
grals of motion). With d integrals of motion, the dynamics is restricted to toroidal motion
in phase space. Examples include the hydrogen atom with d = 6 degrees of freedom, where
translational invariance ensures conservation of total momentum (3 conservations) and the
fact that the interaction is central conserves the angular momentum (3 conservations); note
27
that the total energy is contained in these six conservations. Adding a third body to the
problem destroys integrability, however.
Nonintegrable systems can still exhibit some degree of regularity in their motion. The
famous KAM theorem ensures in fact that for a large class of dynamical systems, a small
perturbation, which strictly speaking destroys integrals of motion (not necessarily all of
them at the same time), only distorts but does not destroy the invariant tori on which the
dynamics is bound. For a very small perturbation, this is true for all but a set of zero
measure of resonant tori. For larger perturbation, however, the system can become chaotic.
What does that mean ?
Chaos is actually defined by a hierarchy, at the bottom of which lie ergodic systems.
The probability to find these systems in a given state/at a given position in real/phase space
is not a function of that state/position, but only given by the inverse of the total available
volume. An example is a point particle travelling on a circle with constant velocity. After a
sufficiently large time, the probability that the particle is measured in [θ, θ + δθ] is δθ/2π,
independently of θ. Obviously, this system is not chaotic - its position at any time is known
with exact precision as θ(t) = θ0 + vt (modulo 2π).
The second level in the chaos hierarchy is occupied by mixing systems, for which
two initially well separated sets of initial conditions eventually mix on all scales (up to a
smallest scale defined by how long one has left the system evolve). (In)Famous examples
include mixtures of vodka and orange juice. Whiskey and coke also does the job (but please
never, ever mix whisky with coke). Mixing systems are ergodic.
The mixing property is not sufficient to ensure chaos, however, and the upper level of the
chaos hierarchy is occupied by chaotic systems which exhibit local exponential instability,
usually quantified by at least one positive Lyapunov exponent. The exact definition relies
on a positive Kolmogorov-Sinai entropy, of which we will say nothing, since in a conservative
Hamiltonian system, the KS entropy is equal to the sum of the positive Lyapunov exponents.
Still, what does that mean ? A positive Lyapunov exponent quantifies the rate at which
two initially close points in phase-space move away from one another with time. For chaotic
systems with a single positive Lyapunov λ exponent one has for the phase-space distance
between two such points
|δx(t)| ∝ |δx(0)| exp[λt]. (2.19)
Mathematically speaking, λ is defined from the asymptotic increase of |δx(t)|, and in fully
28
chaotic systems with no constant of motion, λ does not depend on the initial phase-space
position. For homogeneously hyperbolic systems, the spectrum of Lyapunov exponent cor-
responds to eigenvectors of the linearized time-evolution operator M
δx(t) = M(x0)δx(0), (2.20)
i.e. there are 2d Lyapunov exponents. Two of them must be zero in an autonomous Hamil-
tonian system, corresponding to the direction of the flow and the canonically conjugated
(local) direction perpendicular to the energy surface. For the 2d − 2 remaining exponents,
the Liouville theorem has the important consequence that the sum of the Lyapunov expo-
nents must vanish. The dynamics is thus expanding in certain directions and contracting
in other directions, in such a way that phase-space volumes are conserved. In Hamiltonian
systems, one can go one step further and show that Lyapunov exponents come in pairs,
i.e. for each positive Lyapunov exponent λi corresponds a negative Lyapunov exponent
λ−i = −λi. For a given phase-space point x0, the manifold spanned by the eigenvectors
of M with positive λi is called the unstable manifold, while the manifold spanned by the
eigenvectors of M with negative λi is called the stable manifold. These are of course locally
defined concepts.
There are only few dynamical systems for which chaoticity has been shown with full
mathematical rigor – among them the Sinai billiard (a circle embedded in a square) and the
Bunimovich stadium, which we will discuss shortly. We stress that this is a hierarchy, where
chaotic systems have to be mixing and ergodic, and mixing systems have to be ergodic,
whereas some ergodic systems are not mixing and some mixing systems are not chaotic.
2. The eikonal method and its failure
Here, we discuss the eikonal method in the context of the Helmholtz equation
(
∇2 + n2(q)k2
)
ψ(q) = 0 (2.21)
in two dimensions. In principle this is an equation for the electric/magnetic field in a
medium with local index of refraction n(q), however in two dimensions, and considering a
TM (transversal magnetic) polarized electro-magnetic field, this equation is equivalent to
a Schrodinger equation with standard boundary conditions (continuity of the wave func-
29
tion/electric field and their derivative), and with n(q) playing the role of a potential. Con-
sidering a two-dimensional billiard, the optical and quantum mechanical problems are thus
equivalent.
The eikonal approach uses the asymptotic ansatz (note that we use kS(q) in this para-
graph, where we used S(q) in the previous one; we also set ~ ≡ 1)
ψ(q) ∼ exp[ikS(q)]∞∑
ν=0
Aν(q)
kν(2.22)
in the limit k → ∞. Inserting Eq. (2.22) into Eq. (2.21) one finds to lowest order in the
asymptotic parameter 1/k, the eikonal equation
(∇S)2 = n2(q) (2.23)
and the transport equation
2∇S · ∇A0 + A0∇2S = 0 (2.24)
Note at this point we only assume one eikonal S(q) and one amplitude A(q) at each order
in the expansion. We also specialize to a uniform medium of dielectric constant n. In this
framework, each wave solution ψ(q) corresponds to a family of rays defined by the vector
field
p(q) = ∇S(q) (2.25)
where the field has a fixed magnitude, |∇S| = n. The solution for the function S(q) can be
found by the specification of initial value boundary conditions on an open curve C : q = q(s)
and propagating the curve using the eikonal equation. This is sketched in Fig. 9. Such an
initial value solution can thus be extended until it encounters a point at which two or
more distinct rays of the wavefront converge; at or nearby such a point will occur a focus
or caustic at which the amplitude A will diverge and in the neighborhood of which the
asymptotic representation becomes ill-defined. (A caustic is a curve to which all the rays
of a wavefront are tangent; if the curve degenerates to a point it is a focus). This causes
only a local breakdown of the method and can be handled by a number of methods. At a
distance much greater than a wavelength away from the caustic the solutions are still a good
approximation to the true solution of the initial value problem, much in the same way as
the WKB wavefunction we constructed in the previous paragraph is a good approximation
of the exact wavefunction away from turning points.
30
Figure 9: Eikonal propagation of initial value boundary conditions on an open curve.
In contrast, to find asymptotic solutions on a bounded domain D with boundary value
conditions, one must introduce more than one eikonal at each order in the asymptotic
expansion. We will illustrate the important points here with Dirichlet boundary conditions
on the boundary ∂D, though the basic argument holds for any linear homogeneous boundary
conditions. For the discussion of Dirichlet boundary conditions we will set the index n = 1
for convenience within the domain D. The leading order in the asymptotic expansion of the
solution takes the form
ψ(q) =N∑
m=1
Am(q) exp[ikSm(q)] (2.26)
with N ≥ 2. It is easily checked that there must be more than one term (eikonal) in the
solution in order to have a non-trivial solution; if there were only one term in the expression
for ψ(q) then any solution which vanished on the boundary would vanish identically in D
due to the form of the transport equation.
The question we now address is the following. For what boundary shapes ∂D in two
dimensions do there exist approximate solutions of the form Eq. (2.26) which are valid
everywhere in D except in the neighborhood of caustics (which are a set of measure zero)?
To answer this question we first note that with Dirichlet boundary conditions we have
a Hermitian eigenvalue problem. Therefore solutions only exist at a discrete set of real
31
wavevectors k. In the eikonal theory the quantization condition for k arises from the re-
quirement of single-valuedness of ψ(q) – just as for WKB or Bohr-Sommerfeld quantization.
Here our primary goal is to show that the existence of eikonal solutions to the boundary
value problem is intimately tied to the nature of the ray dynamics within the region D.
Moreover for the case of fully chaotic ray dynamics this connection shows that eikonal so-
lutions do not exist. We will prove this latter statement by showing that a contradiction
follows from assuming the existence of eikonal solutions in the chaotic case. This argument
will be a “physicist’s proof” without excessive attention to full mathematical rigor.
The proposed solution for ψ(q) posits the existence of a finite number N of scalar func-
tions Sm(q) each of which satisfy the eikonal equation, (∇Sm(q))2 = 1 and which, while
themselves not single-valued on the domain D, allow the construction of single-valued func-
tions ψ(q) and ∇ψ(q). Moreover, for the asymptotic expansion to be well-defined, the “rapid
variation” in ψ(q) must come from the largeness of k; i.e. to define a meaningful asymptotic
expansion in which terms are balanced at each order in k, the functions Sn cannot vary too
rapidly in space. From the eikonal equation itself we know that |∇Sm| = 1, but we must
also have that the curvature ∇2Sm ≪ k for the asymptotic solution to be accurate. This
condition fails within a wavelength of a caustic, as one can check explicitly, e.g. for the case
of a circular domain D; but for a solvable case like the circle it holds everywhere else in D.
It is convenient for our current argument to focus on ∇ψ, instead of ψ itself. Consider
an arbitrary point q0 in D where ∇ψ(q0) 6= 0; to leading order in k and away from caustics
∇ψ(q0) = ik
N∑
m
Am(q0)∇Sm(q0)eikSm(q0). (2.27)
The N < ∞ unit vectors ∇Sm(q0) ≡ pm define N directions at q0 which are the directions
of rays passing through q0 in the stationary solution. An important point is that due to
the condition on the curvature just noted, these directions are constant at least within a
neighborhood of linear dimension λ = 2π/k around q0. Choose one of the ray directions, call
it p1 and follow the gradient field ∇S1 to the boundary D. For a medium of uniform index
(as we have assumed) the vector ∇S1 is strictly constant in both direction and magnitude
along a ray. Thus one can find the direction of ∇S1 at the boundary and calculate its “angle
of incidence”, n · ∇S1, where n is the normal to the boundary at the point of intersection.
The condition ψ = 0 on the boundary implies that there is a second term with the eikonal
S2 in the sum, which satisfies S1 = S2 and A1 = −A2 on the boundary. As a result tangent
32
derivatives of S1,2 on the boundary are also equal and together with Eq. (2.23), this implies
that for a non-trivial solution n · ∇S2 = −n · ∇S1. In other words a ray of the eikonal S1
must specularly reflect at the boundary into a ray of another eikonal in the sum, which we
label S2. Hence we know the direction of ∇S2 at the boundary and can follow it until the
next “reflection” from the boundary. Thus each segment of a ray trajectory corresponds to
a direction of ∇Sm for some m in Eq. (2.27). A ray moving linearly in a domain D and
specularly reflecting from the boundary describes exactly the same dynamics as a point mass
moving on a frictionless “billiard” table with boundary walls of shape ∂D. Such dynamical
billiards have been studied since Birkhoff in the 1920’s as simple dynamical systems which
can and typically do exhibit chaotic motion. Thus the problem of predicting the properties
of the vector fields Sm is identical to the problem of the long-time behavior of dynamical
billiards.
One property of any such bounded dynamical system (independent of whether it displays
chaos) is that any trajectory starting from a point q0 will return to a neighborhood of that
point an infinite number of times as t → ∞ – this is the Poincare recurrence theorem[24].
Therefore we are guaranteed that the ray we followed from q0 in the direction p1 will
eventually re-enter the neighborhood of size λ around q0. By our previous argument, each
linear segment of the ray trajectory, corresponds to one of the directions ∇Sm and thus
when the ray re-enters the neighborhood of q0 for ∇ψ to be single valued it is necessary that
the ray travel in one of the directions ∇Sm(q0) = pm. There can be two categories of ray
dynamics: (i) Although the ray enters the neighborhood of q0 an infinite number of times it
only does so in a finite number, N of ray directions. (ii) The number of ray directions grows
monotonically with time and tends to infinity as t→ ∞. We will now show that the general
applicability of the eikonal method depends on which category of ray motion occurs.
Let us first consider a billiard ∂D with fully chaotic dynamics. In the current context
“fully chaotic” means that for arbitrary choice of q0 and the direction p1 (except for sets
of measure zero, such as unstable periodic orbits) the distribution of return directions (mo-
menta) is continuous and isotropic as t→ ∞ – this is so, because a chaotic system is ergodic.
Therefore the number of terms in an eikonal solution of the form Eq. (2.26) would have to
be infinite, contradicting our initial assumption that N was finite. Thus there do not exist
eikonal solutions with finite N for wave equations on domains with fully chaotic ray dy-
namics. A very closely related point was made by Einstein as early as 1917 in what has to
33
be accepted as the first (and for six decades the only) article on quantum chaos [25]. One
may ask whether an eikonal solution with an infinite number of terms could be defined; this
appears unlikely as the amplitudes for the wavefronts are bounded below by (λ/L)1/2, where
L is the typical linear dimension of D, so that only a very special phase relationship between
terms would allow such a sum to converge. The essential physics of this breakdown of the
eikonal method is that in a chaotic system wave solutions exist but do not have wavefronts
which are straight on a scale much larger than a wavelength, hence it is impossible to develop
a sensible asymptotic expansion with smooth functions Sm.
Returning now to the case of a boundary ∂D for which the distribution of return momenta
is always discrete, this means that there exist exactly N ray directions for each point q0
and any choice of p1. In this case the entire spectrum of the wave equation on ∂D can be
obtained by an eikonal approximation with N terms of the form Eq. (2.26). The quantized
values of k are determined by the conditions that the eikonal only advance in phase by an
integer multiple 2π upon each return to q0 and hence the solution is single-valued. The
correct quantization condition must take into account phase shifts which occurs for rays
as they pass caustics. The details of implementing this condition have become known as
Einstein-Brillouin-Keller quantization.
From modern studies of billiard dynamics we know that both of the cases we have just
considered are exceptional. The billiards for which eikonal solutions for the entire spectrum
is possible are integrable ones, and their ray dynamics has one global constant of motion for
each degree of freedom. For example in the circular billiard both angular momentum and
energy are conserved and for each choice of q0 and direction p1 there are exactly two return
directions (see Fig. 10). While the circle is a good and relevant example here, there are
other shapes, such as rectangles and equilateral triangles for which the method also works;
obviously these are shapes of very high symmetry. It is also known that an elliptical billiard
of any eccentricity is integrable; however this is believed to be the only integrable smooth
deformation of a circle. Billiards being considered as generic dynamical systems, it seems
that there is a relatively small class of systems for which eikonal methods work globally.
As already noted, the type of boundary shape which generates continuous return distri-
butions for each choice of q0 and direction p1 correspond to completely chaotic billiards and
such shapes are also quite rare. No smooth boundary (i.e. ∂D for which all derivatives exist)
is known to be of this type. A well-known and relevant example for us of such a shape is the
34
p1
q0
Figure 10: (a) A typical quasi-periodic ray motion in a circular billiard. The two possible ray return
directions for a specific point q0 and initial direction p1 are shown in red. (b) The Bunimovich
stadium, consisting of two semi-circles connected by straight segments, for which ray motion is
completely chaotic. As the schematic indicates, for any point q0 the ray return directions are
infinite, continuously distributed and isotropic, making an eikonal solution impossible.
stadium billiard, consisting of two semi-circular “endcaps” connected by straight sides (see
right panel of Fig. 10). Note that the generation of continuous return distributions would
fail for a point q0 between the two straight walls if we chose p1 perpendicular to the walls
generating a (marginally stable) two-bounce periodic orbit passing through q0. However
this choice represents a set of measure zero of the initial conditions in the phase space. It
follows from our above arguments that eikonal methods fail for the entire spectrum in such
a billiard (except a set of measure zero in the short wavelength limit).
The generic dynamics of billiards arises when the boundary is smooth but there is not
a second global constant of motion. Such a billiard has “mixed” dynamics. Some trajec-
tories exhibit a regular behavior, in that they remain bound to a one-dimensional curve in
phase-space (once the energy and the direction of the flow have been subtracted from its
represention), while others look chaotic – they explore a two-dimensional volume of phase-
space and exhibit local exponential instability. For such a billiard, depending on the choice
of the initial phase space point (q0, p1), one may get either a finite number N of return
directions or an infinite number as t→ ∞. It is not obvious just from our above arguments
that this means that eikonal methods will fail in such a case. We will skip over this point
and simply state that in the case of mixed dynamics in principle only a finite fraction of the
35
spectrum could be calculated by eikonal methods.
We thus see that WKB and related approach fail for generic, higher dimensional systems
– or that they succeed only for integrable systems. Can we circumvent this difficutly, what
other short wavelength approaches exist that remain valid for generic systems ? The devel-
opment of short wavelength approximations for mixed and chaotic systems is precisely the
problem of quantum chaos which has been widely studied in atomic, nuclear, solid-state and
mathematical physics over the past two decades[23, 26]. Powerful analytic methods have
been developed, but with an essentially different character than eikonal methods – these
techniques are typically referred to as semiclassical methods in the quantum chaos litera-
ture. The analytic methods in quantum chaos theory are all of a statistical character and do
not allow one to calculate individual modes. Instead the methods focus on the fluctuating
part of the density of modes and the statistical properties of the spectrum (e.g. level-spacing
distributions). More recently these methods have been extended to treat transport in meso-
scopic systems. It is the purpose of the reminder of this manuscript to introduce, develop
and apply these methods to situations of interest in mesoscopic physics.
III. FROM PATH INTEGRALS TO THE SEMICLASSICAL GREEN’S FUNC-
TION
The fundamental building block of the semiclassical theory to be constructed is the semi-
classical Green’s function. We start with its derivation. As always we do not go into details
of the derivations with full rigor, but instead refer the reader to appropriate references here
and there.
A. The Green’s function
Take the time-dependent Schrodinger equation
i~∂tψ(q, t) = Hψ(q, t) (3.1)
with initial condition ψ0 = ψ(t = 0). The formal solution to this initial value problem reads
ψ(t) = U(t)ψ0, (3.2)
U(t) = exp[−iHt/~]. (3.3)
36
Define next ψE as the Laplace-transform of ψ(t),
ψE = − i
~
∫ ∞
0
dt exp[i
~(E + iǫ)t]ψ(t), (3.4)
with ǫ > 0 ensuring convergence at large t. From ψE we define the retarded Green’s function
G+(E) as
ψE = G+(E)ψ0. (3.5)
Formally one has
G+(E) = − i
~
∫ ∞
0
dt exp[i
~(E + iǫ)t] exp[− i
~Ht] (3.6)
= (E − H + iǫ)−1. (3.7)
Likewise, the advanced Green’s function is G−(E) = (G+(E))†. Using the (assumed) discrete
eigenbasis |n〉 of H one has
U(t) =∑
n
exp[− i
~Ent]|n〉〈n|, (3.8)
G±(E) =∑
n
(E − En ± iǫ)−1|n〉〈n|. (3.9)
Further using Im[(x± iǫ)−1] = ∓ǫ/(x2 + ǫ2) → ∓πδ(x) for ǫ→ 0, one expresses the density
of states as
ρ(E) =∑
n
δ(E − En) = ∓1
πTr[ImG±(E)]. (3.10)
Our program in this section is to derive semiclassical expressions for the propagator U(t),
the Green’s function G(E) and the density of states. In position representation, the former
two will be expressed as a sum over classical trajectories (either of duration t or at energy
E) connecting their two spatial arguments, while the trace over these arguments, together
with a stationary phase conditions will allow us to express the density of states as a sum
over periodic orbits.
B. The Feynman path integral
As this is only an intermediate step towards the construction of the semiclassical prop-
agator, we do not discuss the path integral formalism in details, but instead only mention
the necessary steps to make this manuscript somehow self-contained. For a much nicer and
37
rigorous construction of the path integral formalism we refer the reader to Andreas Wipf’s
lecture notes on http://www.personal.uni-jena.de/∼p5anwi/lecturenotes.html .
The time-evolved wavefunction can be written, somehow formally
ψ(q; t) =
∫
dq′K(q, q′; t)ψ(q′; 0), (3.11)
with the time-evolution kernel K(q, q′; t) = 〈q| exp[−iHt/~|q′〉. We take H = p2/2m +
V (q) ≡ T + V and consider first infinitesimal evolution times δt. It is easy to show that
exp[− i
~δt(T + V )] = exp[− i
~δtT ] exp[− i
~δtV ] exp
[
− δt2
2~2[V , T ]
]
+ O(δt3), (3.12)
so that for sufficiently small δt, one neglects the commutator and writes
K(q, q′; δt) =
d∏
j=1
1
2π~
∫ ∞
−∞
dpj exp
[
− i
~
(
p2jδt
2m− pj · (qj − q′
j)
)]
exp
[
− i
~V (q′)δt
]
.
(3.13)
The Fresnel integrals are easily performed and one gets
K(q, q′; δt) =( m
2πi~δt
)d/2
exp
[
i
~δt
(
m
2
(
q − q′
δt
)2
− V (q′)
)]
. (3.14)
One recognizes the Lagrange function, L[q′, (q − q′)/δt] in the complex exponential. To
move on to a finite-time evolution we take t = Nδt with N → ∞ and δt → 0. We further
use
K(q, q′; t) =
∫
dr1 . . .
∫
drN−1 K(q ≡ rN , rN−1; δt) . . .K(r1, q′ ≡ r0; δt), (3.15)
which follows from the unitarity of the time evolution. With this one gets
K(q, q′; t) =( m
2πi~δt
)Nd/2∫
dr1 . . .
∫
drN−1 exp
[
i
~δt
N∑
j=1
L(rj−1;rj − rj−1
δt)
]
≡∫
D[q] exp
[
i
~
∫ t
0
dt′L(q(t′); q′(t′))
]
. (3.16)
This is the Feynman-Kac formula and is the path integral formulation for the quantum-
mechanical time-evolution kernel / the propagator. The integration element D[q] stands for
”all paths going from q′ to q”. To see why this is called a path integral we connect the
points qi, i = 0, 1, ...N by straight lines so that we have a broken path from q′ to q. This is
sketched in Fig. 11.
38
Figure 11: A broken path of a particle propagating from q0 to qN .
C. The Van Vleck–Gutzwiller propagator and the semiclassical Green’s function
1. Generalization of the stationary phase method
Let f, g be two real-valued function on the n-dimensional real space, and let x(ν) be
the set of isolated zeroes of ∇f , with det[∂2f/∂xi∂xj ]x=x(ν) 6= 0 and g(x(ν)) 6= 0. Then
lim~→0
∫
dxg(x) exp[i
~f(x)] = (2π~)n/2
∑
ν
g(x(ν)) exp[if(x(ν))/~] exp[iπαν/4]√
|det[∂2f/∂xi∂xj ]x=x(ν) |. (3.17)
This generalizes Eq. (2.12) to the case with more than one stationary phase solution.
2. Semiclassical approximation to the path integral propagator
We now apply a stationary phase condition on Eq. (3.15). In the continuous limit,
N → ∞, δt → 0, the sequence of points (rN , rN−1, ...r0) goes to a continuous trajectory
q(t),parametrized by the time coordinate t. Then, the argument of the complex exponential
becomes
δtN∑
j=1
L(rj−1;rj − rj−1
δt) →
∫ t
0
dtL(q(t′); q′(t′)), (3.18)
so that the stationary phase condition, that (this is a functional derivative)
δ
δq(t)
∫ t
0
dtL(q(t′); q′(t′)) = 0 (3.19)
is equivalent to the Euler-Lagrange equation, i.e. it selects classical trajectories q(t), and the
argument of the complex exponential is Hamilton’s principle function evaluated on one of
those trajectories, Rν(q; q′; t) =∫ t
0dtL(qν(t
′); q′ν(t
′)). There are some not necessarily trivial
39
q’
qc
Figure 12: Hamiltonian flow near a conjugate point, where ∂2Rν/∂qi∂q′j|q′=qc= −∂p′/∂qc → ∞.
steps that follow, and we only write the endproduct which is that the propagator is given
by a sum over classical orbits (labelled ν) going from q′ to q in time t,
Ksc(q, q′; t) =
∑
ν
(2πi~)d/2
√
∣
∣
∣
∣
det
(
− ∂2Rν
∂qi∂q′j
)∣
∣
∣
∣
exp
[
i
~Rν(q; q′; t) − iπκν
2
]
, (3.20)
with the so-called Morse index κν counting the number of negative eigenvalues of the matrix
∂2Rν/∂qi∂q′j . This number is also given by the number of conjugate points on ν lying
between q′ and q, where an eigenvalue of ∂2Rν/∂qi∂q′j becomes infinite. These conjugate
points are (usually) isolated, and the eigenvalue is finite again soon after ν has passed
the conjugate point. Going through that point, however, the eigenvalue has changed sign.
Fig. 12 illustrates a flow near a conjugate point.
3. The semiclassical Green’s function
We are only one Laplace transform away from obtaining the semiclassical, van Vleck-
Gutzwiller Greens function. We have
G(q, q′;E) = − i
~
∑
ν
(2πi~)d/2
∫ ∞
0
dt
√
∣
∣
∣
∣
det
(
− ∂2Rν
∂qi∂q′j
)∣
∣
∣
∣
exp
[
i
~(Rν(q, q
′; t) + Et) − iπκν
2
]
.
(3.21)
The classical action Sν = Rν + Et replaces now Hamilton’s principal function. Performing
the t-integral brings down an additional prefactor −1/∂2Rν/∂t2 with ∂2Rν/∂t
2 = −∂E/∂tfrom the stationary phase condition. Additionally, one gets a new Morse index
µν = κν +
1 ∂2Rν/∂t2 = −∂E/∂t < 0,
0 otherwise.(3.22)
40
∂E∂t < 0
∂E∂t > 0
µν = κν µν = κν + 1
Figure 13: Qualitative sketch of a potential for which ∂tE∂t < 0 (left) and ∂tE
∂t > 0 (right). Potentials
steeper than harmonic have ∂tE∂t < 0. For more potentials with polynomial behavior, V (x) =
∑
Vaxa, the sign of ∂tE
∂t may depend on the energy range considered.
What does ∂E/∂t exactly means, considering that one has autonomous systems only ? It
gives the answer to the question : fix the endpoints q′ and q of your trajectory, how does
the time to travel between these endpoints change as one varies the energy slightly ? Let us
go through three examples
• Billiard with hard walls. The trajectories do not change when the energy is varied, but
if E increases, so does the velocity, and the time between two endpoints becomes shorter.
for such systems one has ∂E/∂t < 0.
• Harmonic oscillator. The period is the same for all orbits, regardless of their energy,
hence ∂E/∂t = 0.
• Pendulum. The period increases as the energy increases, hence ∂E/∂t > 0.
Two ”generic” situations are qualitatively sketched in Fig. 13.
With all these considerations, the semiclassical Green’s function reads
Gsc(q, q′;E) =
∑
ν
2π
(2πi~)(d+1)/2
√
|Dν(q, q′;E)| exp[iSν(q, q′;E)/~ + iπµν/2], (3.23)
with
Dν ≡ det
∂2Rν
∂q∂q′
∂2Rν
∂E∂q′
∂2Rν
∂q∂E∂2Rν
∂E∂E
(3.24)
41
D. The trace formula
We have seen that the density of states can be rewritten in terms of the Green’s function
as
ρ(E) = −1
πIm
∫
dqG+(q, q;E). (3.25)
It can be shown that ρ can be splitted into a smooth and an oscillating part, ρ = ρ+ ρosc,
with the smooth part given by the Thomas-Fermi formula
ρ(E) = (2π~)−d
∫ ∫
dqdp δ[E −Hcl(q,p)]. (3.26)
The oscillating part contains more information, and it turns out in particular that it is
strongly affected by the system’s periodic orbits. Using the semiclassical expression for the
Green’s function one has
ρosc(E) = − 2
(2πi~)(d+1)/2Im
∫
dq∑
ν
|Dν|1/2 exp[iSν/~ − iπµν/2]
. (3.27)
Our strategy is again to enforce a stationary phase condition on that expression, which
means that the action Sν(q, q;E) must be stationary upon variation of its initial and final
point, i.e.(
∂Sν
∂q′+∂Sν
∂q
)
q=q′
= 0 (3.28)
which is equivalent to p = p′, i.e initial and final momenta on ν are equal. In other words,
this stationary phase condition selects periodic orbits only. Specifying to chaotic systems,
where periodic orbits are dense in phase space, yet are isolated and form a set of zero
measure, and skipping some calculational steps, one obtains Gutzwiller’s trace formula [26]
ρosc(E) =1
π~
∑
ν∈p.p.o
∑
n
Tν√
|det(Mnν − 1)|
cos(nSν/~ − πnσν/2). (3.29)
This expression differentiates primitive periodic orbits (labeled ν) from general periodic
orbits, the primitive ones being orbits travelled only once. Accordingly, n in Eq. (3.29) gives
the winding number, i.e. the number of times one goes around the primitive periodic orbit.
The matrix Mν is the (2d−2)× (2d−2) monodromy matrix, giving the linearized dynamics
42
around the periodic orbit ν in one period, i.e.
Mν
q′⊥
p′⊥
=
q⊥(Tν)
p⊥(Tν)
, (3.30)
M =∂(p⊥, q⊥)
∂(p′⊥, q
′⊥). (3.31)
For fully chaotic systems, the eigenvalues of the monodromy matrix are given by Lyapunov
exponents so that for a 2-dimensional chaotic system one has
√
|det(Mnν − 1)| =
√
|(exp[nλTν ] − 1)(exp[−nλTν ] − 1)| = 2| sinh(nλTν/2)|. (3.32)
The trace formula has been successfully used to relate spectra of chaotic systems to short
periodic orbits. It is remarkable in that it connects a system’s quantal property – the
oscillating part of its density of states – to its classical dynamics, via its set of periodic
orbits.
E. Beyond the trace formula: two-point correlations
Most physical situations of interest require the knowledge of quantities which, unlike the
density of states, are quadratic (or higher order) in Green’s functions. Examples we will
encounter below include the conductance and its fluctuations, as well as the shot-noise power
through mesoscopic systems and persistent currents in mesoscopic metallic rings. Before we
start the analysis of such quantities with the two-point correlation function
K(E, ǫ) = ρ(E + ǫ/2)ρ(E − ǫ/2), (3.33)
let us first make some quick remarks on the time scales we will encounter in our analysis.
1. Time scales
In mesoscopic systems one differentiates between classical and quantal time scales. Classi-
cal time scales include the diffusion time τD = L2/D, with the diffusion constant D = vFℓ/d.
This time is much larger than the elastic mean free time τe = ℓ/vF. In ballistic systems, the
diffusion time is replaced by the dwell time τD spent by an electron inside an open cavity.
For small openings, the latter time is much larger than the time of flight τf = L/vF. Other
43
important classical time scales are the ergodic time τerg measuring the time it takes for an
initial condition to have visited most of its available phase-space, and the Lyapunov time
λ−1, roughly giving the time it takes for local exponential instability to set in. In most
ballistic systems/billiards one has λ−1 ≈ τf and we will not differentiate them.
Quantal time scales include the Heisenberg time τH = ~/∆, i.e. the inverse of the level
spacing, which is the time it takes for the system to resolve the discreteness of its spectrum
– hence the name Heisenberg time, as τH is set by an uncertainty relation between level
spacing and time. The importance of τH is that it sets an upper limit to the validity of
semiclassical methods. A second quantal time scale is the Ehrenfest time τE = λ−1 lnL/λF,
which is the time it takes for the underlying classical chaotic dynamics to exponentially
stretch an initial narrow wave packet, of spatial extension given by the Fermi wavelength
λF, to the linear system size L. The name refers to the Ehrenfest theorem which breaks
down at times longer than τE in the sense that the center of mass of the wavepacket is no
longer well defined/representative of the dynamics. This time is important in semiclassics as
it signals the moment when nontrivial quantum interference effects set in. For times shorter
than τE, quantum mechanics is, roughly speaking, well approximated by the Liouville flow.
For disordered systems with short-range impurities, the Ehrenfest time vanishes. It has
meaning only in ballistic systems or in presence of smooth, long-range disorder.
The hierarchy of time scales we consider is the following
τf ∼ λ−1 < τerg . τD ≪ τH, (3.34)
where by τerg . τD we mean that τerg can be smaller (sometimes much smaller, as in the
ballistic systems we will consider) than τD but that in other instances the two times scale can
be of a similar order of magnitude (like in diffusive systems). A lot depends on the position
of the Ehrenfest time in this hierarchy, and we will consider both the universal regime, when
τE is so short that it can be neglected, and the deep semiclassical regime, where τE becomes
comparable to τD.
44
2. The diagonal approximation
We want to calculate the two-point correlation function of Eq. (3.33), and its partial
Fourier transform, the form factor
K(E, t) =1
2π~
∫
dtK(E, ǫ) exp[−iǫt/~] . (3.35)
Our strategy is to insert Gutzwiller’s trace fomula for the density of states in the expression
for K(E, ǫ), as one does not expect any nontrivial correlation arising from the smooth part
ρ of the density of states. We use the short-hand notation
ρ(E) =∑
ν
Aν exp[iSν(E)/~] , (3.36)
where it is implicitly assumed that only the oscillatory part of the density of states is
considered. Here and there, this way of writing things requires that one takes the real part
of the calculated expressions.
We first note that, since Tν = dSν/dE, one has, for not too large ǫ,
ρ(E ± ǫ/2) =∑
ν
Aν exp[iSν(E) ± ǫTν/2~] . (3.37)
One thus has, for the form factor,
K(E, t) =∑
ν,ν′
Aν(E)A∗ν′(E) exp[i(Sν(E) − Sν′(E))] δ[t− (Tν + Tν′)/2] , (3.38)
where, as will become usuall, we neglect the ǫ-dependence of the stability amplitude Aν . We
want to identify the semiclassically dominant contributions to this double sum over classical
periodic orbits, and to this end enforce a stationary phase condition. The rigorous motivation
for doing so is that one is interested in quantities averaged over a classically small (so that
the dynamics does not change) but quantum mechanically sizable (larger than the level
spacing) energy window. Contributions that survive this averaging are those whose phase,
Sν(E) − Sν′(E) does not depend on E. For not too long times, only ν = ν ′ and νT = ν ′
satisfy this condition (νT indicates the time-reversed of ν). For longer times however, there
are many other stationary phase solutions, which we however neglect in this section (we will
discuss such contributions when constructing a semiclassical theory of transport). We thus
evaluate the form factor within the diagonal approximation, ν = ν ′ (and νT = ν ′ in presence
45
of time-reversal symmetry at the classical level), to get
K(E, t) =∑
ν,ν′
|Aν(E)|2δ(t− Tν) ×
2 with TRS
1 without TRS.(3.39)
This expression looks rather simple, to evaluate it however we have to replace the sum
over periodic orbit by an integral. To this end we need to construct a sum rule. This
step is, besides the identification of stationary phase solutions, one of the main steps in all
semiclassical calculations.
3. Semiclassical sum rules
The first construction of a semiclassical sum rule was the work of Hannay and Ozorio de
Almeida. Here we follow the derivation of Ref. [27].
We take R = (q,p) as a 2d-dimensional vector in phase space. Starting from a given
initial condition R0, a classical system evolves towards a single phase-space Rt after time t.
The phase-space distribution is thus δ(2d)(R − Rt), which, taking energy conservation into
account, we formally rewrite as δ(2d−1)(R−Rt) δ(H(R)−H(R0)). In phase space, Rt = R0
implies periodic motion. We can therefore write, for the average classical probability to have
a periodic orbit of period t
Ppo(t) =
(
dΩ
dE
)−1 ∫
dR0 δ(2d−1)(R− Rt) δ(H(R) −H(R0)), (3.40)
with the volume of the energy hypersurface
dΩ
dE=
∫
dR0 δ(H(R) −H(R0)). (3.41)
We next define the contribution to Ppo(t) coming from a given periodic orbit ν as
Pν(t) =
∫
Γν
dR0 δ(2d−1)(R− Rt) δ(H(R) −H(R0)) , (3.42)
which, for later convenience we do not normalize. Here,∫
Γνindicates that one integrates
over initial points R0 lying on ν only – modulo some finite resolution. Accordingly Pν(t)
vanishes unless t = Tν .
To evaluate Pν(t), we first define a local system of coordinate specific to ν. We first take
d−1 spatial and momentum coordinates Q and P that are perpendicular to the periodic orbit
46
ν
P
Q0
0
T
ν
(P’,Q’)
(P,Q)=F (P’,Q’)
Figure 14: A periodic orbit ν, the local system of coordinate (Q,P, T ) for one point on ν and a
neighboring orbit going from (Q′,P′, 0) to (Q,P, Tν) after one period.
and the energy hypersurface. We also define a time coordinate T ∈ [0, Tν ], with Tν the period
of the primitive periodic orbit, which labels any point on ν, starting from an arbitrarily
chosen initial point. This is sketched on fig. 14. With this system of coordinates we also
define a Poincare surface of section, which records the coordinates of passage (Q0,P0, t =
nTν) in the hypersurface perpendicular to ν and the energy surface, which interesects ν
at the chosen initial point. This defines an application Fν , according to which one has
(Fν(Q,P);H, T ) = (Q,P;H, T + Tν). The stability matrix in the vicinity of the periodic
orbit for one period of the orbit, the so-called monodromy matrix, is associated with the
linearized form of Fν ,
Mν
QT
PT
=
QT+Tν
PT+Tν
, (3.43)
M =∂F (PT ,QT )
∂(PT ,QT ). (3.44)
It can equivalently be seen as the Jacobian matrix of a volume-conserving (by Liouville)
coordinate transformation. One then rewrites
Pν(t) =
∫
Γν
dQ0dP0dT0dH0 δ(H0 − E) δ(T0 − Tt) δ(d−1)(Q0 − Qt) δ
(d−1)(P0 − Pt) . (3.45)
We are almost there. We finally rewrite Tt = T0 + t and (Qt,Pt) = Fν(Q0,P0) (and thus
δ(d−1)(Q0 −Qt) δ(d−1)(P0 −Pt) → δ(2d−2)[(Q0,P0)− Fν(Q0,P0)]), and perform the T0 and H0
integral (trivially). The integrals over Q0 and P0 finally deliver the inverse determinant of
47
the derivative matrix of the argument of the corresponding δ-function. We finally get
Pν(t) = δ(t− Tν′)Tν |det(Mν′ − 1)|−1 . (3.46)
Note that here ν stands for the primitive orbit, while ν ′ refers to any integer winding around
that primitive orbit. Inserting this into the form factor, Eq. (3.39), one obtains
K(E, t) =∑
ν
Tν
(2π~)2Pν(E, t) . (3.47)
Since the total probability to perform periodic motion is P (E, t) = (dΩ/dE)−1∑
ν Pν(t) one
finally obtains the sum rule
K(E, t) =t
(2π~)2
dΩ
dEP (E, t) . (3.48)
To be useful, this sum rule still requires to know something about P (E, t). We discuss
two cases. First, consider a bounded chaotic dynamical system. In the semiclassical limit
~ → 0, all classical time scales discussed above become much smaller than the Heisenberg
time. For a closed chaotic system, the largest classical time scale is the ergodic time, and so
for τerg ≪ t≪ τH (this interval is semiclassically large) one can invoke the ergodicity of the
dynamics to write
δ(2d−1)(Rt −R) ≃(
dΩ
dE
)−1
. (3.49)
Invoking further the principle of uniformity according to which, long enough periodic orbits
are homogeneously distributed over the energy hypersurface [the above ergodicity then also
applies to P (E, t)] one has
P (E, t) ≃(
dΩ
dE
)−1
, (3.50)
and one finally gets
K(E, t) =t
(2π~)2. (3.51)
This must be multiplied by two for classically time-reversal symmetric systems. This result
agrees with the leading order (in time) behavior of the form factor computed from random
matrix theory.
We next discuss diffusive systems, for which the phase-space probability distribution is
uniform in momentum space but is a spreading Gaussian in real-space. For long times the
48
system becomes ergodic, and one recovers the above results – this requires t & L2/D. But
for times shorter that L2/D, Eq. (3.49) is replaced by
δ(2d−1)(Rt − R) ≃(
dΩp
dE
)−11
(4πDt)dexp[−(x − x0)
2/4Dt], (3.52)
with Ω = ΩpΩx. This almost directly leads to
K(E, t) =Ωx
(2π~)2 (4πD)dt1−d . (3.53)
This must also be multiplied by two in presence of time reversal symmetry.
IV. PERSISTENT CURRENTS
We saw above how a magnetic flux affects the behavior of charged quantum mechanical
particles, even in cases when the wavefunction of the latter are not threaded by (i.e. do not
touch) the flux. Quantal nonlocality arises from the Hamiltonian formulation of the problem
which makes the vector potential (and not the magnetic field itself) affect the phase of the
wavefunction. Consequently, a particle enclosing n times a flux Φ accumulates a phase
2πnΦ/Φ0. This leads, in a metallic ring closed onto itself, to the appearance of a nonzero
equilibrium current when Φ 6= 0. At zero temperature, this current is carried by the ground-
state itself, therefore, it cannot decay (in absence of decoherence processes). Even in absence
of decoherence processes, the current disappears when the temperature is increased, as many
excited states become populated, which carry currents of alternating signs – this is not (yet)
decoherence, rather it is called thermal averaging. It is important to realize right away that
this equilibrium persistent current is very different from the electronic currents measured in
transport experiments. In particular, the persistent current is not dissipative, and in this
sense is similar to supercurrents occuring in superconductors — this analogy motivated early
authors to denote persistent currents by Josephson effects in normal metallic rings [28].
Before we look at these points in some more details we stress right away that giving
a complete overview of the theory of persistent currents goes beyond the scope of this
class. Moreover, and despite very recent developments [29] it is yet not clear to this day
(November 2008) why the experimentally observed currents in diffusive samples are so big,
their magnitude exceeding the theoretical prediction by at least an order of magnitude. The
introductory part of our discussion somehow follows the discussion presented in Ref. [30],
49
while the more advanced, semiclassical calculation of the typical and average persistent
current follows Ref. [27].
A. Generalities
An applied magnetic field acts on the electrons in a metal – it has an orbital and a
Zeemann effect. We neglect the latter here. The magnetization generated by applying a
uniform field B is given by
M(B) = −∂Ω
∂B
∣
∣
∣
∣
µ,T
, (4.1)
with the grand canonical potential Ω(T, µ,B) = −T ln ZG and ZG =∑
N exp[µN/T ]∑
i exp[−Ei/T ]. From now on, we set the Boltzmann constant equal
to one. We define the magnetization per unit volume as M = M/V and the magnetic
susceptibility as χ = ∂M/∂B.
We then consider a one-dimensional or quasi one-dimensional ring of radius R pierced by
a magnetic flux. The flux is localized well inside the ring and we neglect orbital effects due
to the field piercing the metal. The current I = M/S is then only due to the flux
I(φ) = −∂Ω∂φ
∣
∣
∣
∣
µ,T
, (4.2)
with the surface S = πR2 of the ring. This current is called a persistent current.
To fix ideas one might consider the following Hamiltonian for electrons confined to the
ring
H = − ~2
2m(∇ +
ie
~A(r))2 + V (r). (4.3)
It has a spectrum ǫn which depends on A, hence on φ. The grand canonical potential
reads
Ω(T, φ) = −2T
∫ ∞
−∞
ρ(ǫ, φ) ln[1 + exp[(ǫF − ǫ)/T ]dǫ, (4.4)
with the density of states ρ(ǫ, φ) =∑
n δ[ǫ− ǫn(φ)]. One writes the magnetic moment as
M(φ) = 2ST
∫ ∞
−∞
(∂ρ(ǫ, φ)/∂φ) ln[1 + exp[(ǫF − ǫ)/T ]dǫ, (4.5)
which, after integration by parts gives
Ω(T, φ) = −2
∫ ∞
−∞
N(ǫ, φ)f(ǫ, T ) dǫ (4.6)
= 2
∫ ∞
−∞
N(ǫ, φ)(∂f(ǫ, T )/∂ǫ) dǫ. (4.7)
50
Here, f is the Fermi function and
N(ǫ, φ) =
∫ ǫ
−∞
ρ(ǫ′, φ) dǫ′ (4.8)
N(ǫ, φ) =
∫ ǫ
−∞
∫ ǫ′
−∞
ρ(ǫ′′, φ)dǫ′ dǫ′′ (4.9)
At zero temperature, one gets Ω(φ) = −2N(ǫ, φ) and
M(T, φ) = −2S∑
ǫn<ǫF
∂ǫn∂φ
. (4.10)
The magnetization becomes a sum of contributions from each occupied level. The semiclas-
sical treatment is to replace such expression by an integral containing the density of states,
and express the latter by a sum over periodic orbits, according to the trace formula.
B. Persistent currents in a clean one-dimensional ring
We rewrite the persistent current as
I(φ) = 2∂
∂φ
∫ ∞
−∞
N(ǫ, φ)
(
−∂f∂ǫ
)
dǫ, (4.11)
which becomes
I(φ) = 2∂N(ǫF, φ)
∂φ, (4.12)
at zero temperature. For a clean, one-dimensionall ring of perimeter L, the vector potential
in the Hamiltonian, Eq. (4.3) only has one nonzero, azimuthal component A = eθφ/L. The
Schrodinger equation then reads
− ~2
2m
(
∂
∂x+ 2πi
ϕ
L
)2
ψn(x) = Enψn(x), (4.13)
with ψn(x+ L) = ψn(x), and ϕ = φ/φ0. The spectrum is given by
ǫn =~
2
2m
(
2π
L
)2
(n− ϕ)2, (4.14)
with n ∈ Z. We next use Poisson’s summation formula∞∑
n=−∞
f(n− ϕ) =
∞∑
m=−∞
cos(2πmϕ)
∫ ∞
−∞
f(y) exp[2πimy] dy, (4.15)
to rewrite the density of states as
ρ(ǫ, φ) =L
2π
∞∑
p=−∞
∫ ∞
−∞
δ(ǫ− ǫ(k)) exp[ipkL] dk cos(2πpϕ), (4.16)
51
Figure 15: Persistent current in a one-dimensional clean ring for (left) even and (center) odd
number of electrons on the ring and (right) change between even and odd as the flux is varied.
Figures taken from Ref. [31].
with ǫ(k) = ~2k2/2m. This gives a harmonic expansion for the integrated and doubly
integrated density of states N(ǫ, φ) and N(ǫ, φ), from which one gets a harmonic expansion
for the zero-temperature current
I(φ) =2evF
πL
∞∑
p=1
p−1
[
cos(pkFL) − sin(pkFL)
pkFL
]
sin(2πpϕ). (4.17)
The pth harmonics corresponds to the motion of an electron p times around the ring, thereby
enclosing a flux pφ.
C. Even-odd effect, finite temperature, disorder and all that
In our regime of interest, kFL≫ 1, one has
I(φ) =2evF
πL
∞∑
p=1
p−1 cos(pkFL) sin(2πpϕ). (4.18)
This is valid at fixed kF. For fixed number N of electrons, one has instead (kF = Nπ/L)
I(φ) =2evF
πL
∞∑
p=1
(−1)pN
psin(2πpϕ). (4.19)
There is thus a parity effect, where the odd harmonics are different for even or odd N . This
is illustrated in Fig. 15.
52
Figure 16: Persistent current in a one-dimensional clean ring for even number of electrons on the
ring, for different temperature T/T ∗, with T ∗ = ∆/2π2 is given by the mean level spacing at zero
flux. Figure taken from Ref. [31].
What is next the effect of a finite temperature ? One goes back to Eq. (4.11), and expand
to linear order in ǫ− ǫF to get an expression valid to leading order in T/ǫF
I(φ) =2evF
πL
∞∑
p=1
p−1 sin(2πpϕ)πpLT
~vF
1
sinh(πpLT/~vF). (4.20)
One sees that each harmonics is damped by a prefactor T/Tp
/
sinh(T/Tp) with Tp = δ/pπ2,
with the one-dimensional level spacing ∆ = π~vF/L. The pth harmonics is thus strongly
damped as soon as T becomes comparable to ∆/p. This is illustrated in Fig. 16. This
reduction of the persistent currents by thermal averaging is easy to understand qualitatively.
From the ϕ-derivative of Eq. (4.14), we see that the currents carried by two consecutive
levels have opposite signs. Second, the current magnitude is larger for higher levels. Taking
Eq.(4.12) into account, one concludes that the zero-temperature current is dominated by the
contribution from the last occupied level. At finite temperature, however, one has a number
∼ T/∆ of partially occupied levels – the current is thermally averaged. The reduction given
in Eq. (4.20) follows from the fact that levels with the smaller occupancy are those carrying
the larger currents.
Let us finally discuss the effect of static impurity disorder on the persistent currents.
From the topology of the system, we still have a periodic potential, and thus can apply
53
Bloch theory – this is all one needs to have persistent currents. However, the presence of
disorder broadens the Fermi level. To take this into account for weak disorder (i.e. within the
Born approximation/Fermi golden rule, and again linearizing in ǫ−ǫF) we replace cos(pkL)/p
by∫ ∞
−∞
cos(pkL)
p
~/2πτe(ǫ− ǫF)2 + (~/2πτe)2
dǫ = exp[−pL/2ℓ]cos(pkFL)
p(4.21)
There is an exponential damping of the pth harmonics once pL is bigger than the elastic
mean free path ℓ = vFτe. For fixed number of electrons one (naively – see below) obtains
I(φ) =2evF
πL
∞∑
p=1
(−1)pN
pexp[−pL/2ℓ] sin(2πpϕ) (4.22)
where I is an average over disorder realizations.
D. Average and typical current in the canonical ensemble
Since we are doing mesoscopics we have to differentiate the current carried by a single ring
from the average current taken over an ensemble of many rings. The former is standardly
referred to as the typical current (assuming that the ring considered is typical...), which one
defines as
Ityp(φ) ≡ 〈I2(φ)〉1/2, (4.23)
where 〈...〉 stands for a mesoscopic ensemble average (over energy or disorder realizations
for instance at fixed classical parameters L, ℓ aso).
There is a subtle but extremely important difference between the average persistent cur-
rents in the grand canonical and in the canonical ensemble. The experiments are performed
on closed rings with fixed number of electrons - the appropriate ensemble is thus the canon-
ical ensemble. It turns out that the current is significantly bigger and less disorder-sensitive
in the canonical than in the grand canonical ensemble. As we are going to see, the aver-
age current in the canonical ensemble becomes quadratic in Green’s functions, thus pairing
of trajectories allows for long-range correlations to exist, which in their turn enhance the
average current by significant amounts. In particular, the disorder dependence is no longer
exponential, but becomes algebraic.
Fixing the number N of electrons gives a φ-dependence to the Fermi energy ǫF in order
for the number of electrons to be φ-independent. We write (the subindex N meaning that
54
the number of electrons is fixed inside a given ring; it does not mean, however that all rings
in our ensemble have the same number of electrons)
IN = −∂Ω∂φ
∣
∣
∣
∣
ǫF(φ)
= −∂Ω∂φ
∣
∣
∣
∣
ǫF
− ∂2Ω
∂φ∂µ
∣
∣
∣
∣
ǫF(φ)
(ǫF(φ) − ǫF) . (4.24)
The first term on the right-hand side is the grand canonical current. It is exponentially
damped by disorder, ∝ exp[−L/ℓ]. We next note that ∂Ω/∂µ = −N and ǫF(φ)−ǫF = δµ(φ),
and use the thermodynamic relation δµ|N = −∆δNµ to write
IN = Iµ − ∆
2
∂(δN)2
∂φ
∣
∣
∣
∣
ǫF
(4.25)
where Iµ is the grand canonical current, at fixed chemical potential. This is where semiclas-
sics enters the game. The number of electrons in the system is (from now we set ǫF = 0)
is
N(ǫ, φ) =
∫ 0
−∞
ρ(ǫ′, φ)dǫ′ , (4.26)
so that
〈(δN)2〉 =
∫ ∫ 0
−∞
〈ρosc(ǫ′, φ)ρosc(ǫ
′′, φ)〉 dǫ′dǫ′′ . (4.27)
One has the product of two density of states, hence the product of two Green’s functions
leading to a double sum over trajectories. Stationary phase contributions to that double
sum are much more numerous than to a single sum over trajectories – thanks to diagonal
trajectory-pairing. This results in a strong enhancement of the average canonical current
compared to the grand canonical one. Neglecting the grand canonical contribution, one can
easily express the average canonical current in terms the two-point correlation function,
IN(φ) = −∆
2
∂
∂φ
∫ ∫ 0
−∞
K(ǫ, ǫ′;φ), (4.28)
K(ǫ, ǫ′;φ) = 〈ρosc(ǫ, φ) ρosc(ǫ′, φ)〉 . (4.29)
E. Semiclassical approach to persistent currents in normal metallic rings
The two quantities that have been calculated semiclassically are the typical current,
I2typ ≡ 〈I2(φ)〉 = c2
∂
∂φ1
∂
∂φ2
∫ 0
−∞
dǫ1dǫ2ǫ1ǫ2
⟨
ρosc(ǫ1, φ1)ρosc(ǫ2, φ2)⟩
µ
∣
∣
∣
∣
φ1,2=φ
, (4.30)
(this comes from the zero-temperature expression Ω =∫
ǫρ(ǫ, φ)dǫ) and the average canon-
ical current, Eqs(4.28) and (4.29).
55
Here, we consider the case of zero temperature, but finite temperatures can be dealt
with as well. All one needs to do is to extend the integrals up to +∞ and introduce Fermi
functions f(ǫ1,2). We first discuss Ityp. Inserting Gutzwiller’s trace formula, Eq. (3.29), into
Eq. (4.30) one gets
I2typ = c2
∂
∂φ1
∂
∂φ2
∫ 0
−∞
dǫ1dǫ2ǫ1ǫ2∑
ν1,ν2
Tν1Tν2√
(det[Mν1 − 1])(det[Mν2 − 1])exp[i(Sν1 − Sν2)/~]
× exp[2πi(nν1φ1 − nν2φ2)/φ0] exp[−iπ(σν1 − σν2)]∣
∣
∣
φ1,2=φ. (4.31)
To calculate this expression, the same steps as in section III E are taken. First, one expands
the actions Ss(ǫ1) = Ss(ǫ+) + ǫ−Ts, with ǫ± = (ǫ1 ± ǫ2)/2. Second, one works out Eq. (4.31)
within the diagonal approximation, setting ν1 = ν2 or ν1 = νT2 . This is justified under the
assumption that the system is chaotic, i.e. that periodic orbits are isolated and unstable,
and for long enough orbits, meaning that they accumulate widely varying action integrals,
in the limit of small enough electronic wavelength (thus Ss/~ ≫ 1). Third, the sum over
periodic orbits is replaced by an integral over time (the duration of the orbit) and a sum
over winding numbers by means of the Hannay/Ozorio de Almeida sum rule. One has
∑
ν
T 2ν exp[iTν(ǫ− ǫ′)/~]/|det[Mν − 1]| =
∫ ∞
−∞
dt|t|∑
n
Pn(t) exp[it(ǫ− ǫ′)/~] , (4.32)
and finally,
〈I2(φ)〉 =
∫ ǫF
−∞
dǫ
∫ ǫF
−∞
dǫ′ǫǫ′∫ ∞
−∞
dt|t|
(2π~)2exp[it(ǫ− ǫ′)/~]
×∑
n
Pn(t)
(
2πn
φ0
)2
(1 − exp[4πinφ/φ0]). (4.33)
In the last parenthesis, the first term originates from ν1 = ν2 and the second one from
ν1 = νT2 . In complex systems, the return probabilities Pn are usually assumed to be Gaussian
Pn(t) = exp[−n2/2n20(t)]/
√
2πn20(t), (4.34)
with a linearly increasing variance n0(t) = t/τ0. Pn(t) gives the probability that a particle
returns to its initial position (modulo a quantum-mechanical resolution) at time t, after
travelling n times around the ring. Two limiting cases can be considered: (i) the diffusive
regime, with τ0 = L2/D [27], and (ii) the ballistic regime, with τ0 = αL/vF , α = O(1) being
a system-dependent constant [32, 33]. In both instances, the integrals can be performed to
56
yield
〈I2(φ)〉 =
(
e
τ0
)2∑
n
6
π2n3sin2(2πnφ/φ0). (4.35)
In this way one reproduces the standard Ityp ∝ (evF/L)(ℓ/L) result obtained from the Im-
purity Green Function technique (with the classical diffusion constant D = vF ℓ/d emerging
from Cooperon-Diffusion diagrams) [34], as well as the ballistic result Ityp ∝ (evF/L). Of
particular interest is the fact that the typical current has both even and odd harmonics.
This is so, because one paired periodic orbits with themselves and with their time-reversed.
We are going to see now that the odd harmonics disappear from the average current because
only strictly diagonal pairing of trajectories with themselves is possible there.
For the average canonical current one has
〈I(φ)〉 = − ∆
2φ0
∫ 0
−∞
dǫ1dǫ2∑
ν1,ν2
2πi(nν1 − nν2)Tν1Tν2√
(det[Mν1 − 1])(det[Mν2 − 1])exp[i(Sν1 − Sν2)/~]
× exp[2πi(nν1 − nν2)φ)/φ0] exp[−iπ(µν1 − µν2)]. (4.36)
This time, only time-reversed pairs of orbits ν1 = νT2 contribute (for ν1 = ν2, nν1 −nν2 = 0).
Following the same procedure as for the typical current one gets
〈I(φ)〉 =π∆
φ0
∫ 0
−∞
dǫ1dǫ2∑
ν
nνT2ν
det[Mν − 1]sin[4πnνφ/φ0] exp[iTν(ǫ1 − ǫ2)/~]
=2∆
πφ0
∑
n>0
sin[4πnφ/φ0]. (4.37)
This obviously delivers a paramagnetic current, and the odd harmonics have disappeared.
All the semiclassical calculations presented above can be extended to the case of a sys-
tem with spin-orbit interaction. In the weak coupling regime (meaning that the classical
trajectories are not affected by spin-orbit interaction), and for spin-1/2 particles, the trace
formula (3.29) is modified by considering a double sum over initial position of the spin (i.e.
up or down) and projecting the final position of the spin at the end of the periodic orbit
onto the initial one. One gets, instead of (3.29)
ρosc(ǫ, φ) = (2π~)−1∑
σ=↑,↓
∑
ν
Tν cos(α(σ)ν )
√
det(Mν − 1)exp[iSν(ǫ)/~ − iπµν/2 + 2πinsφ/φ0], (4.38)
The angle α(σ)ν gives the precession angle of the spin, initially in the position σ =↑, ↓, after
the particle has travelled along the periodic orbit ν.
57
The first line in Eq.(4.37) becomes
〈I(φ)〉 =π∆c
φ0
∫ 0
−∞
dǫ1dǫ2∑
σ,σ′=↑,↓
∑
ν
nνT2ν cos(α
(σ)ν ) cos(α
(σ′)
νT )
det[Mν − 1]
× sin[4πnνφ/φ0] exp[iTν(ǫ1 − ǫ2)/~]. (4.39)
A generic form for the spin-orbit interaction Hamiltonian is ~C(r,p)·~σ, where ~σ is the vector
of Pauli matrices and one has C(r,−p) = −C(r,p). This results in α(σ)ν = −α(−σ)
ν = −α(σ)
νT .
Thus the presence of spin-orbit interaction reduces the current, but does not change its sign.
Assuming that any spin orientation is equally likely at the end of a periodic orbit, the sum
rule (4.32) is multiplied by a prefactor 〈cos2 α(σ)s 〉 = 1/2.
V. SEMICLASSICAL THEORY OF TRANSPORT: RAY OPTICS FOR THE XXIst
CENTURY
We next present our semiclassical theory of transport in ballistic chaotic systems. First
attempts at developing such a theory were made by Barange, Jalabert and Stone [35].
They however failed to capture coherent effects such as weak localization and universal
conductance fluctuations. Major steps forward were made by Sieber and Richter [36, 37],
and the theory was completed simultaneously by Brouwer and Rahav [38] and Whitney and
Jacquod [39]. The group of Fritz Haake claims to have constructed a method to get quantum
corrections to all orders in the number of channels connected to the cavity, however it is
not clear at what order diffraction effects (not included in their theory) have to be included.
The range of validity of their theory is in any case restricted to exactly zero Ehrenfest time.
The theory presented here is the one constructed by Whitney and Jacquod [39].
Our starting point is the scattering approach to transport, according to which the dimen-
sionless conductance (in units of 2e2/h) reads g = Tr[t†t].
A. Diagonal approximation: the Drude conductance
Our starting point is the semiclassical expression for the transmission matrix [35, 40],
tji = −(2πi~)−1/2
∫
L
dy0
∫
R
dy∑
γ
(dpy/dy0)1/2γ 〈j|y〉〈y0|i〉 exp[iSγ/~ + iπµγ/2] , (5.1)
58
Figure 17: Scattering trajectory γ through a ballistic cavity contributing to the transmission
amplitude, Eq.(5.1).
where |i〉 is the transverse wavefunction of the ith lead mode. This expression sums over
all trajectories γ (with classical action Sγ and Maslov index µγ) starting at y0 on a cross-
section of the injection (L) lead and ending at y on the exit (R) lead. Contributions to Tji
are sketched in Fig. 17. We do not discuss the construction of Eq. (5.1), but only mention
that the transmission from i to j proceeds by means of a Green’s function, which, when
expressed in real-space representation, requires the projection of lead modes onto transverse
position vectors on the cross-sections of the leads. Once the Green’s function is replaced by
its semiclassical expression (in terms of classical scattering trajectories) one gets Eq. (5.1).
Inserting Eq. (5.1) in the Landauer-Buttiker formula, one gets a double sum over tra-
jectories and over lead modes, |i〉 and |j〉. We make the semiclassical approximation
that∑
n〈y′|n〉〈n|y〉 ≃ δ(y′ − y), which would be exact, if one had a complete orthogo-
nal mode basis. This is not the case here. Still, for an ideal lead, with N lead modes of
the form 〈y|n〉 = (2/W )1/2 sin(πyn/W ) for 0 ≤ y ≤ W , one finds that∑
n〈y′|n〉〈n|y〉 =
(2W )−1[sin[(z′ − z)(N + 1/2)]/ sin[(z′ − z)/2]− sin[(z′ + z)(N + 1/2)]/ sin[(z′ + z)/2]] where
z = πy/W . This function is strongly peaked at y′ = y with peak width ∼ λF and height
∼ λ−1F , and in the semiclassical limit we can calculate all quantities to lowest order in ~ by
approximating this function with a Dirac δ-function. Note however that this approximation
breaks down if only few modes are carried by the leads.
With this approximation, the conductance simplifies to a double sum over trajectories
59
which both go from y0 on lead L to y on lead R,
Tr[t†t] =1
(2π~)
∫
L
dy0
∫
R
dy∑
γ1,γ2
Aγ1Aγ2eiδS/~ . (5.2)
Here, Aγ = [dpy/dy0]1/2γ . Reflection, R = Tr[r†r], is given by the same expression, with both
y0 and y on lead L.
We are interested in quantities averaged over variations in the energy or the cavity shape.
For most γ1, γ2 the phase of a given contribution, δS/~, will oscillate wildly with these
variations, so the contribution averages to zero. The most obvious contributions that survive
averaging – that satisfy a stationary phase condition – are the diagonal ones with γ1 = γ2.
These contributions give the Drude conductance,
〈g〉D = NLNR
/
(NL +NR), (5.3)
as we now proceed to show.
We define P (Y,Y0; t)δyδθδt as the product of the initial momentum along the injection
lead, pF cos θ0, and the classical probability to go from an initial position and momentum
angle Y0 = (y0, θ0) to within (δy, δθ) of Y = (y, θ) in a time within δt of t. Then the sum
over all trajectories γ from y0 to y is
∑
γ
A2γ [· · · ]γ =
∫ ∞
0
dt
∫ π/2
−π/2
dθ0
∫ π/2
−π/2
dθ P (Y,Y0; t) [· · · ]Y0 . (5.4)
This is the sum rule we need that replaces the Hannay/Ozorio de Almeida sum rule for the
transport problem at hand here (reminder: the HOdA sum rule applies to closed systems and
relates spectral correlators to periodic orbits). The meaning of this sum rule is the following.
On the left-hand side we have the sum of all currents carried by trajectories γ from one side
to the other of the cavity, with fixed initial and final points y0 and y. This is replaced, on
the right-hand side, by a probability to go from y0 to y in time t with any injection and
exit angle, which is late integrated over all durations and angles. That probability has to
be multiplied by pF cos θ0 because one deals with currents here, and that more current is
injected into the cavity if the incident particle flows perpendicularly to the entrance.
For an individual system, P has δ-functions for all classical trajectories – this follows
from classical determinism. However averaging over an ensemble of systems or over energy
gives a smooth function
〈P (Y,Y0; t)〉 =pF cos θ0 cos θ
2(WL +WR)τDexp[−t/τD] . (5.5)
60
How does one get this formula ? The presence of a term pF cos θ0 has already been dis-
cussed. One has next to introduce a current measure at exit – properly normalized, this
measure is cos(θ)/2. Then, the probability at time t is proportional to the survival probablity
τ−1D exp[−t/τD] inside the cavity, with the average dwell time given by
τD =πA
vF
1
WL +WR
. (5.6)
Finally the probability to exit at y is ergodically distributed over the total exit volume
WL +WR. For a rigorous derivation of such sum rules, see e.g. [36, 37, 41].
This latter expression (5.5) is valid as long as no restriction is imposed on the trajectory
inside the cavity. Using Eqs. (5.4) and (5.5) to calculate the conductance within the diagonal
approximation, one recovers the Drude conductance (5.3),
gdiag = 〈g〉D =NLNR
NL +NR, (5.7)
Rdiag =N2
L
NL +NR. (5.8)
One also sees that at the level of the diagonal approximation, there is unitarity, i.e. gdiag +
Rdiag = Tr[t†t]+Tr[r†r] = NL. We stress that, unlike in Ref. [37], we do not include coherent
backscattering in the diagonal contribution, it is dealt with separately below.
Once the philosophy of the scattering approach to transport is accepted, Eq. (5.7) has
an easy to understand, probabilistic interpretation : the conductance is the product of the
number NL of ways one can inject an electron into the cavity times the probability that the
particle exits via the right lead. Assuming ergodicity, this latter probability isNR/(NL+NR).
B. Beyond the diagonal approximation I : weak localization for transmission
To go beyond the Drude conductance, we need to identify less trivial pairings of trajec-
tories than the diagonal one. Unlike for persistent currents, setting γ1T = γ2 in Eq. (5.2) is
not an option here, since both trajectories have to go from left to right. A pair of trajectories
giving the leading order correction to the Drude conductance is shown in Fig. 18. The tra-
jectories are paired almost everywhere except in the vicinity of an encounter. This ensures
that their action difference is minimal, and hopefully stationary under variation of energy.
Going through an encounter, one of the trajectories intersects itself, while the other one
avoids the crossing. The encounter is connected on one side to two legs, each of them going
61
L 00
R
~L
~L
1/2heff
W y θ
θy W
~W
γ2γ1
Figure 18: Sketch of the leading order quantum correction to the conductance. Trajectory γ1
(solid line) is injected at Y0 = (y0, θ0), crosses itself and escapes at Y = (y, θ). Its first visit to
the crossing (the open dot) occurs at R1 = (r1, θ1), where r1 is the position in the cavity, and θ1
is the angle of the momentum to a reference axis (not shown). Trajectory γ2 (dashed line) starts
and ends at the same positions as γ1, however it avoids the crossing.
through its own lead (when calculating the weak localization correction for transmission),
and on the other side to a weak localization loop. It is easy to see that the two trajectories in
Fig. 18 travel along the loop they form in opposite direction. It has been shown in Ref. [36]
that (i) trajectories intersecting themselves are numerous (at least in the two-dimensionall
context we consider here), they are actually generic when considering long trajectories; and
(ii) for any self-intersecting trajectory with a small enough crossing angle ǫ, there exists one
and only one partner, crossing-avoiding outer trajectory. To see this just consider that γ1
intersects itself at a real-space point r inside the cavity. The crossing defines two phase-space
points R1,2 = (r, φ1,2), denoting the point of first and second passage through the crossing.
In a chaotic system, both a stable and an unstable manifold are associated to each of these
phase-space points, so that there are many phase-space points near the crossing which will
be attracted by the loop/one of the legs as one moves forward/backward in time. We are
looking for a smooth partner of γ1 which avoids the crossing, but goes rapidly towards γ1
as one moves away from the crossing in any of the four directions defined by γ1. Thus
one searches for real-space points A and A′ (see Fig. 19) that lie at the intersection of the
unstable (A in Fig. 19) and stable (A′ in Fig. 19) manifolds of R1 and R2 (A in Fig. 19). In
62
A
A’
L1
L2
φR
φL
Figure 19: Classical trajectory γ1 which intersects itself (black), thereby forming a loop on the
right-hand side and having two legs (L1 and L2 on the left-hand side). To prove the existence and
unicity of a smooth partner, staying very close to γ1 most of the times, but avoiding the crossing,
we search for a point A near the crossing, with the property that, for well chosen momentum φR
(φL) directed to the right (left) the phase-space point (A,φR) [(A,φL)] lies on the unstable manifold
for the loop (unstable manifold for L1). In two dimensions, the intersection of these two manifolds
gives a single real-space point A. The search for A′ proceeds in the same way, except that the
phase-space points must lie on stable manifolds.
two dimensions these intersections are unique. Connecting the trajectory going through A
with the one going through A′ somewhere along the loop is facilitated by the finite quantum-
mechanical resolution one tolerates – exponential accuracy ensured by the classical dynamics
is good enough. This proves the existence and unicity of a partner trajectory to γ1, which
avoids the crossing, but stays close to γ1 otherwise.
For another proof this existence and unicity based on the linearized dynamics close to
γ1, see e.g. Ref. [36].
For the relevant case of small ǫ, the probability to find a weak localization pair is
thus given by the probability to find a self-intersecting trajectory. The two trajectories
are always close enough to each other that their stability is the same, i.e. one can set∑
γ1,γ2 Aγ1Aγ2 →∑
γ1 A2γ1. To evaluate the weak localization correction to conductance, we
63
perform a calculation similar to Ref. [37], adding the crucial fact that pair of trajectories
such as depicted in Fig. 18 have highly correlated escape probabilities due to the presence
of an encounter [42]. The situation is depicted in more detail in Fig. 20.
The presence of the encounter introduces two new ingredients, both of these were over-
looked in Ref. [37]. First, pairs of trajectories leaving an encounter escape the cavity in
either a correlated or an uncorrelated way. Uncorrelated escape occurs when the perpen-
dicular distance between the trajectories is larger than the width WL,R of the leads. This
requires a minimal time TW (ǫ)/2 between encounter and escape, where, neglecting the lead
asymmetry
TW (ǫ) = λ−1 ln[ǫ−2(W/L)2] . (5.9)
The two pairs of trajectories then escape in an uncorrelated manner, typically at completely
different times, with completely different momenta (and possibly through different leads).
Correlated escape occurs in the other situation when the distance between the trajectories
at the time of escape is less than WL,R. Then the two pairs of trajectories escape together,
at the same time through the same lead. This latter process affects coherent backscattering
(see Fig. 20). The second new ingredient is that the survival probability for a trajectory with
an encounter is larger than that of a generic trajectory. This is so, because the encounter
duration affects the escape probability only once. In other words, if the trajectory did not
escape in its first passage through the encounter, neither will it during its second passage
(this was first noticed in Ref. [42]).
To calculate the contribution from pairs of transmitting trajectories sketched in Fig. 20a
we still need to evaluate the action difference between the two trajectories. This difference
is only due to the encounter. Linearizing the dynamics near the encounter, and assuming
uniformly hyperbolic dynamics the matrices for the linearized dynamics in both directions
close to the encounter read
MR,L(t) =
cosh λt (mλ)−1 sinhλt
mλ sinh λt coshλt
, (5.10)
with the mass m of the particle and the Lyapunov exponent λ of the classical dynamics. One
gets that, for preserved time-reversal symmetry, the action difference for this contribution is
δSwl = EFǫ2/λ [36]. To write the sum over crossing trajectories in terms of the probability
64
y0y0
L
correlated region
W Wεγ2γ1
y
correlated regionγ1
W Wεγ2
y
L
correlated region
y
γ2γ1 WW
(a) Weak−localization for transmission (conductance)
(b) Weak−localization and coherent back−scattering for reflection
y0
L
Figure 20: Sketches of the trajectory pairing which give the leading off-diagonal contributions to
(a) transmission (conductance) and (b) reflection. All contributions involve a trajectory γ1 crossing
itself at an angle ǫ, and a trajectory γ2 which avoids the crossing. The action difference between
the two trajectories is thus small and does not fluctuate under averaging. For transmission, y0 is
on L lead and y is on R lead, for reflection both y0 and y are on L lead. There are two reflection
contributions. On the left is weak localization, and on the right is coherent backscattering (details
of the latter are in Fig. 22).
P introduced above, we use
P (Y,Y0; t) =
∫
C
dR2dR1P (Y,R2; t− t2)P (R2,R1; t2 − t1)P (R1,Y0; t1) . (5.11)
The integrals over R1,2 are over the energy surface C of the cavity phase-space. We then
restrict the probabilities inside the integral to trajectories which cross themselves at phase-
space positions R1,2 = (r1,2, φ1,2) (φ1,2 defining the direction of the momenta) with the first
(second) visit to the crossing occurring at time t1 (t2). Using Fig. 21, we write dR2 =
v2F sin ǫdt1dt2dǫ and set R2 ≡ (r2, φ2) = (r1, φ1 ± ǫ). Next, we note that the duration of the
loop must exceed TL(ǫ) = λ−1 ln[ǫ−2], because for shorter times, two trajectories leaving an
encounter remain close enough to each other that their relative dynamics is hyperbolic, and
the probability of forming a loop is zero. Then the probability that a trajectory starting at
Y0 crosses itself at an angle ±ǫ and then transmits, multiplied by its injection momentum
65
δxδt2F+v
δt1
δ
Fx,y( )
x+ x,y+ yδ δ( )
−v
yε+δε ε
complete ergodicity
Figure 21: Sketch of a trajectory (solid line) which crosses itself at point r1 = (x, y), visiting this
point first at time t1 and second at time t2. Superimposed is an infinitesimally different trajectory
(dashed line) which also visits the point (x, y) at t1, but is at point r2 = (x + δx, y + δy) at time
t2. This trajectory also intersects itself, however it visits the self-intersection (which is no longer
(x, y)) at times t1 + δt1 and t2 + δt2.
pF cos θ0, is
I(Y0, ǫ) = 2v2F sin ǫ
∫ ∞
TL+TW
dt
∫ t−TW /2
TL+TW /2
dt2
∫ t2−TL
TW /2
dt1
×∫
R
dY
∫
C
dR1 P (Y,R2; t− t2)P (R2,R1; t2 − t1) P (R1,Y0; t1) , (5.12)
where TW , TL are shorthand for TW (ǫ), TL(ǫ). Thus
gwl = (π~)−1
∫
L
dY0dǫRe[
eiδSwl/~]⟨
I(Y0, ǫ)⟩
. (5.13)
We perform the average of the P ’s as follows. Within TW (ǫ)/2 of the crossing the two
legs of a self-intersecting trajectory are so close to each other that their joint escape prob-
ability is the same as for a single trajectory. Self-intersecting trajectories thus have an
enhanced survival probability compared to non-crossing trajectories of the same length,
i.e. the duration of the crossing must be counted only once in the survival probabil-
ity [42]. Outside the correlated region, the legs can escape independently through either
lead at anytime. Furthermore, the probability density for the trajectory going to a given
point in phase-space is assumed to be uniform. Thus the probability density for leg 1
gives 〈P (R1,Y0; t1)〉 = (2πA)−1 exp(−t1/τD) × pF cos θ0, and the loop’s probability den-
sity is 〈P (R2,R1; t2 − t1)〉 = (2πA)−1 exp−[t2 − t1 − TW(ǫ)/2]/τD. Finally the con-
ditional probability density for leg 2 (given that leg 1 exists for a time t1 > TW (ǫ)) is
66
〈P (Y,R2; t− t2)〉 = [2(WL +WR)τD]−1 cos θ exp−[t− t2 − TW(ǫ)/2]/τD. One gets
〈P (Y,R2; t− t2) P (R2,R1; t2 − t1) P (R1,Y0; t1)〉 =
1
(2πA)2
pF cos θ cos θ02(WL +WR)τD
exp[−(t− TW (ǫ))/τD], (5.14)
so that 〈I(Y0, ǫ)〉 becomes
⟨
I(Y0, ǫ)⟩
=(vFτD)2
πApF sin ǫ cos θ0
NR
NL +NR
exp[−TL(ǫ)/τD] . (5.15)
We insert this into Eq. (5.13). The ǫ-integral is dominated by contributions with ǫ ≈(λ~/EF)1/2 ≈ (λF/L)1/2 ≪ 1, so that we write sin ǫ ≃ ǫ and push the upper bound for the
ǫ-integration to infinity. The ǫ-integral can then be computed to give an Euler Γ-function
[43]. To leading order in (λτD)−1 it equals −π~(2EFτD)−1. The integral over Y0 yields a
factor of 2WL. Finally noting that NL = (π~)−1pFWL and (NL +NR)−1 = (mA)−1~τD, the
weak localization correction to the conductance reads
gwl = − NLNR
NL +NRexp[−τ cl
E /τD] . (5.16)
We see that weak localization is exponentially suppressed with τ clE /τD in term of the closed
cavity Ehrenfest time τ clE ≡ λ−1 ln[~−1
eff ] with the effective Planck’s constant, ~eff = λF/L.
The meaning of this suppression is that in the semiclassical limit of very short wavelengths,
~eff ≪ 1, the onset of the Ehrenfest time strongly reduces the number of short orbits that
intersect themselves with a semiclassically small intersection angle ǫ ≈ (λF/L)1/2. Only
orbits longer than the Ehrenfest time have a chance to contribute to weak localization, and
the latter is damped by the measure∫ τE0τ−1D exp[−t/τD] = exp[−τE/τD] of these trajectories.
C. Beyond the diagonal approximation II : quantum corrections to reflection
The above result (5.16) has already been derived in Ref. [43] by a similar approach.
We go beyond that by showing explicitly that our semiclassics preserves the unitarity of
the scattering matrix. There are two leading-order off-diagonal corrections to reflection.
They are shown in Fig. 22. The first is weak localization while the second is coherent
backscattering. The former reduces the probability of reflection to arbitrary momentum,
while the latter enhances the probability of reflection to the time-reverse of the injection
state. The distinction between these two contributions is whether the trajectories escape
67
0
0
0
bγ1
γ1ay
a
bγ2
γ2r
y
θθ
L lead cavity
Figure 22: Trajectories for the backscattering contributions to reflection. Trajectory γ1 (solid
black line) start on the cross-section of the L lead at position y0 with momentum angle θ0 and ends
at y with momentum angle θ. In the basis parallel/perpendicular to γ1 at injection, the initial
position and momentum of path γ1 at exit are r0⊥ = (y0 − y) cos θ0, r0‖ = (y0 − y) sin θ0 and
p0⊥ ≃ −pF(θ − θ0).
while correlated or not: for weak localization the trajectories escape independently, while
for coherent backscattering the trajectories escape together in a correlated manner.
1. Weak localization
The weak localization contribution to reflection, Rwl, is derived in the same manner as
gwl, replacing however a factor of WR/(WL +WR) with WL/(WL +WR). One obtains
Rwl = − N2L
NL +NRexp[−τ cl
E /τD] (5.17)
2. Coherent backscattering
Contributions to coherent backscattering are shown in Fig. 20b, with Fig. 22 showing the
trajectories in the correlated region in more detail. These contributions require special care
because (i) their action phase difference δScbs is not given by the expression used so far and
(ii) injection and exit positions and momenta are correlated.
From Fig. 22, and noting that γ2b decays exponentially towards γ1a, we find the action
difference between these two path segments to be
S2b − S1a = pF(y0 − y) sin θ0 + 12mλ(y0 − y)2 cos2 θ0 . (5.18)
We have dropped cubic terms which only give ~-corrections to the stationary-phase integral.
The action difference between γ2a and γ1b has the opposite sign for y0 − y and θ0 replaced
68
by θ. We get for the total action difference, in terms of (r0⊥, p0⊥),
δScbs = −(p0⊥ +mλr0⊥)r0⊥ . (5.19)
The coherent backscattering contribution to the reflection reads
Rcbs = (2π~)−2
∫
L
dY0dY
∫ ∞
0
dt 〈P (Y,Y0; t)〉 Re[
eiδScbs/~]
. (5.20)
To perform the average we define T ′W (r0⊥, p0⊥) and T ′
L(r0⊥, p0⊥) as the times for which the
perpendicular distance between the γ1a and γ1b is W and L, respectively. For times less
than T ′W (r0⊥, p0⊥) the escape probability for two trajectories is the same as for one, while
for times longer than this the trajectories evolve and escape independently. For Rcbs we
consider only those trajectories that form a closed loop, however they cannot close until the
two trajectory segments are of order L apart. The t-integrals must have a lower cut-off at
2T ′L(r0⊥, p0⊥), hence
∫
R
dY
∫ ∞
2T ′
L
dt〈P (Y,Y0; t)〉 = pF cos θ0NL
NL +NRexp[−T ′(r0⊥, p0⊥)/τD], (5.21)
where T ′((r0⊥, p0⊥) = 2T ′L((r0⊥, p0⊥)− T ′
W ((r0⊥, p0⊥). For small (p0⊥ +mλr0⊥) we estimate
T ′(r0⊥, p0⊥) ≃ λ−1 ln
[
W (p0⊥ +mλr0⊥)
mλL2
]
. (5.22)
We substitute the above expression into Rcbs, write pF cos θ0dY0 = dy0d(pF sin θ0) =
dr0⊥dp0⊥ [35], and then make the substitution p0 = p0⊥ + mλr0⊥. We evaluate the
r0⊥−integral over a range of order WL, take the limits on the resulting p0-integral to in-
finity and write it in terms of Euler Γ-functions. Finally we systematically drop all terms
O(1) inside logarithms. The result is that
Rcbs =NL
NL +NRexp[−τ cl
E /τD] . (5.23)
Thus we see that coherent backscattering is also suppressed exponentially in exactly the same
manner as weak localization. Furthermore Rcbs +Rwl = −gwl and unitarity is preserved.
3. The off-diagonal nature of coherent backscattering
Continuous families of trajectories that are present in open chaotic systems, such as γ2
and γ1 in Fig. 22, have an action difference given in Eq. (5.19). This action difference
69
does not fluctuate under energy or sample averaging, moreover, these contributions are not
diagonal in the lead mode basis. The stationary phase integral over such trajectories is
dominated by p0⊥ ≃ −mλr0⊥ where r0⊥ is integrated over the total lead width. Thus
p0⊥ varies over a range of order mλW ≫ 2π/W , coupling to many lead modes. Such
contributions were not taken into account in the previous analysis of coherent backscattering
[37]. This caused the erroneous belief (of the authors of the present manuscript amongst
others) that coherent backscattering originates from trajectories that return to any point in
the L lead with θ ≃ ±θ0, which would imply that coherent backscattering is independent of
the Ehrenfest time.
Once we correctly sum the many off-diagonal contributions to∑
nm |rnm|2 which have an
encounter near the L lead, we conclude that coherent backscattering approximately doubles
the weight of all returning trajectories in a strip defined by
θ − θ0 ≃ −p−1F mλ(y − y0) cos θ0 (5.24)
across the lead. This strip sits on the stable axis of the classical dynamics, with a width in
the unstable direction of order ~(pFW )−1. Therefore, trajectories in the strip first converge
toward each other, and only start diverging at a time of order τ opE /2. Such trajectories
cannot form a loop on times shorter than τ opE + τ cl
E .
D. Magnetoconductance
A weak magnetic field B has very little effect on the classical dynamics. Its dominant
effect is to generate a phase difference between two trajectories that go the opposite way
around a closed loop. This phase difference is AloopB/φ0 where Aloop is the directed area
enclosed by the loop, and φ0 is, as usual, the flux quantum. To incorporate this in the
theory we must introduce a factor of exp[iAloopB/φ0] into I(Y0, ǫ) in Eq. (5.12) and inside
the average in Rcbs in Eq. (5.20). To average the additional phase 〈exp[iAloopB/φ0]〉, we
divide the loop into two parts — the correlated part (within TL(ǫ)/2 of the crossing), and
the uncorrelated part (the rest of the loop). We average the two parts separately.
For the uncorrelated part, we use the fact that the distribution of area enclosed by classical
scattering trajectories in a chaotic system is Gaussian with zero mean and a variance which
70
increases linearly with time [35]. One then has,
〈eiAuncorrBφ0〉 = exp[−A2B2 (t2 − t1 − TL(ǫ))/αφ20τf ], (5.25)
where, A is the area of the cavity, α a system-dependent parameter of order unity, and τf
the time of flight between two consecutive bounces at the cavity’s wall.
We first briefly comment on the correlated part, i.e. we analyze the part of the area
enclosed by a loop-forming trajectory when it is in the correlation region close to the crossing,
i.e. within TL(ǫ/2) of the crossing. The situation is depicted in Fig. 23. We consider a loop
formed after N bounces at the cavity’s wall. In the correlation region, the segment of the
trajectory between the (n− 1)th and nth collision with the cavity walls, is highly correlated
with the segment between the N − n+ 1st and the N − n+ 2nd collision. We consider the
directed area An enclosed by these two segments (dashed region in Fig. 23). We assume
that |An| is uncorrelated with |Am6=n|, and take each such area from a Gaussian distribution.
The typical perpendicular distance between the trajectories at time t (measured from the
crossing) is ±vFǫλ−1 sinh(λnτ0). Thus we assume 〈An〉 = 0 and
〈A2n〉 =
[
v2Fǫλ
−1
∫ nτ0
(n−1)τ0
dt sinh(λt)]2
. (5.26)
This grows exponentially with n. Thus the sum over the An’s is dominated by the largest
of them with nmax = TL(ǫ)/2τf , whose variance is ∼ A2. Anticorrelations in the signs of
consecutive directed areas in the correlated region further reduce the total directed area.
The flux enclosed in the correlated region is thus at most ≃ A2B2/φ20. This is smaller than
the flux enclosed in the uncorrelated region by a factor τf/τD ≪ 1.
The correlated part thus provides at most only small corrections O(τf/τD) which we
henceforth ignore. Multiplying the integrand in Eq. (5.12) with (5.25), and integrating over
t1, t2 gives
⟨
I(Y0, ǫ)⟩
=(vFτD)2
πApF sin ǫ cos θ0
NR
NL +NR
exp[−TL(ǫ)/τD]
1 + α−1(φ/φ0)2(τD/τf). (5.27)
After a similar analysis for Rcbs, we conclude that for finite flux, the quantum corrections
to the average conductance acquire a Lorentzian shape
71
flux
ε
~
bounce 1bounce 2
bounce 3
wheren= 3
bounce Nbounce N−1
n
bounce N−2
Figure 23: A sketch of the area enclosed by the correlated part of the weak localization loop. The
area An is defined by the segment of the loop-forming trajectory between the (n − 1)th and nth
collision and the segment between the N −n + 1st and the N −n + 2nd collision. (dashed region).
gwl(Φ) = − NLNR
NL +NR
exp[−τ clE /τD]
1 + α−1(φ/φ0)2(τD/τf), (5.28)
Rwl(Φ) = − N2L
NL +NR
exp[−τ clE /τD]
1 + α−1(φ/φ0)2(τD/τf), (5.29)
Rcbs(Φ) =NL
NL +NR
exp[−τ clE /τD]
1 + α−1(φ/φ0)2(τD/τf). (5.30)
Interestingly enough, there is no Ehrenfest dependence in the width of the Lorentzian. At
zero Ehrenfest time, these results reproduce the random matrix theory result, Eq. (1.26).
E. Spin-orbit interaction and weak antilocalization
Recent experiments on lateral quantum dots have shown the presence of spin-orbit inter-
actions in chaotic ballistic systems based on semiconductor heterostructures. It is therefore
of interest to try and investigate the effect of spin-orbit interaction on transport. There
are (at least) two goals one might pursue. The first one is to understand the effect of spin-
orbit interaction on charge transport (for instance weak localization), and the second one to
capture spin-dependent transport properties (spin currents and accumulations for instance).
Here we restrict ourselves to the effect that spin-orbit interaction has on charge transport.
72
Figure 24: Semiclassical transmission of an electron across a ballistic cavity with spin orbit inter-
action. The electron bounces back and forth at the cavity’s walls, each bounce changing its mo-
mentum direction. Spin-orbit interaction gives an effective constant magnetic field around which
the electron’s spin precesses between two bounces, however, the direction of this field changes at
each bounce. For large enough dwell time through the cavity, the electron has spent enough time
subjected to spin-orbit interaction and has made enough bounces that its spin is randomized.
Our discussion follows more or less faithfully Refs. [44, 45].
The presence of spin-orbit interaction introduces a new time scale, the spin-orbit time
τso, giving the time it takes for spin-orbit interaction to randomize the electronic spin. We
consider the regime τf ≪ τso ≪ τD, in which case one can assume, in semiclassical language,
that the only effect of spin orbit interaction is to rotate the spin along otherwise unchanged
classical trajectories. Then the semiclassical transmission amplitude of Eq. (5.1) is extended
to
tji = −(2πi~)−1/2
∫
L
dy0
∫
R
dy∑
γ
(dpy/dy0)1/2γ 〈j|y〉〈y0|i〉 exp[iSγ/~ + iπµγ/2] Uγ , (5.31)
where the SU(2) matrix Uγ rotates the electron’s spin along the trajectory γ. Under our
standard semiclassical assumptions [as discussed above Eq. (5.2)] one next gets the conduc-
tance as
Tr[t†t] =1
(2π~)
∫
L
dy0
∫
R
dy∑
γ1,γ2
Aγ1Aγ2eiδS/~ Tr[Uγ1 U
†γ2] . (5.32)
Here, g gives the conductance expressed in units of e2/h and not 2e2/h.
73
Calculating the conductance proceeds along the same lines as without spin-orbit inter-
action. The key point is that an average over Tr[Uγ1 U†γ2] is performed, where γ1 and γ2
are related by the pairing under consideration. There are two limits that are analytically
tractable (i) in absence of spin-orbit interaction, both matrices Uγ1,2 = 1, and (ii) for strong
enough spin-orbit interaction that the spin rotational symmetry is broken (meaning that
τso ≪ τD), the average is performed over the SU(2) group. In both instances, the average
over the spin part factorizes from the calculation of the orbital part. It is important to
note that the semiclassical limit we have in mind here affects only the orbital part of the
electronic’s wavefunction, i.e. we are not considering the limit of large spin. Our electrons
still have spin s = 1/2.
One first considers the diagonal approximation. From the unitarity of the matrix Uγ one
has Tr[Uγ U†γ ] = 1 for electrons (more generally 2s+1, with the spin s of the charge carriers).
For s = 1/2, one recovers the Drude conductance of Eq. (5.3),
〈g〉D = NLNR/(NL +NR). (5.33)
It is seen that spin-orbit interaction has no effect on the classical part of the conductance.
For the weak localization correction, the orbital part is calculated in the same way as
above, and the resulting contribution has to be multiplied by the spin part which reads
Tr[Uγ1U†γ2] = Tr[U2
loop], (5.34)
since Uγ1 = UγLUloopUγR and Uγ2 = UγLU†loopUγR, with UγL,R giving the spin rotation along
the left/right leg part of the weak localization trajectory. For unbroken spin rotational
symmetry (absence of spin-orbit interaction), one has Tr[U2loop] = 2s+1. However, once spin
rotational symmetry is broken, the average over the spin SU(2) group reads [45]
Tr[U2loop] = −1. (5.35)
The sign of the weak localization correction to the conductance is reversed, and its amplitude
is divided by two for s = 1/2 particles.
Of particular interest is of course the crossover regime when the spin-orbit interaction
is not quite strong enough to completely destroy spin rotational symmetry. In this case,
one can still semi-analytically use semiclassics as an input formula for a trajectory-based
numerical calculation of the conductance where the computer determines trajectories to take
74
Figure 25: Left panel: numerical simulations showing the crossover from weak localization (topmost
curve) to weak antilocalization (bottommost curve); right panel: experimental data illustrating
the presence of spin-orbit interaction in ballistic quantum dots (the original figure has been turned
upside-down to facilitate comparison with the left panel). The bottom, largest dot has a dwell
that exceeds/is comparable to the spin-orbit time. Consequently, its magnetoresistance exhibits a
nonmonotonous behavior typical of systems in the crossover regime between β = 1 and β = 4. Left
figure adapted from Ref. [44], right figure adapted from Ref. [46]. Both are courtesy of K. Richter.
into account, and calculates the conductance from a semiclassically derived sum over those
trajectories. Some results are shown on Fig. 25
F. Universal conductance fluctuations
Conductance fluctuations are perhaps, besides weak localization, the most celebrated
quantum corrections to transport, and we here briefly comment on their semiclassical cal-
culation. A full, rigorous derivation can be found in Ref. [47]. Here we restrict ourselves to
the universal regime with τE/τD ≪ 1, and will comment on the deep semiclassical limit at
the end of this chapter.
75
γ1 γ1
γ2γ2γ2γ2
γ3
γ3γ4γ4γ4γ4
y01 y01
y03
y03
y1 y1
y3
y3
WW L L
Figure 26: Four-trajectory pairings giving the two dominant contributions to the conductance
fluctuations in a ballistic system with time reversal symmetry. When time-reversall symmetry is
broken, the right contribution vanishes, since on the loop γ1 and γ4, on one hand, and γ2 and γ3,
on the other hand, travel in opposite directions, and the total contribution becomes sensitive to a
flux piercing the loop.
The scattering approach to transport expresses the variance of the conductance as
varg =∑
ij
∑
mn
〈|tji|2|tnm|2〉 − 〈∑
ij
〈|tji|2〉2. (5.36)
There are four trajectories to take care of – the problem is significantly more complicated
than weak localization. Without much ado, we show in Fig. 26 the four trajectory pairing
one has to consider.
At first glance it looks like a hard task to try and calculate the contribution from the
diagram shown on Fig. 26. Life gets much easier, however, once one realizes that these
contributions factorize into products of two weak-localization contributions. Indeed, legs
coming from each encounter have the ergodic probability ∝ (2πA)−1 to connect after a
time TL(ǫ) – but then it also follows from ergodicity that this probability is the same as
〈P (R2,R1; t)〉, given above Eq. (5.14). One thus gets that both contributions on Fig. 26
gives a contribution[
NLNR
(NL +NR)2
]2
, (5.37)
in presence of time-reversal symmetry. Breaking time-reversal symmetry with an external
magnetic field does not affect the left-hand contribution on Fig. 26 –the flux phase accu-
mulated by γ1 is cancelled by γ4 and the same applies to the pair (γ2, γ3) (in that sense,
this contribution is not exactly the product of two weak localization contributions). The
situation is different for the second contribution to varg, where (γ1, γ3) and (γ2, γ4) form
closed loops (up to the encounter) in opposite direction. Taking complex conjugation into
76
+
+ = 0
γ1
γ1
γ1
γ2γ2
γ2γ2
γ2γ2
γ3
γ3
γ3
γ4γ4
γ4γ4
γ4γ4
y01
y01
y01
y03
y03
y03
y1
y1
y1
y3
y3
y3
W
W
W
L
LL
Figure 27: Correlated four-trajectory pairing that exactly cancel in a ballistic system ideally cou-
pled to external leads. The cancellation is no longer exact when the coupling between cavity and
leads is non-ideal (tunnel barriers).
account, one concludes that this second contribution has the square of the flux sensitivity
of weak localization, Eqs. (5.28)–(5.30). The variance of the conductance finally reads
varg =
[
NLNR
(NL +NR)2
]2[
1 +
(
1
1 + α−1(φ/φ0)2(τD/τf)
)2]
. (5.38)
This result is correct and agrees with random matrix theory, Eqs. (1.24) and (1.27). One
sees that breaking time-reversal symmetry only reduces the conductance variance by a factor
of two – unlike weak localization, there is a ”flux-resistant” contribution to varg. Two
important questions have to be asked (and answered!) here. First, the contributions we
just calculated are not the only ones, and Fig. 27 gives other contributions to be calculated.
Does that modify our result ? Second, what happens at finite Ehrenfest time ?
The first question is straightforwardly answered. Under our ergodic assumptions, the
three contributions of Fig. 27 factorize, two of them as product of a coherent backscattering
contribution times a weak localization correction, and the third one as the product of two
backscattering contributions. From Eqs (5.28) - (5.30) one directly obtains that these three
contributions cancel exactly.
The answer to the second question is that nothing happens to the conductance fluctu-
ations at large Ehrenfest time – at least not if one considers parametric fluctuations, i.e.
under variation of a quantum parameter such as magnetic field or chemical potential. This
is puzzling, given that weak localization disappears ∝ exp[−τ clE /τD]. As a matter of fact,
77
all contributions just discussed disappear as ∝ exp[−2τ clE /τD] – this follows from the fac-
torization argument – but this damping is exactly compensated by the emergence of new
contributions, coupling transport to density of states fluctuations. At zero Ehrenfest time,
and for ideal coupling between the leads and the cavity, density of states and conductance
do not couple. This is no longer the case at finite τE/τD, even without tunnel barriers. The
semiclassical theory for conductance fluctuations, including the finite Ehrenfest time regime
has been developed by Brouwer and Rahav [47] and we refer the reader to that article for
more details.
G. Shot-noise
Shot noise has been introduced above in paragraph IE. We are interested here in the
zero-frequency shot-noise in a two-terminal set-up. This is quantified by the Fano factor
F ≡ P (ω = 0)/P0, giving the ratio of shot-noise to Poissonian noise. According to the
scattering approach to transport the Fano factor reads
F ≡ P (ω = 0)
P0
= Tr[t†t(1 − t†t)]/Tr[t†t] . (5.39)
In random matrix theory, one obtains
Frmt =NLNR
(NL +NR)2(5.40)
It was predicted by Agam et al. [48] that this value is further reduced when the Ehrenfest
time is finite,
F = Frmt exp(−τE/τD). (5.41)
The RMT value has been observed by Oberholzer et al. in shot-noise measurements on
lateral quantum dots [49]. The same group later observed that the shot noise deviates from
the universal RMT result when the system is opened up (which reduces τD/τE) [50]. We
here confirm Eq. (5.41) and present the semiclassical theory of Ref. [51] for the Fano factor
F .
To calculate the F , one needs to calculate the conductance g = Tr[t†t], as well
as Tr[t†tt†t]. Compared to the conductance, the main new difficulty is to calculate
Tr[t†tt†t] =∑
i,j,q |tj,i|2|tq,i|2 +∑
i,j,p |tj,i|2|tj,p|2 +∑
i6=p;j 6=q t∗j,itj,pt
∗q,ptq,i.
78
01
y03 y1
y3
D2 D3 D4
y01y03
y3
y1
y03
y01y3
y1
(a) contributioncorrelated region
γ4
γ2γ3
γ1
W Wε
W Wγ4
γ2 γ1γ3
correlated region
(b) & contributions (c) contribution
γ4 γ3γ1γ2
W
correlated region
W
D1y
Figure 28: (Color online) The four dominant contributions to Tr[t†tt†t]. Paths are paired every-
where except at encounters where two of them (γ1, γ3) cross each other (solid lines) while the
other two (γ2, γ4) avoid the crossing (dashed lines). (a) Contribution D1 has uncorrelated escape
on both sides of the encounter. (b) Contribution D2 and D3 have correlated escape only on one
side of the encounter. (c) Contribution D4 has correlated escape on both sides.
In our semiclassical derivation, following the lines of Ref. [51], we find the dominant
contributions to F from the path pairings shown in Fig. 28. These pairings are similar to
those considered in Ref. [52] for quantum graphs. However, unlike quantum graphs, chaotic
systems have continuous families of scattering trajectories with similar actions, which means
in particular that we cannot make a diagonal approximation to evaluate the contributions
D2 and D3 shown in Fig. 28.
Semiclassically, we write Tr[t†tt†t] as a sum over four paths, γ1 from y01 to y1, γ2 from
y03 to y1, γ3 from y03 to y3 and γ4 from y01 to y3,
Tr[t†tt†t] =1
(2π~)2
∫
L
dy01dy03
∫
R
dy1dy3
∑
γ1,···γ4
Aγ4Aγ3Aγ2Aγ1 exp[iδS/~] . (5.42)
Here, Aγ = [dpy/dy0]1/2γ and δS = Sγ1−Sγ2 +Sγ3−Sγ4 (we absorbed all Maslov indices into
the actions Sγi). We are interested in quantities averaged over variations in the energy or
the system shape. For most contributions, δS/~ oscillates wildly with these variations. The
dominant contributions that survive averaging are those for which the fluctuations of δS/~
79
are minimal. They are shown in Fig. 28. Their paths are in pairs almost everywhere except
in the vicinity of encounters. Going through an encounter, two of the four paths cross each
other, while the other two avoid the crossing. They remain in pairs, though the pairing
switches, e.g. from (γ1; γ4) and (γ2; γ3) to (γ1; γ2) and (γ3; γ4) in Fig. 28a. Paths are
always close enough to their partner that their stability is the same. Thus, for all pairings
in Fig. 28,∑
γ1,...γ4
Aγ4Aγ3Aγ2Aγ1 →∑
γ1,γ3
A2γ3A
2γ1. (5.43)
We define P (Y,Y0; t)δyδθδt as the product of the momentum along the injection lead,
pF cos θ0, and the classical probability to go from an initial position and angle Y0 = (y0, θ0)
to within (δy, δθ) of Y in a time within δt of t. Then the sum over all paths γ from y0 to y
is
∑
γ
A2γ [· · · ]γ =
∫ ∞
0
dt
∫
dθ0
∫
dθ P (Y,Y0; t) [· · · ]Y0 . (5.44)
For an individual system, P has δ-functions for all classical trajectories. However averaging
over an ensemble of systems or over energy gives a smooth function
〈P (Y,Y0; t)〉 =pF cos θ0 cos θ
2(WL +WR)τDexp[−t/τD] . (5.45)
Using Eqs. (5.44) and (5.45) to calculate the conductance within the diagonal approximation
directly leads to the Drude conductance 〈Tr[t†t]〉 ≃ gD = NLNR/(NL + NR). This level of
approximation for 〈Tr[t†t]〉 is sufficient to obtain F to leading order in N−1L,R. We now use
Eqs. (5.42), (5.43) and (5.44) to analyze the contributions in Fig. 28.
There are two things that can happen to two pairs of paths as they leave an encounter.
The first is uncorrelated escape. The pairs of paths escape when the perpendicular distance
between them is larger than WL,R, which requires a minimal time TW (ǫ)/2 = λ−1 ln[ǫ−1W/L]
between encounter and escape. The two pairs of paths then escape in an uncorrelated
manner, typically at completely different times, with completely different momenta (and
possibly through different leads). The second is correlated escape. Pairs of paths escape
when the distance between them is less than WL,R, then the two pairs of paths escape
together, at the same time through the same lead.
Taking into account the two escape scenarios just described, we write 〈Tr[t†tt†t]〉 =
80
D1 +D2 +D3 +D4. Each of these four contributions, sketched in Fig. 28, can be written as
Di =1
(2π~)2
∫
L
dY01 dY03
∫
R
dY1 dY3
∫
dt1 dt3
× 〈P (Y1,Y01; t1) P (Y3,Y03; t3)〉 exp[iδSDi/~] , (5.46)
where subscripts 1, 3 make the connection to Fig. 28. When evaluating Eq. (5.46) the joint
exit probability for two crossing paths has to be computed.
To evaluate D1, we have to take into account that paths in the same region of phase-
space (shaded areas in Fig. 28) have highly correlated escape probabilities (this point was
already mentioned in our discussion of weak localization). Here the action difference is
δSD1 = EFǫ2/λ [36, 37], where ǫ is the crossing angle shown in Fig. 28a. We write
P (Yi,Y0i; ti) =
∫
dRiP (Yi,Ri; ti − t′i)P (Ri,Y0i; t′i) ,
where P is the probability for the classical path to exist (not multiplied by the injection
momentum), and Ri is a point in the system’s phase-space (ri, φi) visited at time t′i, with
φi giving the direction of the momentum. We choose R1 and R3 as the points at which the
paths cross, so R3 = (r1, φ1 ± ǫ) and dR3 = v2F sin ǫdt′1dt
′3dǫ. Thus
D1 = 2(2π~)−2
∫
L
dY01dY03
∫ π
0
dǫ Re[
eiδSD1/~]⟨
I(Y01,Y03; ǫ)⟩
. (5.47)
I(Y01,Y03; ǫ) is related to the probability that γ3 crosses γ1 at angle ±ǫ. Its average is
independent of Y01,03, so 〈I(Y01,Y03; ǫ)〉 = 〈I(ǫ)〉. For D1, injections/escapes are more
than TW (ǫ)/2 from the crossing, so
〈I(ǫ)〉 = 2v2F sin ǫ
∫
R
dY1dY3
∫
dR1
∫ ∞
T
dt1
∫ t1−T/2
T/2
dt′1
∫ ∞
T
dt3
∫ t3−T/2
T/2
dt′3
×⟨
P (Y1,R1; t1 − t′1)P (R1,Y01; t′1) P (Y3,R3; t3 − t′3)P (R3,Y03; t
′3)⟩
, (5.48)
where T is shorthand for TW (ǫ). We next note that within TW (ǫ)/2 of the crossing, paths
γ1 and γ3 are so close to each other that their joint escape probability is the same as for a
single path. Elsewhere γ1, γ3 escape independently through either lead at anytime, hence
⟨
I(ǫ)⟩
=p4
FτDπ~m
N2R cos θ01 cos θ03 sin ǫ
(NL +NR)3e−TW (ǫ)/τD , (5.49)
where we used NR = (π~)−1pFWR, and assumed, as usual, that the probability that γ3 is
at R3 at time t′3 in a system of area A is (2πA)−1 = m[2π~τD(NL + NR)]−1. Then the
81
Y01,03-integral in Eq. (5.47) gives (2WL)2, while the ǫ-integral is dominated by ǫ ≪ 1 and
yields a factor of −π~(2EFτD)−1e−τopE /τD
1 + O[(λτD)−1]
[43]. Thus
D1 = −N2LN
2R(NL +NR)−3 exp[−τ op
E /τD] . (5.50)
The contribution D2 is shown in Fig. 28b, with Fig. 22 showing the paths in the correlated
region in more detail (the correlated region is the same for weak localization as for theD2 and
D3 contributions to shot-noise). In the same way as in section VC2 for weak localization,
we express the action difference in terms of (r0⊥, p0⊥) as
δSD2 = −(p0⊥ +mλr0⊥) r0⊥ , (5.51)
with the same level of accuracy as in Eq. (5.19). We next perform the average in Eq. (5.46).
We define T ′W (r0⊥, p0⊥) as the time for which γ1 and γ3 are less than W apart, and insist
that the paths are more than W apart before they escape to the right. Hence we must
evaluate
∫
R
dY1dY3
∫ ∞
T ′
W
dt1dt3〈P (Y1,Y01; t1)P (Y3,Y03; t3)〉
=N2
Rp2F cos θ01 cos θ03(NL +NR)2
exp[−T ′W (r0⊥, p0⊥)/τD] . (5.52)
Inserting this into Eq. (5.46), we change integration variables using pF cos θ03dY03 =
dr0⊥dp0⊥ [35], and then define p0 ≡ p0⊥ +mλr0⊥. In the regime of interest T ′W (r0⊥, p0⊥) ≃
λ−1 ln[(mλW )−1p0]. Evaluating the integral over r0⊥ leaves a p0-integral which we cast as
Euler Γ-functions. To lowest order in (λτD)−1 we find,
D2 = NLN2R(NL +NR)−2 exp[−τ op
E /τD] . (5.53)
Substituting NL ↔ NR in the derivation of Eq. (5.53) gives,
D3 = N2LNR(NL +NR)−2 exp[−τ op
E /τD] . (5.54)
The contribution D4 is shown in Fig. 28c, with Fig. 22 showing the paths in detail at the
L lead. This contribution can be evaluated in a way similar to D2, the difference being that
the paths escape before time T ′W (r0⊥, p0⊥), i.e. before becoming a distance W apart. The
paths are always correlated, so the escape probability for the two paths equals that for one.
82
Moreover, both paths will automatically escape through the same lead, hence
∫
R
dY1dY3
∫ T ′
W
0
dt1dt3⟨
P (Y1,Y01; t1)P (Y3,Y03; t3)⟩
=NRp
2F cos θ01 cos θ03NL +NR
(
1 − e−T ′
W(r0⊥,p0⊥)/τD
)
. (5.55)
Performing the same analysis as for D2 we find that
D4 = NLNR(NL +NR)−1(1 − exp[−τ opE /τD]) . (5.56)
The Fano factor is given by F = 1 − g−1D (D1 +D2 + D3 + D4). Our results of Eqs. (5.53),
(5.54) and (5.56) show that D2 + D3 + D4 = gD. One hence gets F = −D1/gD. From
Eq. (5.50), one finally recovers Eq. (5.41).
We finally confirm that our approach preserves unitarity. The unitarity of the scat-
tering matrix ensures that t†t + r†r = 1 and hence the Fano factor can be written as
F = g−1D 〈Tr[t†tr†r]〉. To calculate this expression, we first note that there is no contribution
D3 nor D4 to Tr[t†tr†r]. We are left with the calculation of two contributions, D′1 and D′
2,
obtained from D1 and D2 shown in Fig. 28a,b with y01, y03 and y3 on the left lead and y1 on
the right lead. The calculation proceeds as for D1 and D2, with one factor of NR/(NL +NR)
replaced by NL/(NL + NR) in both contributions. The sum of these two contributions is
D′1 + D′
2 = e−τopE /τDN2
LN2R(NL + NR)−3, the Fano factor is then F = (D′
1 + D′2)/gD, which
reproduces Eq. (5.41).
VI. CONCLUSION: REGIME OF APPLICABILITY OF THESE SEMICLASSICS
To this day it remains somehow underestimated that all trajectory-based semiclassical
methods used so far in the theory of transport (including in the present manuscript) are
only applicable in the regime W ≥ ~1/2eff L. Dominant off-diagonal contributions such as
those discussed above have encounters of a typical size ∼ ~1/2eff L. When W < ~
1/2eff L, the two
non-crossing paths at an encounter are a distance apart greater than W . The probability
that one of the four paths escapes while the other three paths remain in the system is of
order τE/τD, where τE ∼ λ−1 ln[~eff(L/W )2] is the time over which this path is a distance of
order W from any of the other paths. The methods presented here fails once this is taken
into account, suggesting that diffraction effects may become important. We believe that the
regime ~eff < (W/L) ≤ ~1/2eff is well described by RMT, and thus suspect this diffraction may
83
be the microscopic source of RMT universality in this regime. Clearly this regime merits
further study.
Acknowledgments
My gratitude goes to M. Buttiker and F. Mila, for inviting me to teach the series of
lectures from which these notes emerge. It is a pleasure to thank Peter Schlagheck for
very helfpul discussions on topics presented in Section III. In my works on semiclassical
theories, I have been collaborating with I. Adagideli, C. Petitjean, H. Schwefel, D. Stone
and H. Tureci – thanks to all of them. Most of all, I would like to express my friendship
and gratitude to R. Whitney, my main collaborator in these endeavors. Many thanks also
to Markus Buttiker, Rafael Sanchez and Janine Splettstosser for proofreading part or all of
this manuscript. This work has been supported by the National Science Foundation, under
grant No DMR-0706319.
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