presented at the moshe flato lecture series, ben gurion ...haldane/talks/topolint_flato.pdf · r,j...

Topological Insulators

• From abstract mathematics, to novel materials with exciting and unexpected properties that may prove to be technologically important!

• A journey guided by a series of intuitive leaps by theorists

F. Duncan M. Haldane, Princeton University,

Presented at the Moshe Flato Lecture Series, Ben Gurion University, Beer-Sheva Israel, March 10, 2011

• Starting around 1982 new mathematical ideas of quantum geometry and topology ideas have penetrated into condensed matter theory and led to new insights and ways of looking at condensed matter

• Recently they led to the discovery of the topological insulators, a new exiting class of materials that shocked material scientists and practical band-structure calculators who had studied these materials for years without noticing these properties!

Band insulators(pedagogical)• In the “clean, non-interacting limit”, a Band Insulator is

described by a one-particle Bloch bandstructure with an energy gap between the highest occupied one-particle Bloch state and the lowest empty Bloch state

H|Ψn(k) = εn(k)|Ψn(k) T (R)|Ψn(k) = eik·R|Ψn(k)

H = H†, [H,T (R)] = 0 T (R)† = T (−R)T (R)T (R) = T (R+R)

H is a Hermitian Hamiltonian that commutes with the set of Unitary translation operators T(R), where R is a d-dimensional Bravais Lattice

R ∈d

i=1

miai, m ∈ Zd

k is defined modulo a reciprocal vector G in the Brillouin zone, which is a manifold equivalent to the d-torus Td.

εn(k) ≤ εn+1(k), n = 1, 2, . . .

“tight-binding” models

• It is convenient to simplify the model by simplifying the Hilbert space in which H acts so it has a finite number of “orbitals” (basis states) in the unit cell, and only has matrix elements between “nearby” orbitals

• is embedded in d-dimensional Euclidean space at position |ψR,j

T (R)|ψR,j = |ψR+R,j

• This makes the number of Bloch bands finite. Need Na> 1 to get a non-trivial bandstructure.

xR,j = R+ aj , j = 1, Na

ψR,j |H|ψR,j = tjj(R−R)

“bonds” = non-zeromatrix elements

of the Hamiltonian

Two different choicesof unit cell

(like a gauge choice)

H(E,B) =

R,j,R,j

tjj(R−R)eiΦ(xRj ,xRj )|ψRjψRj |

eiΦ(x,x) = exp ie

tE · (x− x) + 1

2B · (x× x)

time-dependent

coupling to uniform electromagnetic field

• Two distinct pieces of information characterize the tight-binding model:

• The matrix elements of the Bloch Hamiltonian

• The embedding of the discrete real-space basis set in d-dim Euclidean space

ψR,j |H|ψR,j = tjj(R−R)

xR,j = R+ aj , j = 1, Na

• The energy bands of eigenvalues of H depend only on the matrix-elements of H, and are independent of the embedding

• The Berry curvature in the Brillouin zone of the Bloch states characterizes the response to uniform electric and magnetic fields, which cannot be included in the Bloch Hamiltonian. It depends on the embedding.

• The “Berry curvature” in k-space is a little different from its usual form, as it depends not only on the eigenstates of the Hamiltonian, but also on how it is embedded in the Euclidean space in which the electromagnetic fields are supported.

• significance of the Berry connection:

• As a connection it is not gauge invariant under the U(1) ambiguity

|Ψn(k) → eiχn(k)|Ψn(k)• The Berry curvature is invariant:

Fabn (k) = −Fba

n (k) ≡ ∇akAb

n(k)−∇bkAa

n(k)

DakΦn(k)|Db

kΦn(k) = 12

Gabn (k) + iFab

n (k)

Fubini-Study metric Berry curvature

• The Berry phase (factor) is invariant

eiφΓ = exp i

ΓAa

n(k)dka

closed path in k-space

Semiclassical dynamics of Bloch electrons

write magnetic flux density as an antisymmetric tensorFab(r) = !abcB

c(r)

Note the “anomalous velocity” term! (in addition to the group velocity)

Karplus and Luttinger 1954

!dka

dt= eEa(r) + eFab

drb

dt

!dra

dt= !a

k!n(k) + !Fabn (k)

dkb

dt

Fabn (k) is the Berry curvature tensor in k-space.

non-commutative geometry induced byBerry curvature

usual commutation of momentum in presence of real-space magnetic field[ka, kb] = ieFab(r)/

[ra, rb] = −iFabn (k)

Band insulators (clean case)

• Bulk energy spectrum does not distinguish non-topological from topological insulators, both have a gap.

• The topology of the Berry curvature distinguishes them

• The Berry curvature depends on the embedding, but its topological properties do not.

Fermilevel

Filledbands

gap

emptybands

The (first) Chern Invariant• Integer Quantum Hall Effect and the First

Chern invariant of a band that is non-degenerate everywhere in the BZ.

1

2π

BZd2kF12

n (k) = Cn

integer invariant

2-manifold=2D Brillouin zone

Fabn (k) = Fab

n (k)

Fabn (k) = Fba

n (k)

Fabn (k) = 0

(even), if spatial inversion symmetry is present

(odd), if time-reversal symmetry is present

if both symmetries are present

(odd), if time-reversal symmetry is present

vanishes if time-reversal symmetry is presentBerry curvature

• Landau levels plus a periodic potential

• non standard Bloch states, when flux through unit cell of periodic potential is rational p/q ( tiny Brillouin zone, 1/q x 1/q)

E

Magnetic flux quanta through unit cell

0 0.5 1

Hoffstadter“butterfly”

lowest Landau level

TKNN asked what was QHE when Fermi

level in these gaps

n=0

n=1n=2

• Integer QHE

σH =e2

2π

occ

Cn

The integer Hall quantization is the sum of Chern numbers of occupied bands

• The “Bloch bands” in this application were non-standard, because the unit cell must contain an integer number of flux quanta.

2D zero-field Quantized Hall Effect

• 2D quantized Hall effect: !xy = "e2/h. In the absence of interactions between the particles, " must be an integer. There are no current-carrying states at the Fermi level in the interior of a QHE system (all such states are localized on its edge).

• The 2D integer QHE does NOT require Landau levels, and can occur if time-reversal symmetry is broken even if there is no net magnetic flux through the unit cell of a periodic system. (This was first demonstrated in an explicit “graphene” model shown at the right.).

• Electronic states are “simple” Bloch states! (real first-neighbor hopping t

1, complex second-neighbor

hopping t2ei#, alternating onsite potential M.)

FDMH, Phys. Rev. Lett. 61, 2015 (1988).

The “Chern insulator”

• Also called “quantum anomalous Hall effect” or “zero-field” Hall effect. (FDMH 1988)

• No net magnetic flux through the unit cell, so standard Bloch states.

23

Topological Insulators in Three-Dimensions

23

Topological Insulators in Three-Dimensions

starts withtwo Dirac points, at the

two distinct cornersof the Brillouin zone (graphene)

• edge states of finite region with edges

EBerry flux π neareach gapped Dirac point at corners of Brillouin zone

gapless edge stateswhere local cherninvariant changes• effect of

disorder

localized

localized

localized

extended

extended

momentum along edge

bulk

edgevery unusualedge modeconnectingvalence and

conduction bands

empty

filled

(a conduit for spectral flow)

• The 2D chern insulator has not yet been found, but there are indications that it may be realized by spontaneous T-reversal breaking in Bilayer graphene

• However the edge state effect (not the QHE) has been demonstrated (with microwaves) in 2D photonic bandstructures with non-trivial chern invariant :

• “one-way light!”

(what you can do with electrons you can do with photons!)

3

vector from the the BZ corner, the three “free photon”TE plane waves with speed c0 split into a “Dirac-point”doublet with ! = !D ± vD|"k| + O(|"k|2), where !D =c0|K|(1!#/4 + O(#)2), vD = c0/2 + O(#), and a singlet! = !0 + O(|"k|2), !0 = c0|K|(1 + #/2 + O(#2)).

We now perturb the Dirac points by a Faraday term(which explicitly breaks time-reversal symmetry), withan axis normal to the xy plane, added to the permittivitytensor: $xy = !$yx = i$0$%(r, !), where

%(r, !) = %0(!) + %1(!)VG(r); (7)

%0(!), %1(!) are real odd functions of !. We assume that,for ! " !D, |%0(!)|, |%1(!)| # |#| # 1, with negligiblefrequency-dependence. The Dirac points now split, withdispersion ! = !D ± vD(|"k|2 +&2)1/2, where, to leadingorder in %, & = |K|(3

2%1(!D) ! 3#%0(!D)).For small &, the Berry curvatures of the upper and

lower kz = 0 bands near the split Dirac points are

F xy± ("k) = ± 1

2&!

|"k|2 + &2"!3/2

. (8)

There is a total integrated Berry curvature of ±' neareach Dirac point, giving total Chern numbers ±1 for thesplit bands. By inversion symmetry, the Berry curvaturesat the two Dirac points have the same sign; if the gapwas opened by broken inversion symmetry, with unbro-ken time-reversal invariance, they would have oppositesign, and the Chern number would vanish.

We now consider an adiabatically spatially-varyingFaraday term parameterized by a &(r) that is positivein some regions and negative in other regions. The split-ting of the Dirac points vanishes locally on the line where&(r) = 0. It is necessary that, in the perfectly periodicstructure with & = 0, there are no photonic modes atother Bloch vectors that are degenerate with the modesat the Dirac points.

Such frequency-isolation of the Dirac points cannot oc-cur in the weak-coupling “nearly-free photon” limit, butcan be achieved, at least for kz = 0 modes, in hexagonalarrays of infinitely long dielectric rods parallel to the zaxis. An example can be seen in Fig.(1a) of Ref.[9]. Thatfigure was exhibited to demonstrate a frequency gap be-tween the first and second TE bands, but incidentallyalso shows that the second and third TE bands are sep-arated by a substantial gap except in the vicinity of theBZ corners, where they touch at Dirac points. The cor-responding TM bands were not given in Ref.[9], but wefound that the Dirac-point frequency !D is also insidea large gap of the TM spectrum (see Fig.(1)). When aFaraday term is added, the bands forming the Dirac pointin Fig.(1) split apart, and each now non-degenerate bandwill have associated with it a non-zero Chern number (seeRef. [3]).

The Faraday e!ect incorporated to the hexagonal ar-ray of rods explicitly breaks time-reversal symmetry on

FIG. 1: Photon bands for kz = 0 electromagnetic waves prop-agating normal to the axis of a hexagonal 2D array of cylin-drical dielectric rods; a is the lattice constant. As in Fig.(1a)of Ref.[9], the rods fill a fraction f = 0.431 of the volume,with dielectric constant ! = 14, and are embedded in an ! =1 background. The lowest five 2D bands are well-separatedfrom higher bands, except near a pair of “Dirac points” atthe two distinct Brillouin zone corners (J).

the scale of the unit cell of the metamaterial: the permit-tivity tensor acquires an imaginary, o!-diagonal compo-nent having the periodicity of the unit cell, as describedabove. A hexagonal array consisting of a material hav-ing a large Verdet coe"cient, such as a rare-earth garnetwith ferromagnetically-ordered domains would give riseto such an e!ect.

While these kz = 0 Dirac-point modes are not degen-erate with any other kz = 0 modes, they are degeneratewith kz $= 0 modes. To fully achieve a “one-way” edge-mode structure, the light must also be confined in thez-direction, with Dirac points at a frequency that is non-degenerate with any other modes. To design such struc-tures, it will be necessary to vary the filling factor of therods along the z-direction, so that light remains confinedto regions of relatively larger filling factors. The technicalchallenge would be to vary the filling factors without in-troducing any modes into the bulk TE gaps surroundingthe Dirac points.

Let |u!(±kD)%, ( = ±, be the degenerate solutionsof (1) at a pair of isolated Dirac points, normalizedso &u!(±kD)|B0(!D)|u!!(±kD)% = B0"!!! . Now adda Faraday perturbation "B(r, !): in degenerate pertur-bation theory, normal modes with small "! = ! ! !D

have the form#

!,± )±! (r)U(±kD, r)u!(±kD, r). For

slow spatial variation, there is negligible mixing betweenmodes at di!erent Dirac points, and )±

! (r) is the solutionof

$

!!

(!iJa"'a ! !D"B(r))±!!! )±

!! (r) = "!B0)±! (r),

(9)

4

where Ja! and !B(r) are 2 ! 2 matrices given by

(Ja!)±!!! = "u!(±kD)|Ja|u!!(±kD)#, a = x, y,

(!B(r))±!!! = "u!(±kD)|!B(r, "D)|u!!(±kD)#.(10)

For a straight-line interface, this equation has the formvDK|## = !"|##, with vD > 0, and

K = $i!x%x + !k"!y + $(x)!z , (11)

where !a are Pauli matrices. Here kDy + !k" is the con-served Bloch vector parallel to the interface; we take $(x)to be monotonic, with $(x) & ±$# as x & ±'.

It is instructive to first consider the exactly-solvablecase $(x) = $# tanh(x/%), % > 0, where K2 is essen-tially the integrable Poschl-Teller Hamiltonian[10]. Thespectrum of modes bound to the interface is

"0(!k") = "D + s"vD!k", s" ( sgn($#), (12a)

"n±(!k") = "D ± vD

!

!k2" + $2

n

"1/2, n > 0, (12b)

with |$n| < |$#|; for the integrable model, $2n is given

by 2n|$#|/%, n < |$#|%/2. There is always a unidirec-tional n = 0 mode with speed vD and a direction deter-mined by the sign of $#; in the small-% (or sharp-wall)limit |$#|% < 2, this is the only interface mode.

Let &($2) be the dimensionless area in the x-kx

phase-plane enclosed by a closed constant-frequency orbit(kx)2 + ($(x))2 = $2 < |$#|2, corresponding to a boundstate. For the integrable model, this has the simple form&($2) = '$2%/|$#|; the n > 0 bidirectional modes thussatisfy a constructive-interference condition

&($2n) = 2'n. (13)

This contrasts with the usual “semiclassical” condition &= 2'(n + 1

2 ); the change is needed for the n = 0 “zeromode” (12a) to exist, and can be interpreted as derivingfrom an extra Berry phase factor of $1 because the orbitencloses a Dirac degeneracy point at (x, kx) = (0, 0). Forgeneral $(x), the n = 0 eigenfunction is

#0!(r) ) (!(s") exp

#

i!k"y $ s"

$ x

$(x$)dx$

%

, (14)

!y"(s) = s"(s). For slowly-varying $(x), the condition(13) will determine $2

n for any n > 0 interface modes.Since there are two Dirac points, there are two such

unidirectional edge modes at a boundary across whichthe Faraday axis reverses. The crucial feature is thatboth modes propagate in the same direction, and cannotdisappear, even if the interface becomes sharp, bent, ordisordered. As in the QHE, the di!erence between thenumber of modes moving in the two directions along theinterface is topologically determined by the di!erence ofthe total Chern number of bands at frequencies below

the bulk photonic band gap in the regions on either sideof the interface; in this case |"C(1)| = 2.

For |!"| < vD|$#| a Faraday interface has no counter-propagating modes into which elastic backscattering cantake place, so the “one-way waveguide” that it forms isimmune to localization e!ects, just like electronic trans-port in the QHE. In the QHE, the number of electrons isstrictly conserved; in photonics, the photons only propa-gate ballistically if absorption and non-linear e!ects areabsent. These e!ects do allow degradation of the elec-tromagnetic energy current flowing along the interface,so the analogy with the QHE is not perfect.

Even if a 2D metamaterial with isolated Dirac pointscan be designed, the problem of finding a suitablemagneto-optic material to provide the Faraday e!ectmust be addressed. The e!ect must be large enough toinduce a gap that overcomes the e!ect of any inversion-symmetry breaking. The parameter |$#| is the inverselength that controls the width of the unidirectionally-propagating channel (and the unidirectional frequencyrange); in order to keep the wave confined to the inter-face, and prevent leakage, the Faraday coupling must bestrong enough so that this width is significantly smallerthan the physical dimensions of the sample of metama-terial.

In summary, we have shown that analogs of quantumHall e!ect edge modes can in principle occur in two di-mensional photonic crystals with broken time-reversalsymmetry. The electromagnetic energy in these modestravel in a single direction. Explicit theoretical examplesof such modes have been constructed in Ref. ([3]). Suchquasi-lossless unidirectional channels are a novel possibil-ity that might one day be physically realized in “photonicmetamaterials” with non-reciprocal constituents.

This work was supported in part by the U. S. NationalScience Foundation (under MRSEC Grant No. DMR02-13706) at the Princeton Center for Complex Materials.Part of this work was carried out at the Kavli Institutefor Theoretical Physics, UC Santa Barbara, with supportfrom KITP’s NSF Grant No. PHY99-07949.

[1] B. I. Halperin, Phys. Rev. B 25, 2185 (1982).[2] X.-G. Wen, Phys. Rev. B 43, 11025, (1991).[3] S. Raghu and F.D.M. Haldane, cond-mat/0602501.[4] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M.

den Nijs, Phys. Rev. Lett. 49, 405 (1982).[5] B. Simon, Phys. Rev. Lett. 51, 2167 (1983).[6] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).[7] M. V. Berry, Proc. R. Soc. London A392, 45 (1984).[8] M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev.

Lett. 93, 083901 (2004).[9] M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565

(1991).[10] See e.g., L. D. Landau and E. M. Lifschitz, Quantum Me-

chanics: Non-Relativistic Theory (Pergamon Press, Ox-ford, 1977) p. 73, Problem 5.

• Proof of concept: structure was a lattice of dielectric rods; add some Faraday effect to break T-reversal

• Look at an internal edge between C= +1 and C= -1 regions with opposite Faraday effect (get 2 edge modes)

12

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

10 20 30 40 50 60 70

!a/

2"c

Kparallel

FIG. 10: The spectrum of the composite system consisting30 copies of a single hexagonal unit cell duplicated along adirection R!. Both inversion and time-reversal symmetriesare present, and the Dirac points are clearly visible. Whilethe composite system has a spectrum containing many bands,only two bands touch at the Dirac point. The dispersion iscomputed in k space along the direction parallel to the wall.

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

10 20 30 40 50 60 70

!a/2"c

Kparallel

FIG. 11: The same system as above, but with broken time-reversal symmetry without a domain wall. There is a singleFaraday axis in the rods of the entire system.

to the wall, and the states of the composite system of30 unit cells can be labeled by k!, Bloch vectors in thedirection parallel to the wall. Figures 10, 11,and 12 con-sist of a spectral series of a system without any brokentime-reversal symmetry (Fig. 10), with uniformly bro-ken time-reversal symmetry (Fig 11), and a domain wallconfiguration (Fig. 12) for the 30 unit cell composite sys-tem. The bands are plotted along a trajectory in k-spacein the k! direction which contains the two distinct Bril-louin zone corners. It is clear that in the Domain wall,there are two additional modes formed in the bandgapthat arose from the Faraday coupling. Since the domain

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

10 20 30 40 50 60 70

!a/2"c

Kparallel

FIG. 12: Same system as above, but with a domain wall in-troduced corresponding to maximum separation of the wallson the torus. The two additional modes present in the gapcorrespond to edge modes with a “free photon” linear disper-sion along the wall. There are two modes, since across thedomain wall, the Chern number of the band just below theband gap changes by 2.

walls are duplicated on the torus, the spectrum of edgemodes will also be doubled; in Fig. 12, only the two non-equivalent modes are shown. Each mode in the band gaphas a free photon linear dispersion along the direction ofthe wall; moreover, both have positive group velocities,and therefore propagate unidirectionally.

To be certain, however, that these “chiral” modes areindeed localized near the interface, we have numericallycomputed !u(r)|B"1|u(r)", the electromagnetic energydensity (the B matrix, defined in section II, is not tobe confused with the magnetic flux density), the pho-ton probability density in real space. We have computedthis quantity along with all the spectra of the compositesystem using the real space bandstructure algorithms de-scribed in Appendix B. As shown in Fig. 13, the energydensity is a gaussian function, peaked at the position ofthe domain wall, decaying exponentially away from thewall. From this calculation, we extract a localizationalso approximately 5 unit translations in the directionperpendicular to the interface.

We have therefore shown here using explicit numeri-cal examples that photonic analogs of the “chiral” edgestates of the integer quantum Hall e!ect can exist alongdomain walls of Hexagonal photonic systems with brokentime-reversal symmetry. We have studied the unphysicalcase in which such domain walls are abrupt changes in theaxis of the Faraday coupling. However, due to the topo-logical nature of these modes, a smoother domain wallin which the Faraday axis slowly reverses over a lengthscale much larger than a unit cell dimension would alsoproduce such modes. The most important requirementfor the existence of these modes, is that at some spa-tial location, the Faraday coupling is tuned across its

“one-way”edge-modes

FDMH and S. Raghu arXiv:cond-mat/0503588v2 [cond-mat.mes-hall] 23 Oct 2008

Possib

leR

ealiz

atio

nofD

irectio

nalO

ptic

alW

aveguid

es

inP

hoto

nic

Cry

stals

with

Bro

ken

Tim

e-R

eversa

lSym

metry

F.

D.

M.

Hald

ane

and

S.

Raghu

Depa

rtmen

tof

Physics,

Prin

ceton

University

,Prin

ceton

NJ

08544-0

708

(Dated

:A

ugu

st30,

2007)

We

show

how

inprin

ciple

tocon

struct

analogs

ofquan

tum

Hall

edge

statesin

“photon

iccry

stals”m

ade

with

non

-reciprocal

(Farad

ay-e!

ect)m

edia.

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form“on

e-way

wavegu

ides”

that

allowelectrom

agnetic

energy

toflow

inon

edirection

only.

PA

CS

num

bers:

42.7

0.Q

s,03.6

5.V

f

Inth

isletter,

we

describ

ea

novel

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involving

aninterface

betw

eentw

om

agneto-op

ticphoton

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(period

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etamaterials”

that

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agnetic

waves)

which

canth

eoreticallyact

asa

“one-w

ayw

aveg-uid

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ach

annel

along

which

electromagn

eticen

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canprop

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asin

gledirection

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ithno

possib

ilityof

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atben

ds

orim

per-

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irectional

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odes

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med

iath

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symm

etryin

the

metam

aterial.

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asin

the

electronic

case,every

two

dim

ension

alphoton

icban

dis

characterized

bya

topological

invariantkn

own

asth

eC

hern

num

ber[5],

aninteger

that

vanish

esid

enticallyunless

time-reversal

symm

etryis

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.If

the

material

contains

aphoton

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dgap

(PB

G),

the

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num

ber,

sum

med

overall

ban

ds

below

the

gap,

plays

arole

similar

toth

atof

the

same

quantity

sum

med

overall

occupied

ban

ds

inth

eelectron

iccase.

Inparticu

-lar,if

the

totalChern

num

ber

changes

acrossan

interfacesep

arating

two

PB

Gm

edia,

there

necessarily

will

occur

stateslocalized

toth

einterface

havin

ga

non

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current

along

the

interface[1,2].

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ephoton

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such

statesw

ould

comprise

our

“one-w

ayw

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e”.

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betw

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be

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nid

irectional

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provid

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ds

with

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ctphoton

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ithnon

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ahexagon

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ofdielctric

rods

with

aFarad

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eth

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atas

acon

sequen

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topology

ofth

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gle-particle

photon

ban

ds

inth

eB

rillouin

zone,

the

edge

statesof

lightoccu

ralon

gdom

ainw

alls(w

here

the

Farad

aye!

ectvan

ishes).

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ayseem

surp

rising

that

the

physics

ofth

eQ

HE

canhave

analogs

inphoton

icsystem

s.T

he

QH

Eis

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incom

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ds

canbe

describ

edclassically,

interm

sof

Maxw

ell’sequ

ations.

How

ever,th

einteger

QH

Ecan

inprin

ciple

occurw

ithou

tany

uniform

magn

eticflux

den

sity(ju

stw

ithbroken

time-reversalsym

metry)

ashas

explicitly

show

nby

one

ofus

ina

graphen

e-likem

odel

ofnon

-interacting

Bloch

electrons[6];

thus

Lan

dau

-levelqu

antizationis

not

anessentialrequ

irement

forth

equ

an-

tum

Hall

e!ect.

We

have

transcrib

edth

ekey

features

ofth

eelectron

icm

odel

ofR

ef.[6]to

the

photon

iccontext.

The

edge-states

area

prop

ertyof

aon

e-particle

eigenstate

prob

lemsim

-ilar

toth

eM

axwell

norm

al-mod

eprob

lem,

soare

repli-

catedin

the

photon

icsprob

lem.

(The

QH

Eitself

has

no

photon

ican

alog,as

itfollow

sfrom

the

Pau

liprin

ciple

offillin

gall

one-p

articlestates

below

the

Ferm

ilevel.)

The

Maxw

ellnorm

al-mod

eprob

lemin

loss-freelin

earm

edia

with

spatially-p

eriodic

localfrequ

ency-d

epen

dent

constitu

tiverelation

sis

agen

eralizedself-con

sistentH

er-m

itianeigen

prob

lem,som

ewhat

di!

erentfrom

the

stan-

dard

Herm

itianeigen

prob

lem.

The

non

-reciprocal

parts

ofth

elocal

Herm

itianperm

ittivityan

dperm

eability

ten-

sors!(r

,!)

and

µ(r

,!)

areod

dim

aginary

function

sof

frequen

cy,so

frequen

cy-dep

enden

ceis

unavoid

able.

The

generalized

eigenprob

lemhas

the

structu

re

U†(k

)AU

(k)|u

n(k

)!=

!n(k

)B(!

n(k

))|un(k

)!,(1)

where

U(k

)is

aunitary

operator

that

defi

nes

the

Bloch

vectork;

Aan

dB

(!)

areH

ermitian

operators,

with

the

real-eigenvalue

stability

condition

that

the

Herm

i-tian

operator

B0 (!

)"

("/"

!)(!

B(!

))is

positive

defi

-nite

(this

assum

esth

atth

eperiod

icm

ediu

mcou

pled

toth

eelectrom

agnetic

field

shas

alin

earresp

onse

describ

edby

harm

onic

oscillatorm

odes,

non

eof

which

have

natu

-ral

frequen

cy!

n(k

)-a

detailed

derivation

has

been

pre-

sentedin

Ref.

[3]).T

he

eigenfu

nction

s#r|u

n(k

)!are

the

spatially-p

eriodic

factorsof

the

Bloch

states.T

he

elec-tron

icban

d-stru

cture

prob

lemis

asim

plifi

cationof

(1),w

ithA

replaced

byth

eon

e-electronH

amilton

ian,B

byth

eid

entityop

erator11

,an

d!

nby

the

energy

eigenvalue.

Inth

isform

ulation

ofM

axwell’s

equation

s,th

eeigen

-fu

nction

un (k

,r)"

#r|un(k

)!is

the

6-compon

entvector

, Phys. Rev. Lett.100, 013904 (2008)

Analogs of quantum Hall edge states in photonic crystals

• Predicted theoretically that using magnetooptic (time-reversal-breaking) materials, photonic analogs of electronic quantum Hall systems could be created where topologically-protected edge modes allow light to only travel along edges in one direction, with no possibility of backscattering at obstacles!

• Effect was experimentally confirmed recently at MIT (Wang et al., Nature 461, 775 (8 October 2009).

• Obvious potential for technological applications! (one-way loss-free waveguides)

Haldane and Raghu, Phys. Rev. Lett.100, 013904 (2008)

Before we discuss the results of our measurements, we will firstdescribe how we arrived at this particular choice of experimentalsystem. We chose rods in air for the basic photonic-crystal geometrybecause of ease of fabrication.We thenperformed a series of numericalsimulations for a variety of rod sizes and lattice constants on a model2D photonic-crystal system to optimize the band structure andcompute corresponding band Chern numbers using materialparameters appropriate to a low-loss ferrite (Methods).Ournumericalsimulations predicted that when the ferrite rods in this photoniccrystal are magnetized to manifest gyrotropic permeability (whichbreaks time-reversal symmetry), a gap opens between the secondand third transverse magnetic (TM) bands. Moreover, the second,third and fourth bands of this photonic crystal acquireChern numbersof 1,22 and 1, respectively. This result follows from theC4v symmetryof a non-magnetized crystal17. The results of our simulations for thephotonic crystal withmetallic cladding are presented in Fig. 2. (Similarnumerical results were obtained in ref. 7, albeit using a differentmaterial system and geometry.) Here we show the calculated fieldpatterns of a photonic CES residing in the second TM band gap(between the second and the third bands). Because the sum of theChern numbers over the first and second bands is 1, exactly one CESis predicted to exist at the interface between the photonic crystal andthe metal cladding. The simulations clearly predict that this photonicCES is unidirectional. As side-scattering is prohibited by the bulkphotonic band gaps in the photonic crystal and in the metalliccladding, the existence of the CES forces the feed dipole antennas(which would radiate omnidirectionally in a homogeneous medium)to radiate only towards the right (Fig. 2a, c). Moreover, the lack ofany backwards-propagating mode eliminates the possibility ofbackscattering, meaning that the fields can continuously navigatearound obstacles, as shown in Fig. 2b. Hence, the scattering from the

obstacle results only in a change of the phase (compare Fig. 2a andFig. 2b) of the transmitted radiation, with no reduction in amplitude.

For CESs to be readily measurable in the laboratory (where it isnecessary to use a photonic crystal of finite and manageable size) theymust be spatially well localized, and this requires the photonic bandgaps containing the states to be large. The sizes of the band gaps thatcontainCESs (and the frequencies atwhich theyoccur) are determinedby the gyromagnetic constants of the ferrite rods constituting thephotonic crystal. Under a d.c. magnetic field, microwave ferritesexhibit a ferromagnetic resonance at a frequency determined by thestrength of the applied field18. Near this frequency, the Voigtparameter, V5 jmxyj/jmxxj (where mxx and mxy are diagonal and off-diagonal elements of the permeability tensor, respectively), which isa direct measure of the strength of the gyromagnetic effect, is of orderone. Such ferromagnetic resonances are among the strongest low-lossgyrotropic effects at room temperature and subtesla magnetic fields.Using ferrite rods composed of vanadium-doped calcium–iron–garnet under a biasing magnetic field of 0.20T (Methods andSupplementary Information), we achieved a relative bandwidth of6% for the second TM band gap (around 4.5GHz in Fig. 3b). Asdiscussed earlier, this is the gap predicted to support a CES at theinterface of the photonic crystal with the metallic wall. We emphasizeagain that band gaps with trivial topological properties (that is, forwhich the Chern numbers of the bulk bands of lower frequencies sumto zero), such as the first TM band gap (around 3GHz in Fig. 3b), donot support CESs. All of the insight gained from the model 2D photo-nic-crystal system was then incorporated into the final design (Fig. 1).To emulate the states of the 2D photonic crystal, the final design

a

by x

z

4 cm

Antenna A

Antenna B

CES waveguide

Metal wall

Scatterer ofvariable length l

Figure 1 | Microwave waveguide supporting CESs. a, Schematic of thewaveguide composed of an interface between a gyromagnetic photonic-crystal slab (blue rods) and a metal wall (yellow). The structure issandwiched between two parallel copper plates (yellow) for confinement inthe z direction and surrounded with microwave-absorbing foams (greyregions). Two dipole antennas, A and B, serve as feeds and/or probes. Avariable-length (l) metal obstacle (orange) with a height equal to that of thewaveguide (7.0mm) is inserted between the antennas to study scattering. A0.20-T d.c. magnetic field is applied along the z direction using anelectromagnet (not shown). b, Top view (photograph) of the actualwaveguide with the top plate removed.

a

b

c

A

A

B

l

a

Ez0Negative Positive

Figure 2 | Photonic CESs and effects of a large scatterer. a, CES fielddistribution (Ez) at 4.5 GHz in the absence of the scatterer, calculated fromfinite-element steady-state analysis (COMSOL Multiphysics). The feedantenna (star), which is omnidirectional in homogeneous media(Supplementary Information), radiates only to the right along the CESwaveguide. The black arrow represents the direction of the power flow.b, When a large obstacle (three lattice constants long) is inserted, forwardtransmission remains unchanged because backscattering and side-scatteringare entirely suppressed. The calculated field pattern (colour scale) illustrateshow the CES wraps around the scatterer. c, When antennaB is used as feedantenna, negligible power is transmitted to the left, as the backwards-propagating modes are evanescent. a, lattice constant.

NATURE |Vol 461 |8 October 2009 LETTERS

773 Macmillan Publishers Limited. All rights reserved©2009

microwaves goaround obstacle!

Kane and Mele 2005• Two conjugate copies of the 1988 spinless

graphene model, one for spin-up, other for spin-down

At edge, spin-up moves one way, spin-down

the other way

If the 2D plane is a plane of mirror symmetry, spin-orbit coupling preserves the two kind of spin. Occupied spin-up band has chern number +1,

occupied spin-down band has chern-number -1.

E

k k

B=0Zeeman coupling

opens gap

• This looks “trivial”, but Kane and Mele found that the gapless “helical” edge states were still there when Rashba spin-orbit coupling that mixed spin-up and spin-down was added.

• They found a new “Z2” topological invariant of 2D bands with time-reversal symmetry that takes two values, +1 or -1. The invariant derives from Kramers degeneracy of fermions with time-reversal symmetry.

• This launched the new “topological insulator” revolution when an experimental realization was demonstrated.

An explicitly gauge-invariant rederivation of the Z2 invariant

• If inversion symmetry is absent, 2D bands with SOC split except at the four points where the Bloch vector is 1/2 x a reciprocal vector. The generic single genus-1 band becomes a pair of bands joined to form a genus-5 manifold

• This manifold can be cut into two Kramers conjugate parts, each is a torus with two pairs of matched punctures. In each pair, one puncture boundary is open one is closed.

FDMHunpub.

• on a punctured 2-manifold

exp i

d2kF12(k) =

i

eiφi

product of Berry phase-factors of puncture boundaries• without punctures,

d2kF12(k) = 2πC

• punctures come in Kramers pairs:

2n

i=1

eiφi =

n

i=1

eiφi

2

exp i

1

2

d2kF12(k)

n

i=1

e−iφi = ±1 a perfect square, sowe can take asquare root!

• If inversion symmetry is present, the bands are unsplit and doubly-degenerate at all points in k-space, so the Berry curvature is undefined.

• Fu and Kane found a beautiful formula

n

k∗

In,k∗ = ±1

occupied bands

T+I-invariant k-points

= the Z2 invariant

Inversion quantum number (about any inversion center)

±1

Mercury Telluride Inverted Topological Insulator

(-)

+

+−

−

Quantum Spin Hall insulator: quantized conductance in the gap

Trivial band insulator: zero conductance in the gap

Molenkamp group

A further surprise: the 3D generalization:

• In the 3D BZ there are 8 T-invariant points, which can be decomposed in many ways to two sets of 4 coplanar T-invariant points on two parallel T-invariant planes in the BZ

• For all such decompositions, the product Z1Z2 has the same value, +1 or -1.

• Strong TI’s have Z1Z2 = -1. Weak TI’s (+1) are stacks of 2D TI’s

2D plane 2

2D plane 1Moore and BalentsRoy

Fu and Kane k-space

special k-space points invariant under time-reversal

Strong TI’s have an odd number of 2D Dirac points on any facet of a crystal

• Dirac points can only annihilate each other in pairs

• Disorder cannot destroy the metallic properties of the 2D surface state of a 3D strong TI, providing time-reversal symmetry remains unbroken. (Topological stability)

• Localization/antilocalization with SOC

dirty clean

g-2

anti-localizationlocalization

scattering from A to A’ cancels.

scattering from A to B only when sufficiently dirty

spin-split Fermi

surfaces ink-space

STI surface:nothing to backscatter into

full sigma model calculation byRyu and Ludwig

conventional renormalization group

analysis

modified renormalization group

analysis

Z2 Topological Order in 3D bulk solids

Bi2Se3 class as TIs : KITP 08Xia et.al., (Hasan) NATURE PHYS 09, arXiv (2008)

Hsieh et.al. (Hasan) NATURE 09

Materials challenges

• get Fermi level into band gap to make true TI• (Bi,Sb)2(Se,Te)3 • large amount of DARPA funding ($60M)• Thermoelectrics• Seeing Landau levels, FQHE in magneLc fields• for strong enough SOC, expect 50% of non-‐magneLc insulators to be TI.

• without SOC, all are trivial.

33

Magnetoelectric effect.

• apply an electric to a magnetoelectric material

• generate a magnetic field parallel to the applied electric field!

• material breaks both time-reversal and spatial inversion symmetry.

• Break time-reversal invariance at the surface of a 3D TI, get a half-integer θ = π “integer Hall effect” (Pauli-principle-based QHE)!

• Interpret it as a “magnetoelectric” effect

σH =e2

2πθ

2π

exp iθ is fixed by the band structures of the insulating 3D regions on either side of the “Hall surface”

“generalized integer QHE”

replacementfor integer

a surprising property of a Hall surface:

• Magnitude of the effect is controlled by the dimensionless small parameter

I

II II

I

• Electric charges on or near a “Hall surface” produce (virtual) magnetic monopole images!

• FDMH and L. Chen, PRL 53, 2591 (1984)

• (rediscovered by Qi et al 2009 in TI Hall surfaces)

Quantum Geometry and the Hall Viscosity : a fundamental characteristic of

incompressibility in the Fractional Quantum Hall Effect

F. D. M. HaldaneDepartment of Physics, Princeton University, Princeton NJ 08544-0708

(Dated: March 28, 2010, v0.1)

PACS numbers:

I. THE HALL SURFACE

The “Hall surface” wll be a locally-flat two-dimensional(2D) surface that is an interface between two 3D regions“+” and “−” that may have different physical properties.These regions will be taken a non-conducting, but elec-trons bound to the Hall surface will give rise 2D chargeand current densities confined to the interface.

At each point s on the surface, two tangent vectorsea(s¯

), a = 1,2, and a normal vector n(s) form an orthor-mal basis of directions in Euclidean space, and

ea(s)× eb(s) = abn(s), (1)

where ab is the 2D antisymmetric Levi-Civita symbol.The boundary conditions at the surface on the electro-magnetic flux densities B, D, and fields E, H are thatB

n = n ·B and Ea = ea ·E are continuous (these maybe called the surface electromagnetic fields, and that

∆Ha ≡ H+a−H

−a

= abja, ∆D

n = ρ, (2)

where ρ and j = jaea are the surface charge and cur-

rent densities. Note the use of covariant (lower) and con-travariant (upper) indices, which distinguish componentsof vectors related to directions (upper indices) and thoseof “dual vectors” related to derivatives along directions(lower indices). Only upper/lower pairs of indices can becontracted in Einstein summation convention, and an ex-plicit metric tensor that lowers indices is needed to formthe dot product between two vectors. The Euclideanmetric tensor for tangent-vector indices is

ea · eb = ηab (3)

where setting ηab to be numerically equal to the metric-independent Kronecker symbol δa

brepresents the choice

to use a Cartesian basis. This choice also means that ηaband ηab are numerically equal, as are ab and ab.

The surface electromagnetic fields satisfy the Faradaylaw

∂tBn + ab∂aEb = 0, (4)

I will define a “Hall surface” as a surface on which the2D surface charge ρ and surface current density j

a arelinear responses to the the surface electromagnetic givenby

ρ = σHBb, j

a = σHabEb, (5)

where σH is a (local) “Hall conductivity”, that is constanton the surface so that the continuity equation

∂fρ+ ∂aja = 0 (6)

is consistent with the Faraday law (4). Note that σH isodd under time-reversal, so σH = 0 implies time-reversalsymmetry is locally-broken at the surface Combining thiswith (2) gives the interesting boundary condition

∆Dn = σHB

n, ∆Ha = σHEa, (7)

which, as apparently first pointed out by Haldane andChen? , and rediscovered recently in connection withtopological insulator surfaces? induces a small (non-Dirac) magnetic-monopole image on an electric chargeplaced on the surface (with a sign dependent on whichside of the surface it is “viewed” from), in addition to theusual electric image charges that accompany a change indielectric properties across the surface.

If the regions on each side of a flat Hall surface are filledwith uniform isotropic non-conducting dielectric and dia-magnetic materials with static permittivities ε0ε± andpermeabilities µ0µ

±, and thus with effective “vacuumimpedances” for electromagnetic waves travelling normalto the surface given (in the low-frequency limit) given by

R±vac =

µ0µ

±/ε0ε

±1/2, (8)

a point charge q placed on the Hall surface generatesstatic electromagnetic fields at a displacement r from itgiven in regions ± by

D± = f

± qer

4πr3, f

± =

2ε±

ε+ + ε−

(9)

B = ± qmr

4πr3(10)

qm = βR0qe, qe = q/(1 + β2), (11)

β = R0σH ,1

R0=

1

R+vac

+1

R−vac

, (12)

where R0 is the combined vacuum impedance given byaddition in parallel of the separate vacuum impedances ofthe two semi-infinite media. If the charge is displaced offthe surface into the± region, the fields in the± region arethose of the bare electric charge q at the location of thephysical charge, plus virtual electric and magnetic imagecharges (f±

qe)−q and ±qm at the mirror position in the∓ region, while the fields in the ∓ region are those of vir-tual electric and magnetic charges f∓

qe and ∓qm at the

R0σH

• R0 is the combined vacuum impedances of regions I and II, added in parallel. (units = Ohms) qm < 10−4qDirac

m (tiny!)

Radiation-damping of currents on the Hall surface

• vacuum impedances combine in parallel

H

EE × H Axially-symmetric media

II

I Da = 0⊥gabEb Rvac =

µ0µ⊥0⊥

1/2

Ba = µ0µ⊥g

abHb

J spatially-uniform surface current

J(t) = limk→0

Ja0 eae

i(kara−iωt)

surface electric field

1

R(ω)=

1

Rrad(ω)+

1

R2D(ω)

Ea(t) = R(ω)gabJb0e

−iωt

radiation dissipationon surface

1

R0=

1

RIvac

+1

RIIvac

limω→0

Rrad(ω) = R0 R0 characterizes coupling of 2D surface currents to 3D E&M

2D Euclidean metric

∼ 100Ω

Topological Superconductors, “10fold way”

• Unitary symmetries are non-‐topologocal, only anLunitary symmetries can have significanct.

• T = 0, + 1, -‐1• Add “C” for Bogoliubov-‐deGennes for quasiparLcle spectrum in superconductors.

• C = 0 ,+1, -‐1, T*C gives 9 possibiliLes, in fact 10 since C, T = (0,0) can have TC = -‐1 or 0.

38

Periodic Table of Topological Insulators and superconductorsDim/Symmetry

(0+1)d(1+1)d(2+1)d(3+1)d(4+1)d(5+1)d(6+1)d(7+1)d

BDIZ2

Z000Z0Z2

DZ2

Z2

Z000Z0

DIII

0Z2

Z2

Z000Z

AIIZ0Z2

Z2

Z000

CII0Z0Z2

Z2

Z00

C00Z0Z2

Z2

Z0

CI000Z0Z2

Z2

Z

AIZ000Z0Z2

Z2

AZ0Z0Z0Z0

AIII

0Z0Z0Z0Z

Kitaev: Adv. in Theoretical Phys. 2009, Schnyder,Ryu,Furusaki,Ludwig: PRB (2008), Qi, Hughes, Zhang: PRB(2008).

“Classification by symmetric spaces”.table repeats when dimension of space increases by 8!

Where is the subject going theoretically?

• Classification of possibilities based on one-particle physics seems done. Seach for examples of various possible Topological superconductors.

• Big question: topological band insulators (and probably superconductors) are pertubatively stable against disorder and interactions. Are there any new type that (like FQHE) are not adiabatically related to one-body physics?

presented at the moshe flato lecture series, ben gurion ...haldane/talks/topolint_flato.pdf · r,j...

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