11

Consider Evolution of a system when adiabatic theorem holds(discrete spectrum, no degeneracy, slow changes)

Adiabatic theorem and Berry phase

set of "slowly changing"parameters.

, time-independent sol

{ [ ( )]} completeset ofi nstantaneous

utions: [

eig

] [ ] [ ] [ ]

ensta

( ) [

te

(

s

) )

:

] ( ,

n

n n n

a R

R

t H R a R

t

E R

i

a R

t H R t tt

Assume thatat 0 system is in instantaneous eigenstate,

( 0) [ (0)]; then at time t ( ) ( ) is a wave-packet:

( ) ( ) [ ( )] [ ( )] (0) 1

Then Katoadiabatic theorem ( ) 1 in adiab

n n

n n n m m nm n

n

t

t a R t t

t c t a R t c a R t with c

c t atic limit.

To find the Berry phase, we start from the expansion on instantaneous basis

( ) ( ) [ ( )] [ ( )] with (0) 1n n n m m nm n

t c t a R t c a R t c

2

( ) ( ) ( ) ( )and the ansatz ( ) ( ) ( ( ) ( ))

and plug into using the fac (t tha )t

n n n n

n R

i t i t i t i tn n n n

n nn

c t

dRa t

e c t i

i H at d

t e

t

t

t

( ) ( )

0

( ) 1 ( ) , where by definition

( ) ' [ ( ')] dynamical phase, while

( ) geometric phase = Berry phase.

n ni t i tn n

t

n n

n

c t c t e

t dt E R t

t

0( ) ' [ ( ')]

( )t

n ni

i t

n R m

dt E R t

n n n m mm n

m

iE i a a

tct eR c a

Negligible because second order (derivative is small, in a

small amplitude)

. . .

( ) [ ( )] ( ) ( ) [ ( )]n n n m m mm n

n E a R t

r

c t c t E

s

H t a t

h

R

33

[ ( )] [ ( )]

[ ( )] [ ( )] Berry connection

( ) phase collected at time t

n n R n

n R n

n

iR a R t a R t

a R t a R t

t

Now, scalar multiplication by an

removes all other states!

Professor Sir Michael Berry

0'[ [ ( ')] ( )]

0 ( ) [ ]t

n ni

dt E R t i t

n R n m m mm n

i R a e c E a

4

( ) is a topological phase

and vanishes in simply connected parameter spaces where C can collapse

to a point but in a multiply connected spaces yields a quantum number

n n R nC

C i a a dR

0

[ ( )] [ ( )]

The matrix element looks similar to a momentum average, but the gradient

is in parameter space. The overall phase change in the transformation is a line integral

n n R n

T

n

iR a R t a R t

i dt

0

. .

This has no physical meaning, it's a gauge, but if C is c

(

losed it bec

) which is gauge inveriant like a magnetic fl

o es

u

m

xn n R n

T

n R n R n

C

n

C

a

i a a dR

a R i a a dR

C

Vector Potential Analogy

One naturally writes ( ) · , | . |n n n n R nCC A dR A i a a

introducing a sort of vector potential (which depends on n, however). The gauge invariance arises in the familiar way, that is, if we modify the basis with

( )[ ] [ ], ,i Rn n n n Ra R e a R A A

and the extra term, being a gradient in R space, does not contribute. The Berry phase is real since

| 1 | 0 | | 0

| , . 0 |n n R n n R n n n R n

n R n n R n

a a a a a a a a

a a c c a a is imaginary

| | , |Im |n n R n n n R nA i a a is real A a a

We prefer to work with a manifestly real and gauge independent integrand; going onwith the electromagnetic analogy, we introduce the field as well, such that

( ) · · .n n nS SC rot A ndS B ndS

5

Im ( | ) Im ( | ) Im[ ( | ) ( | )].i ijk j k ijk j k ijk j kiB a a a a a a a a

The last term vanishes,

and, inserting a compl

( | ) ( | ) ( | ) ( | | )

ete set

Im ( | | ) Im

,

.

ijk j k ijk j ki i

n n n n m m nm n

a a a a a a a a

B a a a a a a

6

| | ,n n R n n nA i a a B rot A and omitting n

To avoid confusion with the electromagnetic field in real space one often speaks about the Berry connection

and the gauge invariant antisymmetric curvature tensor with components

1 233 , , .

Y

In d B Y etc

i

Imn n m m nm n

B a a a a

The m,n indices refer to adiabatic eigenstates of H ; the m=n term actually vanishes (vector product of a vector with itself). It is useful to make the Berry conections appearing here more explicit, by taking the gradient of the Schroedinger equation in parameter space:

H R a R E R a [R]     

( H R )a R H R a R E R a [R]  ( E R )a [R] 

R n R n n

R n R n n R n R n n

7

Taking the scalar product with an orthogonal am

a H a a H a E a a   a ( E ) a  

a H a a a E a a  

a aa a divergence if degeneracy occurs along C.

E

m R n m R n n m R n m R n n

m R n m m R n n m R n

m R nm R n

m n

E

H

E

Formula for the curvature

A nontrivial topology of parameter space is associated to the Berry phase, and degeneracies lead to singular lines or surfaces

Quantum Transport in nanoscopic devices

If all lengths are small compared to the electron mean free path the transport is ballistic (no scattering, no Ohm law). This occurs in Carbon Nanotubes (CNT) , nanowires, Graphene,…

Ballistic conduction - no resistance

A graphene nanoribbon field-effect transistor (GNRFET) from Wikipedia

This makes problems a lot easier (if interactions can be neglected). In macroscopic conductors the electron wave functions that can be found by using quantum mechanics for particles moving in an external potential lose coherence when travelling a mean free path because of scattering . Dissipative events obliterate the microscopic motion of the electrons .

In ballistic transport the quantum propertes of the electrons are revealed.

8

Number of conducting channels due to

transverese degrees of freedom .

Electrons available for conduction are those

between the Fermi levels

FM k W

9

WClassically, the conductance wolud be G=

it should increase without limit for small L.

This fails for L < mean free path

L

Ballistic conductor between contacts

Complication: quantum reflection at the contacts

( )k

k

Fermi level right electrode

Fermi level left electrode

W

left electrode right electrode

k

For transport across a junction with M conduction modes, i.e. bands of the unbiased hamiltonian at the Fermi level one measures a finite conductivity. If V is the bias, eV= difference of Fermi levels across the junction,

212 12.9

Conductance quantum G= per mode resistance=Ge

kh M

2

, hopping time

the current is per spin per mode

hophop

hop

heV t

t

e e Vi

t h

This quantum can be measured.

10

W

left electrode right electrode

k

B.J. Van Wees experiment (prl 1988)A negative gate voltage depletes and narrows down the constriction progressively

Conductance is indeed quantized in units 2e2/h11

1 2 1 2

2

2

1 2

Linear Response Current: I=G V, where:

, , electrochemical potentials,

2 Conductance: G= ,

number of modes, quantum transmission probability

2I=

V

eM T

hM T

eMT

h

Current-Voltage Characteristics: Landauer formula (1957)

2

, 1,2

Extension to finite bias and temperature:Current-Voltage characteristics

I= ( )

2I(E)= [ ( ) ( ) ( ) ( ) ( ) ( )]

( ) ( ) f=Fermi function.

L L L R R R

L R

dEI E

eM E T E f E M E T E f E

hf E f E

12

Phenomenological description of conductance at a junction

1 1FE2 2FE

Rolf LandauerStutgart 1927-New York 1999

13

Quantum system

13

More general formulation, describing the propagation inside a device. Let

represent leads with Fermi energy EF, Fermi function f(e), density of states ( )r e

FEFE

( ),Quantum System:eigenstates | , retarded Green's

Quantum System-leads hopping: ( ), ( )

rmn

L Rm m

m g

V V

FE

bias

1414

current-voltage characteristic J(V)

*

, , , 1,2,

with ( ) ( ) f=Fermi function.L R mn mn L Rmn

eJ d ff t t f E f E

( ) ( ) * ( ), ,transmission amplitudes 2 ( ) ( ) ( )L R R L r

mn m n mnt V V g

Quantum system

This scheme was introduced phenomenologically by Landauer but later confirmed by rigorous quantum mechanical calculations for non-interacting models.

2*

, ,,

Linear response: 2 .mn mnmn

eJ V t t

Multi-terminal extension (Büttiker formula)

,,

voltage between contacts i and j

i i j ij jmn

ij

eJ dE T f E eV f E

V

*

, , , 1,2,

with ( ) ( ) becomesL R mn mn L Rmn

eJ d ff t t f E f E

1616

† †1 1

Thecurrent operator at site m (Caroli et al.,J.Phys.C(1970))

is physically equivalent.hoppingm m m m m

etJ c c c c

i

†hoppin

† † † †1 1 1 1 , 1 1

g 1

hopping

( . .)

hopping integral

Heisenberg EOM:

Chain or wire Ha

ˆˆ[ , ] [ ] (

miltoni

[ ( ), ( )]

an:

)

i

mm hopping m m m m m m m m m m m

ii

m

dnie e n H et c c c c

H t c c h c

t

d

c c c c i

Ai A A t H t i

dt t

J Jdt

J

† †, 1 1 1

hoppingm m m m m m

eti c c c c

Microscopic current operator

device

J

Traditional partitioned approach (Caroli, Feuchtwang): fictitious unperturbed biased system with left and right parts that obey special boundary conditions

device

this is a perturbation (to be treated at all orders = left-right bond

† †1 1 1 1

† †

Time-independent partitioned framework for the calculation of characteristics

ˆ ( ) ( ) ,2

( ) ( ') ( ) , ( ) ( ) ( ')

hopping hoppingm m m m m m mm mm

et et dJ c c c c J J g g

i

g t c t c t g t c t c t

17Drawback: separate parts obey strange bc and do not exist.

=pseudo-Hamiltonian connecting left and right

Pseudo-Hamiltonian Approach

18

Simple junction-Static current-voltage characteristics

J

chemical potential 1

2-2 0

1

U=0 (no bias)

no current

Left wire DOS

Right wire DOS

no current

U=2

current

U=1

19

2 2

0

22

2

Half-filled 1d system left wire right wire

hopping 1

( ) 1 (1 )

( ) ( )8 2( )( ( ) ( ))

2 4V

g

Vg ge

J V dV V

g g

1 2 3 4V

0 .1

0 .2

0 .3

0 .4

J

( ) , 0

quantum conductivity

( ) 0, 4 bands mismatch

eUJ V U

e

J V U

Static current-voltage characteristics: example

J

Meir-WinGreen FormulaConsider a quantum dot ( a nano conductor with size of angstroms, modeled for example by an Anderson model) connected with wires

† †, , ,

, ,

†, , ,

, ,

( . .)

( . .)

d Lk Lk L Lkd dk k

Rk Rk R Rkk k

H d d Un n n V d c h c

n V d c h c

where L,R refers to the left and right electrodes. Due to small size, charging energy U is important. If one electron jumps into it, the arrival of a second electron is hindered (Coulomb blockade)

20

Quantum dot

U

LV RV

Meir and WinGreen in 1992 have shown, using the Keldysh formalism, that the current through the quantum dot is given by

( )2( ) ( ) ( Im )

rL R

L R dL R

eI d f f G

This has been used for weak V also in the presence of strong U.21

2

2 2

Let ( ) ( ) ( ), where:

( ) Re ( ) , ( ) Re ( ) ,

kL R

k k

Lk RkL L L R R R

k kLk Rk

V

i

V Vi i

i i

and the like, even in the presence of the bias.

Quantum dot

U

LV RV

*

, ,,

ThisissimilartoLandauerformula L R mn n mmn

eJ d ff t t

2222

General partition-free framework and rigorous Time-dependent current formula

Partitioned approach has drawbacks: it is different from what is done experimentally, and L and R subsystems not physical, due to specian boundary conditions. It is best to include time-dependence!

2323

† †1 1

Current operator at site (Caroli et al.,1970)

, hopping integral

Poses a genuine many-body problem even without interactions,

because of Pauli principle depends on Fermi l

hoppingm m m m m hopping

m

m

etJ c c c c t

i

J

evel.

Time-dependent Quantum Transport System is in equilibrium until at time t=0 blue sites are shifted to V and J starts

device

J

24

, 1 , 1' 0

, 1

, 1 , 1

Current in discrete model

lim ,

of course depends on Fermi level;

but 1-body Schrodinger equation yields ,

hopping T Tm m m m m

t t

Tm m

advanced retardedm m m m

etJ G G

i

G

G G

†' '

'

( )' '' 0 '

'

For anyone-bodydensity ( ) ( ) ( ) ( '),

( ) lim lim ( ) ( , ' ').T

t t x x

f x f x x x

f x i f x G xt x t

( ) †' '( , ' ') ( ) ( ' ') depends on Fermi levelTG xt x t i T xt x t

( ) †' '( , ' ') ( ) ( ' ') ' independent of Fermi level (if no interactions)rG xt x t i xt x t t t

Use of Green’s functions

25

† † †' ' ' '( , ') ( ) ( ') ( ') ( ) ( ') ( ' ) ( ') ( )T

kk k k k k k kiG t t Ta t a t t t a t a t t t a t a t

' ' ' ' ''

( , ') ( ') ( , ') ( , ')T T Tkk kk k kk kk kk

kk

i G t t t t G t t V G t tt

' ' ' ' ''

( , ') ( ') ( , ') ( , ')r r rkk kk k kk kk kk

kk

i G t t t t G t t V G t tt

General formulation for independent-electron problems

† †0 1 0 1 ' '( ) ( ) ( )k k k kk k k

k k

H H H t H a a H t V t a a

Equation Of Motion for time-ordered GF

EOM for retarded (advanced) GF

EOM is the same, but initial conditions differ

26

( ) †' '( , ' ') ( ) ( ' ') ' independent of Fermi level (if no interactions)rG xt x t i xt x t t t

Initial conditions for time-ordered: GF

† † †' ' '

' '

'

' '

' '

' '

( , ') ( ) ( ') ( ') ( ) ( ') ( ' ) ( ') ( )

(0,0 )

( , )

(1 ),

(1 ) ( ,

,0

)

(0 )

T Tkk k

Tkk k k

k k kk kk k

k k k k

T Tkk kk k kk kk k

iG t t Ta t a t t t a t a t t t

G t t i f G t t i

a t a t

iG

f

fi G f

Initial conditions for retarded: GF

' '( 0 , )rkk kkG t t i

Define ( , ') (1 ) ( ') ( ' ) ( ')q q q qt t f t t f t t t t f

With H constant up to t=0, here is the solution for t>0,t’>0

' 'Then, ( , ') ( ,0) (0, ') ( , '), 0 , 'T r akk kq qk q

q

iG t t G t G t t t t t

(Blandin, Nourtier, Hone (1976) had derived this formula by the Keldysh formalism in a paper on atom-surface scattering

This allows to write a rigorous time-dependent current formula

27

'

'

'

' '

' '

One can also write:

(1 ) ( ,0) (0, '), '

( , ')( ,0) (0, '), '

obeys the EOM and

(1 ) (0,0) (0,0) (1 ) , ' 0,

(0,0) (0,0) , 0, '

r aq kq qk

qTkk r a

q kq qkq

r aq kq qk k kk

q

r aq kq qk k kk

i f G t G t t t

G t ti f G t G t t t

i f G G i f t t

i f G G if t t

q

Note:Occupation numbers refer to H before the time dependence sets in! System remembers initial conditions!

†1 ' '

0

( ) ( ) ( ) i.e. bias is on a

eigenstates for 0,

Fermi functio un for system.

The current at bond is

t tim

nb

:

d

e0

iase

kk

q

q

k kk

Let H q

H t V t

q V

t

n

a

f

a

m

Rigorous Time-dependent current formula

*, ,

( ') †

2( ) Im ( ) ( )

where m and n are connected by a bond .

and ( , ') ( ') vac ( ')

hopping retarded retardedmn q m q n q

q

hopping

retarded iH t tij j i

etJ t f G t G t

t

iG t t t t c e c t vac

28

, 1 , 1' 0limhopping T T

m m m m mt t

etJ G G

i

29

† † †1 1

This formula can also be rewritten in terms of evolution operator,

ˆ ( ) ( ) ( ) with .

( ( ) for constant H but this is quite general ).

Also, onecan put in

hoppingm q m m m m m m

q

iHt

etJ t f q U t J U t q J c c c c

i

U t e

f

† ˆˆside, ( ) ( ) ( ) .

The states that were occupied initially are those that evolve and carry the current.

m mq

J t q U t J U t f q

0( )

In terms of Heisenberg operators,

1ˆ( ) [ ( )]1

.

mn mn HJ t Tr fJ t fe

Current-Voltage characteristics

In the 1980 paper I have shown how one can obtain the current-voltage characteristics by a long-time asyptotic development. Recently Stefanucci and Almbladh have shown that the characteristics for non-interacting systems agree with Landauer

30

2 2

0

22

2

half-filled 1 system left wire right wire

hopping 1

( ) 1 (1 )

( ) ( )8 2( ) in units( ( ) ( ))

2 4V

d

g

Vg g e

J V dV V

g g

1 2 3 4V

0 .1

0 .2

0 .3

0 .4

J

( ) , 0

quantum of conductivity

( ) 0, 4 bands mismatch

eVJ V V

e

J V V

Long-Time asymptotics and current-voltage characteristics are the same as in the earlier partitioned approach

31

Current in the bond from site 0 to -1

Transient current

asymptote

0.5*hopping integralV

In addition one can study transient phenomena

M. Cini E.Perfetto C. Ciccarelli G. Stefanucci and S. Bellucci, PHYSICAL REVIEW B 80, 125427 2009

10

0, t<0 ( ) ( ) , ( )

0.5, t>0

= =0

hopping bias mm

a b

t H t V t n V t

M. Cini E.Perfetto C. Ciccarelli G. Stefanucci and S. Bellucci, PHYSICAL REVIEW B 80, 125427 2009

10

0, t<0 ( ) ( ) , ( )

0.5, t>0

= =0

hopping bias mm

a b

t H t V t n V t

34

G. Stefanucci and C.O. Almbladh (Phys. Rev 2004) extended to TDDFT LDA scheme

TDDFT LDA scheme not enough for hard correlation effects: Josephson effect would not arise

Keldysh diagrams should allow extension to interacting systems, but this is largely unexplored.

Retardation + relativistic effects totally to be invented!

Bloch states e ( , ) Parameters q,t FL in gap Adiabatic theorem holds

Berry A =-i( , )

Berry

Pumped cur

iqxn

n nn n n

n n n n nqt

u q t

u uvector potential u u

q t

u u u ucurvature i

q t t q

0bands n

rent: j=- charge in cycle c =- 2 2

Tn nqt n qt

BZ BZ

dq dqdt

A B A B A B A B A B A B

Pumping in 1d insulator with adiabatic perturbation periodic in space and time H(t+T)=H(t) (Thouless Phys. Rev. B (1983) )

Perturbation such that Fermi level remains within gap

Pumped charge for band n over a cycle: first Chern number=integer nc

Niu and Thouless have shown that weak perturbations, interactions and disorder cannot change the integer.

2 Berry phase

1c =- ( ,0) (1, ) ( ,1) (0, )

2

n

B C D A

n x y x yA B C D

c

dxA x dyA y dxA x dxA y

1 1 1

0 0 0

( ,1) and ( ,0) are physically equivalent can only differ by a phase: ( ,1) ( ,0)exp( ( )) :

[ ( ,0) ( ,1)] [ ( ,0) ( ,0) ( ,0) ( ( ,0) ) ] (1) (0)x x

x

i ix x x x x x x x

u x u x u x u x i x

dx A x A x dx u x i u x u x e i u x e dx

1

Thererfore, c =- (1) (0) (1) (0) .2n x x y y

(1)(0)

(1)(0) (1)

(1)(0) (1)

Matching relations at the corners A,B,C,D

u(0,0)=e (0,1) but since (0,1) e (1,1),

u(0,0)=e e (1,1) but since (1,1) e (1,0),

u(0,0)=e e e (1,0)

yx

yx x

yx x

ii

ii i

ii i

u u u

u u u

u

(0)

(1) (0)(0) (1)

but since (1,0) e (0,0),

u(0,0)=e e e e (0,0)

y

y yx x

i

i ii i

u u

u

Example taken from P.W.Brouwer Phys. Rev.B 1998

Two parameter pumping in 1d wire

U

Circuit in parameter space

0x x L

Phase 1: no potential

( )V x

length along wire

back to phase 1

0x x L

Phase 2: very high tunneling barrier for x (0, x)

( )V x

length along wire

0x x L

Phase 3:raise barrier U for x ( x,L); carriers flow to right

( )V x

U

length along wire

0x x L

Phase 4:open barrier for x (0, x)

( )V x

U

length along wire

2*

, ,,

Linear response in analogy with Landauer formula:

I 2 mn mnmn

eV t t

*

, ,1 2

, 1 2

Brouwer formula: X parameters, phase difference, =repetition frequency

sin I Im

2

i

mn mn

mn

t teX X

X X

There is a clear connection with the Berry phase (see e.g. Di Xiao,Ming-Che-Chang, Qian Niu cond-mat 12 Jul 2009). The magnetic charge that produces the Berry magnetic field is made of quantized Dirac monopoles arising from degeneracy. The pumping is quantized.

Bouwer formulation for Two parameter pumping assuming linear response to parameters X1, X2

Mono-parametric quantum charge pumping ( Luis E.F. Foa Torres PRB 2005)

quantum charge pumping in an open ring with a dot embedded in one of its arms. The cyclic driving of the dot levels by a single parameter leads to a pumped current when a static magnetic flux is simultaneously applied to the ring.

The direction of the pumped current can be reversed by changing the applied magnetic field.

20 proportional to j

0

The response to the time-periodic gate voltage is nonlinear.

time-periodic gate voltage

The pumping is not adiabatic.No pumping at zero frequency.

The pumping is not quantized.