1

1

Topologic quantum phasesPancharatnam phase

The Indian physicist S. Pancharatnam in 1956 introduced the concept of a geometrical phase.

Let H(ξ ) be an Hamiltonian which depends from some parameters, represented by ξ ; let |ψ(ξ )> be the ground state.

Compute the phase difference Δϕij between |ψ(ξ i)> and |ψ(ξj)> defined by

This is gauge dependent and cannot have any physical meaning.Now consider 3 points ξ and compute the total phase γ in a closed circuitξ1 → ξ2 → ξ3 → ξ1; remarkably,γ = Δϕ12 + Δϕ23 + Δϕ31

is gauge independent!

Indeed, the phase of any ψ can be changed at will by a gauge transformation, but such arbitrary changes cancel out in computing γ. This clearly holds for any closed circuit with any number of ξ. Therefore γ is entitled to have physical meaning.

There may be observables that are not given by Hermitean operators.

.ijii j i je

22

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 Rt

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:

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

Then Katoadiabatic theorem ( ) 1 in adiab

n n

n n n m m nm n

n

tt a R t tt c t a R t c a R t c

c t atic limit.

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

( ) ( ) [ ( )] [ ( )] with (0) 1 and (t) smalln n n m m n mm n

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

3

( ) ( ) ( ) ( )

and plug into .

in the . . . : ( ( ) [ ( )]) ( ) [ ( )] ( ) [

( ) ( )

( )]

( ) )

(

(

)

( )n n n ni t i t i t i tn n n

n n

n n n n

n

n

n R n

n

i Ht

L H S c t a R t c t a R

dRa t

t c t a R tt

c t e c t i

at d

et

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

4

0( ) ' [ ( ')]

( )t

n nii t

n R m

dt E R t

n n n m mm n

mi E i a a

tct eR c a

Negligible because second order (derivative is small, in a

small amplitude)

r.h.s.( ) [ ( )] ( ) ( ) [ ( )]n n n

mn m m m

n

E a R t c t cH t t E a R t

0'[ [ ( ')] ( )]

All together,

0 ( ) [ ] [ ( )]t

n ni dt E R t i t

n R n m m mm n

i R a e c E a R t

Now, scalar multiplication by an removes all other states!

[ ( )] [ ( )]

[ ( )] [ ( )] 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

55

Professor Sir Michael Berry

6

( ) 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 gradientis 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

77

Relation of Berry to Pancharatnam phases

( )n n R nC

C i a

B r y

dR

e r

a

Idea: discretize path C assuming regular variation of phase and compute Pancharatnam phase

differences of neighboring ‘sites’.

arg

:

ij

ij

i

ii j i

i j

j

j

Pancharatnam phase de

e

fined by

C

8

1

, 11

Imi ii C C

d8

Pancharatnam phase defined by: ij i ji

i j

e

Pancharatnam for nearby point 1s: i

Limit:

2

thusneglec

Panchar

Wemay conclude

indeed denominator | 1 | 1 ,2

for nearby pointatnam phas

tingsecond ord

s

r,

.e

e

: 1

i

i

i

999

Discrete(Pancharatnam)

1

11

arg ( ) ( )M

i ii

Continuous limit(Berry)

( ) ( )C

i d

Berry’s connection

i

The Pancharatnam formulation is the most useful e.g. in numerics.

Among the Applications:Molecular Aharonov-Bohm effectWannier-Stark ladders in solid state physicsPolarization of solidsPumping

Trajectory C is in parameter space: one needs at least 2 parameters.

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 | is im aginary

n n R n n R n n n R n

n R n n R n

a a a a a a a aa a c c a a

is rea| | I m |l, |n n R n n n R nA i a a 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

10

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

11

| and omittin| g,n n R n n nA i a a B nrot A

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 23In 3 parameter spaces, , .

Y

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

12

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

HE

Formula for the curvature (alias B)

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

13

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

W

left electrode right electrode

k

Quantum Transport in nanoscopic devicesBallistic conduction - no resistance. V=RI in not true

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

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.

14

Number of conducting channels due totransverese degrees of freedom .Electrons available for conduction are those between the Fermi levels

FM k W

Complication: quantum reflection at the contacts

( )k

k

Fermi level right electrode

Fermi level left electrode

Particles lose coherence when travelling a mean free path because of scattering . Dissipative events obliterate the microscopic motion of the electrons . For nanoscopic objects we can do without the theory of dissipation (Caldeira-Leggett (1981). See Altland-Simons- Condensed Matter Field Theory page 130)

If V is the bias, eV= difference of Fermi levels across the junction,How long does it take for an electron to cross the device?

212 12.9Conductance quantum G= per mode resistance=Ge k

h M

2

hopping time given by .

the current is per spin per mode

hophop

hop

ht eVt

e e Vit h

This quantum can be measured! 15

W

left electrode right electrode

k

junction with M conduction modes, i.e. bands of the unbiased hamiltonian at the Fermi level

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/h16

1 2 1 2

2

2

1 2

Response Current: I=G V, where:, , electrochemical potentials,

2 Conductance: G= ,

number of modes, quantum transmission probability

2

Line

I=

arV

eM Th

M T

e MTh

Current-Voltage Characteristics J(V) of a junction :

Landauer formula(1957)

17

Phenomenological description of conductance at a junction

1 1FE2 2FE

Rolf LandauerStutgart 1927-New York 1999

18

, 1,

2

2

Extension to finite bias and temperature: Current-Voltage characteristics J(V)

given by J= ( ),

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

where ( ) ( ) f=F

( )]L L L R R R

L R

e M E T E f E M E T E f

dEI E

f Eh

f

E

E

ermi function.

Phenomenological description of conductance at a junction

More general formulation, describing the propagation inside a device.

Quantum system

FEFE

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

1919

What is the transmission amplitude for electron incoming to eigenstate m and outgoing from eigenstate n?

Quantum system

( ),

Quantum System:eigenstates | , retarded Green's r

mnm g

Quantum System-leads hopping:( ), ( )L R

m mV V

r

r

( ) *

( )

probability to find electron in left wire: ( ). hopping amplitude probability to find final state for electron in right wire: ( ), hopping amplitude amplitide to go from m to

L Ln

R Rn

VV

( ),n : ( )r

mng

*why ? It is the time reversal of L Ln nV V

2020

*

, , , 1,2,

Characteristics:

with ( ) ( )

f=Fermi function.

L R mn mn L Rmn

eJ d ff t t f E f E

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

r

r

r r

( )

( )

( ) ( ),

probability to find electron in left wire: ( )probability to find final state for electron in right wire: ( ) incoming to eigenstate m and outgoing from eigenstate n:

2 ( ) ( )

L

R

L Rmnt V * ( )

, ( )R L rm n mnV g

Multi-terminal extension (Büttiker formula)

,,

voltage between contacts i and j

i i j ij ji j

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

2222

†hopping 1

hopping

( . .)

C

hain or wire

hopping integr

Hamiltonian

l

:

a

i ii

H t c c h c

t

Microscopic current operator

device

J

, 1 1

† † † †1 1 1 1

,

Continuity equation: 0. Here t

Heisenberg EOM:

ˆ ˆ

[ ( ), (

[ , ] [ ]

(

)]

m m m m

mm hopping m m m m m m m m

m m

J J

dnie e n

dJdivJ divJdx

d Ai

H et c c c c c c c c

A A t H t idt t

dti J

r

† † † †1 1 1 1 1 1) [ ]m m hopping m m m m m m m mJ et c c c c c c c c

2323

† †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

†hopping 1

hop

† †,

p ng

1 1 1

i

( . .)

Chain or w

hopping integra

ire Hamilto i

l

n an:

hoppingm m m m m m

i ii

H t c c h c

t

etJ i c c c c

Microscopic current operator

device

J

Partitioned approach (Caroli 1970, Feuchtwang 1976): fictitious unperturbed biased system with left and right parts that obey special boundary conditions: allows to treat electron-electron and phonon interaction by Green’s functions.

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 gi

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

24Drawback: separate parts obey strange bc and do not exist.

=pseudo-Hamiltonian connecting left and right

Pseudo-Hamiltonian Approach

25

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

26

2 2

0

22

2

Half-filled 1d system left wire right wirehopping 1

( ) 1 (1 )

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

2 4V

gVg geJ V d

V Vg 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