The geometric phase, or Pancharatnam-Berry phase is known in classical and quantum mechanics. Initially it has been studied in adiabatic processes, where it is simplest, but such a limitation to adiabatic transformations does not exist in general.

In many physical problems when the state of the system depends on parameters and cyclic transformations are done, one observes phenomena, which depend on the geometry and topology of the abstract parameter space.I have introduced this argument on part 11 of this course. The physical effect in question can be a phase shift of a classical wave , a quantum phase shift, or an ordinary angle in the case of a classical mechanics system; there is a great variety of situations in which such phases can be observed experimentally.

1

Below I discuss phenomena where the Hamiltonian has a parameter space R, and performs a closed cycle in it. Depending on the properties of the Hamiltonian in parameter space, the system does not return in the original state. More generally its y acquires a phase, which is related to the singularities of the Hamiltonian encircled by the loop in parameter space. To this end it is necessary that at least 2 parameters are varied and the return to the original point is not done by undoing the first steps in a self-retracing back-and-forth variation but is done by encircling singularities.

Geometric phases (again)

1

2

( )

where is on instantaneous eigenket of H, and C a path in R space,

is a phase, invariant under continuous deformations of C.

I

top

t v

ologic

anishes in simply o

l

a

c

n n R n

C

n

C i a a dR

a

nnected parameter spaces where C can collapse

to a point but in a multiply connected spaces it yields

a good quantum number, which does not arise from any operator.

C

Professor Sir Michael Berry

Pancharatnam-Berry phase

2

Parameter space

R1

R2

R3

Vector Potential Analogy

One naturally writes ( ) · , | . |n n n n R nC

C A dR A i a a introducing a sort of vector potential (which depends on the H eigenstate n, however). The gauge invariance of the Berry phase arises in the familiar way, that is, if we modify the basis with

( )[ ] [ ], ,i R

n n n n Ra R e a R A A

then the extra term, being a gradient in R space, does not contribute to the integral over C.The Berry phase is real since

| 1 | 0 | | 0

| . . 0 | is pure imaginary;

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

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

33

4

To avoid confusion with the physical electromagnetic field in real space one often calls An the Berry connection; calling xa the parameters,

.n n ni x x y x y x

4

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

if the parameter space has dimension 3 the Berry phase is

( ) · · .n n nS S

C rot A ndS B ndS

5

Im ( | ) Im ( | )

Im[ ( | ) ( | )].

The last term vanishes, since it must be even and odd when    .

( | ) Im[ ( | )] Im ( | | ) .

ni n n ijk j n k ni

ijk j n k n ijk n j k n

ni ijk j n k n ijk j k n n i

n

B a a a a

a a a a

j i

B a a a a a a

B

Im ( | | ) .n na a

If the parameter space ,

Im an

is 3d,

| | d

n n

n n R n

B rotA

A a a

and, inserting a complete se

Im ( | | ) Im .

t,

n n n n m m n

m n

B a a a a a a

3-dimensional case

6

If the parameter space has dimension d different from 3 the Berry curvature is a d by d antisymmetric matrix

2Im ( | )n nBa a y x y x

and the Berry phase is given b the generalized Stokes theorem

In general, for R of any number >1 of dimensions, from | | ,

one defines the curvatu

.

The real point is that the Stokes theorem

extends to any dimensio

re ten

ns.

sor

n n R n

n

n n

A i a a

A A

d-dimensional case

7

generalized Stokes theorem

( ) .

Here is an antisymmetric tensor with component

and component - .

n C d d B

d d d d

d d

a

a

a a

a

x x

x x a x x

a

x x

We transform Imn n m m n

m n

B a a a a

Take of the Schroedinger Equation

H R a R E R a [R]:   

(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 n n

R n R n n

R n R n R n n n R n

8

Taking the scalar product with an orthogonal am

  a H a a a E a a  

a aa a divergence of B if degeneracy occurs along C.

E E

m R n m m R n n m R n

m R n

m R n

m n

E

H

Singular curvature

A nontrivial topology of parameter space is associated to

the Berry phase, and degeneracies lead to singular lines or

surfaces, like wizard’s hat.

There must be a singularity!

where m,n indices refer to adiabatic eigenstates of H, and the term with m=n vanishes (vector product of a vector and itself).

8

9

Topological invariant

Integrating the flux of Im

on a closed surface

1.

2

one can show that C Z, that is, C is integer. It is called a Chern number.

n n m m n

m n

n

B a a a a

B nd C

This is clearly related to the Gauss-Bonnet formula

10

Gauss and Charles Bonnet formula

12(1 )

2

curvature, genus

SKdA g

K g

The curvature of a sphere is positive, The curvature of a saddle is negative, in a torusit depends on the point. Chern produced a quantum generalization of the Gauss-Bonnetformula.

11

Aharonov-Bohm effect

The electron(s) see no magnetic field. The phase difference between beams on either side of solenoid is

, magnetic flux in solenoid.

Dimensions: [ ] [ ] [ ] 1

q

c

hc

q

12

Parameters : radius of solenoid,r B

Berry connection : A

curvature : B

Berry phase :

singularity : solenoid

12

0B

Molecular Aharonov-Bohm effect –recall ex

Longuet-Higgins H C, O¨ pik U, Pryce M H L and Sack R A 1958 Proc. R. Soc. A 244 1

The first discovery occurred early:

although systematic understanding occurred after the Berry paper.

H.Christopher Longuet-Higgins (1923-2004)

In the BO approximation, the molecular wavefunction is factorized:

Assume , , where nuclear coordinates, electron coordinates,el nuclx xy x y x x

Let component of momentum.

| , | | , | .

nuclear

el nucl el nucl

p i

p p x x p

a

a

a a a

ax

y x y x y x y x

The effective nuclear momentum acting on ,

averaged over electron degrees of freedom, is therefore :

,

, , .

nucl

nucl el

nucl el el nucl

x i

i x x

a

a

a a

y x

y x y xx

y x y x y x y xx x

13

2

, , , 0.el el elA dx x x dx xa

a a

y x y x y xx x

If the electron wave functions can be taken real, the Berry connection vanishes since

The curvature (magnetic field of the Berry connection) also vanishes.

All OK? It depends on the irreps of electron wave functions and nuclear vibrations. In some cases like eX molecules the electron wave functions cannot be taken real.

The above result

, ,

means that there is an effective magnetic field for the nuclei, in this way:

( )

, , vector potential: it

nucl nucl el el nucl

nucl nucl

el el

i x x

p eA

A x x

a

a a

a a a

a

a

y x y x y x y x y xx x

y x y x

y x y xx

has the form

of a Berry connection.

14

15

To save the situation, the electron wave functions must be taken single-valued but complex, in the absence of a true magnetic field . The Berry connection then does not vanish. The gauge invariant curvature still vanishes, except at singularities, but the flux of the Berry connection through the surface bounded by curves in parameter space does not generally vanish. Again, this leads to the abovespecified observable consequences.

The trouble is that the wave functions change sign for an adiabaticrotation of the molecule by 2 they fail to be single-valued, as in ex example.

Since the total wave function must be single valued, the nuclear wavefunction must also change sign. This is equivalent to a change of the boundary conditions. In turn, this hasobservable consequences, like rotovibrational levels withhalf integer rotational quantum numbers, since the nuclear coordinates behave in a way that resembles a spin, with the typical (-) sign after 2.Rotation.

15

1616

Open path Pancharatnam phase

Then, open path geometric phase

=discrete open path C with

‘equivalent’ initial and final points:

y x y x

1

1

1,2 2,3 2, 1 1,

Fixing a gauge such that ( )=U ( ) we may define

Pancharatnam phase along open path.

This is gauge invariant, . . independent of the initial choice of a gauge,

since a gauge c

n

n n n n

i e

hange would affect in the same way the starting

and the end points of the path.

Two points in parameter space can sometimes correspond to symmetric situations.

x x 1

1

This happens if

unitary U: ( ) U ( )Un

H H

16

x1 x

2

1717

1

Continuous version : , open curve

with equivalent end points .

nk k nk i fC

f i

i dk u u C

U

x x

y x y x

Single point Berry phase: n=2 equivalent points

17

x x 1

2 1

At the limit, one can get a Pancharatnam phase

with 2 equivalent points: ( ) U ( )U.H H

y x y x y x y x y x y x

y x y x

121 2 1

1 1 1 1

1 2

Im log argi

ije U U

Single point Pancharatnam phase: n=2 equivalent points

y x y x1

2 1( )=U ( )

1818

One electron in a periodic potential with pbc.

How to define the mean position within the cell of a cell-periodic

wavepacket

?

y

2

with a Bloch wave does not exist

and does not mean anything!

x dx x x

We want a new definition. 18

Use of Open path Pancharatnam phase:

xa0-a 2a

n(x)

1919

Selloni et al. defined arg( ) Imlog( ), where2 2

a ax z z

y y y

2

0

2| ( )| , reciprocal lattice.

aiGx iGx

cellz dx e x e G

a

y 2

0 0

0Assume ψ perfectly localized at x in first cell and periodic,

( ) . A sensible definition should yield .m

x x x ma x

19

xa0-a 2a

n(x)

2020

One electron in a periodic potential with pbc: position of

wavepacket in cell

2

0Indeed, if ( ) , the definitionm

x x x may

0

0

2 ix

iGx az e e

0

x x

y

2

0

2| ( )| , yieldsarg(

2:),

aiGxz dx e Gx x

a

az

Then, if wave packet is localized in length <<L, this works fine

For poor localization it fails, but then failure is acceptable.

02( )

xArg z

a

20

2121

y y y

y y

2

0

2

2| ( )| , reciprocal lattice

= single-point Pancharatnam phase,

phase difference between and .

The unitary transformation is U= .

This Unitary transformation applied to Blo

aiGx iGx

iGx

i xa

z dx e x e Ga

z

e

e

y y

( )

ch states

U : , ( ) ( )

shifts each k to equivalent k point:

( ) ( ) u ( ) ( )

ikx i k G x

k k

iGx

k k G k k G

k k G e u x e u x

x x x u x e

BZ

arg( ) ; has the form of a Pancharatnam phase.

2

ax z z

21

y x y x y x y x y x

y x y x

12

1 2

1 2 1

1 2

Recall: phase difference between and .i

ije U

Bloch oscillations and the Wannier–Stark ladder

Within the semiclassical approach, an electron in a given band is a wave packet centered at r, whose crystal momentum is q. Neglecting collisions, the following semiclassical picture in an external electric field E seems almost obvious:

2( )1speed: , the classical v=

2

constant acceleration:

since ( ) ( ) is periodic, ( ( )) is periodic.

is periodic in time periodic orbit in real space!

n

n n n

qdr plike

dt q p m

dq eEteE q

dt

q q G q t

dr

dt

Wannier in 1960 predicted that quantization would lead to closed orbits. He assumed that adding G to q the periodic wave function unk resumed the original value leading to a periodic motion. He predicted the so called Wannier –Stark ladders, with an oscillator-like discrete spectrum.

q

( )n q

22

An external electric field breaks periodicity,Bloch’s theorem, bands and all that, leading to some paradoxical results.

23

( )1But the equation of motion is not fully correct, because

of a curvature which must be accounted for.

n qdr

dt q

, = band-energy average, m=integer, integer4

m m mW eEa ma

a

4 4

Periodic motion implies quantized e

Frequency estimate of Bloch oscillations : , lattice constant.

This is observable if 1, where is the collision time.

For E=10 / , 10 e /

nergy.

B

B

eEa a

V cm eE V Angstro

11 1 1.5 10 .

Normally this is not observable-

Semiconductor Superlattices at low T are needed

Bm s

2Constant acceleration: implies that the period T is given by

and the angular frequency = .B

eEt eETq

a

eEa

2424

one uses a parametric Hamiltonian for periodic part unk, k is the parameter

2( )( ) ( ) ( )

2k k k

p kV x u x u x

m

Bloch states in solids: Berry phase and Zak’s phase

We are allowed to study the cell as if it were isolated, but then—as emphasized in Berry’s original paper [1]—the interaction with the ‘rest of the Universe’ gives rise toaparametric Hamiltonian, hence a non-trivial phase which is observable: indeed, this is a Berry’s phase.

Since we use a k- dependent Hamilt onian for uk t her e is a Ber r y’s

connect ion, w it h K=par amet er. i.e, t he K space is t he par amet er space R.

Berry connection ( )n nk k nkk i u u

0nk k nkC

i dk u u

So, there is a Berry’s phase ante litteram if we can choose

the contour C such that

How can we take contour C? What is the physical content?

To compute Bloch functions ( ) ( ),ikr

nk nkr e u ry

J. Zak,Phys. Rev. Lett. 20, 1477 – Published 24 June 1968

24

2525

nkknkC

uudki

G

( )

, ,

, ,

n,k

( , ) ( , ). This requires

( ) ( ),

Natural

( ) ( )

(Thus, u is periodic in r and up to a p

gauge: no phase fa

hase in

t

k)

c or n n

i k G r ikr

n k G n k

iGr

n k n k G

k G r k r

e u r e u r

u r e u r

y y

C

Open path geometric phase

Zak’s phase

C connects k with k+G where G is a reciprocal space vector.

Bloch's function (k, ) ( ) with ( ) ( ).

Adding G to k, the Bloch function can pick a phase.

ikr

n nk nk nk

n

r e u r u r u r Ry

y

25

. 1 for R Bravais latticeiG Re

But what happens to the many-electron state?

2626

Berry phase of Slater determinant (Hartree-Fock or Kohn-Sham) in

terms of Berry phases of orthogonal spin-orbitals

No general result is known for Pancharatnam phases, but taking the continuum limit the problem simplifies. One can show:

In the continuum limit the closed path Berry phase is the sum of

individual spinorbital Berry phases- (This is not granted in general

for finite systems)

26

We can rewrite it as a Pancharatnam phase discretizing with M+1 points q0….qM

Consider transporting over C a determinant

with a number of bands: the Berry phase is

0bandsn

nk k nkCn

i dk u u

is related to the bulk polarization of insulators!bandsn

nk k nkCn

i dk u u

27

G.Sundaram and Q.Niu in1  999 . . 59, 14 915

discovered that the seemingly obvious equations of motion

( )1, ,

are not complete.

n qdq dreE

Phys Rev

dt dt

B

q

Actually,

2Berry curvature of band, clearly is a length and L ,

so is a velocity.

is right but

( )1 , so there is a drift term.

.

q nq q nq q

n

q i u u q

dqq

dt

dqeE

dt

qdr dqq

dt q dt

' phase is obviously the Berry phase

u( ) u( ) q

C

Zak s

i q q dq

They show that .4 2

m mW eEa ma

28

More applications and derivations in Raffaele Resta, J. Phys.: Condens. Matter 12 (2000) R107–R143. and J. Phys.: Condens. Matter 14 (2002) R625–R656

28

29

In ordinary bulk materials these Stark oscillations cannot be seen, because collisions dephase the coherent motion of electrons on a time-scale which is much shorter than TB = 2π/ωB. Eventually they have been seen in semiconductorsuperlattices.

30

3131

Dipole moment of insulators

0 0

1

For a molecule, one defines the dipole moment

functional of the density

N

ii

d e rdV e R

R r

- +

Polar molecule

This definition does not apply to periodic solids!

The operator r takes outside Hilbert space of periodic

functions; it makes surfaces crucial

The results of calculations in a slab geometry depend on how the slab is defined 31

3232

- - - -- + + + - + + +

-+

Periodic array of polar molecules: same pattern can be obtained e.g. by repeating

The polarization of metals is mainly due to screening at the surface, but polarization of piezoelectric insulators is much more a bulk (not surface) property.The dipole moment of the unit cell is not well defined.

- +

and the results of slab calculations depend on the way one terminates the lattice

32

Simple-minded definition of dipole averaged over a determinant

y y

31 ( ) .

Since it is a one-body operator, we may sum over spin-orbitals.

2 ( , ) ( , ) , where .

The plane wave fac

el elcrystal

el n n cellnk

P d r r rV

eP k r r k r V NV

V

tors cancel and

2 ( , ) ( , ) .

el n nnk

eP u k r r u k r

V

3333But result depends on origin, which is arbitrary!

To characterize a piezoelectric crystal we may use an arrangement in which the crystal is uniaxially strained in a shorted capacitor and the current in the external circuit is measured. E and H are negligible.

Amperometer

J

Indeed, the mechanical action varies the polarization P and produces a current.

dispacement current 4 but is negliglible 4 D E P

J D E P Jt t t

but H is negligibleD

H Jt

34

z

35

Note:the phase of the wave function is involved in the current (J=0 if y is real); the mechanical action modulates the polarization P. The phenomenon is not a surface effect. We need a pressure gauge parameter.

The pressure shifts the positive charges relative to the negative charges, thereby changing P and produces the current. Besides, the pressure can also reduce the lattice parameter a, but we disregard this fact because it does not change P.

3636

Introduce a deformation parameter l such that the real solid

corresponds to l=1

l=0.5

l=0.

the dipole is halved for

the dipole vanishes for

36

l l l y l y l

l l l

31 2( ) ( , ) ( ) ( , , ) ( , , ) ,

and removing the plane wave factors,

2 ( ) ( , , ) ( , , ) .

el el el n ncrystalnk

el n nnk

eP d r r r P k r r k r

V V

eP u k r r u k r

V

3737

The current arises from a variation of pressure and of the pressure gauge parameter. Therefore set

Thus we focus on the derivative of dipole

(where bulk contributions dominate)

l ll l

2( , , ) ( , , )eln n

nk

P eu k r r u k r

V

l l l ll l

l ll

2( ( , , ) ( , , ) ( , , ) ( , , ) )

22Re ( , , ) ( , , ) .

n n n nnk nk

n nnk

eu k r r u k r u k r r u k r

V

eu k r r u k r

V

and doing the derivative

3838

l ll l

22Re ( , , ) ( , , ) .el

n nnk

P eu k r r u k r

V

Thus, the object we want is the Bloch function contribution :

l l l l ll l

2Re ( , , ) ( , , ) 2Re ( , , ) ( , , ) .

nk n n n nB u k r r u k r u k r r u k r

a aa

a

l

al

l l l ll l

l l

ˆdoes not mix different k, since k is eigenvalue of , and a is not changed

however mixes different bands,and we expand over bands :it holds that

( , , ) ( , , ) ( , , ) ( , , )

Thus, 2Re ( , , )

a

n n

nk n

T

u k r u k r u k r u k r

B u k r ra a

l l ll

( , , ) ( , , ) ( , , )

nu k r u k r u k r

ll

2elnk

nk

P eB

V

3939

......but r is a tricky operator for a periodic solid.

We must replace it.

ll

2elnk

nk

P eB

V

a aa

l l l l ll

2Re ( , , ) ( , , ) ( , , ) ( , , )

nk n nB u k r r u k r u k r u k r

y y

( , ) ( , )

( Appendix: tedious but elementary consequence of Bloch's theorem)

m n m n m nr u r u u k r i u k r

k

see

Recall: we met : it is Zac's phase (page 20)nk k nkC

i dk u u

summarizing: 4 P

Jt

a aa

a aa

l l l l ll

l l ll

y y

Embellish

2Re ( , , ) ( , , ) ( , , ) (

using: ( , ) ( , ) , and get

, , )

2Re ( , , ) ( , ) ( , , ) (

m

nk

n m n m

n

n

n n

nk n

B u k r r u k r u k r u k r

B u

r u r u u k r i u k rk

k r i u k r u k r u kk

l, , ) .r

40

a

Now, we wish to do and to this need to shiftend we :

k

aa al

( , ) ( , )

where the blue stuff 0. N

( , , ) ( , ) ( , )

o

(

,

,

w

)n n nu k r i u k r i u ki u k r u rk r

kr u

kk

k

a aa

l l l l ll

2Re( ) ( , , ) ( , , ) ( , , ) ( , , ) ,

and we can sum over the complete set.

nk n nB i u k r u k r u k r u k r

k

40 l l ll

2Re( ) ( , , ) ( , , )

nk n nB i u k r u k r

k

41

l l ll

2Re( ) ( , , ) ( , , )

nk n nB i u k r u k r

k

l ll

2Im ( , ) ( , , )nk n nB u k r u k r

kthat is:

l ll l

22Im ( , , ) ( , , ) .

Next, we sh

Summary: we have shown tha

ow that this is a Berry phase by a model calculat

t

ion.

eln n

nk

P eu k r u k r

V k

41

ll l

2Modelcalculationof 2Im ( , ) ( , , ) .el

n nnk

P eu k r u k r

V k

3

3

First, weassume that only the nth band contributes,and

is inserted backat the end.

For

Brilloui

simplici

n Zo

ty assume(2 )

2 2but neglecting dependence on

ne su

, ,

mmationn

a b c

x y zk BZ

a b c

x yk

Vd k dk dk dk

k ka b

c

z

c

dk

l

l l

3

Then,

2 2 22Im ( , ) ( , , )

(2 )

cel

z n z n z

z

c

P e Vdk u k r u k r

V a b k

42

42

l

l l

l

3

z

2 2 2Integrating 2Im ( , ) ( , , )

(2 )

overλ with ( 0) 0, and writing k for k

cel

z n z n z

z

c

el

P e Vdk u k r u k r

V a b k

P

l l

l

1

30

2 2 2 2 Im ( , ) ( , , ) .

(2 )

c

el n n

c

e VP d dk u k r u k r

V a b k

43

k

l

c

c

1

This is the change of P from no dipole (l=0) to actual dipole, so it is the actual dipole. We want to transform this surface integral to the integral of a curl 43

*The identity 2Im Im Im , with z ( , ) ( , , )n nz z z u k r u k rk

ll

l l ll l l

2Im ( , ) ( , , ) Im ( , ) ( , , ) Im ( , , ) ( , )

n n n n n nu k r u k r u k r u k r u k r u k rk k k

44

l l l

l l

1

0

Putting all toget

Im ( , ) ( , , ) ( , , ) ( , )

her:el

c

n n n n

c

P

ed dk u k r u k r u k r u k r

ab k k

l l

l

1

30

2 2 22Im ( , ) ( , , ) .

(2 )

c

el n n

c

e VP d dk u k r u k r

V a b k

The l and k derivatives on the bra must be taken out the matrix elements 44

l l ll l l

l l ll l l

l ll l

l

( , ) ( , , ) ( , ) ( , , ) ( , ) ( , , )

( , , ) ( , ) ( , , ) ( , ) ( , , ) ( , )

( , ) ( , , ) ( , , ) ( , )

( , ) (

n n n n n n

n n n n n n

m n n m

m n

u k r u k r u k r u k r u k r u k rk k k

u k r u k r u k r u k r u k r u k rk k k

u k r u k r u k r u k rk k

u k r u kk

l ll

, , ) ( , , ) ( , )n mr u k r u k r

k

l l l l l

1

0Im ( , ) ( , , ) ( , , ) ( , )

Now it is a curl of the following vector:

c

el n n n n

c

eP d dk u k r u k r u k r u k r

ab k k

4545ll l

l

( , ) ( , , ) ( , ) , ( , ) ( , , ) .

k n m n nV V u k r u k r u k r u k r

k

ll l

l

( , ) ( , ) ( , , ) ( , ) , ( , ) ( , , ) .

x y k n m n nV V V V u k r u k r u k r u k r

k

For any vector , ( ) . In this case,

y x zV V V rot V

x y

l l l l l

1

0Im ( , ) ( , , ) ( , , ) ( , )

c

el n n n n

c

eP d dk u k r u k r u k r u k r

ab k k

l l l

l

1

0Im ( , ) ( , , ) , ( , ) plane

is indeed the fluxof the curl of ( , ) ( , , ) .

c

el n nz

c

n n

eP d dk curl u k r u k r z k

ab

u k r u k r

k

l

c

c

1

46

l l ( , ) ( , , ) in ( , ) space, andn n

V u k r u k r k

46

4747

k

l

c

c

1

l

Stokes theorem Im ( , ) ( , , ) . ( )

el n n n

C

e eP u k r u k r ds C

ab ab

George Gabriel Stokes

x

x

x x x

l

( ) ( ) ( ) ,

( , )

n n n

C

C i u u d

k

Berry phase

of the periodic functions u ( , )

along the

( ) is a

rectan

ctually a

gular circuit.

n

n

C

k r

47

.

4848

x x x x ( ) ( ) ( )n n n

C

C i u u d

( )

,

Along vertical tracks, . Since commutes with e , one can

replace periodic part by full Bloch function:

( ) ( ) ( ) ( ) .

Recall Zac phase and Natural gauge:

( )

ikx

n n n n

i k G r

n n k G

d d

d d

u u d d

k G e u

x

x x

l l

x x x y x y x x

y

,( ) ( ) ( ),

vertical tracksareintegrated in oppositesenseand cancel.

is given by the difference of the horizontal tracks, which are Zac's phases.

becomes

ikr

n k n

el

r e u r k

P

d

dkx

y

Dipole = ( )

el n

eP C

ab

k

l

c

c

1

48

4949

nkknkC

uudki

G

C

Recall the open path geometric phase

(Zak’s phase)

C connects k with k+G

l

31

1

For =0 there is no dipole.

General formula:

2(King-Smith and Vanderbilt,1993)

2

sum over bands

b

b

n

nq nqn c

n

n

ieP dq u u

q

49

l l

0 1

( ) ( ) ( ) ( ) ( )n n n n n

d dC i u k u k dk i u k u k dk

dk dk

50

Mathematical digression: Topological space

A topological space is (X,N(x)) where X represents a set of points x and N(x) is a neighborhood topology of x:neighborhood topology of x means that

If a topological space is such that two different points have at least two distinct neighbourhoods than it is a Hausdorff space.

and n(x) (x) n(x)

that is a point belongs to all its neighbourhoods

The intersection of two neighbourhoods is a neighbourhood

n(x) is a neighbourhood of all points of some neighbourhood of x

x X N x

51

Mathematical digression: Fiber bundle

If E and F are topological spaces, the product ExF is a trivial fiber bundle.One may take E=base space and F=fiber (fibre in UK English).More generally, a fiber bundle is a structure that locally is like ExF whileglobally can be more complex.

The moebius strip is a bundle which is locally the product of a circle times a segment, but is nontrivial because it has one face.One can also make a cylinder which is a trivial fiber bundle.

Fiber bundles obtained from two circles

Cats always fall on their legs. They manage to control the rotation angle while conserving angular momentum

Falling cats

52

53

Falling cat model as a problem in fiber bundle theory with base space M parametrized by a, and the fibre is the rotation group SO(3) or SO(2) parametrized (in 2d) by q The fibre specifies the orientation of the cat.

The shapespace isparamertizedby a,

angle between 1 and 2

angle between 2 and 3

a

angle between 1 and x axis

defines orientation in a fixed frame

q

53

Masses 1, 2 and 3 are taken equal to m.

3

2

1

x

y

qa

Simple model by Surya Ganguli

54

A deformable body taken through the sequence of shapes a,b,c,d.There is a net rotation at the end of the sequence

55

This is an example of a nonholonomic system; this means that its state depends on the path taken in order to achieve it. The system depends by a set of parameters subject to differential constraints. It evolves along a continuous path in its space of parameters. When it returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state. Nonholonomicsystems in mechanics cannot be represented by a potential energy and is called non-integrable. A sphere rolling on a plane can return to its starting point with a modified orientation. The Foucault pendulum is another nonholonomic system, and others can examples arise in optics (polarized light in an optica fibre can be made to change the polarization plane by twisting the fibre).

Change in shape rotation because of conservation of angular momentum

1 1 1 1 2 2 2 2 3 3 3 3

z component of angular momentum of the cat

( )

can be written in terms of angles

zL m x y y x x y y x x y y x

1 1cos y sinx R Rq q

2 2cos y sinx R Rq a q a

3 3cos + cos y sin + sinx R R R Rq a q a q a q a 56

0 (4 2cos ) (3 2cos ) (1 cos ) 0

,

3 2cos 1 cos

4 2cos 4 2cos

zdL d d d

d A d A d

A A

a

a

q a

q a

57

zL

q

zL

a

zL

Impose dL 0z z zz

L L Ld d dq a

q a

58

A closedcircuit in shape spaceleading to net rotation

2

a

1

2

3

2

1

x

y

qa

( ) angle of rotation of cat in fixed frameA d A da q a

,

3 2cos 1 cos

4 2cos 4 2cos

d A d A d

A A

a

a

q a

59

A closedcircuit in shape spaceleading to net rotation

0 02 2

0 02 2

The rotation of the cat during the motion shown in the

right panel

( )

7.5 degrees

A d A d

A d A d A d A d

a

a a

q a

a a

2

a

1

2

59

3

2

1

x

y

qa

This may be amusing but similar problems are of key interest in Robotics.

60

2

0

Selloni et al. found how to compute the mean value .

arg( ),2

We shall also need to compute off-diagon

2| ( ) | , .

al .

ax

m

G

n

iz dx e x Ga

x

ax z

r

y

y

y

Appendix: Coordinate and Momentum matrix elements

between Bloch functions

[ , ] [ , ] ( ( ) ( ))m n n m m n m n

i iH r p H r E k E k r p

m my y y y y y

so since , m n we must work out( ( ) ( ))

the momentum matrix elements.

m n m n

n m

ir p

m E k E ky y y y

( )One finds ( , ) ( , ) and off-diagonal n

n n

E kpk r k r

m ky y

( , ) ( , ) ( ( ) ( )) ( , for) ( , ) , y y

m n n m m n

pk r k r E k E k u k r u k r m

kn

m

61

2 2( )[ ( )] ( , ) ( )] ( , ) [ ( )] ( , ) ( ) ( , )2 2

ikr ikr

n n n n n n

p p kV r e u k r E k e u k r V r u k r E k u k r

m m

2 2

so ( ) ( , ) ( ) ( , ), where ( ) ( ) . .2 2

n n n

p kH k u k r E k u k r H k V r k p

m m m

We need the equation for the periodic function un(k,r)

From the previous result ( ( ) ( ))

one finds ( , ) ( , ) for .

n m m n m n

m n m n

iE k E k r p

m

r u k r i u k r nk

m

y y y y

y y

Proof

62

2( )Apply on Schrodinger equation [ ( )] ( , ) ( ) ( , ) :

2n n n

p kV r u k r E k u k r

k m

( ) ( , )( )( , ) ( ) ( , ) ( , ) ( )n n

n n n n

E k u k rp ku k r H k u k r u k r E k

m k k k

( , )

( )(

but red terms van

Le

( ) ( , ) ( , )

( ) ( , ) ( , ) ,

( , ) real ( , ) ( , ) 0

t us take matrix elements, multiplying b

, ) ( , )

(

y . On diagonal, m

ish

si

)

=n

nce

,

m

n n n n

n n nn

n n n

n E k u k r u k rk

E k u k r u k rk

u k r u

u k r

p

k r u

ku k

k

r u k rm

rk

E k

k

Thus , we are left with:

( )( )(

on diag

, )

a

( , )

on

.

l

nn n

E kp ku k r u k r

m k

( )In terms of ( , ) we may write ( , ) ( , ) .n

n n n

E kpk r k r k r

m ky y y

( )( , ) ( ) ( , ) (

here red stuff 0 by orthogonality, a

, )

Off diagonal fo

n

r

d oneis le

(

) ( ,

ft wi

,

() ;

)

th

m n m n nnEp

u k r H k u k r u k r E k uk

kk

m

rm k k

n

k

( , ) ( ) ( , ) ( , ) ( ) ( , )m m n m n n

pu k r E k u k r u k r E k u k r

m k k

63

that is,

( , ) ( , ) ( ( ) ( )) ( , ) ( , ) .m n n m m n

pu k r u k r E k E k u k r u k r

m k

F

for .

rom the previous result ( ( ) ( ))

one finds ( , ) ( , )

n m m n m n

m n m n

iE

m

k E k r pm

r u k r i u k rk

n

y y y y

y y

We may also write

( , ) ( , ) ( ( ) ( )) ( , for) ( , ) , y y

m n n m m n

pk r k r E k E k u k r u k r m

kn

m

since the extra term generated by differentiating the plane wave is killed by

orthogonality.

( ) ( , )( )( , ) ( ) ( , ) ( , ) ( )n n

n n n n

E k u k rp ku k r H k u k r u k r E k

m k k k

63