Eschrig H. Topology and Geometry for Physics

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ðRðsÞ; tÞi ¼ jWðRð sÞÞiexp icðtÞi

Eð Rðs

ÞÞ

ð8:74Þ

into the equation (8.72) and ﬁnd straightforwardly after multiplication from the

left with hW

ðRðsÞ; tÞj

¼ i WðRðsÞÞ



WðRðsÞÞ



cðtÞ¼i

WðRðsÞÞ



WðRðsÞÞ



¼ i

RðtÞ

Rð0Þ

dR WðRÞ

WðRÞ

¼ i

hWðRÞjdjWðRÞi; ð8:75Þ

where C is the considered path RðsÞ through M: The phase cðtÞ is called Berry’s

phase (Berry, 1984).

It is in many instances a measurable quantity, and it took

nearly 60 years since the foundation of the Hilbert space representation of quan-

tum theory to realize that not every dynamical quantum observable is represented

as a Hermitian operator.

B. Simon was the ﬁrst to realize that the last integrand is a local connection

form on ðP; M; p; Uð1ÞÞ:

A¼

¼hWðRÞjdjWðRÞi ¼ ðdhWðRÞjÞjWðRÞi: ð8:76Þ

The last relation is a direct consequence of the normalization of jWðRÞi: It shows

that A is anti-Hermitian, it is called the Berry–Simon connection . To see that it is

a local connection form, consider two local sections s

ðRÞ¼jWðRÞi

and s

ðRÞ¼

jWðRÞi

with the transition function w

¼ expðivÞ; jWðRÞi

¼jWðRÞi

ðRÞ:

Then,

ðRÞ¼

hWðRÞjdjWðRÞi

¼ w

1

hWðRÞjdjWðRÞi

þ w

1

hWðRÞjWðRÞi

¼ w

1

ðRÞw

þ w

1

¼A

ðRÞþidvðRÞ;

which proves the required property. (In the ﬁrst term on the second line d is meant

to operate on jWðRÞi

only.) The corresponding curvature form F¼DA¼dA

(the latter since Uð1Þ is Abelian) is called Berry’s curvature, it is given by

A collection of most of the relevant original papers on the subject is gathered in the volume [1].

8.6 Geometric Phases in Quantum Physics 277

F¼ðdhWðRÞjÞ ^ðdjWðRÞiÞ ¼

ohWðRÞj

ojWðRÞi

^ dR

: ð8:77Þ

A phase difference of two quantum states or two classical waves can be measured,

if both waves are brought to interference. This happens, if parts of the wave

propagate along different paths between the same start and end points or, equiv-

alently, if a wave circuits along a closed loop C: In the latter case it interferes with

itself according to the phase difference (8.75). Clearly, if x 2 M is a base point of

loops, then all possible phase differences cðCÞ for all possible loops C based on x

and running through M constitute the holonomy group H

related to the connection

on ðP; M; p; Uð1ÞÞ provided by the local connection form A:

Let S be a two-dimensional surface in M bounded by C ¼ oS: Then, Stokes’

theorem yields

cðCÞ¼i

C¼oS

A¼i

F; ð8:78Þ

that is, Berry’s phase equals i times the ﬂux of Berry curvature through the surface

S: This is suggestive of magnetism and of Aharonov–Bohm physics, but is much

more general.

The expressions (8.76, 8.77) point out yet another important generalization of

these considerations: As a connection in a physical parameter space, the Berry–

Simon connection A and also Berry’s phase (8.75) between distinct points of the

parameter space is a gauge potential and hence gauge dependent and in general

not measurable. On the contrary, Berry’s curvature (8.77)isagauge ﬁeld and

hence has physical relevance leading not only to a measurable quantity (8.78) but

is also measurable locally along any path through the parameter space, closed or

not.

8.6.2 Degenerate Case

Shortly after Berry’s and Simon’s papers, Wilczek and Zee pointed out that this

concept has a relevant non-Abelian generalization. It happens that a quantum state

has an isolated energy level which however is globally, that is on a whole

parameter manifold M; N-fold degenerate. Think for instance of a Kramers

degenerate doublet state of a molecule (see p. 337). Instead of (8.73), now

a¼1

ðRÞihW

ðRÞj; hW

ðRÞjW

ðRÞi ¼ d

ð8:79Þ

is the adiabatic quantity, where locally the orthonormalized states jW

ðRÞi can

always be chosen smoothly depending on the parameter set R ([7]).

278 8 Parallelism, Holonomy, Homotopy and (Co)homology

With the ansatz

jWðtÞi ¼

a¼1

ðtÞjW

ðRðsÞÞi ð8:80Þ

one ﬁnds after projection on hW

ðRðsÞÞj

ðtÞ

a¼1

ðRðsÞÞ



ðRðsÞÞ



þ iEðRðsÞÞd



ðtÞ¼0 ð8:81Þ

with the formal solution

ðtÞ¼

a¼1

T exp

dt AðRðsÞÞ iEðRðsÞÞ1

ðÞ

ð0Þ; ð8:82Þ

where now the simple power series expressed by the exponentiation is to be

replaced by a series which observes the order of factors with ascending time

from right to left. This is formally expressed by the time-ordering operator T

physicists are familiar with. In the adiabatic limit, the time integration can again

be expressed as a path integration along the path parameter s in the parameter

space, leading to

ðRÞ



ðRÞ



¼hW

ðRÞjdjW

ðRÞi ð8:83Þ

for which the transition between local patches U

and U

of the m-dimensional

parameter manifold M, jW

ðRÞi

ðRÞ

; in complete analogy to

the case (8.76) yields

¼ w

1

þ w

1

; A

; ð8:84Þ

that is, A is again a local connection form of a connection on ðP; M; p; GÞ:a

1-form on M, which is g-valued, where g is the Lie algebra to the Lie group

G 3 w

providing the degeneracy of quantum states. Note that due to (8.79) G is

unitarily acting on the space C

of wave functions jWðRÞi at given R: The geo-

metric change of state along a closed loop is given by

wðCÞ¼Pexp 



; ð8:85Þ

where P means path ordering from right to left surviving from the time ordering

T : For loops based on x 2 M it is again an element of the holonomy group H

related to the connection on ðP; M; p; GÞ provided by the local connection form

8.6 Geometric Phases in Quantum Physics 279

A: In gauge ﬁeld theory the corresponding quantity is called Wilson’s loop

integral.

The integral

wðCÞ is gauge covariant. Indeed, take any gauge transformation

wðgðRÞÞ on M, where wðgÞ as on p. 261 is an element of the unitary representation

of G in C

. It now corresponds to a smooth (on M) transition to new states

ðRÞi ¼

ðRÞiw

ðgðRÞÞ alternative to (8.79). The corresponding change

of the connection is A

¼ w

1

Aw þ w

1

dw: Exploit expðB

1

ABÞ¼B

1

ðexp AÞB,

which holds for arbitrary matrices A; B: Consider the transformed loop integral

wðCÞ¼

exp w

1

Aww

1



1

expðAÞexp dR



wðgðRÞÞ

1

expðAÞwðgðRdRÞÞ¼wðgðR

ÞÞ

expðAÞ

wðgðR

ÞÞ;

where the product is understood in path order which precisely leads to cancellation

of the intermediate products wðgðR  dRÞwðgðR  dRÞÞ

1

, and R

is the base point

of the loop C. Hence

ðCÞ¼wðgðR

ÞÞ

1

wðCÞwðgðR

ÞÞ: This also means that

wðCÞ is gauge dependent and hence not directly measurable.

Recall from (8.70) that i=ð2pÞtr A (with the trace taken in g) is the Chern–

Simons form of the ﬁrst Chern character i=ð2pÞF of the connection provided by

A; since F ¼ DA ¼ dAþA^A; it follows that tr F¼d tr A; because tr A^

A¼0: Hence, if one takes the trace under the integrals of (8.78), one gets again a

gauge invariant measurable Berry phase:

cðCÞ¼i

C¼oS

tr A¼i

tr F: ð8:86Þ

The above considerations show that

wðCÞ¼tr P exp

Að8:87Þ

is another gauge-independent quantity which can be measured.

Finally, Aharonov and Anandan generalized the concept to general non-adia-

batic situations. Although this seems not to lead to new measurable quantities, it

provides a general classiﬁcation of UðNÞ principal ﬁber bundles and hence of all

possible cases of geometric phases in quantum physics [2].

Nowadays there is a wealth of applications of this concept in solid state physics.

The interested reader is referred to [1, 2] and citations therein. We only select a few

typical examples.

280 8 Parallelism, Holonomy, Homotopy and (Co)homology

8.6.3 Electrical Polarization

For details see the reviews [8, 9] and [10] and citations therein. This presentation

follows closely [8]. Consider the bulk electric dipole density of a material, that is,

the dipole density which is independent of the shape and the surfaces of a piece of

material. This quantity is what is described by the thermodynamic limit, where the

volume is let go to inﬁnity with all average densities kept constant. To get rid of

surface effects one uses periodic boundary conditions, that is, one replaces a

volume L

by a 3-torus x

 x

þ L; x

 x

þ L; x

 x

þ L: Any charge

density is forced to be periodic. For the sake of simplicity consider just one

dimension. The electric charge density is qðxÞ¼qðx þ LÞ: Let it be represented by

a generating function RðxÞ; qðxÞ¼dRðxÞ=dx: For a neutral case, it must be

aþL

dxqðxÞ¼

aþL

dxðdR=dxÞ¼Rða þ LÞRðaÞ¼0 for arbitrarily chosen a.

Hence RðxÞ is also periodic. Of course, an additive constant to R has no physical

consequence and hence no physical meaning. Now calculate the ‘average dipole

density’ with the help of integration by parts: ð1=LÞ

aþL

dx xqðxÞ¼

ð1=LÞ

aþL

dxðRðxÞRðaÞÞ ¼ ð1=LÞ

aþL

dxRðxÞþRðaÞ: Due to periodicity

of RðxÞ the ﬁrst term is independent of a: Hence, via the second term the result

depends on the physically irrelevant reference position a: Although formally a

‘bulk dipole density’ seems to be deﬁned, it can be given a quite arbitrary value, it

is not at all related to the physics at hand. This ﬂaw has entered many textbooks. In

fact, the dipole density anticipated in physics, although a bulk property, is ﬁxed by

the surface of the sample which destroys periodicity. Opposite charges move in an

applied electric ﬁeld in the bulk in opposite directions and accumulate only at the

surface, although the bulk determines how far charge is moving.

Consider a reference situation of an inﬁnite crystal with zero electrical polar-

ization for physical reasons, for instance since the crystal has a center of inversion.

Let the system polarize by destroying this symmetry in an adiabatic process with

keeping the periodicity ﬁxed (that is, retaining some ﬁxed periodicity without

which the thermodynamic limit can hardly be dealt with), for instance by letting a

ferroelectric slowly polarize by moving a (charged) sublattice of nuclei in some

direction or by applying a spatially periodically oscillating electric ﬁeld.

To treat these cases, the notion of lattices L

3 R and L

3 G inverse to each

other is adopted and of the corresponding three-tori T

and T

as introduced in

Sect. 5.9 on p. 160 ff to be the unit cells of those lattices. (Here, the notation

k ¼ p=h is used and

f ðRÞ is written instead of

f ðR

Þ, likewise for G:)

Recall that in inﬁnite three-space

dðkÞ¼

ð2pÞ

ikr

ð2pÞ

ikr

ikR

¼ FðkÞGðkÞ:

ð8:88Þ

8.6 Geometric Phases in Quantum Physics 281

Here, F is clearly smooth and ﬁnite while the inﬁnite sum G is a distribution like

dðkÞ, with the obvious property Gðk þGÞ¼GðkÞ due to RG ¼ 2p  integer

(5.102). Moreover, for k ! 0 obviously dðkÞ¼ðjT

j=ð2pÞ

ÞGðkÞ with the cell

volume jT

j; while FðGÞ is easily found by direct calculation, together with a

corresponding integral over the reciprocal cell,

iGr

¼jT

;

ikR

¼jT

; ð8:89Þ

with the Kronecker symbol d

on the lattice L

and d

on L

; respectively.

Altogether one has

ikR

¼jT

dðk GÞ;

iGr

¼jT

dðr RÞ; ð8:90Þ

where we also added the analogous relation for the reciprocal lattice, and jT

j¼

ð2pÞ

=jT

j: If one limits the variables k and r to the corresponding tori only

(considering periodic functions), then only the single item with G ¼ 0 and R ¼ 0,

respectively, survives on the right hand sides.

The electron charge density of a crystal may in principle rigorously be obtained

from an effective one particle equation, the Kohn–Sham equation of density

functional theory (e.g. [11]). The crystal orbitals themselves being eigenfunctions

of the Kohn–Sham Hamiltonian, H ¼ðh

=2mÞr

þ U with Uðr þRÞ¼UðrÞ,

are not lattice periodic; according to Bloch’s theorem they carry a phase e

ikr

and

are obtained from Hw

ðrÞ¼w

ðrÞe

with the orthonormality condition



¼ d

dðk k

Þ: Comparison of this condition with the ﬁrst equality

(8.88) tells that in a constant potential U the state is w

¼ð2pÞ

3=2

ikr

which

means jT

j=ð2pÞ

¼ 1=jT

j electrons per cell T

: The states may, however, be

represented as

ðrÞ¼

ikr

1=2

ðrÞ; u

ðr þRÞ¼u

ðrÞ; u

n;kþG

ðrÞ¼u

ðrÞ; ð8:91Þ

where the periodic functions u

are obtained as eigenfunctions of the Hamiltonian

¼ e

ikr

ikr

¼ðh

=2mÞðirþkÞ

þ U which acts on functions on the torus

and depends parametrically on k 2 T

;

n;k

ðrÞ¼u

n;k

ðrÞe

; ðu

Þ¼



ðrÞu

ðrÞ¼d

: ð8:92Þ

The last orthonormality relation results from the orthonormality condition for the

with (8.91) and the ﬁrst equality (8.90). The functions u

still carry an

arbitrary k-dependent but now r-independent phase as seen from the last

282 8 Parallelism, Holonomy, Homotopy and (Co)homology

eigenvalue problem; it is, however, essential for the following that this phase is

chosen to be a periodic function on T

(and hence is the same for k and k þ G).

This assumption has already been made in the last relation (8.91).

In a semiconductor, there is an energy gap between all occupied energies e

and all unoccupied energies. Then, for all occupied bands the n-non-ordered sets

g and fu

g are smooth functions of k 2 T

(in an appropriate topology of a

functional space of set-valued functions, [7]).

Instead of the periodically repeated functions u

, multi-band Wannier func-

tions

ðrÞ¼

ikðrRÞ

occ:

ðkÞu

ðrÞ; U

ðkÞ¼U

1

ðkÞ; ð8:93Þ

with a unitary-matrix function UðkÞ may be introduced for the occupied bands.

The matrix function UðkÞ must again be periodic as function of k 2 T

but may

otherwise be rather arbitrary. It is well known that, depending on its choice, for an

energy-gap separated band group the Wannier functions can be exponentially

localized in r-space. With the relations above one easily veriﬁes (exercise)

ða

Þ¼



ðrÞa

ðrÞ¼d

n;n

: ð8:94Þ

It is another simple exercise to show that (n runs over the bands per spin and over

the spin quantum number)

occ:

ðrÞj

occ:

k w

ðrÞ

¼ qð rÞð8:95Þ

is the total electron density of the crystal, in the left expression written as a sum

over the unit cells of the lattice. This lattice sum may be used to express a change

of the average electron dipole density of the crystal as

¼

occ:

rrDja

ðrÞj

i

occ

rðUuÞ

ðUuÞ

ð8:96Þ

where e is the positive electric charge unit (proton charge). The last expression is

obtained by inserting (8.93) into the previous one, using re

ikr

¼ir

ikr

; inte-

grating per parts, and again using

ikr

FðrÞ¼

ikR

ikr

FðrÞ¼

jdðkÞ

ikr

FðrÞ for a periodic function F (exercise).

8.6 Geometric Phases in Quantum Physics 283

By applying the Leibniz rule, the unitarity of the matrix UðkÞ and (8.92), the

last r-integral in (8.96) is easily transformed,

occ:

ððUuÞ

ðUuÞ





occ:

Þþtr U

1

ðkÞr

UðkÞ



: ð8:97Þ

With (5.19) in the form det U ¼ det exp lnU ¼ exp tr lnU and the linearity of the

trace the last term is cast into tr rln U ¼rtr ln U ¼rln det U ¼ ir#; since U

is unitary its determinant is det U ¼ e

: Now, recall that UðkÞ was supposed

periodic in k, hence the same must hold for e

which implies #ðkÞ¼

aðkÞþ

k  R with again a periodic function aðkÞ and some finite selection of

lattice vectors R; here,

means a sum over ﬁnitely many items. We mention

without proof (because this would go off to far from our subject) that for

exponentially localized Wannier functions aðkÞ must be smooth. Finally, when

put into (8.96), application of Stokes’ theorem yields

aðkÞ¼

kaðkÞ¼0 since the torus T

has no boundary o T

(or equivalently, to

each point on a face of a reciprocal cell there is an identical point on the

opposite face with the same value of aðkÞ but opposite surface normal vector

k). There remains, however, a term

e;R

R ð8:98Þ

undetermined (per spin; if spin degeneracy holds as in normal ferroelectrics, then

there always appears twice this term). Judged from (8.95, 8.96) this undetermined

integer multiple of ‘dipole quanta’ P

e;i

¼ðe=jT

jÞa

; i ¼ 1; 2; 3; where the a

form

a basis of the lattice L

of an inﬁnite crystal, appears quite natural because the

assignment of a Wannier function to a lattice position R has this arbitrariness. If a

surface for a ﬁnite crystal is introduced, then a change of this assignment means a

change of the surface contribution too which cancels the change of (8.98) ren-

dering the total dipole moment unique. The remaining dipole density of the ﬁnite

crystal is normally much smaller than the quanta P

e;i

, and the term (8.98) may be

skipped when calculating DP

In order to reveal the algebraic-topological structure of the obtained results we

introduce lattice adapted (in general non-orthogonal) coordinates given by r ¼

; k ¼

where the a

and b

form bases of the lattices L

and L

;

respectively; a

 b

¼ 2pd

; that is, r

¼ð2pÞ

1

 r and k

¼ð2pÞ

1

 k: The

cell volumes expressed in these bases are jT

j¼ða

; a

Þ and jT

j¼ðb

; b

with the triple scalar products ð; ; Þ (4.46). On the tori the coordinates run from 0

to 1, so that the volume element in T

is d

k ¼jT

jdk

^ dk

while r

ð2pÞ

1

o=ok

: We have to cope with two dualities here, that between position

and momentum and that between tangent vectors dk and forms on the torus of

284 8 Parallelism, Holonomy, Homotopy and (Co)homology

quasi-momenta k: This is why here tangent vectors have lower indices and forms

upper.

Consider now an adiabatic parameter k changing from 0 to 1 with P

ðkÞ

changing from P

ð0Þ¼0 to P

ð1Þ¼P

: Since it was important for the

Wannier representation analysis to have energy-gap separated occupied bands

in order that the unitary transformation matrices Uðk; kÞ and the set of occupied

states fu

nkk

g resulted in a smoothly k-dependent phase aðk; kÞ, the crystal must

remain semiconducting all the way along the k-path. Combine k and k in a

four-dimensional manifold M ¼½0; 1T

; with the volume form jT

jdk ^

^ dk

: The boundary of M is oM ¼ð1; T

Þð0; T

Þ where the

minus sign indicates that the surface normal at k ¼ 0 points into the negative

k-direction. (T

itself has no boundary.) Also, introduce the notation of a

1-form

nkk

; ðdu

nkk

: ð8:99Þ

Then, the result (8.96) may be cast into

¼i

occ:

ðu

nkk

jðdu

nkk

Þdk

^ dk

¼ i

occ:

ððdu

nkk

Þdk

^ dk

ð8:100Þ

Though the expression for A

is multiplied with the imaginary unit i, it is clear

from the derivation that, like P

, the A

are real. The last equality holds because of

the constancy of normalization, ðu

nkk

Þ¼1: Comparison to (8.76) clearly

shows that the polarization density component in a

-direction is given by A

; the

wedge-product of a Berry–Simon connection form in b

-direction with the volume

form of a two-dimensional section in T

perpendicular to the a

-direction (spanned

by b

and b

in the case of a

). The integration domain oM contains the cycle from

¼ 0 to k

¼ 1 while the k-path in the adiabatic parameter space is not closed

here. This particular case of a Berry phase in an adiabatic change of a band

structure was ﬁrst observed by Zak.

Phys. Rev. Lett. 62, 2747–2750 (1989).

8.6 Geometric Phases in Quantum Physics 285

The expression with Berry’s curvatures F

¼ dA

corresponding to (8.100)is

;

¼2i

occ:

ððdu

nkk

jðdu

nkk

Þdk ^ dk

^ dk

; ðdu

nkk

ð8:101Þ

(As usually o

nkk

=ðokok

Þdk ^ dk

¼ 0 since the second derivative is symmetric;

the prefactor 2 is introduced since the matrix element in F

is understood as

element of an alternating tensor ðÞ

 ðÞ

, see below and (4.18)). While the

connection A

is only determined modulo an item (8.98), the curvature F

does

not have this ambiguity; since it must be smooth in M from k ¼ 0 to k ¼ 1 it

cannot jump by discrete values (8.98).

Equation 8.101 allows for another physical interpretation. Let (with some

adiabatic rescaling as in 8.72) k ¼ t denote time. Then, (8.101) may be understood

as P

dtJ

ðtÞ with the electronic charge current density J

¼ðe=jT

jÞ

ða

=2pÞ

averaged over the unit cell of the crystal. This ﬂowing charge

against the lattice during the time T makes up the polarization density P

.This

clearly shows that Berry’s curvature is uniquely connected to a physical

observable and hence not dependent on the gauge center R of the Wannier

function.

To pursue the latter aspect further, a lattice periodic electric ﬁeld Eðr; tÞ¼

ðoA=cot r

Þðr; tÞ is applied (cf. (5.80, 5.99)). Use a gauge with A

¼ 0, then

E ¼oA=cot and the Hamiltonian in (8.92) becomes (with k ¼ t ¼ k

, the

electron charge e and the canonical momentum operator p)

h

ir

þ k þ

hc

Aðr; tÞ



þ UðrÞ¼

þ UðrÞð8:102Þ

implying the commutation relations

½r

; H

¼h

;

; H



¼

E  p ð8:103Þ

and, with k

¼ðk

; k

Þ,

; H







¼ðe

 e

Þ u





¼ðe

 e

Þðu

jðdu

Þ: ð8:104Þ

What enters the expression F

in (8.101) is the alternating tensor

286 8 Parallelism, Holonomy, Homotopy and (Co)homology