First Chern class of lattice magneto-Bloch bundles

Vol. 38 (1996)

FIRST

REPORTS

CHERN

CLASS

ON

MATHEMATIC4L

OF LATTICE

PHYSICS

MAGNETO-BLOCH

BUNDLES

V. A. GEYLER Department

of Mathematics,

Mordovian State University,

Saransk 430000,

Russia

(Received February 15. 1996)

The magneto-Bloch bundle of a lattice Hamiltonian in a uniform magnetic fic,ld is considered. It is shown that the first Chern class of the bundle vanishes.

1.

Introduction Topological

dimensional

investigations

Schrodinger

of the magneto-Bloch

operators

of S. P. Novikov [I]. In particular, associated its Chern

with a fixed Landau number

collaborators mathematical

with a uniform

point

the effect

before

level have an irregular

has its origin

In the case of a rational the fulfilment satisfy

in the paper

cell of periods

vector bundle is nontrivial

[l], K. von Klitzing

quantization

[Z].

in the equality

associated

From

the

(‘1 = 1 (see

with magnetic

If the number

and with

subbands

7 of the magnetic

flux

Al, then the numbers (i) satisfying the conservation law C c1 = 1 (see [I ~ 4, 71). number 7 (n = M/N, where M and N are coprime integers),

of an additional

the Diophantine

behaviour.

operator

the article

of the Hall conductance

of view this quantization

quanta through an elementary ci” may be arbitrary integers

for t,wo- and three-

field originated

level ~1 of a two-dimensional

[3--61). The Chern classes c1(1) , . . . , cl(nl of subbundles of a fixed Landau

bundles

Novikov showed that the magneto-Bloch

cl is equal to one. Shortly

discovered

vector magnetic

condition

equation

is equal to an integer

is required.

Namely,

the numbers

(T = c\“’ must

[3, 81:

Mo-tNm

=

1.

(1)

In the last few years the Hall conductance of lattice models become a matter of considerable interest (see e.g. [9915]). It is shown for tight-binding Hamiltonian on the square lattice that the sum of contributions from all magnetic subbands to the Hall conductance is zero [9]. The related result for the lattice Hamiltonian with next-nearest,neighbour hopping is obtained in [lo]. Our purpose is to show that these results are special cases of a general topological assertion. More precisely, we consider an arbitrary lattice Hamiltonian H which is invariant with respect to magnetic generalized eigenvectors of H which are transformed by irreducible magnetic

translation

group W, form a vector 13331

bundle

translations [16]. The representations of the

E (if W, is commmativc,

then

E

334

V. A. GEYLER

is precisely the magneto-Bloch bundle analysed in [6-91. We prove that the first Chern class cl of E vanishes. Some of physical applications of this statement are discussed in 1171. 2.

Preliminaries

Let us introduce the basic notation. We shall denote by A a two-dimensional lattice with basis vectors al, a2 (al, a2 E R2), thus (1 = {Xlal+ &as : X1, X2 E 25). We denote by Q the elementary cell of A: Q = {tlal + t2az : 0 < tl, t2 < I}. Let us consider a discrete n-invariant subset I’ of the plane IR2; we shall suppose without loss of generality that 0 E r. Thus r has the form r = ASK, where K is a finite subset of Q containing 0; we denote by k the number of elements of K. The state space we consider is the sequence space ‘FI = 12(r). Let 77 E R, v > 0; we denote by IV, the discrete group of magnetic translations [16]. The group IV, consists of all the pairs (X,5), where X E n and < is a number of the form exp (ring) , n E Z. The multiplication in W, is defined as follows: = (A + X’, cc’ exp (&5X A X’)) Here the symbol A denotes the standard (al A az)-% We define a representation on I’ as follows [18, 191:

symplectic

multiplication

(2) in lR2, and 6 =

D of the group W, in the space of complex functions

D(h C)$(r) = Sexp(~0 A Y)~(Y- A).

4

(3)

The representation D is in some sense an analogue of the continuous Zak representation [16]; the parameter 77plays the role of the number of magnetic flux quanta through the cell Q. It is evident that Eq. (3) defines a faithful unitary representation of W, in the space Ft. The group W, is a group of type I (see [20]), if and only if 77is a rational number [21]. Furthermore, W, is a commutative group if and only if q is an integer. In connection with this we shall consider rational values of the flux 77only: 77= M/N. We recall that a unitary representation T of the group W, is called the physical one if T(0, <) = [I. Obviously, D is a physical representation. All irreducible physical representations of the group WV are described as fo!lows [16]: Such representations of W,, are N-dimensional and are parameterized in a unique way by points of the torus Tz = [0, N-l) x [0,1). The operators DP(X,<) of these representations

in the standard

basis of the space CN on the generators (al,

1) and (a2, 1)

are given by the matrices e-2ripl

0 .

Dp(al,1)= L

0

... 0 e-wPl+17) .,.

0 0 0

0i

.... .

e-24~1+(N--lh)

(4)

FIRST

CHERN

CLASS OF LATTICE

(here

and below I,

o

the n x n-identity

denotes

integral

of physical

irreducible

335

BUNDLES

>

matrix).

the group W, is of type I, we can decompose

Because direct

IN-1 e-Ztp2 (

1) =

D&2,

MAGNETO-BLOCH

representations

D into the

the representation

[22]:

6

IFi =

‘FIY

Each fibre g(p) representation kN).

D-L?

%(p)dp:

J’ T2,

in (6) is isomorphic

.I’ T%

&dp.

(6)

to the space @” @ CN @ @” and is a carrier

of W, which is a multiple

The transformation

=

.Fq which realizes

of a

D, (with the multiplicity

of the representation the isomorphism

of the spaces

3-t and G in

(6) has the form

x exp[7ri
of the natural

s T2,

pz~[O,l),

Statement

3.

of the direct

j,mE{O,l,...,

Let H be a self-adjoint integral

@ @” @ C” @

integral N-l},

fi

C” ,

as functions

(8)

of five variables:

KEK.

of results operator

D.

of the representation direct

7?(p) dp = L2(T;)

here the elements

pi~[0,N-‘),

+ m + 2j))].

isomorphism $

we consider

+ X2p2 + (r]/2)Xi(NX2

Then

in ‘l-i which is invariant

[22] HZ

with respect

I? = F,,H.F;l

the operator

to the operators

is decomposable

into the

@ s

G(P)

dP >

(9)

Tf “;l(P_), and G(P) where every fibre H(p) is an operator in the finite dimensional-space is invariant with respect to the operators of the representation D,. Thus N(p) has kN eigenvalues: El(P)

each of which is N-fold continuous

functions

I

degenerate.

&2(P) i

. ‘. 5 &V(P)

We shall suppose

that the dispersion

of p. Let El-l(P)

< &I(P)

>

< &1+1(P)

(10) laws EL(P) are

336

V.A.GEYLER

The corresponding eigenspace El(p) is an N-dimensional subspace of 7?(p). In virtue of (7), to the eigenspace El(p) there corresponds an N-dimensional vector space El(p) of generalized eigenvectors of the operator H. It is easy to show that El(p) c l""(F),and El(p) is invariant with respect to the representation D. The elements of El(p) are called magneto-Bloch eigenfunctions of the quasimomentum p for the I-th band of the spectrum a(H). The space El = U{El(p); p E T:} admits in an obvious way the structure of a vector bundle over the torus TG. If all the vector bundles El are defined, then the direct sum E = $ El is independent on the operator H. Formally, the fibre E(p) of the bundle E over the point p E Tg is the subspace Ftl

G(p) of the space l”(r). We shall call ( > bundle. It is self-evident that the correct definition of the

E the lattice magneto-Bloch

fibre E(p) requires to replace the map J=q in the expression Fql

by its extension > onto an appropriate space of distributions. We omit the tedious details; the scrupulous reader may accept the result of formal calculations by formula (11)as the definition of the fibre E(p). THEOREM. COROLLARY.

(

e(p)

The first Chern class cl of the vector bundle E vanishes. Let the bundle E be represented as a direct sum of subbundles El

l,..., n < AN), and let c$l) be the first Chern number of the bundle El. Then cl’) +

(l= ...+

CI”’= 0.

Sketch of the proof of the theorem: It is a matter of direct verification to prove that for a function f Eti = L2(T;)@CN@@“@C” the vector C$= .F;‘f

has the form

qb(Alal + (NX2 + m)a2 + +niMA~b]

n) = exp

x Nc exp (2riAl{vj}) j=o

[ - 7ri<(lf.+ ma2) A (Ala1 + (X2N + m)a2)+ / f (p,j, m, K) exp[2ni(Ml+

~zpz)] dp,

(11)

T:,

where {z} denotes the fractional part of a number Z. Using (11)we can obtain that the sequences $j,,,,,,, (p) (j’, m’ E (0, 1, . . , N - l},K' E K), where ti.g~,m~,n~(p): (Alal t-+

Lrz~&K

I

+

(NA2

exp [-nit

+

m)a2

+

K)

(K’ + m’a2) A (Ala1 + (&IV + m’)az) + dbfX1A~] x x exp Pri (Xl (PI + {17j’)) +

~2~2)1

,

(12)

form a basis in the fibre over the point p E Ti. Fix m’ E (0, 1, . . , N - l}, n’ E K and p E Tt. Then the vectors q!~~,~~,~l(p)(j runs from 0 to N - 1) span the subspace Em,+!(p) of the fibre E(p) over p. In this manner we obtain a vector subbundle E,!,,! of the bundle E. Relation (12) shows that all these subbundles are mutually isomorphic and their direct sum coincides with E. We denote for simplicity Eo = Eo,o. To prove the

FIRST

theorem,

CHERN

it remains

CLASS

OF LATTICE

to show that

MAGNETO-BLOCH

the first Chern

class cl(&)

337

BUNDLES

vanishes.

Let, us introduce

the notation N-l

([uo,

,~Lv-1];Pl~P2)

“’

=

[13)

UjE@. a,ti'j.o.o(P) 1

c

j=o As can be seen from (la),

the gluing laws of the bundle

([(co. “’ .Q-1l;m ([ao. ..’

fllN,Pz)

,aN-1];Pl,P2+1)

where B is the permutation

Let us consider

if

an N-sheeted

covering

([UO,

4.

([a07

=

ran-~l;m

+

.aA-l]:pl,p2)

(14)

:

N from the relation

= {rU1+

(1:i)

l/N.

(l(i)

permutation.

Hence from (14) and (16) we

over T2 is defined by the trivial gluing functions

l/N,p2)

g

(Lao,

”

([a07

.~N-I]:P~,P~)

.

(17)

. raN-l];PlrP2

vanishes.

(g*(EO))

‘..

({NP~)IN,P~).

(15) implies that crN IS the identity

([ao. .

~1

”

9 : T2 4 Tz given by the formula

see that the induced vector bundle g*Ee

Hence

([am(o), ‘.. >%(,V~1)l;Pl,P2) 1

{r/j’)

dm,~2)

Formula

”

which is defined modulo

cr(j) = j’

Ee have the form

+

1)

Since cl (g*(Ee))

.

= Nci(&)

>a,V-l];pl.p2).

[23], we have cl(&)

= 0.

•I

Remarks (1) The magneto-Bloch

different

bundle

the sublattice form

of A generated

where

A by A’ and K by K’ we obtain

E and E’ are different.

in [9; lo] (we denote

Moreover,

it by E’)

is rather

E and E’ is as follows. Consider al and Nap. Then r is represented in the between

by the elements

r = A’ + K’ , Replacing

considered

E. The relationship

from the bundle

K’ = K + (0.1,

, N - 1).

(13)

E’ as above. If n # A’. the bundles E’ is always isomorphic to a t,rivial one

the bundle

the bundle

[241’ (2) Although

the bundle E’ is trivial, we can obviously decompose it into a direct For example! let 77 = l/3 and let H he the tightsum of nontrivial vector bundles. binding Hamiltonian H = C cIcp exp idxP (the summation is taken over all the nearest,neighbour sites); the phase factor 0x,, is defined on a link and represents the magnetic field ~1, i.e., c

o+

plaquette

= 27q

and

HA,, = -0,x

338

V.A. GEYLER

The spectrum of H consists of three isolated magnetic subbands and each subband Bj defines a subbundle Ei of the magneto-Bloch bundle E’. It may be shown by the straightforward calculation that the Chern numbers cl(Ei) of the corresponding subbundles are as follows: cl (E;) = cl (E;) = 1, cl (Ei) = -2 (see also numerical calculations in [9]).

Acknowledgements I am grateful to V. A. Margulis and I. Yu. Popov for useful remarks. The work is partially supported by RFFR and the Ministry of Higher Education of Russian Federation. REFERENCES [l] Novikov S.P.: Sov. Math. Dokl. 23 (1981), 298. [2] von Klitzing K., Dorda G. and Pepper M.: Phys. Rev. LeU. 45 (1980), 494. [3] Thouless D.J., Kohmoto M., Nightingale M.P. and den Nijs M.: Phys. Rev. Lett. 49 (1982), 405. [4] Avron J.E., Seiler R. and Simon B.: Phys. Rev. Lett. 51 (1983), 51. [5] Thouless D.J.: J. Math. Phys. 35 (1994), 5362. [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20]

Bellissard J., van Elst A. and Schulz-Baldes H.: J. Math. Phys. 35 (1994), 5373. Lyskova A.S.: Theor. Math. Phys. (USSR) 65 (1986), 1218. Dana I., Avron Y. and Zak J.: J. Phys. Cl8 (1985), L679. Kohmoto M.: Phys. Rev. B39 (1989), 11943. Hatsugai Y. and Kohmoto M.: Phys. Rev. B42 (1990), 8282. Bellissard J., Kreft C. and Seiler R.: J. Phys. Math. Gen. A24 (1991), 2329. Barelli A. and Fleckinger R.: Phys. Rev. B46 (1992), 11559. Hatsugai Y., Kohmoto M. and Wu Y.-S.: Phys. Rev. Lett. 73 (1994), 1134. Geyler V.A. and Popov I.Yu.: Zeikchr. Phys. B93 (1994), 437. Geyler V.A. and Popov I.Yu.: Zeitschr. Phys. B98 (1995), 473. Zak J.: Phys. Rev. 134 (1964), A1602. Geyler V.A. and Popov I.Yu.: JETP Lett. 63 (1996), 381. Geyler V.A.: St. Petersburg Math. J. 3 (1992), 489. Geyler V.A. and Popov I.Yu.: Phys. Lett. A201 (1995), 359. Mackey G.W.: UnitanJ Group Representations in Physics, Probability, and Number Theory, Benjamin, Boston 1978. [21] Boon M.H.: J. Math. Phys. 13 (1972), 1268. [22] Maurin K.: General Eigenfunction Expansions and Unitary Representations of Topological Groups, PWN, Warszawa 1968. [23] Bott R. and Wu L.W.: Differential Forms in Algebraic Topology, Springer, Berlin 1982. [24] Geyler V.A. and Popov I.Yu.: paper submitted to J. Phys. A.