The role of a form of vector potential — normalization of the antisymmetric gauge Wojciech FLOREKa) and Stanis law WAL CERZb) A. Mickiewicz University, Institute of Physics, Computational Physics Division, ul. Umultowska 85, 61–614 Poznan´, Poland (July 30, 1997) 8 ResultsobtainedfortheantisymmetricgaugeA=[Hy,−Hx]/2byBrownandZakarecompared 9 with those based on pure group-theoretical considerations and corresponding to the Landau gauge 9 A = [0,Hx]. Imposing the periodic boundary conditions one has to be very careful since the 1 first gauge leads to a factor system which is not normalized. A period N introduced in Brown’s n and Zak’s papers should be considered as a magnetic one, whereas the crystal period is in fact a 2N. The ‘normalization’ procedure proposed here shows the equivalence of Brown’s, Zak’s, and J other approaches. It also indicates the importance of the concept of magnetic cells. Moreover, 0 it is shown that factor systems (of projective representations and central extensions) are gauge- 1 dependent, whereas a commutator of two magnetic translations is gauge-independent. This result indicates that a form of thevector potential (a gauge) is also important in physical investigations. ] l l PACS numbers: 02.20-a, 03.65.Bz a h - s e I. INTRODUCTION m . t ThediscoveryofthequantumHalleffect1,2 ledtoremarkableinterestintwo-dimensionalelectronsystemssubjected a m to a magnetic field.3 Since 1980 authors working in different fields — from applied to mathematical physics — have considered related problems and many new features have been observed and discussed.4 One of the most interesting - d questionsis the dynamic of two-dimensional electrons in a periodic potential and an external magnetic field.5 The n first results, in the tight binding approximation, were presented by Peierls,6 shortly after Landau’s7 discovery of the o quantization of electron states in a magnetic field. A new impact was due to Brown8 and Zak9,10 who independently c introduced magnetic translation operators in two different, but equivalent, ways. Both approaches were based on [ group-theoreticalconsiderationsandledtothebroadeningoftheLandaulevelsandquantizationofamagneticfield.8,11 1 Although more than thirty yearshave passed, their papers are still considered as fundamental ones.5 Brownand Zak v proved that the problem considered is in fact two-dimensional and their investigations confirmed the importance of 8 projectiverepresentationsandcentralextensionsinquantumphysics.12 Onthe otherhand,Zak’sandBrown’sresults 8 were not gauge-independent — only a completely antisymmetric vector potential was considered by both authors. 0 An attempt to consider gauge-equivalent vector potentials leads to some ambiguities and misconceptions if it is not 1 done carefully. A bit simpler and more clear results can be obtained from pure group-theoretical considerations. 0 8 For example, Divakaran and Rajagopal did not consider gauges at all and they worked with central extensions and 9 projectiverepresentationsonly.13 However,puremathematicaldescriptionmaynotprovideuswithanintuitiveimage / of the physical phenomena. Moreover,many experiments and theories indicate the importance of vector potential,14 t a so it is necessary to include gauges and potentials in considerations. m The aim of this paper is to show sources of misconceptions, ambiguities, and unexpected gauge-dependence of the - problem. In particular, factor systems of projective representations and central extensions introduced by Brown and d Zakhavebeencarefullycheckedandcomparedwiththoseobtainedfrompuregroup-theoreticalconsiderations.13,15,16 n It occurs that they can be considered as standard but they are not normalized.12,17 This last fact is the main source o c of differences between Brown’s and Zak’s approaches. Moreover, it indicates points at which a form of the vector : potential is important, i.e., the points at which the problem is not gauge-independent. v In this paper we propose a procedure of ‘normalization’of those factor systems, which enables us to identify irreps i X introduced by Brown and Zak. A comparison of these irreps with those obtained for central extensions of finite r translation groups leads to a concept of the so-called magnetic cells10 and shows that Brown and Zak considered in a fact finite lattices with a period 2N not N. For the sake of clarity, the following simplifications arising from the quoted papers are assumed. Position (r, R), momentum (p), and vector potential (A) are considered to be two-dimensional vectors. Note that r = (x,y) is any vector of R2, whereas R = (X,Y) ∈ Z2 denotes a vector of a square lattice with a = xˆ and a = yˆ, so the area of 1 2 the elementary cell is equal to 1. The magnetic field is perpendicular to the x-y plane and H = Hˆz. The periodic boundary conditions are imposed on representations of Z2 and the periods are equal, i.e., N = N = N; the finite 1 2 translation group and its representations can be considered equivalently. The paper is organized as follows. In Sec. II the most fundamental formulas of Brown’s and Zak’s papers are recalled and equivalence of their approaches are indicated. Basic properties of projective representations are briefly 1 presented,too. TheroleoffactorsystemsisbrieflydiscussedinSec.III. Thenextsectionisdevotedtodetermination of the equivalence of different approaches. From the physical point of view it is done by introducing the concept of magnetic cells. The results obtained are discussed in Sec. V. II. DIFFERENT DESCRIPTIONS OF MAGNETIC TRANSLATION GROUPS Fromthealgebraicpointofviewtherearetwoequivalentdescriptionsofthemagnetictranslationoperators. Brown8 investigated a projective representation of the translation group T then imposed the magnetically periodic boundary conditions on it. On the other hand, Zak9 introduced a closed set of noncommuting operators which, in fact, form a covering group T′ of T so its standard (vector) representations are projective representations of T.17 The finiteness of these representations was again achieved by imposing the periodic boundary conditions. These two approaches are related by a formula which follows from the induction procedure if one constructs representations of the covering ′ group. Since T is a central extension of T by the group U(1) (or its subgroup referred to hereafter as a group of factors and denoted F)15,16 then its (vector) representations can be written as Ξ[α,R]=Γ(α)D(R), (1) where α ∈ F ⊂ U(1), R ∈ T, Γ is a vector representation of F and D is a projective representation of T. A factor system m(R,R′) of this representations is determined by the relation17 ′ ′ ′ D(R)D(R)=m(R,R)D(R+R), (2) ′ whereas the multiplication rule for T reads ′ ′ ′ ′ ′ [α,R][α,R]=[ααµ(R,R),R+R] (3) ′ with µ(R,R) being a factor system of a central extension. These factor systems are related to each other by the formula ′ ′ m(R,R)=Γ[µ(R,R)]. (4) This relation establishes the equivalence of both approaches. Moreover, both authors assumed the antisymmetric vector potential (gauge) A=(H×r)/2=[Hy,−Hx]/2 and were not able to generalize their considerations to other gauges, in particular their approaches did not include the Landau gauge. On the other hand, their results and some conclusions are different in some points which will be discussed here18 and compared with the results obtained for the Landau gauge. All considerations and formulas given above are also valid for a finite group T and its (finite-dimensional) repre- N sentations. In addition, we can apply to this case a version of the Burnside theorem which reads that nonequivalent irreducible projective representations of T with the same factor system m(R,R′) satisfy the following condition17 N [jD]2 =|T |=N2, (5) N j X where j labels nonequivalent representations (there is no expression for a number of these representations) and [jD] denotes the dimension of jD. Since F is an Abelian group then it has |F| irreducible nonequivalent representations ′ and each of them determines different (nonequivalent) factor system m(R,R) according to (4). It follows from (5) ′ that irreducible representations of T determined by (1) satisfy the Burnside theorem. N Brown8 defined a magnetic translation operator as T(R)=exp[−iR·(p−eA/c)/~], (6) where p is the kinetic momentum andbA is the vector potential such that ∇×A = H. These operators form a projective representation of T with a factor system8 m(R,R′)=exp[−πi(R×R′)·H/ϕ ] (7) 0 where ϕ = ch/e. Brown showed that one can impose the periodic boundary conditions T(Na )ψ = ψ if (notice 0 j simplifications assumed in this paper — in fact, H denotes hereafter the magnetic flux through one primitive cell) b 2 l H = ϕ , (8) 0 N where l is mutually prime with N, i.e. gcd(l,N) = 1. Hence a factor system of a finite projective representation lD for H satisfying (8) is given as ′ ′ ′ ′ m ([X,Y],[X ,Y ])=exp[−πil(XY −YX )/N]. (9) l Brown showed that there is the unique (up to equivalence) irreducible projective representation with a dimension N and matrix elements8 lX 1lDjk[X,Y]=exp πiN (Y +2j) δj,k−Y , (10) (cid:20) (cid:21) where j,k =0,1,...,N −1 and δ is calculated modulo N (Brown labeled rows and columns by j,k =1,2,...,N). j,k Zak9 considered a covering group of the translation group consisting of operators τ(R|R ,...,R )=T(R)exp[2πiφ(R ,...,R )/ϕ ], (11) 1 j 1 n 0 where R = R and φ(R ,R ,...,R ) is the flux of the magnetic field through a polygon enclosed by a loop i i 1 2 j b consisting of the vectors R ,R ,...,R ,−R. The periodicity condition was the same as (8) for even N but for odd 1 2 j P N Zak proved that the condition l H =2 ϕ (12) 0 N should be satisfied. This condition implies that for even N the number of different factors is 2N, whereas it equals N for odd N.9,10 The result obtained by Zak agreed with Azbel’s considerations19 who showed that wave functions had to be periodic functions of H with the period 2ϕ . It is worthwhile noting that Azbel also worked with the 0 antisymmetricgauge. Zakdidnotintroduceafactorsysteminanexplicitway(itwasnotnecessaryinhisconstructions ofrepresentations)butitcanbeeasilyfoundbyconsideringmultiplicationofcosetrepresentativesτ(R|R),17,20 which simply are equal to T(R) [see (11)]. Therefore the factor system is also given by (9), but now it is the factor system of the covering group being a central extension so it should be denoted as µ(R,R′). Zak10 also showed that matrix l elements of an irredubcible N-dimensional representation should be (only the coset representatives τ(R,R) are taken into account here)21 lX lD τ([X,Y]|[X,Y]) =exp 2πi (Y +2k) δ , (13) 2 jk bN j,k+Y h i (cid:20) (cid:21) where b=1,2 for N odd and even, respectively. It is obvious that this representation corresponds to the irreducible ′ ′ representation Γ(α) = α of the factor group F, so m (R,R) = µ(R,R). According to (8) and (12) changes of H l l are related to changes of l but they were interpreted in different ways. In Brown’s considerations H determines a factorsystemofprojectiverepresentationsinadirectway—differentvaluesofH satisfying(8)leadtononequivalent projective representations. On the contrary, Zak considered different (nonequivalent) central extensions of T with factor systems µ . However, Zak assumed that only the representations Γ(α) = α were physical whereas the others l were rejected as nonphysical in further considerations.10,22 It means, according to Zak, that for a central extension with a factor system µ one has to find projective representations lD with a factor system m = Γ(µ ) = µ. The l l l l same result can be obtained while considering only the factor system µ and next all irreducible representations 1 Γ of F such that Γ (µ ) = m . Thus all representations necessary in physical applications, considered by Zak as l l 1 l representations of different although isomorphic groups, can be obtained by use of ‘nonphysical’ representations Γ l withl>1. Nevertheless,itseemsthatthe representationsintroducedbyBrown(10)couldbe usedinZak’sapproach ′ to construct (vector) representationsof T (finite or not) according to (1). A comparisonof (10) and (13) shows that for odd N Brown and Zak used different representations. However, for even N (b=2) we have lD [X,−Y]= lD τ([X,Y]|[X,Y]) , (14) 1 jk 2 jk h i wherethe sign‘−’originatesfromadifferentchoiceofthesignofeassumedbyZak18 (inZak’sapproacheigenvectors of D[1,0] are permuted by D[0,1] in the opposite direction than that assumed in Brown’s definition). The third approach is based on pure group-theoretical considerations and consists in determination of all possible centralextensionsofafinitegroupZ2 (ingeneralZ ⊗Z )byaninfinite(U(1))orfinite(C ={α∈C|αN =1}) N N1 N2 N 3 group of factors F.13,15,16 It was shown, by means of the Mac Lane method, that all nonequivalent factor systems corresponding to finite magnetic translation groups can be written as15,16) ′ ′ ′ µ ([X,Y],[X ,Y ])=exp(2πikYX /N) (15) k with k =0,1,...,N −1. Some important facts have to be mentioned: • This formula resembles the Landau gauge A=[0,Hx]; recently it has been shown that this convergence is not accidental.23 • The fractionk/N canbe interpretedas H/ϕ ,15,16 so the resulting numbers constitute a periodic function ofH 0 with the period ϕ in agreement with Brown’s result but contrary to Zak’s and Azbel’s results. 0 • In both Brown’s and Zak’s approaches a (group-theoretical) commutator of two magnetic translations, corre- ′ ′ sponding to vectors [X,Y] and [X ,Y ], is equal to ′ ′ ′ ′ c([X,Y],[X ,Y ])=exp[−2πi(XY −YX )H/ϕ ] (16) 0 and it is the same as obtained in the cited papers15,16 if H does not satisfy (12) but (8). • There are no additional conditions imposed on k and on N (i.e., the results are valid for both odd and even N and for gcd(k,N)>1). Taking into account only parameters k = l mutually prime with N it can be shown that N-dimensional irreducible projectiverepresentationsofT (or‘physical’vectorrepresentationsoftheextensionofT byC )havethefollowing N N N matrix elements l 3lDjk[X,Y]=exp 2πiNXj δj,k−Y . (17) (cid:18) (cid:19) Representations with gcd(k,N) > 1 were briefly discussed elsewhere,22 but the difference between odd and even N was not considered there. III. PROJECTIVE REPRESENTATIONS — STANDARD AND NORMALIZED FACTOR SYSTEMS Tocomparedifferentdescriptionsofthemagnetictranslationgroupswehavetodiscussnotonlyprojectiverepresen- tationsthemselvesbutalsotheirfactorsystems. Tobeginwithwerecallnowsomedefinitionsrelatedtofactorsystems andtheirproperties.17,20,24,25 Asonecanseefactorsystemsappearinthe definitionofaprojectiverepresentation(2) and in the multiplication rule for a central extension of groups (3). Factor systems m are determined directly (as in l Brown’sapproach)orviafactorsystems µ forcentralextensionsbymeans ofthe ‘physical’representationsΓ(α)=α l and the formula (4). A factor system m:T ×T →C has to satisfy the following condition20,25 ′ ′ ′′ ′ ′′ ′′ m(R,R)m(R+R,R )=m(R,R )m(R,R+R ) (18) for all R,R′,R′′. A trivial factor system t(R,R′) is determined by any mapping f:T →C according to ′ ′ ′ t(R,R)=f(R)f(R)/f(R+R). (19) ′ ′ Since T is Abelian then each trivial factor system is symmetric, i.e., t(R,R)=t(R,R). If ′ ′ ′ ′ m(R,R)=t(R,R)m(R,R) (20) then factor systems m and m′ are called equivalent. Notice, however, that the projective representations determined by equivalent factor systems are nonequivalent.17 Since all factor systems for a givenT form an Abelian groupΦ and a set of trivial factor systems Θ is its normal subgroup then elements of the factor group M = Φ/Θ (known as the Schur multiplicator) correspond to representatives of classes of equivalent factor systems. A factor system is called standard if it satisfies m(R,0)=m(0,R)=1, ∀R. (21) 4 A factor system of an N-dimensional projective representation is normalized if ′ ′ m(R,R)∈C , ∀R,R (22) N (i.e., each factor is the Nth root of 1). It is well known that the Schur multiplicator of Z2 is C ,26 so the factor system (15) is normalized and standard N N ′ sincem ([0,0],[X,Y])=1. Moreover,itisperiodicwithrespecttoY andX —theperiodisequaltoN. Inparticular k we have m ([N,0],[X,Y])=m ([X,Y],[N,N])=1, etc. On the other hand, the factor system (9) of N-dimensional k k representations (10) or (13) is not normalized because some of factors do not belong to C but to C instead. It N 2N also means that this system is standard, because m ([0,0],[X,Y]) = 1, but it appears that N does not serve as a l period because, for example, m ([0,N],[1,0])=exp(πil)=(−1)l. This fact stirs up a conflict between the conditions l obtained by Brown and those obtained by Zak and, moreover, leads to difficulties in studying magnetic translations for the antisymmetric gauge. Of course, one may work with factor systems (and, hence, representations) which are neither standard nor normalized, but such considerations have to be carried very carefully and results obtained have to be carefully interpreted, too.17 Brown and Zak did not check normalizationof their factor systems and this led to ambiguity of their results [cf. (8) and (12)]. At first let us notice that Brown took into account one requirement only, namely8 T(Na )T(R)=T(R)T(Na ), (23) j j i.e., that T(Naj) would commute with anby otherboperatorb. Onbthe other hand, Zak demanded in addition that9 m(Na ,Na )=1, (24) b j k i.e., that lD(Na ) should behave as a constant factor. As follows from (9) j m ([N,0],[0,N])=(−1)lN, l so for odd N the magnetic field H has to be twice as high as in (8). Note that representations(17), correspondingto the Landau gauge, satisfy, for both odd and even N, the following stronger condition lD(Na )lD(R)= lD(R+Na )= lD(R), (25) 3 j 3 3 j 3 i.e., both Na and T(Na ) are equal to the unit element in the translation group T and in the group of magnetic j j N translation operators, respectively. The condition (12) (for odd N) removes these problems but, however, leads to another question whby odd and even N should be considered separately while both cases can be evidently treated as one with the use of the Landau gauge. While considering restrictions imposed on H by the periodic boundary conditions with the Landau gauge, i.e., the standard factor system (15), one can see that the condition H = kϕ /N is sufficient. So, it seems that Brown’s 0 approach is well-supported. Moreover, it should be noted that Zak weakened his requirements and later on he considered only Brown’s condition.27 To enlighten the problem we have to check whether the factor systems (9) and (15) are equivalent or not. At first note that the group-theoretical commutator of operators of any projective representations (of an Abelian group T) is equal to ′ c(R,R′)=D(R)D(R′)[D(R′)D(R)]−1 = m(R,R), (26) ′ m(R,R) then it is the same for allequivalent factor systems. Since the equivalence of factor systems means the equivalence of vector potentials (gauges),2,23 then the above commutator does not depend on A but rather on H and in this sense this commutator only (not a factor system) has the physical meaning — if D(R) represents a lattice translation in ′ ′ the presence of a magnetic field then the commutator corresponds to a loop determined by vectors R,R,−R,−R and its value depends on the flux through a nonprimitive, in general, cell determined by these vectors. So, Brown’s requirement (23) leading to the condition (8) was based on a reasonable assumption. However, the factor system considered was not normalized which led to a disagreement with Zak’s results. IV. EQUIVALENCE OF FACTOR SYSTEMS AND REPRESENTATIONS Let us consider a mapping f [X,Y]=exp(2πiwXY), w ∈R, which determines the following trivial factor system w 5 ′ ′ ′ ′ t ([X,Y],[X ,Y ])=exp[−2πiw(XY +YX )]. (27) w The factors obtained belong to C , i.e., t is standard and normalized, if w =j/N. For example for j =k the factor N system (15) is transformed to ′ ′ ′ ′ ′ µ ([X,Y],[X ,Y ])=(µ ◦t )([X,Y],[X ,Y ])=exp(−2πikXY /N), (28) k k k/N whichcorrespondstoanotherformoftheLandaugaugeA¯ =[−Hy,0]. Itisimportantthatiftisnotnormalizedthen ′ a new factor system m =tm is not normalized, too. This is, however, the case which leads to the factor system (9) determined by Brown and Zak — one has to put w = k/2N. This, and the previous discussion on the commutator, proves that the stronger condition introduced by Zak following from (24) is superfluous. It can be easily shown for odd N, since for l mutually prime with odd N also gcd(2l,N) = 1 (the mapping l 7→ 2l is an automorphism of Z N which changes the order of elements only). Therefore in the formulas obtained by Brown, (8)–(10), one can replace l<N in the following way 0, for l=0 ′ l = 2k=2l, for even l6=0 (29) (2k−1=2(k+N′)=2l′, for odd l, ′ ′ ′ ′ where N = (N −1)/2, k = 1,2,...,N , and l = 1,2,...,2N = N −1. In this way a relation similar to (14) is obtained lD [X,−Y]= l′D τ([X,Y]|[X,Y]) , (30) 1 jk 2 jk h i ′ where l and l are interrelated by (29). In the same way one can transform the factor system (15) into (9). If ′ gcd(l,N)=1 then l is replaced by l, so ′ ′ ′ ′ m ([X,Y],[X ,Y ])=exp(2πi(2l)YX /N) (31) l ′ and next w is taken to be l/N. The factor system obtained ′ ′ ′ ′ t m ([X,Y],[X ,Y ])=exp[πil(YX −XY )/N] (32) w l is exactly the same as (9). In a sense, we have performed a ‘normalization’ of the factor system used by Brown and Zak. In other words, the projective representations (10) do not satisfy the condition (24) for odd N since they are not given in normalized form. Such a form can be obtained by substitution l →2l′ which leads to the condition (12) determined by Zak. Anyway, this way of normalization is not possible in the case of even N, since in general gcd(l,N) 6= gcd(2l,N). However, we can use a hint given by Zak, who did not exploit it in full. At the end of his paper9 Zak noticed that a finite magnetic translation group contains N3 elements21 for odd N whereas for even N the number of elements is two times bigger. It suggests that a group considered by him was, in fact, an extension of T by C — the factors N 2N obtained were not normalized since they did not belong to the multiplicator of T . N Ina previouspaper15 itwasshownthat centralextensionsofT by C whichcorrespondto magnetictranslation N 2N groups have factor systems πi ′ ′ ′ m ([X,Y],[X ,Y ])=exp 2kYX (33) 2 k N (cid:16) (cid:17) with k=0,1,...,N −1. A mapping f assigns to each [X,Y] an element of C so it is well defined for w =k/2N, w 2N i.e., πi f [X,Y]=exp kXY . w N (cid:16) (cid:17) Note that the product kXY is calculatedmodulo 2N and, therefore,f is a multivalued function: in Z numbers X w N and X +N represent the same element, whereas πi f [X +N,Y]=(−1)kY exp kXY w N (cid:16) (cid:17) is not equal to f [X,Y], in a general case. To calculate a trivial factor system t according to (19) one has to w w ′ determine f (R+R). Let us assume, at this moment only formally, that a sum of vectors in this formula will not w be calculated modulo N. Then 6 πi ′ ′ ′ ′ t ([X,Y],[X ,Y ])=exp − k(XY +X Y) (34) w N h i and πi ′ ′ ′ ′ m t ([X,Y],[X ,Y ])=exp k(YX −XY ) , 2 k w N h i which coincides with (9). It means that in order to obtain the above results we have to treat [X,Y] as an element of T rather than that of T . So, in fact, we haveconsidereda larger,2N×2N, lattice althougha parameterlabeling 2N N nonequivalent central extensions has been taken to be equal 2k. Even for gcd(k,N)=1 we have gcd(2k,2N)=2, so theconditionaccompanying(8)thatl ismutuallyprimewiththeperiod,isnotfulfilledinthiscase. Thisleadstothe magnetic periodicity with a period N though the lattice (crystal) period is 2N. This problem was briefly discussed by Brown8 and Zak10 and its solution is possible through a concept of magnetic cells — if gcd(l,N)= λ > 1 [in (8) and (12)] then the N ×N lattice can be decomposed into (N/λ)×(N/λ) magnetically periodic sublattices, which form a λ×λ lattice of magnetic cells. In the case considered λ =2 and the (2N)×(2N) lattice is decomposed into four N ×N sublattices consisting of points [X,Y], [X +N,Y], [X,Y +N], and [X +N,Y +N], respectively, where X,Y ∈ Z . According to (17) and (33) an irreducible projective representation of T should be N-dimensional in N 2N this case. It follows from the Burnside’s theorem that there are four such representations and they can be chosen as l 3lDjκkx,κy[X,Y]=(−1)κxǫx+κyǫyexp 2πiNXj δj,k−Y , (35) (cid:18) (cid:19) where κ ,κ = 0,1 and ǫ (ǫ ) is equal to 0 for X < N (Y < N) and to 1 otherwise. Therefore the representations x y x y considered by Brown and Zak (14) are equivalent to lD0,0. However, the latter is clearly periodic with the period N 3 and satisfies the condition (25). Since the trivialfactor system (34) is not normalizedin C then the representations N (14) satisfy the condition (24) and do not satisfy (25). V. DISCUSSION AND FINAL REMARKS Summarizing the above discussion on odd and even periods N we can state that the antisymmetric gauge A = [Hy,−Hx]/2, considered by Brown and Zak, corresponds in fact to the crystal period 2N and the magnetic period N. If one, like Brown and Zak, does not take this fact into account then results obtained can lead to erroneous conclusions. For example, in this way the additional condition (12) was derived. Investigations of the standard and normalized factor system (15), corresponding to the Landau gauge, have clearly indicated points at which the magnetic translation groups are ‘gauge-dependent’ and how one can ‘normalize’ factor systems and representations. The magnetically periodic boundary conditions of projective representations, when the Landau gauge is assumed, can be invoked if l H = ϕ , (36) 0 N where l = 0,1,...,N −1. (If gcd(l,N)= λ then the magnetic period is equal to N/λ, whereas the crystal period is still N.) The factor system (15) (and also the representation (17) and the physical properties) is a periodic function of H with a period ϕ . Hence the different magnetic response of the considered system can be observed only for N 0 values of H =lϕ /N. 0 If N is an odd integer then gcd(2l,N) = gcd(l,N) so the magnetic periodicity is the same in both cases and l in ′ (36) can be replaced by 2l, which is equivalent to Zak’s condition (12). However, the successively counted values of HN/ϕ havetobe arrangedinadifferentorder: 0,N+1,2,N+3,...,2N−2,N−1. Ifthesevalueswerearrangedin 0 the increasing order, i.e., 0,2,...,N −1,N +1,...,2N −2, it might suggest that the condition (12) has to be taken into accountandthat the magnetic period is 2ϕ . The only wayto settle this problemis by investigationofa system 0 described by a Hamiltonian with a nonperiodic part, e.g., the paramagnetic term. The case of even N has a quite different nature. As was shown above, the factor systems and representations considered by Brown and Zak describe a lattice with the crystal period 2N and the magnetic period equal N. The condition (36) yields H = lϕ /(2N), with l = 0,1,...,2N −1, but to achieve the magnetic period N only even 0 ′ ′ values of l = 2l are considered, so H = l ϕ /N. The representation (10) [see also (14)] is one of four nonequivalent 0 irreducible representations which can be determined in this case. It can be easily seen that Brown’s considerations for odd N can also be interpreted in this way (since the decomposition of a (2N)×(2N) lattice into four N ×N 7 lattices does not depend on the parity ofN). It means, in particular,that the antisymmetric gaugefor N =2 can be introduced only if one considers the 4×4 lattice with H =0 (a trivial case) or H =ϕ /2. 0 InthisworkthedescriptionsofthemagnetictranslationgroupproposedbyBrownandZakwerecomparedwiththe results obtained by means of the Mac Lane method.15 The first authors assumed the antisymmetric gauge, whereas the Mac Lane method led to the Landau gauge. Due to a factor 1 in the antisymmetric gauge some problems arise 2 when one introduces the magnetically periodic boundary conditions. More careful considerationsput forwardby Zak gave the additional condition (12) for an odd period N. However, the condition (36) obtained for the Landau gauge resembles the Brown’s condition (8) and does not depend on the parity of N. This condition was obtained from the group-theoreticalconsiderationsleadingto the factorsystem(15). Inthe nextstepk/N wasinterpretedasH/ϕ . At 0 firstsight it canbe interpretedas any value proportionalto H, e.g., as 2H/ϕ . However,the firstchoice is confirmed 0 bythe valueofgroup-theoreticalcommutator,whichdoesnotdependonthe gaugeor,intheotherwords,isidentical for all equivalent factor systems. Let us also remind that in this work H in fact denotes the magnetic flux through one primitive cell. Therefore, according to (8) or (36), the total flux through the N ×N lattice is equal to Φ=lNϕ , (37) 0 i.e., toanintegermultiplicity ofthe fluxon. Tointroducethe antisymmetricgaugeone hastoconsidera(2N)×(2N) ′ ′ lattice and even l = 2l. Hence the total flux equals Φ = 4lNϕ , so the flux through one N ×N magnetic cell is 0 ′ equal to l Nϕ , which is consistent with teh previous value (37), and the flux trough one primitive cell is equal to 0 ′ H =l ϕ /N. Ontheotherhand,thefluxthroughoneprimitivecellofthe(2N)×(2N)lattice(assumingtheLandau a 0 gauge)is H =lϕ /(2N), so in generalit is a half of H . Therefore we can set up the certain procedure: For a given L 0 a magneticfieldH the antisymmetricgaugecanbe introducedifthe magneticallyperiodic boundaryconditionsadmit a twotimes smallerH . Forthe sakeofillustrationletusconsiderthe (2N)×(2N)lattice andH =ϕ /N. Then, from L 0 (16), one obtains a commutator corresponding to the primitive vectors [1,0] and [0,1] as c([1,0],[0,1])=exp(−2πi/N). The formula (9) gives the following values of the corresponding factors [see (26)] m([1,0],[0,1])=exp(−πi/N) 1 and m([0,1],[1,0])=exp(πi/N), 1 whereas (15) leads to m([1,0],[0,1])=1 2 and m([0,1],[1,0])=exp(2πi/N). 2 So, the flux through the primitive cell, corresponding to the commutator and independent of the gauge, was decom- posed in two different ways into fluxes through ‘primitive’ triangles. However, the first decomposition (related to the antisymmetric gauge) is not possible for the minimal flux H = ϕ /(2N). If considering any other trivial factor 0 system (27) determined by the parameter w ∈R one can obtain many other decompositions of the commutator into factors. It can be viewed as the decomposition of the flux through the primitive cell into fluxes through the ‘lower’ and‘upper’ triangle. In particular,the other Landaugauge,correspondingto the factor system(28),changesrolesof these triangles since one obtains m([1,0],[0,1])=exp(2πi/N) and m([0,1],[1,0])=1. Despite the fact that the physical properties are gauge-independent we have noticed that the form of the vector potential A has a certain importance in the mathematical description of a system. One has to be especially very carefulconsideringprojectiverepresentationsorextensionsofgroups,sincesomeequivalentfactorsystemsareneither standard nor normalized. However, it may occur that in certain applications or in other descriptions of the same problem it is more convenient to use such a form of A. 8 a) Electronic mail: fl[email protected]. b) Electronic mail: [email protected]. 1K.vonKlitzing,G.Dorda,M.Pepper,Phys.Rev.Lett.45,494(1980);Forrecentreviewssee,e.g.,R.E.Prange,S.M.Grivin (eds.), The Quantum Hall Effect (Springer, New York, 1990); M. Stone (ed.), Quantum Hall Effect (World Sci., Singapore, 1992); M. Janssen, O. Viehweger, U. Fastenrath, J. Hajdu, Introduction to the Theory of the Integer Quantum Hall Effect (VCH,Weinheim, 1994). 2B. Huckestein,Rev.Mod. 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Phys. 6, 167 (1994); M.S. Raghanathan, ibid.6, 207 (1994). 13P.P. Divakaran,A.K. Rajagopal, Int.J. Mod. Phys.B 9, 261 (1995). 14ThemostspectacularandthebestknownistheAharonov–Bohmeffect[Y.Aharonov,D.Bohm,Phys.Rev.115,485(1959); Y.Aharonov,J.Anandan,Phys.Rev.Lett.58,1593(1987)]butsomenewideashavebeenpresentedlatelybyF.Ghaboussi: quant-ph/9702054, cond-mat/9701128, cond-mat/9703080, cond-mat/9705108. 15W. Florek, Rep. Math. Phys. 34, 81 (1994). 16W. Florek, Rep. Math. Phys. 38, 235 (1996). 17S.L. Altmann,Induced Representations in Crystal and Molecules (Academic Press, London, 1977). 18Inthiswork it isassumed, as inBrown’s paper,that thecharge of an electron is −e,i.e.e>0.Zak used e<0, sosome his formulas differin sign from Brown’s ones, but in fact are identical. However, we use H to denotethemagnetic field,as Zak did, whilst Brown used B. 19M.Ya. Azbel, Zh.Eksp. Teor. Fiz. 44, 980 (1963) [Sov. Phys. JETP 17, 665 (1963)]. 20A.O. Barut, R.Ra¸czka, Theory of Group Representations and Applications (Polish Sci. Publ.— PWN, Warsaw, 1977). 21Since a two-dimensional lattice is considered then a term m n /N is omitted in (13) as compared with Zak’s formula and 3 3 thefinite magnetic translation group consisting of bN3 elements is considered [b is defined in the text below (13)]. 22W. Florek, Phys. Rev.B 55, 1449 (1997). 23W. Florek, Acta Phys.Pol. A 92, 399 (1997). 24C.W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras (Interscience, NewYork, 1962). 25A.G. Kurosh, Group Theory (Chelsea, NewYork, 1960). 26A.Babakhanian, Cohomological Methods in Group Theory (M. Dekker,New York,1972). 27J. Zak, Phys. Rev. 136, A1647 (1964). 9