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Wannier functions and Fractional Quantum Hall Effect R. Ferrari Dipartimento di Fisica Universitá di Trento, 38050 Povo, Trento Italy and I.N.F.N. sezione di Padova, gruppo collegato di Trento [email protected] (February 1, 2008) We introduce and study the Wannier functions for an electron moving in a plane under the influence of a perpendicular uniform and constant magnetic field. The relevance for the Fractional QuantumHall Effect is discussed; in particularit shown that an interestingHartree-Fock state can be constructed in terms of Wannier functions. 5 9 PACS numbers:05.30.Fk,72.20.My,73.20.Dx 9 cond-mat/9501055 1 n a I. INTRODUCTION J 3 1 It is widely accepted that Quantum Hall Effect [1,2] can be described by a model of interacting electrons moving in a plane. For many features of the phenomenon the zero temperature limit seems to be a good approximation. 1 Intheattemptto describethe stateintermsofsingleparticleapproximationandsubsequentperturbationsaround v this trial state, the author has introduced [3] a set of functions which are eigenfunctions of the single particle Hamil- 5 tonianandmoreoverareinvariant,upto aphase,under finitetranslationsassociatedto alattice. The phase depends 5 on the coordinates. In subsequentpapers [4,5]an Ansatz was introduced in orderto constructthe trial state. In Ref. 0 1 [5] numerical evidence is given that this state yields a mean Coulomb energy close to the value provided by genuine 0 many-body states (e.g. Laughlin’s [6]). From this property and from the fact that the trial state is given in analytic 5 formintermsofaSlaterdeterminant(orsingleparticleapproximation)theauthorhopesthataperturbativeapproach 9 based on this Ansatz will improve the value of the mean energy and provide a further progress in the understanding / t of the phenomenon. a In this note the Wannier functions are introduced by using the functions invariant under translations. This con- m structionhas some degree ofarbitrarinessrelatedto the phase,which cannotbe fixeda priori. Some possible natural - choices are considered and numerical methods are used in order to decide which is the better one. d n Finally the Wannier functions are used in order to reformulate the Ansatz for the trial state. This analysis shows o that there is at least another natural candidate for the trial state. By considering the particular case of filling factor c 1/3 numerical evidence is given that the original proposal gives the better choice. : v i X II. THE FORMALISM r a In this section some essential points developed in I and III are briefly reviewed. The Hamiltonian for the single electron is 1 y 2 x 2 H = i∂ + i∂ + (1) x y 2(cid:20)(cid:16)− − 2(cid:17) (cid:16)− 2(cid:17) (cid:21) where the unit of length is 1 c¯h 2 λ= (2) (cid:18)eB(cid:19) and the unit of energy eB ¯hω =h¯ . (3) c mc The complex notation is used for the vectors 1 w=w +iw . (4) 1 2 Step down and step operators are defined 1 w∗ a +2∂ w ≡ √2(cid:18) 2 (cid:19) 1 w b +2∂w∗ . (5) ≡ √2(cid:16)2 (cid:17) They obey the algebra [a,a†]=1 [b,b†]=1 [a,b†]=0 [a,b]=0 (6) and moreover 1 H =a†a+ . (7) 2 Let ϕ be the solution of nL a†aϕ =n ϕ bϕ =0. (8) nL L nL nL One gets easily ϕnL(r)=(2πnL!)−21 (cid:18)x√+2iy(cid:19)nLexp(cid:18)−r42(cid:19). (9) It is convenient to introduce the coherent-state operator 1 i S(w)=exp (w∗b wb†) =exp w r ˆz exp(w ∂ +w ∂ ) (10) 1 x 2 y (cid:18)√2 − (cid:19) (cid:18)2 × · (cid:19) i.e. it is the product of a translation and a phase. It has the properties [S(w),H]=0 (11) and i S(c)S(d)=S(c+d)exp c d zˆ (12) (cid:18)−2 × · (cid:19) (Magnetic Translation Group [7] (MTG)). Thus they commute [S(c),S(d)]=0 if c d zˆ=2πu (u integer). (13) × · Then these operators are adequate in order to impose quasi-periodic boundary conditions on a parallelogramA with side vectors L ,L 1 2 S(L )ψ =eiθ1ψ 1 S(L )ψ =eiθ2ψ (14) 2 provided they commute [S(L ),S(L )]=0 i.e. L L zˆ=2πg (g integer). (15) 1 2 1 2 L L × · Then g will be the degeneracy of the Landau levels. The elements of the MTG are defined by the constraint L [S(w),S(L )]=0 j =1,2 (16) j i.e. 2 1 w = ( n L +n L ) with n ,n integers. (17) 1 2 2 1 1 2 g − L Let c,d be any two vectors that satisfy eq. (13) and provide a commensurate tiling of the parallelogram,i.e. L =pc 1 L =p′c+qd (18) 2 where the coefficients are integers. One has g =pqu. (19) L A finer lattice can be introduced generated by the vectors f and g such that c=κf +ι′g d=κ′f +ιg (20) with the condition that the coefficients are integer numbers and moreover f g zˆ=2π (21) × · i.e. κι κ′ι′ =u. (22) − Sets of functions can be introduced which are eigenvectorsof the Hamiltonian and are invariantunder translations given by the sites of the lattice generated by c,d or by f,g. Thus solutions of S(c)ψµν =eiµψµν nL nL S(d)ψµν =eiνψµν (23) nL nL and S(f)φαβ =eiαφαβ nL nL S(g)φαβ =eiβφαβ (24) nL nL can be introduced. The solutions of the above equations are given by +∞ ψnµLν(r)=(pq)−21 [S(c)e−iµ]m[S(d)e−iν]n ϕnL(r) m,nX=−∞ +∞ =(pq)−12 ( )mnuexp[ i(µm+νn)] − − m,nX=−∞ i exp ˆz (mc+nd) r ϕ (r+mc+nd) (25) h2 · × i nL and similarly +∞ φαnLβ =(gL)−21 [S(f)e−iα]m[S(g)e−iβ]n ϕnL(r) m,nX=−∞ +∞ =(gL)−21 ( )mnexp[ i(mα+nβ)] − − m,nX=−∞ i exp ˆz (mf +ng) r ϕ (r+mf +ng). (26) h2 · × i nL The value of the parameters are fixed by the boundary conditions in eq. (14). See eqs. (49), (83) and (84) of I. In particular it is easy to find the following relations as a consistency condition for the eqs. (23) and (24) 3 2πn +µ=πκι′+κα+ι′β 1 2πn +ν =πκ′ι+κ′α+ιβ. (27) 2 The norm of the wave function is (see eqs. (69) and (91) of I) +∞ φαβ 2 = d2r [S(f)]m[S(g)]n ϕ (r) ∗ϕ (r)ei(mα+nβ). (28) k nLk m,nX=−∞ZR2 { nL } nL One gets +∞ 1 φαβ 2 = ( )mnei(mα+nβ)exp( mf +ng 2). (29) k nLk − −4| | m,nX=−∞ A similar result is valid for the other set of functions +∞ 1 ψµν 2 = ( )mnuei(mµ+nν)exp( mc+nd2). (30) k nLk − −4| | m,nX=−∞ The set of functions φαβ form a complete set. We use capital letters to denote the normalized functions { nL} φαβ Φαβ nL nL ≡ φαβ k nLk ψµν Ψµν nL . (31) nL ≡ kψnµLνk For computation of the matrix elements of the Coulomb part (in II and III) it was convenient to introduce a new set of functions Φˆαβ . One considers a reference wave function Φα0β0 and defines (see Appendix in III) { nL} nL Φˆαβ S(w )Φα0β0 (32) nL ≡ αβ nL where w is a standard set of vectors αβ w =(2π)−1[(β β )f (α α )g]. (33) αβ 0 0 − − − By using the composition rules in eq. (12) S(w )S(f)=S(f)S(w )exp( iw f ˆz)=S(f)S(w )exp( i(α α )) αβ αβ αβ αβ 0 − × · − − S(w )S(g)=S(g)S(w )exp( iw g zˆ)=S(g)S(w )exp( i(β β )) αβ αβ αβ αβ 0 − × · − − (34) it is easy to show that Φˆαβ satisfies eqs. (24) and therefore differs from Φαβ by a phase. nL nL III. WANNIER FUNCTIONS In this section Wannier functions [8–16] are introduced for a particle moving in a constant and homogeneous magnetic field. Various sets of Wannier functions can be introduced. By starting from Φαβ one can define (the { nL} index n will be suppressed for convenience of notation) L g−21 e−i(rα+sβ)Φαβ. (35) Wr,s ≡ L Xαβ If one uses the set Φˆαβ a different Wannier functions is obtained { nL} ˆ g−12 e−i(rα+sβ)Φˆαβ. (36) Wr,s ≡ L Xαβ 4 Yet another Wannier function is obtained if Φˆαβ is replaced by nL i S(w )S(w )Φα0β0 =exp( (α α )(β β ))Φˆαβ. (37) α0β αβ0 nL 4π − 0 − 0 nL All these definitions provide a set of orthonormal functions located around the lattice site x=rf +sg. (38) Moreover they have the property (see eq. (24)) i S(r′f +s′g) (x)=exp( (r′f +s′g) x zˆ) (x+r′f +s′g) r,s r,s W 2 × · W =( )r′s′ (r−r′),(s−s′)(x). (39) − W One expects that any different choice of phases for the translation invariant functions provides Wannier functions with different shapes [9]. By numerical inspection one can chose the functions with better localization properties among the three proposal given in eqs. (35), (36) and (37). From Figs. 1, 2 and 3 one sees clearly that the choice in eq. (35) ( ) gives the best localization. W The formalism developed in section II allows the introduction of other kind of Wannier functions, i.e. those associated to the lattice generated by the c,d vectors [17]. 1 u 2 (u) e−i(mµ+nν)Ψµν (40) Wm,n ≡(cid:18)g (cid:19) L Xµν is expected to be localized around the lattice site x=mc+nd. (41) Fig. 4 shows the shape of the Wannier function (u) . It should be noticed that the functions (u) do not form m,n m,n W {W } a complete set of functions. Itis interestingtoelaboratefurtherthedifferencebetweenthe sets and (u) . Itcanbe shown(seeeq. (90) {W} {W } of I) that fixed µν ψµν =u−12 φαβ. (42) nL nL Xαβ On the other side, evaluated on a c,d lattice site r,s {W } r=mκ+nκ′ s=mι′+nι, (43) gives (see eq. (27)) rα+sβ =m(κα+ι′β)+n(κ′α+ιβ) =m[µ πκι′+2πn ]+n[ν πκ′ι+2πn ] (44) 1 2 − − and therefore 1 fixed µν Wr,s =exp(−iπ(mκι′+nκ′ι))(cid:18)guL(cid:19)2 Xµν e−i(mµ+nν)u−21 Xαβ ΦαnLβ (45)   on the sites of the c,d lattice. The conclusionofthis sectionis thatone has two goodcandidates( and (u) )which canbe usedfor the r,s m,n {W } {W } construction of a trial state for the many-body problem. 5 IV. TRIAL STATE FOR FQHE TheoriginalAnsatzfortheFQHEtrialstatepresentedinthepapersIIandIIIwasformulatedintermsofthewave function Ψµν of eqs. (25) and (31). For the filling factor 1/u it is just the Slater determinant of the g /u functions L with n =0. In the second quantization formalism where ψ is fermion field (spin is neglected), the trial state is L Ω = d2xψ†(x)Ψµν(x)0 . (46) | i Z | i Yµν A The equation (40) is a unitary transformation, thus, up to a phase, the state Ω can be written in terms of Wannier | i functions Ω d2xψ†(x) (u) (x)0 (47) | i≃ Z Wm,n | i mY,n A where m,n are the sites of the lattice generated by c,d. From the discussion at the end of section III it follows that one can similarly constructa trial state in terms of the Wannier functions given in eq. (35) r,s W Ω˜ d2xψ†(x) (x)0 . (48) | i≡mY,nZA Wr,s | i(cid:12)(cid:12)(cid:12)(cid:26)sr==mmικ′++nnκι′ BothAnsatzhaveasrationalethe strategyofplacingtheN g /uelectronsonaregularlattice. Moreoverapeaked L ≡ distribution of the electron density around the sites diminishes the energy coming from the Coulomb repulsion. Before examining the numerical consequences of the choices made in eqs. (47) and (48), it is interesting to look closer at the states occupied by the single electrons. In the state Ω the electron with µν quantum numbers is in a | i state given by (see eq. (42)) fixed µν fixed µν ψµν =u−21 φαβ =u−21 φαβ Φαβ, (49) k k Xαβ Xαβ while in the state Ω˜ (see eq. (45)) is described by the wave function | i fixed µν u−12 Φαβ. (50) Xαβ Thus in the first case the probability amplitude for the state Φαβ is proportionalto φαβ and in the second is equal k k for all αβ (fixed µν). The twoAnsatz givenin eqs. (47) and(48)canbe tested in the self-consistentequationbasedonthe Hartree-Fock approximation(see eq. (158) of III). Fig. 5 shows that the state Ω given in eq. (47) yields a better trial state (to a | i good approximation, it is unchanged by the self-consistent procedure). 1 K.von Klitzing, G. Dorda and M. Pepper, Phys.Rev.Lett. 45, 494 (1980). 2 D.C. Tsui, H.L. Störmerand A.C. Gossard, Phys. Rev.Lett. 48, 1559 (1982). 3 R.Ferrari, Phys.Rev. B42, 4598 (1990). This paper will be quoted as I. 4 R.Ferrari, Int.J. Mod. Phys.B 6, 2253 (1992).This paperwill be quoted as II. 5 R.Ferrari, Int.J. Mod. Phys.B 8, 529 (1994).This paper will bequoted as III. 6 R.B.Laughlin,Phys.Rev.Lett.50,1395(1983).SeealsoTheQuantumHallEffecteditedbyR.E.PrangeandS.M.Girvin (Springer–Verlag, Berlin, 1990) 2nd ed. 7 J. Zak, Phys.Rev. 134, A1602 and A1607 (1964). 8 G.H. Wannier,Phys. Rev. 52, 191 (1937). 6 9 W. Kohn,Phys.Rev. 115, 809 (1959). 10 G.H. Wannier,Rev. Mod. Phys. 34, 645 (1962). 11 E.I. Blount, Solid State Phys.13, 305 (1962). 12 J. Des Cloizeaux, Phys. Rev. 129, 554 (1963). 13 J. Des Cloizeaux, Phys. Rev. 135, A685 (1964). 14 J. Des Cloizeaux, Phys. Rev. 135, A698 (1964). 15 A.Nenciu and G. Nenciu,Phys. Rev.B47, 10112 (1993). 16 M.R.GellerandW.Kohn,Phys.Rev.B48,14085(1993). InthispaperthemaindevelopmentsonWannierfunctionsare reviewed. 17 R.A.Evarestov and V.P. Smirnov,Phys. StatusSolidi B 180, 411 (1993). FIG. 1. 3D plot of the modulus of a Wannier function W given in eq. (35). The lattice is regular and triangular (|f| = |g| ∼ 2.69 in units of magnetic length). This picture is a zoom in on a domain whose size appears in Figs. 2 and 3.. FIG.2. 3D plot of themodulus of a Wannierfunction Wˆ given in eq. (36). SeeFig. 1 for details. FIG.3. 3DplotofthemodulusofaWannierfunctionobtainedbyusingtherepresentationoftheMTGgivenbythefunctions in eq. (37). SeeFig. 1 for details. FIG.4. 3D plot of the modulus of a Wannier function W(u) given in eq. (40) ((c,d) lattice, regular and triangular. u = 3 and therefore |c|=|d|∼4.66). This picture is a zoom in on thedomain of definition,whose size appears in Figs. 2. and 3. FIG.5. Convergence patterns in the iteration procedure based on the self-consistent equation in the Hartree-Fock approximation. ”Times”refertotheAnsatzbasedonthefunctionsW(u) (eq. (47))and”circles”totheAnsatzbasedonthefunctions W (eq. (48)). Filling factor is 1/3. 7