Bosonic fractional quantum Hall states on the torus from conformal field theory
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Bosonic fractional quantum Hall states on the torusfrom conformal field theory
Anne E B Nielsen and Germ´an Sierra Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann-Straße 1, D-85748Garching, Germany Instituto de F´ısica Te´orica, UAM-CSIC, Madrid, Spain
Abstract.
The Kalmeyer-Laughlin state, which is a lattice version of the bosonicLaughlin state at filling factor one half, has attracted much attention due to itstopological and chiral spin liquid properties. Here we show that the Kalmeyer-Laughlin state on the torus can be expressed in terms of a correlator of conformal fieldsfrom the SU (2) Wess-Zumino-Witten model. This reveals an interesting underlyingmathematical structure and provides a natural way to generalize the Kalmeyer-Laughlin state to arbitrary lattices on the torus. We find that the many-body Chernnumber of the states is unity for more different lattices, which suggests that thetopological properties of the states are preserved when the lattice is changed. Finally,we analyze the symmetry properties of the states on square lattices.
1. Introduction
In [1], Moore and Read showed that a number of fractional quantum Hall (FQH) statescan be expressed as certain correlators of conformal fields. More recently it has beenshown that such a connection also holds for FQH states in lattice systems [2, 3] andfor the interpolation between lattice and continuum systems [4]. The construction isinteresting because it shows that the states can be seen as special cases of a more generalmathematical framework. It also means that the states fulfil certain mathematicalrelations inherited from conformal field theory (CFT), and these are useful for derivingvarious properties of the states. Examples include the possibility to derive parentHamiltonians, to study quasi particle or edge excitations, to investigate entanglementspectra and to derive correlation functions [1–8].We shall here focus on a lattice version of the bosonic Laughlin state at filling factor1 /
2, which is known as the Kalmeyer-Laughlin (KL) state [9, 10]. On the Riemannsphere, i.e., the complex plane combined with the point at infinity, the KL state takesthe form ψ KL ( Z , Z , . . . , Z M ) = Y i G ( Z i ) Y i 2, except that the possiblevalues of the positions Z i are restricted to the considered lattice. The KL state hasalso been studied on a square lattice on the torus [11], and we provide an expressionfor this state in section 4. The KL state and closely related states have been analyzedin several papers, see e.g. [11–17]. These analyses show that the KL state has the sametopological properties as the bosonic Laughlin state at half filling in the continuum.The investigated properties include, e.g., correlation functions, the fractional statisticsof quasiparticle excitations, the chiral edge states, and topological entanglement entropy.Parent Hamiltonians have been obtained in [3, 17–21].The huge interest in the KL state is due to its topological and chiral spin liquidproperties combined with a relatively simple analytical wavefunction defined on a lattice.To characterize the properties of topological quantum states a number of differentmeasures have been proposed, but more of these requires the state under investigationto be defined with a particular set of boundary conditions, e.g. periodic boundaryconditions that correspond to defining the state on a torus. It is therefore very relevantto study different geometries.In the present paper, we demonstrate that the KL states on the torus can beexpressed as chiral correlators of fields from the SU (2) Wess-Zumino-Witten (WZW)model. It has previously been shown [3,4,14] that this construction gives a KL-like statewhen considered on the Riemann sphere, and the present study extends this result tothe torus geometry, where the relation to the KL state turns out to be exact. The CFTcorrelators can be defined for arbitrary lattices in 2D and are ensured to be singlets byconstruction. The CFT states thus provide a generalization of the KL states on thetorus to arbitrary lattices, and by investigating the topological properties of the stateson different lattices, we provide evidence that the topology remains the same at leastfor a broad class of lattices. For the case of square lattices with L x × L y spins, we alsofind linear combinations of the CFT states that are eigenstates of various symmetryoperators. In particular, this allows us in certain cases to identify linear combinationsof the states that are guaranteed to be orthogonal.The paper is structured as follows. In section 2, we explain how wavefunctions canbe constructed from conformal fields. We also introduce the CFT states that we studyin this work and provide analytical expressions for them on the Riemann sphere andtorus geometries. In section 3, we derive the analytical expressions for the CFT states onthe torus. In section 4, we show that the CFT states are proportional to the KL stateson the torus when defined on a square lattice. In section 5, we study the symmetryproperties of the CFT states for the case of a square lattice. In particular, we find linearcombinations of the two states that are eigenstates of different transformation operatorsand their eigenvalues. In section 6, we discuss the topological properties of the states,and we find among other things that the many-body Chern number of the states is unityfor several different lattices. Section 7 concludes the paper, Appendix A summarizes the osonic FQH states on the torus from CFT 2. Wavefunctions from conformal blocks We first explain how states of spin systems can be constructed from conformal fields.Our starting point is the correlator h φ j ( z , ¯ z ) φ j ( z , ¯ z ) . . . φ j N ( z N , ¯ z N ) i (2)of N primary fields φ j i ( z i , ¯ z i ) of a CFT. Here, h . . . i stands for the vacuum expectationvalue, z i is a coordinate in the complex plane, and ¯ z i is the complex conjugate of z i . Thefield φ j i ( z i , ¯ z i ) has spin j i , and it therefore has 2 j i + 1 components φ j i ,s i ( z i , ¯ z i ), where s i labels the 2 j i + 1 possible values of the third component of the spin. It follows that (2)is a vector with Q Ni =1 (2 j i + 1) components.The correlator (2) can be broken up into a sum over terms corresponding to differentways of fusing the fields present in the correlator to the vacuum state. Utilizing alsothat the field φ j i ( z i , ¯ z i ) separates into a holomorphic and an antiholomorphic part, i.e., φ j i ( z i , ¯ z i ) = φ j i ( z i ) ⊗ ¯ φ j i (¯ z i ), we can write the components of (2) as h φ j ,s ( z , ¯ z ) . . . φ j N ,s N ( z N , ¯ z N ) i = X k f k ( s j , z j ) f k ( s j , z j ) , (3)where f k ( s j , z j ) ≡ h φ j ,s ( z ) φ j ,s ( z ) · · · φ j N ,s N ( z N ) i k (4)is a conformal block (also known as a chiral correlator), k labels the different ways offusing the fields to the vacuum state, and the bar means complex conjugation.The next step is to regard f k ( s j , z j ) as the components of a wavefunction | ψ k i describing the state of N spins with spin quantum numbers j , j , . . . , j N . Explicitly, | ψ k i = C k X s ,...,s N ψ k ( s , . . . , s N ) | s , . . . , s N i , (5)where ψ k ( s , . . . , s N ) ∝ f k ( s j , z j ) (6)and C k is a normalization constant that allows us to choose the normalization of ψ k ( s , . . . , s N ) after convenience. The complex coordinates z j = x j + i y j are fixedparameters of the wavefunction and are naturally interpreted as the physical positions( x j , y j ) of the spins in the two-dimensional plane. In other words, the choice of z j definesthe lattice under consideration. Note that arbitrary lattices can be considered, since theonly restriction on z j is that z j = z l for j = l .In the present paper, we shall take the CFT to be the WZW model based on theKac-Moody algebra SU (2) . This CFT has central charge c = 1 and two primary fields osonic FQH states on the torus from CFT Re( z )Im( z ) ω ω Figure 1. Definition of the torus through the periods ω and ω . The points z and z + nω + mω , where n and m are integers, are identified. The coordinate system ischosen such that ω is real and positive and Im( ω ) > 0. The modular parameter τ isdefined as τ = ω /ω . φ ( z ) and φ / ( z ) with spin 0 and spin 1 / 2, respectively. φ ( z ) has conformal weight h = 0, and φ / ( z ) has conformal weight h / = 1 / 4. Since the spin-0 primary field isthe identity, we shall take all the fields in the correlator to be spin-1 / φ × φ = φ , φ × φ / = φ / , φ / × φ / = φ . (7)The fields can only fuse to the identity if N is even, and we shall therefore assume thisto be the case throughout. The number of conformal blocks on a Riemann surface withgenus g is then 2 g . There is hence one state on the Riemann sphere ( g = 0) and twostates on the torus ( g = 1).The state on the Riemann sphere (complex plane) can be written as [2, 3] ψ k ( s , . . . , s N ) = δ s χ s Y i 2. The factor δ s = ( P Ni=1 s i = 00 otherwise (9)is the delta function factor, and χ s = N Y j =1 ( − ( j − s j +1) / (10)is the Marshall sign factor. This formula is the one used to consider lattices with openboundary conditions.We shall here mostly be interested in periodic boundary conditions, whichcorresponds to defining the theory on the torus. The torus is defined by specifyingtwo complex numbers ω and ω , which are called the periods of the torus, and thenidentifying all points in the complex plane that differ by integer multiples of theseperiods. The z i in (4) are then restricted to lie within one parallelogram with sides ω osonic FQH states on the torus from CFT ω as illustrated in figure 1. ω and ω must be nonzero and have different phases,and we shall choose the coordinate system such that ω is real and positive and ω haspositive imaginary part. It is convenient to define the modular parameter τ = ω /ω and use the scaled coordinates ζ i = z i /ω . In terms of these, the states (6) take theform ψ k ( s , . . . , s N ) = δ s χ s θ (cid:20) k (cid:21) N X i =1 ζ i s i , τ !| {z } Centre of mass factor Y i 3. Conformal blocks of the SU (2) WZW model on the torus In this section, we derive (11). We also demonstrate that if the numbering of the spinsis altered, then (8) and (11) stay the same except that they are multiplied by the sign ofthe permutation needed to go from one numbering to the other. The choice of numberingis thus not important. We first need an expression for the correlator (2). To get this, we utilize the factthat the SU (2) WZW model can be bosonized in terms of a massless free scalar field ϕ ( ζ , ¯ ζ ) = ϕ ( ζ ) + ¯ ϕ ( ¯ ζ ) compactified on a circle of radius R = √ φ j i ,s i ( ζ i , ¯ ζ i ) = φ j i ,s i ( ζ i ) ⊗ ¯ φ j i ,s i ( ¯ ζ i ) can be expressed as the vertex operator φ j i ,s i ( ζ i , ¯ ζ i ) = : e i s i ϕ ( ζ i , ¯ ζ i ) / √ : = : e i s i ϕ ( ζ i ) / √ s i ¯ ϕ (¯ ζ i ) / √ : , (13)where : . . . : denotes normal ordering. We use here the scaled coordinates ζ i , but we notethat this changes the correlator (2) only by a constant factor. Note that the conformalweight of : e i s i ϕ ( ζ i ) / √ : is s i / / 4, as it should be.We can now use the expression * N Y i =1 : e i ν i ϕ ( ζ i )+i¯ ν i ¯ ϕ ( ¯ ζ i ) : + = δ ν δ ν | η ( τ ) | X ( p, ¯ p ) ∈ Γ A p ( ζ i , ν i ) A p ( ζ i , ν i ) (14)for the correlator of a product of generic vertex operators on the torus derived in [23](see also [24]). Here, ( ν i , ¯ ν i ) ∈ Γ, Γ is the lattice of momenta given by p = nR + 12 mR, ¯ p = nR − mR, n, m ∈ Z , (15) osonic FQH states on the torus from CFT δ ν δ ν is one for P i ν i = P i ¯ ν i = 0 and zero otherwise, η ( τ ) = e π i τ/ ∞ Y n =1 (1 − e π i τn ) (16)is the Dedekind eta function, A p ( ζ i , ν i ) = e i πp τ +2 π i p P i ζ i ν i Y i 2, the momenta become p = 1 √ n + m ) , ¯ p = 1 √ n − m ) , n, m ∈ Z . (18)Noting that n + m always has the same parity as n − m , we can write n + m = 2 r + 2 k and n − m = 2 s + 2 k and replace the sums over n and m by a sum over k ∈ { , / } and sums over r ∈ Z and s ∈ Z . Therefore X ( p, ¯ p ) ∈ Γ A p ( ζ i , ν i ) A p ( ζ i , ν i ) = X k ∈{ , / } X r ∈ Z A √ r + k )0 ( ζ i , ν i ) X s ∈ Z A √ s + k )0 ( ζ i , ν i ) . (19)Combining (3), (13), (14), (17), (19), and (A.1), we conclude that the conformal blocksare f k ( ζ i , s i ) = ξ k δ s η ( τ ) θ (cid:20) k (cid:21) X i ζ i s i , τ ! Y i 7= 1 ∂ ζ θ ( ζ , τ ) | ζ =0 X n ∈ Z π i (cid:18) n + 12 (cid:19) ( ζ i − ζ j )e π i τ ( n + ) + π i ( n + ) + O (cid:0) ǫ (cid:1) = ζ i − ζ j + O (cid:0) ǫ (cid:1) . (22)Inserting these expressions in (20), we get f k ( ζ i , s i ) ∝ ξ k δ s Y i 4. Connection to the KL states on the torus The CFT states can be transformed into states describing particles on a lattice byregarding all spins with s i = +1 as occupied sites and all spins with s i = − L x × L y square latticewith lattice constant √ π and P j ζ j = 0. The CFT states thus provide a naturalgeneralization of the KL states to arbitrary lattices. Expressed mathematically, the desired square lattice is obtained by choosing ω = √ πL x , ω = i √ πL y , τ = i L y /L x , (28)and z n + mL x +1 = √ π [ n − l x + i( m − l y )] , (29)where l x = ( L x − / , n ∈ { , , . . . , L x − } , (30) l y = ( L y − / , m ∈ { , , . . . , L y − } . (31)The result of this section is that ψ k ( s , . . . , s N ) = ( − k cψ k ( Z , . . . , Z N/ ) Y i G k ( Z i ) , k = 0 , / . (32)The right hand side of this expression is the KL state on the torus. Specifically, ψ k ( Z , . . . , Z N/ ) is the Laughlin state with Landau level filling factor 1 / N/ ψ k ( Z , . . . , Z N/ ) = θ (cid:20) N/ − k − ( N − (cid:21)(cid:18) P i Z i √ πL x , τ (cid:19) Y i We use four different ways to label the lattice sites. ( n, m ) is the coordinateson the lattice with n = 0 , , . . . , L x − m = 0 , , . . . , L y − j = n + mL x + 1 isthe site index, which numbers the sites from 1 to N as shown with the labels aboveeach site. z j defined in (29) is the position in the complex plane of the site with index j (in units of the magnetic length, when comparing to FQH states), and ζ j = z j /ω ,where ω = √ πL x is the width of the lattice in the x -direction. with the particle positions Z i = X i + i Y i restricted to the considered square lattice,i.e., Z i ∈ { z , . . . , z N } . Particles are identified with spins in the ‘up’ state such that { Z , . . . , Z N/ } = { z i | i ∈ { , , . . . , N } ∧ s i = 1 } . As in (1), Q i G k ( Z i ) is a gauge factor,which can only take values in the set {− , +1 } , and the purpose of this factor is toensure that the right hand side of (32) is a singlet. Finally, c is an overall constant thatonly depends on the choice of L x and L y and is unimportant, since it is absorbed in thenormalization of the state anyway. We find specifically that N/ Y i =1 G k ( Z i ) = ( Q Ni =1 ( − m i q i for L y even Q Ni =1 ( − ( n i + m i ) q i for L y odd . (34)Here, n i = Re( z i ) / √ π + l x and m i = Im( z i ) / √ π + l y as in (29) and q i = ( s i + 1) / s i = +1 contribute to the products. Note that q i is theoccupation number on site i since q i = 1 for spin up and q i = 0 for spin down. In thefollowing, we derive (32) by rewriting each of the factors in (11). The condition P Ni =1 s i = 0 translates into P Ni =1 q i = N/ 2, and the delta function factorthus ensures that the lattice is half filled when a spin up is considered as an occupiedsite and a spin down is considered as an empty site. In other words, the delta functionfactor ensures that there are N/ Z i coordinates. osonic FQH states on the torus from CFT The Marshall sign factor can be rewritten into χ s = N Y j =1 ( − ( j − s j +1) / = N Y j =1 ( − ( j − q j . (36) Let us first note that N X i =1 s i z i = 2 N X i =1 q i z i − N X i =1 z i = 2 N X i =1 q i z i = 2 N/ X i =1 Z i . (37)Using (A.2), (A.3), and the assumption that N is even, it then follows that θ (cid:20) k (cid:21)(cid:18) P i z i s i √ πL x , τ (cid:19) = ( − k θ (cid:20) N/ − k − ( N − (cid:21)(cid:18) P i Z i √ πL x , τ (cid:19) . (38) We write Y i 1, we find f ( ζ n +( m +1) L x +1 ) f ( ζ n + mL x +1 ) = Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n +i m E (cid:16) n +i m − n ′ − i m ′ L x , τ (cid:17)Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n +i( m +1) E (cid:16) n +i( m +1) − n ′ − i m ′ L x , τ (cid:17) = Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n +i m E (cid:16) n +i m − n ′ − i m ′ L x , τ (cid:17)Q ≤ n ′≤ Lx − − ≤ m ′≤ Ly − n ′ +i m ′6 = n +i m E (cid:16) n +i m − n ′ − i m ′ L x , τ (cid:17) = Q ≤ n ′ ≤ L x − E (cid:16) n +i m − n ′ − i( L y − L x , τ (cid:17)Q ≤ n ′ ≤ L x − E (cid:16) n +i m − n ′ +i L x , τ (cid:17) osonic FQH states on the torus from CFT Y ≤ n ′ ≤ L x − e i π − i πτ e π i[ n +i m − n ′ +i] /L x = e i πL x − i πτL x e π i[ n +i m − ( L x − / = − e πL y e − π ( m +1) = − e − π ( m − l y ) − π , (42)where we have used (A.12). For m = L y − 1, the relevant quantity is f ( ζ n +1 ) f ( ζ n +( L y − L x +1 ) = Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n +i( Ly − E (cid:16) n +i( L y − − n ′ − i m ′ L x , τ (cid:17)Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n E (cid:16) n − n ′ − i m ′ L x , τ (cid:17) = Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n +i( Ly − E (cid:16) n +i( L y − − n ′ − i m ′ L x , τ (cid:17)Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n h E (cid:16) n +i L y − n ′ − i m ′ L x , τ (cid:17) e i π − i πτ +2 π i( n +i L y − n ′ − i m ′ ) /L x i = Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n +i( Ly − E (cid:16) n +i( L y − − n ′ − i m ′ L x , τ (cid:17)Q ≤ n ′≤ Lx − − ≤ m ′≤ Ly − n ′ +i m ′6 = n − i h E (cid:16) n +i( L y − − n ′ − i m ′ L x , τ (cid:17) e i π − i πτ +2 π i( n +i( L y − − n ′ − i m ′ ) /L x i = Q ≤ n ′≤ Lx − n ′6 = n E (cid:16) n − n ′ L x , τ (cid:17)Q ≤ n ′≤ Lx − n ′6 = n E (cid:16) n − n ′ L x + τ, τ (cid:17) × e − i π ( N − πτ ( N +1) Q ≤ n ′≤ Lx − − ≤ m ′≤ Ly − e π i( n +i( L y − − n ′ − i m ′ ) /L x = − Q ≤ n ′≤ Lx − n ′6 = n E (cid:16) n − n ′ L x , τ (cid:17)Q ≤ n ′≤ Lx − n ′6 = n h E (cid:16) n − n ′ L x , τ (cid:17) e − i π − i πτ − π i( n − n ′ ) /L x i × e i πτ ( N +1) e π i( n − l x +i( L y +1) / L y = Y ≤ n ′ ≤ L x − e i π +i πτ +2 π i( n − n ′ ) /L x × e i πτN e π i( n − l x +i( L y +1) / L y = − ( − L y . (43)For n = L x − 1, we get f ( ζ n +1+ mL x +1 ) f ( ζ n + mL x +1 ) = Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n +i m E (cid:16) n +i m − n ′ − i m ′ L x , τ (cid:17)Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n +1+i m E (cid:16) n +1+i m − n ′ − i m ′ L x , τ (cid:17) = Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n +i m E (cid:16) n +i m − n ′ − i m ′ L x , τ (cid:17)Q − ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = n +i m E (cid:16) n +i m − n ′ − i m ′ L x , τ (cid:17) = Q ≤ m ′ ≤ L y − E (cid:16) n +i m − ( L x − − i m ′ L x , τ (cid:17)Q ≤ m ′ ≤ L y − E (cid:16) n +i m +1 − i m ′ L x , τ (cid:17) = ( − L y , (44)where we used (A.11) in the last line. For n = L x − 1, the relevant quantity is f ( ζ mL x +1 ) f ( ζ L x − mL x +1 ) = Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = Lx − m E (cid:16) L x − m − n ′ − i m ′ L x , τ (cid:17)Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 =i m E (cid:16) i m − n ′ − i m ′ L x , τ (cid:17) osonic FQH states on the torus from CFT − Q ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = Lx − m E (cid:16) L x − m − n ′ − i m ′ L x , τ (cid:17)Q − ≤ n ′≤ Lx − ≤ m ′≤ Ly − n ′ +i m ′6 = − m E (cid:16) L x − m − n ′ − i m ′ L x , τ (cid:17) = − Q ≤ m ′≤ Ly − m ′6 = m E (cid:16) i m − i m ′ L x , τ (cid:17)Q ≤ m ′≤ Ly − m ′6 = m E (cid:16) L x +i m − i m ′ L x , τ (cid:17) = ( − L y . (45)The solution of these equations is f ( ζ n + mL x +1 ) ∝ ( ( − m e − π ( m − l y ) for L y even( − n + m e − π ( m − l y ) for L y odd . (46)The result (32) is then obtained by combining the delta function factor, (36), (38), (39),(40), and (46). 5. Transformation properties of the CFT states on a square lattice on thetorus In this section, we investigate how the CFT states on the torus transform under differentoperations. This reveals the symmetries of the states and has the additional advantageof enabling us in certain cases to find linear combinations of the two states on thetorus that are necessarily orthogonal. We assume throughout that the spins sit on an L x × L y square lattice as defined in (29), but unless it is required for the consideredtransformation to make sense, we shall not put other restrictions on L x and L y thandemanding N = L x L y to be even. For the sake of generality, we shall allow for twistedboundary conditions in the following. If the twist angles are θ x and θ y , this means thatthe state acquires a phase of e i θ x N/ if the lattice is displaced by √ πL x in the x -directionand acquires a phase of e i θ y N/ if the lattice is displaced by √ πL y in the y -direction.It is already known how to take the twists into account for the Laughlin states on thetorus (see e.g. (5.5) in [27]), and by comparing the CFT states to this expression, weconclude that ψ a,b ( s , . . . , s N ) = δ s χ s θ h ab i N X i =1 ζ i s i , τ ! Y i Eigenstates and eigenvalues in the subspace spanned by ψ = ψ , ( s , . . . , s N ) and ψ / = ψ / , ( s , . . . , s N ) of the following operators: spin flip ( F ),translation by one lattice constant in the x -direction ( T x ), translation by one latticeconstant in the y -direction ( T y ), rotation by 90 ◦ ( R ◦ ), rotation by 180 ◦ ( R ◦ ), timereversal followed by reflection in the x -direction ( M x Θ), and time reversal followed byreflection in the y -direction ( M y Θ). We consider four different types of square latticeson the torus: ‘equal’ stands for L x = L y with L x even, ‘e × e’ means L x even and L y even, ‘e × o’ means L x even and L y odd, and ‘o × e’ means L x odd and L y even (odd-by-odd is not allowed since N = L x L y must be even). Note that ψ and ψ / are definedas in (11) and are hence not normalized. Lattice Eigenstates F T x T y R ◦ R ◦ M x Θ M y Θequal ψ − (1 − √ ψ / ψ − (1 + √ ψ / − × e ψ ψ / × o ψ ( − N ( − Lx − N ψ / ( − N − ( − Lx − N × e ψ + ψ / ( − N − Ly - ( − N ψ − ψ / ( − N − ( − Ly - ( − N θ x = θ y = 0 are summarized in table 1.From this table we conclude that ψ = ψ , ( s , . . . , s N ) and ψ / = ψ / , ( s , . . . , s N )are orthogonal for even-by-odd lattices, whereas ψ + ψ / and ψ − ψ / are orthogonalfor odd-by-even lattices. For L x = L y , it is the combinations ψ − (1 − √ ψ / and ψ − (1 + √ ψ / that are orthogonal. We also note that the operators describingtranslation in the x -direction and in the y -direction can be diagonalized simultaneously,which is not true in the continuum case [25]. This is so because the area of each latticesite is 2 π , and the magnetic flux penetrating an area of 2 π in the fractional quantumHall setting is one flux quantum. The operator ( T y ) − ( T x ) − T y T x thus moves all theparticles (or spin ups) around one flux quantum, and therefore the Aharonov-Bohmphase picked up is a multiple of 2 π . The transformation properties for general θ x and θ y can be found in (56), (66), (77), (91), (95), (102), and (103). Note also (82) and (83)for the transformation properties of a transformed set of translation operators. Most of the transformations that we shall study below involve rearranging the spins onthe lattice. An operator O describing such a transformation is unitary and acts as O σ αj O † = σ αd − ( j ) , j = { , , . . . , N } , α = x, y, z, (50)where σ αj is the α Pauli operator acting on the spin sitting at the site with index j and d − ( j ) is a bijective map from { , , . . . , N } to { , , . . . , N } describing the osonic FQH states on the torus from CFT ψ a,b ( p ( j ) , ζ j , s j , τ ) = ψ a,b ( s , . . . , s N ) , (51)where p ( j ) specifies the choice of ordering of the labels as in (25). The action of O onthe state (49) is O| ψ a,b i = C a,b O X s ,...,s N ψ a,b ( j, ζ j , s j , τ ) | s , . . . , s N i = C a,b O ψ a,b ( j, ζ j , σ zj , τ ) X s ,...,s N | s , . . . , s N i = C a,b O ψ a,b ( j, ζ j , σ zj , τ ) O † O X s ,...,s N | s , . . . , s N i = C a,b O ψ a,b ( j, ζ j , σ zj , τ ) O † X s ,...,s N | s , . . . , s N i = C a,b ψ a,b ( j, ζ j , σ zd − ( j ) , τ ) X s ,...,s N | s , . . . , s N i = C a,b X s ,...,s N ψ a,b ( j, ζ j , s d − ( j ) , τ ) | s , . . . , s N i . (52)We can rewrite ψ a,b ( j, ζ j , s d − ( j ) , τ ) = ψ a,b ( d ( j ) , ζ d ( j ) , s j , τ ) = ( − S ψ a,b ( j, ζ d ( j ) , s j , τ ) , (53)where S is the number of times one has to swap two labels to change the numbering from d ( j ) to j . The first equality in (53) follows from a relabelling of indices, and the secondequality follows from (27), which trivially generalizes to the case of twisted boundaryconditions. To find the transformation properties of the wavefunction under O , we thusonly need to determine S and ψ a,b ( j, ζ d ( j ) , s j , τ ). Since the delta function factor and theMarshall sign factor do not depend on ζ j , it is sufficient to consider the transformationproperties of the centre of mass factor and the Jastrow factor. For later convenience,let us also define κ d through the relation Y i The sets A , B , C , and D for a 5 × Note that the factors C − a,b and C − − a, − b simply remove the normalization factors from | ψ a,b i and | ψ − a, − b i , respectively. Some special cases of (56) are F | ψ , i = ( − N/ | ψ , i , (57) F | ψ / , i = ( − N/ C / , C − / , | ψ − / , i = ( − N/ | ψ / , i ,F | ψ , / i = ( − N/ C , / C , − / | ψ , − / i = ( − N/ | ψ , / i ,F | ψ / , / i = ( − N/ C / , / C − / , − / | ψ − / , − / i = − ( − N/ | ψ / , / i , where we have used (47), (49), (A.2) and (A.3). In particular, ψ , and ψ / , are botheigenstates with eigenvalue ( − N/ as stated in table 1. x -direction The action of the translation operator T x is to move the spins one lattice constant to theright, except for the rightmost column of spins which is wrapped around to the oppositeedge. T x is of the rearrangement type and is defined through the map d ( j ) = ( j − j ∈ A,j + ( L x − 1) for j ∈ B, (58)which moves the lattice one lattice constant to the left. Here, A = { n + mL x + 1 | n = 1 , . . . , L x − ∧ m = 0 , , . . . , L y − } , (59) B = { mL x + 1 | m = 0 , . . . , L y − } , i.e., A is the set containing the indices of the sites in the L x − B is the set containing the indices of the sites in the leftmost column ofthe lattice as illustrated for a 5 × S = L y ( L x − ζ d ( j ) = ( ζ j − /L x for j ∈ A,ζ j + ( L x − /L x for j ∈ B. (60)Therefore N X j =1 ζ d ( j ) s j = X j ∈ A (cid:18) ζ j − L x (cid:19) s j + X j ∈ B (cid:18) ζ j + L x − L x (cid:19) s j = N X j =1 ζ j s j + X j ∈ B s j , (61) osonic FQH states on the torus from CFT θ h ab i N X j =1 ζ d ( j ) s j , τ ! = θ h ab i N X j =1 ζ j s j , τ ! e π i a P j ∈ B s j . (62)Regarding the Jastrow factor, we observe that E ( ζ d ( i ) − ζ d ( j ) , τ ) = ( E ( ζ i − ζ j , τ ) for ( i ∈ A ∧ j ∈ A ) ∨ ( i ∈ B ∧ j ∈ B ) , − E ( ζ i − ζ j , τ ) for ( i ∈ A ∧ j ∈ B ) ∨ ( i ∈ B ∧ j ∈ A ) . (63)The case i ∈ A and j ∈ B follows from E ( ζ d ( i ) − ζ d ( j ) , τ ) = E (cid:18) ζ i − L x − ζ j − L x − L x , τ (cid:19) = E ( ζ i − ζ j − , τ )= − E ( ζ i − ζ j , τ ) , (64)where we have used (A.11), and the case i ∈ B and j ∈ A then follows from (A.10). For κ d (see (54)), we therefore get κ d = ( − P i ∈ A,j ∈ B ( s i s j +1) / = ( − − ( P j ∈ B s j ) / L y ( L x − / = ( − L y ( L x − / . (65)Collecting all the factors, we arrive at T x | ψ a,b i = ( ( − L x / e π i a P j ∈ B σ zj | ψ a,b i for L y odd , e π i a P j ∈ B σ zj | ψ a,b i for L y even . (66)For a ∈ { , / } , i.e., for θ x = 0, we can replace P j ∈ B σ zj by L y , and it follows that ψ ,b and ψ / ,b are eigenstates of T x with the eigenvalues given in table 1. y -direction The translation operator T y in the y -direction is described by the map d ( j ) = ( j − L x for j ∈ C,j + ( L y − L x for j ∈ D, (67)which moves the lattice one lattice constant in the negative y -direction. Here C = { L x + 1 , L x + 2 , . . . , L x L y } , (68) D = { , , . . . , L x } , i.e., C is the set of indices of the L y − D is the setof indices of the lowermost row of the lattice as illustrated for a 5 × S = L x ( L y − ζ d ( j ) = ( ζ j − i /L x for j ∈ C,ζ j + i( L y − /L x for j ∈ D. (69)Therefore N X j =1 ζ d ( j ) s j = X j ∈ C (cid:18) ζ j − i L x (cid:19) s j + X j ∈ D (cid:18) ζ j + i L y − L x (cid:19) s j = N X j =1 ζ j s j + τ X j ∈ D s j , (70) osonic FQH states on the torus from CFT p → p/ τ → τ , it follows that θ h ab i N X j =1 ζ d ( j ) s j , τ ! = θ (cid:20) a + L x b (cid:21) N X j =1 ζ j s j , τ ! e − i πτ ( P j ∈ D s j ) / − i π P j ∈ D s j ( P Nj =1 ζ j s j + b ) . (71)To derive the transformation of the Jastrow factor, we observe that E ( ζ d ( i ) − ζ d ( j ) , τ ) = E ( ζ i − ζ j , τ ) for i ∈ C ∧ j ∈ C,E ( ζ i − ζ j , τ ) for i ∈ D ∧ j ∈ D, e − i π − i πτ − π i( ζ i − ζ j ) E ( ζ i − ζ j , τ ) for i ∈ D ∧ j ∈ C. (72)The case i ∈ C and j ∈ D is not relevant, because this automatically implies i > j according to (68). The case i ∈ D and j ∈ C follows from E ( ζ d ( i ) − ζ d ( j ) , τ ) = E (cid:18) ζ i + i L y − L x − ζ j + i 1 L x , τ (cid:19) = E ( ζ i − ζ j + τ, τ )= e − i π − i πτ − π i( ζ i − ζ j ) E ( ζ i − ζ j , τ ) , (73)where we have used (A.12). κ d (see (54)) is therefore κ d = ( − P i ∈ D,j ∈ C ( s i s j +1) / e − i π P i ∈ D,j ∈ C [ τ +2( ζ i − ζ j )]( s i s j +1) / . (74)Note that X i ∈ D,j ∈ C [ τ + 2( ζ i − ζ j )] = i L y L x L x ( L y − 1) + 2 L x ( L y − X i ∈ D ζ i − L x X j ∈ C ζ j = i L y L x ( L y − 1) + 2 L x L y X i ∈ D ζ i = i L y L x ( L y − 1) + 2 L x L y ( − i) L y − L x L x = 0 . (75)Therefore κ d = ( − P i ∈ D,j ∈ C ( s i s j +1) / e − i π P i ∈ D,j ∈ C [ τ +2( ζ i − ζ j )] s i s j / = ( − L x ( L y − / e i πτ ( P i ∈ D s i ) / π P Ni =1 ζ i s i P j ∈ D s j . (76)Collecting all the factors, we conclude that T y C − a,b | ψ a,b i = ( ( − L y / e − i πb P j ∈ D σ zj C − a + ,b | ψ a + ,b i for L x odd , e − i πb P j ∈ D σ zj C − a,b | ψ a,b i for L x even . (77)It follows that T y transforms states in the subspace spanned by ψ a, and ψ a +1 / , into states in the same subspace, and one can therefore easily diagonalize T y in thesesubspaces. The result for a = 0 is given in table 1. As a side remark, we show that the action of the operators˜ T x = U T x U † , ˜ T y = U T y U † , (78) osonic FQH states on the torus from CFT U is the unitary operator U = exp " i2 L x − X n =0 Ly − X m =0 (cid:18) nθ x L x + mθ y L y (cid:19) σ zn + mL x +1 , (79)is in fact simpler than the action of T x and T y . We first find explicit expressions for˜ T x and ˜ T y by writing ˜ T d = U ( T d U † T † d ) T d , d = x, y , and evaluating the expression inbrackets using (50) with O = T d . This gives˜ T x = exp − i2 θ x X j ∈ B σ zj + i2 θ x L x N X j =1 σ zj ! T x , (80)˜ T y = exp − i2 θ y X j ∈ D σ zj + i2 θ y L y N X j =1 σ zj ! T y . (81)Combining this result with (66) and (77) and remembering (48), we get˜ T x C − a,b | ψ a,b i = ( ( − k ( − L x / C − a,b | ψ a,b i for L y odd , C − a,b | ψ a,b i for L y even , (82)˜ T y C − a,b | ψ a,b i = ( ( − L y / C − a +1 / ,b | ψ a + ,b i for L x odd , C − a,b | ψ a,b i for L x even , (83)where k = a − θ x / (4 π ) ∈ { , / } . We thus conclude that ψ a,b is an eigenstate of ˜ T x and˜ T y for L x even, whereas ψ a,b ± ψ a +1 / ,b are eigenstates for L x odd. ◦ In this subsection, we assume L x = L y and study the action of a rotation by 90 ◦ . Arotation of the lattice by − ◦ is described by the map d ( n + mL x + 1) = m + ( L x − − n ) L x + 1 , (84)where n = 0 , , . . . , L x − m = 0 , , . . . , L x − 1. For this map S = 3 × N/ ζ d ( j ) = − i ζ j = ζ j /τ, τ = i = − / i = − /τ. (85)We thus need to compute ψ a,b ( j, ζ j /τ, s j , − /τ ). The transformation done on thiswavefunction is precisely the modular S -transformation (see, e.g., [22]), which takes ζ j → ζ j τ , τ → − τ , (86)and is defined for general τ . We have θ h ab i(cid:18) ζτ , − τ (cid:19) = X n ∈ Z e − π i τ ( n + a ) +2 π i( n + a )( ζτ + b ) = X k ∈ Z Z ∞−∞ e π i kx − π i τ ( x + a ) +2 π i( x + a )( ζτ + b ) dx = r τ X k ∈ Z e − i2 πak +i πτ ( k + ζτ + b ) = r τ X µ =0 X n ∈ Z e − i2 πa (2 n + µ )+i πτ (2 n + µ + ζτ + b ) osonic FQH states on the torus from CFT r τ X µ =0 X n ∈ Z e π i τ ( n + µ + b ) +2 π i( n + µ + b )( ζ − a )+ π i ζ τ +2 π i ab = r τ 2i e π i ζ / (2 τ )+2 π i ab X µ =0 θ (cid:20) b + µ − a (cid:21) ( ζ , τ ) . (87)A similar computation gives θ (cid:18) ζτ , − τ (cid:19) = − i( − i τ ) / e i πζ /τ θ ( ζ , τ ) , (88)which, together with (12), implies E (cid:18) ζ i − ζ j τ , − τ (cid:19) = τ − e i π ( ζ i − ζ j ) /τ E ( ζ i − ζ j , τ ) . (89)For the L x × L x lattice, we have P j ζ j = 0 and P j ζ j = 0, and therefore Y i 6. Many-body Chern number and other topological properties A main reason for studying FQH states is their nontrivial topological properties. For theKL states, the topological properties derive from the center of mass factor [25, 27]. Bynow a number of measures have been identified that can be used to describe topologicalstates, and some of these have already been used in [3] and [17] to demonstrate thetopological nature of the CFT state on an irregular lattice on the sphere and the CFTstates on square lattices on the torus. For the generalization of the KL states on thetorus to arbitrary lattices presented in the present paper to be useful, it is importantthat the topological properties are preserved when the lattice is deformed away from osonic FQH states on the torus from CFT a) ω ω b) ω ω c) ω ω d) ω ω Figure 4. The many-body Chern number of the two CFT states is one for the 4 × × the square lattice. To test this property, we compute the many-body Chern numberfor the lattices shown in figure 4, and in all cases we get the expected value 1. (Thecomputation is done as described in detail in [29, 30], and we use the wavefunction (47)with twisted boundary conditions.)The number of degenerate ground states on higher genus surfaces is an importantquantity to partially characterize topological states. Degeneracies may, however, arisefor other reasons than topology. To talk about topologically degenerate states, thelocal structure of the states must be the same, i.e., the expectation value of anylocal operator must be the same for all the states in the thermodynamic limit. Localindistinguishability has been demonstrated for the two states in (11) for the case of asquare lattice in [17]. One can observe from (47) that the two states on the torus canbe transformed into each other by changing the twist angles. Since the local structureis changed little by this operation when the number of spins is large, this is also a signof topology. Specifically, if θ x is changed from 0 to 2 π , a increases by 1 / 2, and from(A.2) it then follows that the states are transformed as ψ , ( s , . . . , s N ) → ψ / , ( s , . . . , s N ) , (104) ψ / , ( s , . . . , s N ) → ψ , ( s , . . . , s N ) . (105)This is illustrated in figure 5. If instead we increase θ y from 0 to 2 π , it follows from (47)and (A.3) that the states are transformed as ψ , ( s , . . . , s N ) + ψ / , ( s , . . . , s N ) → ψ , ( s , . . . , s N ) − ψ / , ( s , . . . , s N ) , (106) ψ , ( s , . . . , s N ) − ψ / , ( s , . . . , s N ) → ψ , ( s , . . . , s N ) + ψ / , ( s , . . . , s N ) . (107) 7. Conclusion In conclusion, we have found that the KL states on the torus can be written asconformal blocks of primary fields from the SU (2) WZW CFT. This representation ofthe states reveals an interesting underlying mathematical structure and gives a naturalgeneralization of the KL states on the torus to arbitrary lattices. It also provides aneasy way to ensure that the states are singlet states. In addition, we have found thatthe many-body Chern number for different lattices is unity and that the two states aretransformed into each other under a twist of the boundary conditions. These findings osonic FQH states on the torus from CFT ψ ,b ψ / ,b . . a = k + θ x π mod 1 θ x π k = 0 k = 1 / Figure 5. When θ x increases from 0 to 2 π , a increases by 1 / 2. Since the wavefunctiononly depends on a modulus 1 (see (A.2)), it follows that ψ ,b is transformed into ψ / ,b and vice versa. If θ x is increased to 4 π , we are back to the starting point. suggest that the topological properties are preserved when the lattice is transformedaway from the square lattice. Finally, we have analyzed the symmetry properties ofthe CFT states on the torus for the case of an L x × L y square lattice, which allowedus in certain cases to construct linear combinations of the states that are guaranteedto be orthogonal because they have different symmetries. Our work also shows thatthe KL states in different geometries can be obtained by evaluating the same conformalcorrelator in different geometries, and we believe that this holds more generally. TheCFT representation found in the present paper constitutes an interesting starting pointfor further analytical investigations of the states. In [4], CFT representations of latticeLaughlin states with general filling factor 1 /q , where q is an integer, have been foundon the Riemann sphere, and using these results, we note that the present work could bestraightforwardly generalized to obtain lattice Laughlin states with filling factor 1 /q onthe torus. Higher genus versions of the CFT states fulfilling the CFT fusion rules canalso be obtained. Another interesting perspective is to make similar constructions forother FQH states. Acknowledgments The authors acknowledge discussions with J. Ignacio Cirac, Denis Bernard, and Hong-Hao Tu. This work has been partially funded by EU through SIQS grant (FP7 600645)and by FIS2012-33642, QUITEMAD (CAM), and the Severo Ochoa Program. Appendix A. Properties of the Riemann theta function The Riemann theta function is defined as θ h ab i ( ζ , τ ) = X n ∈ Z e i πτ ( n + a ) +2 π i( n + a )( ζ + b ) , (A.1) osonic FQH states on the torus from CFT a and b are real numbers, ζ is complex, and τ is complex with Im( τ ) > 0. Fromthis definition, one easily derives the following identities θ (cid:20) a + cb (cid:21) ( ζ , τ ) = θ h ab i ( ζ , τ ) , c ∈ Z , (A.2) θ (cid:20) ab + c (cid:21) ( ζ , τ ) = e π i ac θ h ab i ( ζ , τ ) , c ∈ Z , (A.3) θ h ab i ( − ζ , τ ) = θ (cid:20) − a − b (cid:21) ( ζ , τ ) , (A.4) θ h ab i ( ζ + c, τ ) = e π i ac θ h ab i ( ζ , τ ) , c ∈ Z , (A.5) θ h ab i ( ζ + pτ, τ ) = e − i πτp − i2 πp ( ζ + b ) θ (cid:20) a + pb (cid:21) ( ζ , τ ) , p ∈ R . (A.6)Particular cases that we shall need are θ (cid:20) k (cid:21) ( − ζ , τ ) = θ (cid:20) k (cid:21) ( ζ , τ ) , k = 0 , , (A.7) θ (cid:20) k (cid:21) ( ζ ± , τ ) = e ± π i k θ (cid:20) k (cid:21) ( ζ , τ ) , k = 0 , , (A.8) θ (cid:20) k (cid:21) ( ζ ± τ, τ ) = e − i πτ/ ∓ π i ζ θ (cid:20) / − k (cid:21) ( ζ , τ ) , k = 0 , , (A.9)and E ( − ζ , τ ) = − E ( ζ , τ ) , (A.10) E ( ζ ± , τ ) = e ± i π E ( ζ , τ ) , (A.11) E ( ζ ± τ, τ ) = e − i πτ ∓ i π ∓ π i ζ E ( ζ , τ ) , (A.12)where E ( ζ , τ ) is the prime form defined in (12). Appendix B. Singlet property for four spins from Fay’s trisecant identity Consider the case of four spins. From (11), the wavefunctions on the torus read | ψ k i = C k ( ψ ,k | + 1 , − , +1 , − i + ψ ,k | + 1 , − , − , +1 i + ψ ,k | + 1 , +1 , − , − i (B.1)+ ψ ,k | − , +1 , − , +1 i + ψ ,k | − , +1 , +1 , − i + ψ ,k | − , − , +1 , +1 i ) , where ψ ,k = E ( ζ − ζ , τ ) E ( ζ − ζ , τ ) θ (cid:20) k (cid:21) ( ζ − ζ + ζ − ζ , τ ) , (B.2) ψ ,k = − E ( ζ − ζ , τ ) E ( ζ − ζ , τ ) θ (cid:20) k (cid:21) ( ζ − ζ − ζ + ζ , τ ) ,ψ ,k = − E ( ζ − ζ , τ ) E ( ζ − ζ , τ ) θ (cid:20) k (cid:21) ( ζ + ζ − ζ − ζ , τ ) . This state is a singlet if and only if ψ ,k + ψ ,k + ψ ,k = 0 . (B.3) osonic FQH states on the torus from CFT τ as the productof two theta functions with argument τ , θ (cid:20) k (cid:21) ( ζ , τ ) = η (2 τ ) η ( τ ) θ (cid:20) k (cid:21)(cid:18) ζ , τ (cid:19) θ (cid:20) k (cid:21)(cid:18) ζ − , τ (cid:19) , k = 0 , , (B.4)where η is the Dedekind eta function defined in (16). We can therefore write (B.2) as ψ ,k = [ η (2 τ ) /η ( τ )] E ( ζ − ζ , τ ) E ( ζ − ζ , τ ) (B.5) × θ (cid:20) k (cid:21)(cid:18) ζ − ζ + ζ − ζ , τ (cid:19) θ (cid:20) k (cid:21)(cid:18) ζ − ζ + ζ − ζ − , τ (cid:19) ,ψ ,k = − [ η (2 τ ) /η ( τ )] E ( ζ − ζ , τ ) E ( ζ − ζ , τ ) × θ (cid:20) k (cid:21)(cid:18) ζ − ζ − ζ + ζ , τ (cid:19) θ (cid:20) k (cid:21)(cid:18) ζ − ζ − ζ + ζ − , τ (cid:19) ,ψ ,k = − [ η (2 τ ) /η ( τ )] E ( ζ − ζ , τ ) E ( ζ − ζ , τ ) × θ (cid:20) k (cid:21)(cid:18) ζ + ζ − ζ − ζ , τ (cid:19) θ (cid:20) k (cid:21)(cid:18) ζ + ζ − ζ − ζ − , τ (cid:19) . These quantities satisfy (B.3) as a consequence of Fay’s trisecant identity [31, 32] E ( ζ − ζ , τ ) E ( ζ − ζ , τ ) θ h ab i ( ζ , τ ) θ h ab i ( ζ + ζ + ζ − ζ − ζ , τ ) = E ( ζ − ζ , τ ) E ( ζ − ζ , τ ) θ h ab i ( ζ + ζ − ζ , τ ) θ h ab i ( ζ + ζ − ζ , τ )+ E ( ζ − ζ , τ ) E ( ζ − ζ , τ ) θ h ab i ( ζ + ζ − ζ , τ ) θ h ab i ( ζ + ζ − ζ , τ ) . 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