Lattice Laughlin states on the torus from conformal field theory

Lattice Laughlin states on the torus from conformal field theory

Abstract

Conformal field theory has turned out to be a powerful tool to derive two-dimensional lattice models displaying fractional quantum Hall physics. So far most of the work has been for lattices with open boundary conditions in at least one of the two directions, but it is desirable to also be able to handle the case of periodic boundary conditions. Here, we take steps in this direction by deriving analytical expressions for a family of conformal field theory states on the torus that is closely related to the family of bosonic and fermionic Laughlin states. We compute how the states transform when a particle is moved around the torus and when the states are translated or rotated, and we provide numerical evidence in particular cases that the states become orthonormal up to a common factor for large lattices. We use these results to find the -matrix of the states, which turns out to be the same as for the continuum Laughlin states. Finally, we show that when the states are defined on a square lattice with suitable lattice spacing they practically coincide with the Laughlin states restricted to a lattice.

  • Keywords: fractional QHE (theory), conformal field theory (theory), spin chains, ladders and planes (theory).

1 Introduction

In certain two-dimensional systems consisting of many interacting bosons or fermions, it is possible to have quasi-particles with unusual properties like fractional charge and an exchange statistics that is neither bosonic nor fermionic. The properties of such topological systems are very different from other types of matter, and they are therefore attracting a lot of attention.

The quantum states appearing in connection with the fractional quantum Hall (FQH) effect constitute important examples of topological states, because they can be realized experimentally in semiconductor devices under demanding conditions, and because many of the states are believed to be described by simple analytical wave functions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The interest in topological systems has triggered the question whether FQH physics can occur in other settings, and in the last years there have been several proposals for how to obtain FQH behaviour in lattice systems. This research allows for the investigation of new aspects of FQH physics and paves the way towards new possibilities for realizing the effect experimentally [11, 12, 13, 14, 15, 16, 17, 18, 19, 20].

There are mainly two strategies to obtain lattice models displaying FQH physics. One of them is to engineer a band structure that is reminiscent of a Landau level, ensure that the band is partially filled, and add interactions between the particles [21, 22, 23, 24]. The other strategy is to start from an analytical FQH wave function in the continuum, modify it to a lattice wave function, and then analyze its properties and use analytical tools to derive a Hamiltonian for which the state is the ground state [25, 26, 27, 28, 19, 29, 30, 31, 32, 33, 34, 35, 36]. Both approaches have been shown numerically to be successful.

It has long been known that there is a connection between FQH physics and conformal field theory (CFT) and that a number of continuum FQH states can be expressed as conformal blocks of certain CFT correlators [7]. This observation is also helpful for finding analytical expressions of FQH states with periodic boundary conditions [9]. More recently, the CFT formulation has turned out to be very fruitful for constructing FQH models in lattice systems with analytical wave functions and corresponding parent Hamiltonians [37, 38, 28, 19, 29, 31, 32, 33, 34, 35, 36].

A natural way to transform a wave function in the continuum into a lattice wave function is to keep the analytical expression for the state, but to let the state be a sum over all possible distributions of the particles on a lattice rather than an integral over all possible positions of the particles in the plane. Part of the trick used to derive the parent Hamiltonians in [38, 28, 29, 31, 32, 33, 34, 35], however, is not just to do this, but to make an additional change. The Gaussian factor appearing in the continuum FQH wave functions is obtained by including a uniform background charge in the CFT correlator, and the additional change is to restrict the background charge to the same lattice as the particles. This has the further advantage that the CFT correlator takes a particularly simple form: for a lattice with sites, it is the expectation value of a product on CFT operators without any other factors. The analytical expression for the lattice states then differs slightly from the analytical expression for the continuum states, but at least as long as the distribution of the lattice sites is not too far from uniform, this does not change the physical properties of the states significantly.

Apart from a study of the chiral spin liquid Kalmeyer-Laughlin state in [39], the investigations of lattice FQH behaviour obtained from CFT as described above has so far been concerned with systems having periodic boundary conditions in at most one direction, because the case of periodic boundary conditions in both directions is more complicated to handle in CFT. It is, however, very desirable to be able to investigate two-dimensional models with periodic boundary conditions in both directions, since the topology of the space on which a state is defined plays an important role for topological systems. In particular, several computational methods have been developed in recent years to determine particular topological properties of quantum systems [40, 41, 42, 43, 44, 45, 46, 47], but many of these rely on the ability to define the investigated systems on surfaces with periodic, or more generally twisted, boundary conditions. Systems defined on a torus are also appealing because the lack of a boundary makes it easier to reliably investigate the bulk properties of the states.

In the present paper, we take steps towards generalizing the previous work on lattice FQH physics obtained using CFT techniques to the torus. Our starting point is a CFT correlator that has previously [31] been used to derive lattice models with Laughlin-like ground states on the plane and cylinder. By evaluating the same correlator on the torus we obtain analytical expressions for a family of CFT states that is closely related to the family of bosonic and fermionic Laughlin states on the torus. We investigate the boundary conditions of these states when a single particle is moved around the torus, and we study how the states on particular lattices transform under collective rotation and translation operations. We provide numerical evidence in particular cases that the states become orthonormal up to a common factor for large lattices, and from this and the properties under rotation we conclude that the -matrix of the states is the one expected for the Laughlin states. The ’th entry of the -matrix gives the phase factor acquired by the wave function when an anyon of type is moved adiabatically in a closed path around an anyon of type . The result hence shows that the braiding properties of the anyons are those of the Laughlin anyons. Finally, we show that, for the case of a square lattice, the analytical expressions of the states practically coincide with the analytical expressions of the corresponding Laughlin states on the torus.

2 Lattice Laughlin states on the plane from CFT

We first briefly recall the construction of lattice Laughlin states on the plane from CFT proposed in [31]. The starting point is the CFT correlator

(1)

where denotes the vacuum expectation value, the subscript P stands for plane, and

(2)

is a vertex operator [48]. Here, means normal ordering, is a positive integer, , is a complex number, is the complex conjugate of , and is the field of a free, massless boson compactified on a circle of radius .

CFT correlators can be decomposed [48, 49] into conformal blocks that are functions of the holomorphic coordinates , but not of the antiholomorphic coordinates . This decomposition allows us to write (1) in the form

(3)

where are the conformal blocks and are numbers that do not depend on or .

On the plane, there is only one conformal block , and we use this conformal block to define a wave function on a two-dimensional lattice as follows. We interpret as the position in the plane of the ’th lattice site and as the number of particles on the ’th lattice site. We then write

(4)

where

(5)

The factor is defined as

(6)

and it fixes the lattice filling factor to . Since (3) does not fix the phase of the conformal block, we include an unspecified phase factor . Note that cannot depend on the lattice positions due to the requirement that the conformal block is holomorphic, but it may depend on .

As shown in [31], the factor approaches the Gaussian factor of the Laughlin state with filling factor up to a phase factor when if the lattice is regular (with the area per lattice site being the same for all lattice sites) and the boundary of the lattice is a circle. The phase factor can be transformed away if desired, and in practice, is already enough to obtain very good agreement [31]. We thus observe that (5) is practically the Laughlin state with filling factor , except that the possible positions of the particles are restricted to a set of lattice sites. In [31], it has also been shown that these states have the same topological entanglement entropy as the continuous Laughlin states.

3 Conformal blocks on the torus

In this section, we derive the conformal blocks of the correlator (1) on the torus (the final result can be found in section 3.3). We first define the torus and the coordinates, we are using. Let and be two complex numbers. The modular parameter is defined as , and we assume that . The torus is then the parallelogram spanned by and with periodic boundary conditions. In other words, if and are integers, and and are real numbers in the interval , then we identify the point with the point . Let be a point within the parallelogram. We then define the scaled coordinate , which we shall use throughout the paper.

We would like to determine

(7)

where stands for torus. Our starting point for evaluating (7) is the following relation for the correlator of a product of vertex operators on the torus [50] (see also [51])

(8)

On the left hand side of (3), () is the holomorphic (antiholomorphic) part of the field of the free, massless boson and the numbers (, ) , where is the lattice of the momenta consisting of the elements with

(9)

Here, is the compactification radius mentioned above. For the correlator (7), the relevant values of are and , which are obtained for and or , respectively.

On the right hand side of (3), if and otherwise,

(10)

is the Dedekind eta function,

(11)

and is the function obtained by taking the complex conjugate of and replacing with and with . is defined as

(12)

where the Riemann theta function is given by

(13)

The only part of (3) that does not immediately factorize into holomorphic and corresponding antiholomorphic parts is the factor

(14)

where we have defined . We shall consider the cases even and odd separately in the following.

3.1 Even

When is even, and are integers and differ by a multiple of . Therefore, upon division by , they give the same remainder . We can thus write

(15)

where and without any restriction. That is, for a fixed , the summation over is over the whole set . Therefore, (3) becomes

(16)

which is a sum of factorized terms.

3.2 Odd

When is odd, and can take half integer values. In this case we split the sum over in (3) into a sum over even and a sum over odd . When is even, we again write

(17)

but here we note that and are not unrestricted. Particularly, since is even, we must have

(18)

Therefore, for a fixed , say even (odd), the summation over is over even (odd) integers. We can hence rewrite the sum in (3) over even into

(19)

where the last factor is when and are of opposite parity and otherwise.

When is odd, we write instead

(20)

In this case, we have

(21)

and so the summation over has opposite parity to that over . This allows us to rewrite the sum in (3) over odd into

(22)

The expression in (3) then equals the sum of (19) and (22).

3.3 Expressions for the conformal blocks

Collecting the above results, we conclude that the conformal blocks take the form

(23)

where if and otherwise, is an undetermined phase factor that may depend on , but not on , and we have split the index into two indices and , where . For even, can take only the value , and for odd, . The numbers and are defined as

(24)

For even, the factors in (7) are all , and for odd, we have . There are hence conformal blocks for even and conformal blocks for odd.

3.4 Phase factor

Let us finally discuss the relation between the phase factor for the conformal blocks on the torus and the phase factor for the conformal block on the plane. Locally the torus looks like the plane, and we can hence recover the conformal block on the plane by putting all the lattice sites close together. Note that this does not affect , since does not depend on . In this case,

(25)

and

(26)

The right hand side of (26) is zero for , but is nonzero for the other values of and occurring in the centre-of-mass factors. In this limit, we hence have only one conformal block, satisfying

(27)

Comparing this result to (5), we conclude that it is natural to choose to be the same on the torus and on the plane. In the following, we shall take

(28)

For this choice ensures that the state is an SU(2) singlet [39].

4 Wave functions from conformal blocks

We use the conformal blocks on the torus (23) to define the wave functions

(29)

where is shorthand notation for and we have left out some constant factors that only affect the norm of the states. Note that is not normalized.

Since

(30)

we can alternatively express (29) as

(31)

Here, is the total number of particles, is the operator that creates a particle at position , is the vacuum state, and the first sum is over all possible configurations of the particles on the lattice, i.e., all for which for all and for all and . In addition,

(32)

Since , we observe that the states (31) describe bosons for even and fermions for odd. Note also that if we choose two of the coordinates and to be the same in (31), then the amplitude of the term is zero because . Independent of , there is hence at most one particle on each site, which means that the bosons are hardcore bosons. This result reflects the property of the Laughlin states that the amplitude for two particles being at the same point is zero.

In (7), we chose to order the vertex operators so that the one evaluated at the point was the ’th vertex operator in the correlator, and in (29), we chose the ordering of the ’s in the ket to follow the ordering of the vertex operators in the correlator. We could instead have chosen the ordering such that the ’th vertex operator was operator number in the correlator, where is some bijective map from to . In (29), this would change into , it would change the ordering in the ket, and it would change the product over into the product over all and for which . In (31) there is, however, no longer any reference to this ordering, and is hence invariant under reordering. This means that it does not make a difference how we choose to label the lattice sites.

5 Boundary conditions and fluxes

In this section, we investigate the boundary conditions of the state (31) when the ’th particle is moved all the way around the torus and back to its original position, i.e. when or . We also discuss how to generalize the states to fulfil twisted boundary conditions.

5.1

When , we have . Utilizing

(33)

we conclude that the centre-of-mass factor transforms as

(34)

The Jastrow factor transforms like

(35)

where we have used the property , and

(36)

The overall transformation is hence

(37)

This shows that for even, the state is periodic, whereas for odd it can be periodic () or antiperiodic (). The phase factor obtained when taking a particle around the torus in the direction measures the flux through the hole of the torus. We thus observe that increasing by one, increases the flux through the hole by one.

5.2

In this case, . We shall need the properties

(38)
(39)
(40)

Now

(41)

and

(42)

and

(43)

Combining the factors, we get

(44)

The state is hence periodic for even, and it is periodic () or antiperiodic () for odd. We also observe that the flux through the tube of the torus is independent of .

5.3 Twisted boundary conditions

From the above derivations it is clear that the states can be modified to have twisted boundary conditions with twist angles and by modifying the centre-of-mass factor to

(45)

without modifying other parts of the wave functions. In this case

(46)

and