# Exact density matrix for quantum group invariant sector of XXZ model

###### Abstract.

Using the fermionic basis we obtain the expectation values of all -invariant local operators on 8 sites for the anisotropic six-vertex model on a cylinder with generic Matsubara data. In the case when the symmetry is not broken this computation is equivalent to finding the entire density matrix up to 8 sites. As application, we compute the entanglement entropy without and with temperature, and compare the results with CFT predictions.

^{1}

^{1}1Membre du CNRS

## 1. Introduction

This paper is dedicated to my longtime friend Nikolai Reshetikhin, and its main idea has much in common with our join paper [1]. In this paper it was shown that the restriction of the degrees of freedom for the scattering states of the sine-Gordon model with rational coupling constant is a non-violent procedure: if the quantum group invariant operators are considered the contributions form the states which do not satisfy the RSOS restriction vanish in the correlation functions. In the present paper we apply the same logic to rather different situation. To be precise we are talking here about two different quantum groups which can be combined into the modular double [2].

Consider the XXZ spin chain in critical regime:

(1) |

We often use the parameter :

It is well-known that the XXZ model is closely related to the quantum affine group . We consider one of its finite-dimensional subgroups . For finite temperature consider the modified partition function an correlation functions

where is the total spin: . The insertion of is important: it corresponds to the “quantum group invariant” trace (see [3] for relevant discussion). In some sense we have a generalisation of the Witten index used in the SUSY models.

In fact we can consider more general case of generalised Gibbs distribution, in other words we use rather arbitrary Matsubara data (six-vertex model on a cylinder). Really crucial property for our construction is that the Matsubara maximal vector is quantum group invariant (we call this unbroken quantum group symmetry). We explain this in Section 5 . The similarity with [1] is in the fact that for the states of the lattice model which do not satisfy the RSOS restriction do not contribute.

If the quantum group symmetry is unbroken, only the quantum group invariant operators possess non-vanishing expectation values, this is similar to the usual Lie group symmetry. We consider a finite interval of the lattice and introduce the density matrix . Then for any local operator localised on this interval we have

Again, we use the invariant trace.

For

the scaling theory of the lattice model obtained in this way must coincide with the minimal model . In particular, correspond to unitary models, it can be shown that for them all the contributions to the modified partition function are positive. Otherwise we have non-unitary models.

In the present paper we compute the density matrices up to sites. In order to check the agreement with the scaling limit we compute the Von Neumann entropy. For the non-unitary case it has strange features: it may be negative getting more negative with temperature. This is not very surprising in this case. Nevertheless we always find very good agreement with the CFT predictions [4, 5, 6].

Naïve idea is that for rational the restricted case of the six-vertex model on a cylinder coincides completely with the RSOS case. This, however, is not quite simple as we explain in Section 5.1 . Presumably, for that reason our results differ from those of papers [7, 8]: they agree with the CFT formulae for usual, not effective, central charge and depend continuously on .

Our procedure is the same as in [18]: we compute the expectation values of invariant operators for arbitrary Matsubara data and arbitrary twist . However, unbroken quantum group symmetry is possible only for . For broken quantum group symmetry we don not obtain complete density matrix. Considering all the operators, not only the invariant ones, is possible. We do not do it for two reasons: first, it is technically more complicated, second, we find it more interesting to have in the scaling limit the central charge then independently of the coupling constant.

The paper is organised as follows. In Section 2 we briefly recall some definitions concerning the expectation value on a cylinder. Section 3 contains useful for us information from the theory of quantum groups. Section 4 gives an account of our computational procedure. In Section 5 we give general explanations regarding the unbroken quantum group symmetry. In Sections 6,7 we expose our numerical results and comparison to the CFT for the cases of zero and non-zero temperature respectively.

## 2. Matsubara expectation values

Consider the quantum affine group and its universal -matrix . We use usual Cartan generators. Central charge equals to , so , denote . The affine quantum group allows evaluation representations with being a spin, and an evaluation parameter. Fix two integers and define two representations:

with spins and inhomogeneities . Later we shall use

The index stands for “space” and stands for Matsubara.

We have the image of the universal -matrix:

Consider further the commuting family of Matsubara transfer-matrices

and their eigenvector .

The main object of our study is a linear functionals on operators acting on the representation space of :

(2) |

where , in other words is the Cartan generator evaluated on . If is localised on smaller interval that the space shrinks automatically. Clearly, can be considered as such automatic reduction for the case if is the eigenvector with maximal in absolute value eigenvalue of . This explains importance of for physical applications. The main tool of computation is the fermionic basis.

## 3. Quantum group invariant operators

### 3.1. Generalities

Let us start with some simple facts concerning the quantum groups. We use Drinfeld’s notations [10]. Take a basis , . The comultiplication is a homomorphism

We have the -matrix

(3) |

and the antipode

which is an anti-automorphism. We define two adjoint actions

The first is an homomorphism, the second is an anti-homomorphism. In any quantum double is an internal homomorphism:

for certain . The pairing

is an invariant scalar product of operators for being any cyclic functional, for example usual trace over a finite-dimensional representation if any. We have

Below we give some additional formulae which will be used only Section . From now on we consider two representations, for economy of space we not distinguish between the generators of the quantum group and their representations, everything should be clear from the context.

### 3.2. Invariant operators

The quantum group contains two finite-dimensional subgroups isomorphic to . Take one of them, for example the one created by , which we denote respectively by . We use . The defining relations of are well-known, so, we give them without comments just to fix the notations

The coproduct, counit and antipode are given by

Clearly,

Consider an invariant under operator . It is convenient to identify with a vector in , The identification is

(7) |

where acts from to as

For example for there is one invariant vector:

(8) |

which under (7) goes to the unit operator acting in . Generally, we denote the element of the basis for an irreducible representation of spin by .

We are going to give several definitions, for future convenience they are a bit more general than we need at this point. Bratteli diagram is a sequences such that , . Lexicographically ordered totality of Bratteli diagrams (for fixed ) with and will be denoted by , the meaning of the latter argument will be clear later when we shall consider restricted case. With every Bratteli diagram and , (integer of half-integer depending on ) we associate a vector from :

In order to avoid denominators which are dangerous for being a root of unity we use non-normalised symbols

For this gives an decomposition of the tensor product of spaces :

(9) |

We have for the dimensions of multiplicity spaces

We shall use two bases of the space of invariant operators acting on of . The dimension of this space is the Catalan number .

1. To have an orthogonal basis we use the above construction with -symbols. The basis is given by

This basis is orthogonal, but not orthonormal with respect to the scalar product

The normalisation is

where the quantum dimension is

(10) |

Here and later

2. For our computations it is convenient to use a simpler basis. Take points on the real axis: and connect them pairwise by arcs in the upper half plain requiring that the arcs do not intersect. For example,

fig.1 Simple construction of invariant vectors.

With any such design associate the Bratelly diagram writing one half of the number of arcs passing over every interval . For instance, the fig. 1 corresponds to . Denote by the beginnings of the arcs, and by their ends. Then the vector associated with every design is

with given by (8), and define

This is another basis.

Recall that is lexicographically ordered. We have two bases of invariant operators and . It is easy to see that they are related by a triangular transformation

The matrix is not hard to compute inductively.

Considering the interval we are not interested in the translationally irreducible operators, i.e. the operators which are not localised on subintervals of smaller length. It is easy to figure out that the number of translationally irreducible invariant operator is

For the second basis eliminating the translationally reducible operators is simple: it suffices to throw all the vectors containing or .

## 4. Procedure of computation

Fermionic basis for the case of -invariant operators is parallel to the -invariant case for the XXX model which is explained in details in [18]. So, we shall be brief here. We have two sets of fermionic operators , , () with canonical commutation relations, and use notations , for products, being a strictly ordered multi-index . For two multi-indices of the same length we write if for all . We denote by the sum of elements in . Our fermionic operators act on the space of local fields, role of vacuum is played by the unit operator . Consider the space with the basis

(11) |

with , , (the difference with the XXX case is that we do not impose , since there is no -symmetry). Define the operators

Introduce the space defined as above with the condition lifted. The operators act from to . The operator acts from the space (space of charge ), span by the vectors (11) with , , to . We define the subspace of by

It is easy to see that for , so the actual number of requirements is finite.

Denoting basis of by we have , the first one of several matrices used below:

Our goal is to find an analogue of OPE:

(12) |

where is an invariant operator constructed via the Bratteli diagram and the second of the bases above.

As in the previous paper [18] we fix the OPE coefficients considering finite Matsubara chains. For every Matsubara data we have an equation

(13) |

To compute the right hand side we use

here and later

with are the Taylor series coefficients

of a function is defined below. Here and later the latin letters are are squares of the evaluation parameters of representations.

With every eigenvector of the Matsubara transfer-matrix we associate an eigenvalue of the -operator

The information about the Matsubara spin chain encoded is two functions

We have the Bethe equations

In principle we could introduce a twist multiplying and by and respectively, but our goal will be to obtain equations for the OPE coefficients, and practice shows that twist does not produce independent ones.

Our main trick is to take for the input data

Then for the unknowns we have linear equations.

Introduce the measure

the auxiliary function

and kernels

We have “integral” equation

(14) |

with going around . For finite this is equivalent to a system of linear equations for functions . The function [15] is

where encircles, in addition to , the point .

The expectation values of invariant operators is computed exactly as in [18], basically we rewrite in the basis of Young diagrams formulae of [12, 13, 14].

We repeat some definitions. Consider Young diagrams where , is a partition. We set . It is called the length of . We work in the space whose elements are

In the below we will identify with . The symbol denotes the empty diagram. Define the operation which acts from with to erasing all the terms with . Consider the Grassmann space with the basis () . We have the usual isomorphism between the spaces and .

(15) | |||

In the above, means removing all entries equal to . Schur polynomial is the symmetric polynomial

The above formula gives an isomorphism between and .

For a given polynomial of one variable we define the operator multiplying by , this operator is defined as by the isomorphism (15). This definition can be generalised to polynomials of several variables in obvious way. Certainly the polynomials anti-symmetrise themselves automatically.

We shall also need the simplest Littlewood-Richardson formula for multiplication of a Schur polynomial by elementary symmetric function , which translates as action on

where are all vectors of dimension with elements equal to other elements being , “order” means that we have to drop all the tables in which elements happen to be not ordered, and we also drop all zeros in the final table.

In what follows we shall also need the operation which erases all the Young diagrams in with lengths greater than .

For a partition define the coefficients via

This gives rise to an operator

The easiest algorithm for finding consists in the following: express through Jacobi-Trudi formula using the transposed . This formula is given in terms of the elementary symmetric functions . Using

expand the Jacobi-Trudi determinant getting a sum of Schur polynomials.

Slavnov formula for the scalar product of on-shell and off-shell Bethe vectors is a symmetric polynomial of the off-shell with the following representation in terms of the Young diagrams:

where

We shall need also the Gaudin formula for normalisation:

As usual we consider the matrix elements of the Matsubara monodromy matrix. Their action in our framework translates into the the action of the operators described below.

We begin with the operators , which do not change the charge.

The most complicated operator is which raises the number of variables:

(16) |

where , are polynomials of respectively two and one variables:

Finally, the operator which lowers the number of variables is the simplest one:

In order to compute we present as a sum of the operators , and use

(17) |

where is a linear functional on the vector space which maps every Young diagram to corresponding Schur polynomial of arguments . Now we can construct as many equations for the coefficients of OPE as we wish.

The rest of the computations follow closely that of [18]. For example, for we have 324 fermionic vectors , and construct 20 eqs. with , 120 eqs. with ; 100 eqs. with ; 10 eqs. with ; 2 eqs. with ; 50 eqs. with ; 70 eqs. with ; 2 eqs. with . Then we proceed with Gauss triangularisation. This is easy to do for numeric value of , but rather impossible keeping as variable. That is why we would like to proceed with interpolation. But the solutions for the coefficients contain denominators which we have to fix, then interpolating for the numerators is possible, but we also have to estimate the degree of them as functions of . All these data can be guessed considering several numerical examples.

We have for the lowest common denominators () and for the maximal exponent of the numerators ():

(18) | |||

We proceed with interpolation obtaining finally matrices . Now for any Matsubara data we have

(19) |

Here only depends on the Matsubara data.

Let us give an example. For the basis consists of 9 elements

For with which is constructed from we have

By triangularity . Certainly, for the formulae are getting more complicated, but their structure is more inspiring that in XXX case.

## 5. The case of unbroken quantum group symmetry

Consider the functional for particular case :

(20) |

From the identity (6) one concludes that the transfer-matrix preserves the