Abstract. In this article we unveil a new structure in the space of operators of the XXZ chain. For each α we consider the space W α of all quasi-local operators, which are products of the disorder field q α P 0 j=−∞ σ 3 j with arbitrary local operators. In analogy with CFT the disorder operator itself is considered as primary field. In our previous paper, we have introduced the annhilation operators b(ζ), c(ζ) which mutually anti-commute and kill the "primary field". Here we construct the creation counterpart b * (ζ), c * (ζ) and prove the canonical anti-commutation relations with the annihilation operators. We conjecture that the creation operators mutually anti-commute, thereby upgrading the Grassmann structure to the fermionic structure. The bosonic operator t * (ζ) is the generating function of the adjoint action by local integrals of motion, and commutes entirely with the fermionic creation and annihilation operators. Operators b * (ζ), c * (ζ), t * (ζ) create quasi-local operators starting from the primary field. We show that the ground state averages of quasi-local operators created in this way are given by determinants.

Abstract. For the critical XXZ model, we consider the space W [α] of operators which are products of local operators with a disorder operator. We introduce two anti-commutative family of operators b(ζ), c(ζ) which act on W [α] . These operators are constructed as traces over representations of the q-oscillator algebra, in close analogy with Baxter's Q-operators. We show that the vacuum expectation values of operators in W [α] can be expressed in terms of an exponential of a quadratic form of b(ζ), c(ζ).

Abstract. The Grassmann structure of the critical XXZ spin chain is studied in the limit to conformal field theory. A new description of Virasoro Verma modules is proposed in terms of Zamolodchikov's integrals of motion and two families of fermionic creation operators. The exact relation to the usual Virasoro description is found up to level 6.

Recent studies have revealed much of the mathematical structure of the static correlation functions of the XXZ chain. Here we use the results of those studies in order to work out explicit examples of short-distance correlation functions in the infinite chain. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field for various anisotropies in the massless regime −1 < ∆ < 1. It turns out that the new formulae are numerically efficient and allow us to obtain the correlations functions over the full parameter range with arbitrary precision. † The results of this and of the following section are valid for all ∆ > −1. Later on in sections 4 and 5 we restrict ourselves to the massless regime −1 < ∆ < 1.

It was recently shown by Jimbo, Miwa and Smirnov that the correlation functions of a generalized XXZ chain associated with an inhomogeneous six-vertex model with disorder parameter α and with arbitrary inhomogeneities on the horizontal lines factorize and can all be expressed in terms of only two functions ρ and ω. Here we approach the description of the same correlation functions and, in particular, of the function ω from a different direction. We start from a novel multiple integral representation for the density matrix of a finite chain segment of length m in the presence of a disorder field α. We explicitly factorize the integrals for m = 2. Based on this we present an alternative description of the function ω in terms of the solutions of certain linear and nonlinear integral equations. We then prove directly that the two definitions of ω describe the same function. The definition in the work of Jimbo, Miwa and Smirnov was crucial for the proof of the factorization. The definition given here together with the known description of ρ in terms of the solutions of nonlinear integral equations is useful for performing e.g. the Trotter limit in the finite temperature case, or for obtaining numerical results for the correlation functions at short distances. We also address the issue of the construction of an exponential form of the density matrix for finite α.

Riemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of integrable model. We study XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in thermodynamics limit. We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments.

We consider the inhomogeneous generalization of the density matrix of a finite segment of length m of the antiferromagnetic Heisenberg chain. It is a function of the temperature T and the external magnetic field h, and further depends on m 'spectral parameters' ξ j . For short segments of length 2 and 3 we decompose the known multiple integrals for the elements of the density matrix into finite sums over products of single integrals. This provides new numerically efficient expressions for the two-point functions of the infinite Heisenberg chain at short distances. It further leads us to conjecture an exponential formula for the density matrix involving only a double Cauchy-type integral in the exponent. We expect this formula to hold for arbitrary m and T but zero magnetic field.

Abstract. Correlation functions of the XXZ model in the massive and massless regimes are known to satisfy a system of linear equations. The main relations among them are the difference equations obtained from the qKZ equation by specializing the variables (λ 1 , · · · , λ 2n ) as (λ 1 , · · · , λ n , λ n + 1, · · · , λ 1 + 1). We call it the reduced qKZ equation. In this article we construct a special family of solutions to this system. They can be written as linear combinations of products of two transcendental functionsω, ω with coefficients being rational functions. We show that correlation functions of the XXZ model in the massive regime are given by these formulas with an appropriate choice ofω, ω. We also present a conjectural formula in the massless regime.