It is shown that the description of the states of infinite S = 1/2 interacting spin systems with the Hamiltonian H s J Z a2 sinh-Za(J / ) (~ 1) 2 j# t , j , l~Z 2 can be performed by studying the hyperbolic Calogero-Sutherland eigenvalue problem. The construction of multimagnon wave functions in each N-magnon sector is based on solutions of the set of linear algebraic equations which determine also the structure of zonal spherical functions on symmetric spaces X ~ = SL(N, H)/Sp(N) of negative curvature. The usual Bethe Ansatz for the XXX Heisenberg model corresponds to asymptotic forms of these wave functions at small values of a 1 or large distances between spins turned over the ferromagnetic ground state.
We review some recent advances in the computation of exact correlation functions of the XXZ-½ Heisenberg chain. We first give a general introduction to our method which is based on the algebraic Bethe ansatz and the resolution of the quantum inverse scattering problem, leading in particular to multiple integral representations for the correlation functions. Then we describe recently obtained compact formulas for the spin-spin correlation functions of the XXZ-½ Heisenberg chain. We outline how this leads to several explicit results including the known two point functions in the limit of free fermions, the so-called emptiness formation probability at anisotropy Δ=1/2 and its large distance asymptotic behaviour in the massless phase of the model.
We prove that for the spin-1/2 Heisenberg XXZ chain in the range
Δ∈[−1,1/2], the ground state on the torus of length
L converges to an infinite volume ground state
⟨⋅⟩ as
L→∞, and that the spin-spin correlation
⟨S0(1)Sx(1)⟩ decays polynomially fast in
x. In the range
Δ∈[−1,0] we have the stronger results: that the convergence holds for several finite-volume, finite temperature states, that
L and
β=1/T can be taken to infinity in any order, that the convergence holds on (Euclidean) dynamic correlators, and that
⟨S0(1)Sx(1)(t)⟩ decays to zero in
∣(x,t)∣, and cannot decay exponentially fast. The convergence of the ground state in infinite volume is known rigorously by Bethe Ansatz methods, while our correlation decay results (apart from the points
Δ=0,−1), the convergence on dynamic correlators and the interchangeability of limits are new at the rigorous level. Moreover our methods are new and do not use any Bethe Ansatz techniques, using only the following related probabilistic models. In the Lorentz mirror model with loop weight 2, a model of random loops on the square lattice, in a large range of parameters we prove that connection probabilities tend to 0, and do so polynomially fast in the symmetric case of the model. Our proof uses two couplings of this mirror model with the six-vertex model, one of which is new. The main input is the delocalisation of the height function of the six-vertex model proved by several authors. We further prove that the height function of a certain space-time version of the six-vertex model delocalises and then prove through analogous couplings to those mentioned above that connection probabilities converge to 0 in the loop representation of the XXZ model introduced by Ueltschi.