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XXZ Chain Correlation Functions PDF

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XXZ Chain Correlation Functions
PDF Name XXZ Chain Correlation Functions PDF
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XXZ Chain Correlation Functions

Hey folks, here we are providing XXZ Chain Correlation Functions PDF to all of you. We have recently developed a method to compute the so-called elementary blocks of correlation functions for this model, which we refer to as Paper I in the following) in the framework of the (algebraic) Bethe ansatz for integrable boundary systems. The results essentially agree with previous expressions derived from the vertex operator approach.

The purpose of the present paper is to obtain the physical spin correlation functions for this model, in particular, the one-point functions for the local spin operators at a distance m from the boundary as well as several two-point functions (like boundary-bulk correlation functions). There are numerous physical interests in such quantities that can be measured in actual experiments.

XXZ Chain Correlation Functions PDF – Introduction

Beyond the Ising model, a large class of solvable lattice models has been discovered. In the context of quantum integrable systems on the lattice, spin chains are among the most studied examples with applications that range from condensed matter to high-energy physics.

Given a Hamiltonian, finding analytical expressions for the exact spectrum, identifying the structure of the space of the eigenvectors, and deriving explicit expressions for correlation functions are essential steps in the non-perturbative characterization of the system’s behaviour which can be compared with experimental data.

Among the simplest examples considered in the literature, the XXZ spin chain with different boundary conditions has received particular attention. Over the years, different approaches have been proposed in order to understand the Hamiltonian spectral problem and derive the correlation functions.

For models with periodic boundary conditions, the spectral problem can be handled by methods such as the Bethe ansatz (BA), or the corner transfer matrix method (CTM) in the thermodynamic limit. The computation of the correlation functions, however, is a much more difficult problem in general.

Apart from the simplest example – namely the XXZ spin chain with periodic boundary condition – for which correlation functions have been proposed by the quantum inverse scattering method (QISM), arising from the BA, the generalization of this result to models with higher symmetries requires a better understanding of mathematical structures, for instance, of determinant formulae of scalar products that involve the Bethe vectors (see some recent progress in).

However, in the thermodynamic limit, this problem can be alternatively tackled using the q-vertex operator approach (VOA) arising from the CTM. The space of states is identified with the irreducible highest weight representation of Uq(sl2ˆ) or higher rank quantum algebras. Correlation functions can be obtained using bosonizations of the q-vertex operators for Uq(sl2ˆ) or higher rank quantum algebras.

Either within the QISM or the VOA, correlation functions are obtained in the form of integrals of meromorphic functions in the thermodynamic limit. The situation for integrable spin chains with open boundaries is more difficult. On one hand, for the finite XXZ open chain with diagonal boundaries, related non-diagonal boundaries, or q a root of unity, the BA makes it possible to derive the spectrum and the eigenvectors.

In each case, the corresponding models are studied using Sklyanin’s general formulation of the BA applied to open boundary models. On the other hand, in the thermodynamic limit, the VOA has been applied to the half-infinite XXZ spin chain with a diagonal boundary. Although the hidden symmetry of this model was still unknown at that time,2 the diagonalization of the Hamiltonian could still be achieved.

Based on these results, the computation of correlation functions has been achieved for diagonal boundary conditions either using the BA or using the VOA in the thermodynamic limit. Note that generalizations to models with higher symmetries have been studied for Uq(slNˆ), etc. In the thermodynamic limit, when the comparison is feasible the expressions obtained by both approaches essentially coincide.

In spite of these important developments, the computation of correlation functions of the XXZ open spin chain for more general boundary conditions has remained, up to now, essentially problematic. On one hand, even in a simpler case such as the XXZ spin chain with triangular boundary conditions for which the construction of the Bethe vector is feasible, it remains an open problem in the QISM.

On the other hand, in the thermodynamic limit, the application of the VOA requires prior knowledge of the vacuum eigenvectors of the Hamiltonian. Since 1994, even for the simpler case of triangular boundary conditions, the solution to this problem has been unknown.

However, a breakthrough was recently made, which has opened the possibility of computing correlation functions in the thermodynamic limit of the XXZ open spin chain. Namely, based on the so-called Onsager’s approach the structure of the eigenvectors of the finite XXZ spin chain for any type of boundary conditions was interpreted within the representation theory of the q-Onsager algebra.

In the thermodynamic limit, the q-Onsager algebra is realized by quadratics of the q-vertex operators associated with Uq(sl2ˆ) (see also for an alternative derivation). As a consequence, vacuum eigenvectors of the Hamiltonian for a triangular boundary were constructed using the intertwining properties of the q-vertex operators with monomials of the q-Onsager basic generators. The latter being expressed in terms of Uq(sl2ˆ) generators, the VOA can be applied in a straightforward manner.

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