Correlation functions of the half-infinite XXZ spin chain with a triangular boundary

Abstract

The half-infinite XXZ spin chain with a triangular boundary is considered in the massive regime.Two integral representations of correlation functions are proposed using bosonization. Sufficient conditions such that the expressions for triangular boundary conditions coincide with those for diagonal boundary conditions are identified. As an application, summation formulae of the boundary expectation values σ a 1 with a = z, ± are obtained. Exploiting the spin-reversal property, relations between n-fold integrals of elliptic theta functions are extracted.Beyond the Ising model, a large class of solvable lattice models have been discovered. In the context of quantum integrable systems on the lattice, spin chains are among the most studied examples with applications which 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 behavior 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's 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) [1], or the corner transfer matrix method (CTM) in the thermodynamic limit [2]. 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) [4,5] 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 [7] (see some recent progress in [8]). However, in the thermodynamic limit, this problem can be alternatively tackled using the q-vertex operator approach (VOA) [3] arising from the CTM. The space of states is identified with the irreducible highest weight representation of U q ( sl 2 ) or higher rank quantum algebras. Correlation functions can be obtained using bosonizations of the q-vertex operators for U q ( sl 2 ) or higher rank quantum algebras [3,9,10,11,12,13]. 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 [14], related non-diagonal boundaries [15, 16] or q a root of unity [17], the BA makes it possible to deriv...

A bispectral q-hypergeometric basis for a class of quantum integrable models

“…For instance, in the literature, the two-dimensional Ising and superintegrable Potts models have been studied in details for q = 1 using the explicit relation between the Onsager algebra and a fixed-point subalgebra of sl 2 (under the action of a certain automorphism of sl 2 [47]). For q = 1, the explicit relation between the q−Onsager algebra and a certain coideal subalgebra of U q ( sl 2 ) has been used to analyze the finite open XXZ spin chain [59] or its thermodynamic limit analog for various types of boundary conditions [58,85]. According to the size of the system -finite or semi-infinite -, finite or infinite dimensional representations (q−vertex operators) of the q−Onsager algebra have been considered.…”

“…It appears that an explicit characterization of the Hamiltonian's eigenfunctions as polynomials offers the possibility of studying the correspondence between Bethe roots and solutions of algebraic equations [101]. On the other hand, in the thermodynamic limit, either from the algebraic Bethe ansatz setting [103] or the q−vertex operator approach [104,85], correlation functions and form factors are always given in terms of multiple integral representations. Although formally satisfying, these are difficult objects to handle for practical applications.…”

Non-Abelian symmetries of the half-infinite XXZ spin chain

“…Based on the representation theory of the U q ( sl 2 ) quantum affine algebra at level one and its current algebra, an explicit characterization of the Hamiltonian's spectrum, corresponding eigenstates, as well as multiple integral representations of correlation functions and form factors of local operators has been given [DFJMN92]. This approach has been later on extended to lattice systems with periodic boundary conditions associated with higher spins [IIJMNT92] or higher rank affine Lie algebras [Koy93], as well as for certain class of boundary conditions, see for instance [JKKKMW95,BB13,BKo13,BKo14,Koj10].…”

“…For instance, in the thermodynamic limit N → ∞ of the finite open XXZ spin chain it was expected that non-Abelian infinite dimensional symmetries emerge, that are associated with certain coideal subalgebras of U q ( sl 2 ). Recall that the Hamiltonian of the half-infinite XXZ spin chain is formally defined as (see also [JKKKMW95,BB13,BKo13,BKo14]):…”

A representation basis for the quantum integrable spin chain associated with the su(3) algebra

“…Moreover, it has also provided valuable insight into important universality class in condensed matter physics [3] and cold atom systems [4]. In the past several decades, the integrable quantum spin chains with U(1)-symmetry (with periodic boundary or with diagonal open boundaries [5]) and with some constrained open boundaries [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] have been extensively studied by various Bethe ansatz methods for a finite lattice and by the vertex operator method [22] in an infinite or a half-infinite lattice [23][24][25][26][27][28].…”