Diagonalization of theXXZ Hamiltonian by vertex operators

Abstract

We diagonalize the anti-ferroelectric XXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of U q sl(2) . Our method is based on the representation theory of quantum affine algebras, the related vertex operators and KZ equation, and thereby bypasses the usual process of starting from a finite lattice, taking the thermodynamic limit and filling the Dirac sea. From recent results on the algebraic structure of the corner transfer matrix of the model, we obtain the vacuum vector of the Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex operators, which act as particle creation operators in the space of eigenvectors.We check the agreement of our results with those obtained using the Bethe Ansatz in a number of cases, and with others obtained in the scaling limitthe su(2)-invariant Thirring model.

Exact two-spinon dynamical correlation function of the one-dimensional Heisenberg model

“…The main point of Refs. [12,13] is that the action of H XXZ on W is not well defined due to the appearance of divergences. However, this model is symmetric under the quantum group U q ( sl (2)), and therefore the eigenspace is identified with the following level 0 U q ( sl(2)) module:…”

Notes on Highest Weight Modules of the Elliptic Algebra $\mathfrak{A}_{q, p}(\hat{sl_2})$

“…This relation uniquely determines the action of the vertex operator Φ (l±1,l) ε (ζ): M k,l → M k,l±1 . The vertex operator in the trigonometric case [2] satisfies a quadratic commutation relation (see also [3] for the level-one case). The determination of the coefficients in these relations is based on an analysis of the qKZ equation, of which an elliptic counterpart is yet unknown.…”

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

“…Moreover, direct calculation shows that actions of the monodromy matrix elements on this basis become drastically simple (see below (3.22)-(3.26)). Here we list some of them relevant for us to construct eigenstates of the transfer matrix in the next section, 24) JHEP05(2016)119…”

“…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].…”