arXiv:hep-th/0006081 · Journal of Physics A Mathematical and General

Fusion and analytical Bethe ansatz for theA𝒩-1(1)open spin chain

Anastasia Doikou

Published Jun 22, 2000• 19 citations

Abstract

We generalize the fusion procedure for the A 𝒩-1 (1) open spin chain (𝒩>2) and we show that the transfer matrix satisfies a crossing property. We use these results to solve the A 𝒩-1 (1) open spin chain with U q ( SU (𝒩)) symmetry by means of the analytical Bethe ansatz method. Our results coincide with the known ones obtained via the nested Bethe ansatz.

DOI: 10.1088/0305-4470/33/26/303

Related papers

Analytical Bethe Ansatz for quantum-algebra-invariant spin chains

Luca Mezincescu, Rafael I. Nepomechie

We have recently constructed a large class of open quantum spin chains which have quantum-algebra symmetry and which are integrable. We show here that these models can be exactly solved using a generalization of the analytical Bethe Ansatz (BA) method. In particular, we determine in this way the spectrum of the transfer matrices of the Uq[su(2)]-invariant spin chains associated with A1(1) and A2(2) in the fundamental representation. The quantum-algebra invariance of these models plays an essential role in obtaining these results. The BA equations for these open chains are “doubled” with respect to the BA equations for the corresponding closed chains.

arXiv:hep-th/9110050Nuclear Physics BMar 1, 1992113 citations

THERMODYNAMIC BETHE ANSATZ EQUATION FROM FUSION HIERARCHY OF osp (1|2) INTEGRABLE SPIN CHAIN

KAZUMITSU SAKAI, ZENGO TSUBOI

The thermodynamic Bethe ansatz (TBA) and the excited state TBA equations for an integrable spin chain related to the Lie superalgebra osp (1|2) are proposed by the quantum transfer matrix (QTM) method. We introduce the fusion hierarchy of the QTM and derive the functional relations among them (T-system) and their certain combinations (Y-system). Their analytical property leads to the nonlinear integral equations which describe the free energy and the correlation length at any finite temperatures. With regard to the free energy, they coincide with the TBA equation based on the string hypothesis.

arXiv:math-ph/9912014International Journal of Modern Physics AJun 20, 200011 citations

Fusion procedure for open chains

Luca Mezincescu, Rafael I. Nepomechie

The authors have generalized Sklyanin's approach of constructing open integrable quantum spin chains to the case of PT-invariant R matrices. They formulate a fusion procedure for such chains. In particular, they show that the fused transfer matrix can be expressed in terms of products of the original transfer matrix and products of certain quantum determinants which can be explicitly evaluated. Applications of these results include constructing open integrable higher-spin chains, as well as obtaining functional equations for transfer-matrix eigenvalues, which may be solved by an analytical Bethe ansatz.

Journal of Physics A Mathematical and GeneralMay 7, 1992126 citations

The nested Bethe ansatz for ‘all’ open spin chains with diagonal boundary conditions

S. Belliard, E. Ragoucy

We present in a unified and detailed way the nested Bethe ansatz for open spin chains based on(super)algebras, with arbitrary representations (i.e., 'spins') on each site of the chain and diagonal boundary matrices (K + (u), K -(u)).The nested Bethe ansatz applies for a general K -(u), but a particular form of the K + (u) matrix.The construction extends and unifies the results already obtained for open spin chains based on the fundamental representation and for some particular superspin chains.We give the eigenvalues, Bethe equations and the explicit form of the Bethe vectors for the corresponding models.The Bethe vectors are also expressed using a trace formula.

arXiv:0902.0321Journal of Physics A Mathematical and TheoreticalApr 29, 200928 citations

The Bethe ansatz

Fedor Levkovich-Maslyuk

Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.

arXiv:1606.02950Journal of Physics A Mathematical and TheoreticalJul 20, 201670 citations

Fusion procedure for the Belavin model with open boundary conditions

Heng Fan, Kangjie Shi, Bo-Yu Hou

We have recently used unitarity and cross-unitarity properties of an symmetric R-matrix to construct the transfer matrix t(u) for an N-site open spin chain. Here we give a fusion procedure for such a chain, and we prove that the fused transfer matrix is commutative with the original transfer matrix t(u).

Journal of Physics A Mathematical and GeneralSep 21, 19960 citations