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arXiv:0707.2803 · Journal of Physics A Mathematical and Theoretical

Deformed strings in the Heisenberg model

Rob Hagemans, Jean-Sébastien Caux

Published Nov 21, 2007• 58 citations
arXivHTMLPDF

Abstract

We investigate solutions to the Bethe equations for the isotropic S = 1/2 Heisenberg chain involving complex, string-like rapidity configurations of arbitrary length. Going beyond the traditional string hypothesis of undeformed strings, we describe a general procedure to construct eigenstates including strings with generic deformations, discuss general features of these solutions and provide a number of explicit examples including complete solutions for all wavefunctions of short chains. We finally investigate some singular cases and show from simple symmetry arguments that their contribution to zero-temperature correlation functions vanishes.

DOI
10.1088/1751-8113/40/49/001

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We investigate solutions to the Bethe equations for the isotropic S = 1/2 Heisenberg chain involving complex, string-like rapidity configurations of arbitrary length. Going beyond the traditional string hypothesis of undeformed strings, we describe a general procedure to construct eigenstates including strings with generic deformations, discuss general features of these solutions and provide a number of explicit examples including complete solutions for all wavefunctions of short chains. We finally investigate some singular cases and show from simple symmetry arguments that their contribution to zero-temperature correlation functions vanishes.
We observe a different type of complex solutions in the isotropic spin-1/2 Heisenberg chain starting from N = 12, where the central rapidity of some of the odd-length strings becomes complex so that not all the strings self-conjugate individually. We show that there are at most ( N − 2)/2 singular solutions for M = 4, M = 5 down-spins and at most ( N 2 − 6 N + 8)/8 singular solutions for M = 6, M = 7 down-spins in an even-length chain with N ⩾ 2 M . Correspondence of the non self-conjugate string solutions and the singular string solutions to the rigged configurations has also been shown.
We examine the question of whether Bethe's ansatz reproduces all states in the periodic Heisenberg XXZ and XXX spin chains. As was known to Bethe himself, there are states for which the Bethe momenta knk_nkn​ diverge: these are in fact the simplest examples of ``string'' solutions. The coefficients of the Bethe wavefunction, too, diverge. When there are only two down spins in the system (the case considered by Bethe), we can renormalize these coefficients to get a sensible (and correct) wavefunction. We show that this is not always possible when there are more than two down spins. The Bethe equations have several such divergent solutions, and some of these correspond to genuine eigenfunctions of the Hamiltonian, but several do not. Nor do they reproduce the correct energy eigenvalues. Moreover, we point out that the algebraic Bethe ansatz, an alternative way to construct the wavefunctions proposed by Faddeev, Takhtajan et al., leads to vanishing wavefunctions for all these solutions. Thus, the Bethe ansatz solution of the Heisenberg model must be regarded as either incomplete, or inaccurate.
Abstract Calculations of low lying excited states in even length L → ∞ chains begun in 2019 J. Phys. Commun. 3 025007 in the total spin S = 0 sector are here extended to S > 0. Bethe string configurations that were observed to transition from their definition at small L to different ‘apparent’ configurations at large L are now understood to be a manifestation of the approximate chiral symmetry that develops with increasing chain length. It is conjectured that when chiral symmetry is built into the configuration definitions a complete explicit labelling of all low lying excited states by strings is obtained. The structure is of the form of a product of Bethe strings defining left and right chiral states of spin sL and sR and a central complex ‘string’ necessary to bind <?CDATA s⃗L{\vec{{\rm{s}}}}_{L}sL​?> s ⃗ L and <?CDATA s⃗R{\vec{{\rm{s}}}}_{R}sR​?> s ⃗ R into total <?CDATA S⃗.\vec{S}.S.?> S ⃗ . Rules based on the string content of a state yield its momentum and asymptotic energy, including the amplitude of the leading ln(L) correction.
We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum spin chain with periodic boundary conditions. We formulate a conjecture for the number of solutions with pairwise distinct roots of these equations, in terms of numbers of so-called singular (or exceptional) solutions. Using homotopy continuation methods, we find all such solutions of the Bethe equations for chains of length up to 14. The numbers of these solutions are in perfect agreement with the conjecture. We also discuss an indirect method of finding solutions of the Bethe equations by solving the Baxter T-Q equation. We briefly comment on implications for thermodynamical computations based on the string hypothesis.
Using the algebraic Bethe ansatz, we consider the correlation functions of the integrable higher spin chains. We apply a method recently developed for the spin ½ Heisenberg chain, based on the solution of the quantum inverse problem. We construct a representation for the correlation functions on a finite chain for arbitrary spin. Then we show how the string solutions of the Bethe equations can be considered in the framework of this approach in the thermodynamic limit. Finally, a multiple integral representation for the spin 1 zero-temperature correlation functions is obtained in the thermodynamic limit.
Abstract This chapter presents the extension of the Bethe ansatz to finite temperature, the thermodynamic Bethe ansatz, for the antiferromagnetic isotropic Heisenberg quantum spin chain, the XXX quantum spin chain. It discusses how the added complications of this model arise from the more complicated structure of excitations of the quantum spin chain, the complex string excitations, which have to be included in the Bethe ansatz thermodynamics. It derives the integral equations of the thermodynamic Bethe ansatz for the XXX quantum spin chain and mentions explicit formulas for the free energy of the quantum spin chain and some interesting physical quantities, especially making contact with predictions of conformal symmetry.