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Physics of Atomic Nuclei

String limit of the isotropic Heisenberg chain in the four-particle sector

A. G. Antipov, Igor V. Komarov

Published May 1, 2008• 0 citations
Publisher

Abstract

The quantum method of variable separation is applied to the spectral problem of the isotropic Heisenberg model. The Baxter difference equation is resolved by means of a special quasiclassical asymptotic expansion. States are identified by multiplicities of limiting values of the Bethe parameters. The string limit of the four-particle sector is investigated. String solutions are singled out and classified. It is shown that only a minor fraction of solutions demonstrate string behavior.

DOI
10.1134/s1063778808050037

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The quantum method of variable separation is applied to the spectral problem of the isotropic Heisenberg model. The Baxter difference equation is resolved by means of a special quasiclassical asymptotic expansion. States are identified by multiplicities of limiting values of the Bethe parameters. The string limit of the four-particle sector is investigated. String solutions are singled out and classified. It is shown that only a minor fraction of solutions demonstrate string behavior.
The method of separation of variables (SoV) is employed for the spectral problem of the XXZ model. The Baxter difference equation is resolved by means of a special isotropic asymptotic expansion. States are identified by multiplicities of limiting values of the Bethe parameters. As an application, the statistical properties of integral spectra are investigated. It is shown that the power function gives the more correct description of nearest-neighbour spacing distribution density at intermediate spacings as compared with the exponential.
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.
Evidence is presented that the solutions of the Bethe ansatz equations for spin-½ isotropic Heisenberg chains in fixed total spin and momentum sectors are the roots of single variable polynomials with integer (or integer based) coefficients. Such solutions are used as a starting point for investigation of long chain (critical region) properties. In the total spin S = 0 sector I conjecture explicit formulae for the Bethe string configuration labelling of all left and right tower excitations in the k = 1, SU(2) Wess-Zumino-Witten model.
It is found that the Hamiltonian of the S= 1/2 isotropic Heisenberg chain with N sites and elliptic non-nearest-neighbour exchange is diagonalized in each sector of the Hilbert space with magnetization N/2-M, 1<M<or=(N/2), by means of double quasiperiodic meromorphic solutions to the M-particle quantum Calogero-Moser problem on a line. The spectrum and highest-weight states are determined by the solutions of the systems of transcendental Bethe-ansatz-type equations which arise as restrictions on particle pseudomomenta.
By circumventing the difficulty of obtaining exact string state solutions to Bethe ansatz equations, we devise a truncated string state space approach for investigating spin dynamics in a nonintegrable spin-12\frac{1}{2}21​ Heisenberg chain subjected to a staggered field at various magnetizations. The obtained dynamical spectra reveal a series of elastic peaks at integer multiples of the ordering wave vector QQQ, indicating the presence of multi-QQQ Bethe string states within the ground state. The spectrum exhibits a separation between different string continua as the strength of the staggered field increases at low magnetization, reflecting the confinement of the Bethe strings. This approach provides a unified string-state-based framework for understanding spin dynamics in low-dimensional nonintegrable Heisenberg models, which has a successful application to observations across various phases of the quasi-one-dimensional antiferromagnet YbAlO3{\mathrm{YbAlO}}_{3}YbAlO3​.