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arXiv:1501.00532 · Journal of Physics A Mathematical and Theoretical

Rigged configurations approach for the spin-1/2 isotropic Heisenberg model

Reiho Sakamoto

Published Apr 2, 2015• 1 citation
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Abstract

We continue the rigged configurations analysis of the solutions to the Bethe ansatz equations for the spin-1/2 isotropic Heisenberg model. We analyze the non self-conjugate strings of Deguchi–Giri and the counter example for the string hypothesis discovered by Essler–Korepin–Schoutens. In both cases clear discrete structures appear.

DOI
10.1088/1751-8113/48/16/165201

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We continue the rigged configurations analysis of the solutions to the Bethe ansatz equations for the spin-1/2 isotropic Heisenberg model. We analyze the non self-conjugate strings of Deguchi–Giri and the counter example for the string hypothesis discovered by Essler–Korepin–Schoutens. In both cases clear discrete structures appear.
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 provide a conjecture for the following two quantities related to the spin- isotropic Heisenberg model defined over rings of even lengths: (i) the number of solutions to the Bethe ansatz equations which correspond to non-zero Bethe vectors; (ii) the number of physical singular solutions of the Bethe ansatz equations in the sense of Nepomechie and Wang 2013 J. Phys. A: Math. Theor. 46 325002. The conjecture is based on a natural relationship between the solutions to the Bethe ansatz equations and the rigged configurations.
The main purpose of this report is a thorough analysis of completeness of solutions of the one-dimensional Heisenberg Hamiltonian through the hypothesis of strings. A somehow astonishing conclusion emerges from studying of the structure of the classical configuration space of this system. Namely, all allowed information concerning quantum states, which are exact solutions of the Bethe equations, encoded in quantum numbers, are predictable via a bijection between the set of the magnetic configurations and the string configurations. This startling and beautiful observation constitutes the proof of the completeness of the eigenstates of the Heisenberg Hamiltonian, deduced in a purely combinatorial way. We interpret the set of all magnetic configurations with a fixed number r of spin deviations as the classical configuration space of a hypothetic system of r Bethe pseudoparticles, which move, in a stroboscopic manner, on the magnetic ring. The geometry of this configuration space, induced by the action of Heisenberg Hamiltonian and the translation symmetry group of the ring, implies the structure of a locally r -dimensional hypercubic lattice with well defined F -dimensional boundaries, 1 ⩽ F ⩽ r . We demonstrate that rigged string configurations originate from these boundaries, depending upon the island structure of spin deviations. We show that a relatively simple combinatoric definition of rigged strings reproduces completely exact results of Bethe Ansatz. It is expressed in terms of a combined bijection: Robinson-Schensted with Kerov- Kirillov-Reshetikhin (RSKKR) which produces a geography of exact Bethe Ansatz solutions on the classical configuration space.
The Bethe ansatz provides an exact solution framework for integrable quantum spin chains, yet the detailed structure of its solutions remains subtle. In this work, we solve the Bethe ansatz with SU(N) global symmetry numerically from two perspectives. First, we systematically investigate approximate solutions known as strings, derived from the so-called string hypothesis. These solutions can be labeled by rigged configurations. Second, we compute physically admissible Bethe roots using the Wronskian Bethe equations, whose solutions admit a natural labeling by standard Young tableaux (SYT). These two labeling schemes, SYT and rigged configurations, arise independently in combinatorics. A key combinatorial result, the Kerov-Kirillov-Reshetikhin (KKR) bijection, establishes a one-to-one correspondence between them. While this bijection is well understood in a purely combinatorial setting, its role within the physics of integrable systems has not been systematically examined. To investigate the KKR bijection within integrable spin chains, we perform numerical computations of Bethe roots for su(N) chains up to length L=20. Remarkably, our numerical results show that the SYT labeling of solutions obtained from the Wronskian Bethe equations closely matches the rigged configuration labeling prescribed by the KKR bijection when compared using a nearest neighbor distance measure. Since the two labeling schemes were developed independently and, in particular, the SYT labeling does not rely on the string hypothesis, this correspondence reveals a deep structural relation between the combinatorial and analytic descriptions of integrable systems.