Non Self-conjugate Strings, Singular Strings and Rigged Configurations in the Heisenberg Model

TL;DR

This paper investigates complex solutions in the isotropic spin-1/2 Heisenberg chain, revealing non self-conjugate strings and their correspondence to rigged configurations, with specific bounds on singular solutions for various spin states.

Contribution

It introduces and analyzes non self-conjugate string solutions and establishes their correspondence to rigged configurations in the Heisenberg model.

Findings

Identification of non self-conjugate string solutions starting from N=12.

Derivation of bounds on the number of singular solutions for different spin states.

Establishment of the correspondence between non self-conjugate strings, singular solutions, and rigged configurations.

Abstract

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 making 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^2-6N+8)/8 singular solutions for M=6, M=7 down-spins in an even-length chain with N \geq 2M. Correspondence of the non self-conjugate string solutions and the singular string solutions to the rigged configurations has also been shown.