Exact regimes of collapsed and extra two-string solutions in the two down-spin sector of the spin-1/2 massive XXZ spin chain

Abstract

We derive exactly the number of complex solutions with two down-spins in the massive regime of the periodic spin-1/2 XXZ spin chain of $N$ sites. Here we remark that every solution of the Bethe ansatz equations is characterized by a set of quantum numbers. We derive them analytically for all the complex solutions in the sector, which we call two-string solutions. We show that in a region of $N$ and $\Delta$ the number of two-string solutions is by two larger than the number due to the string hypothesis, i.e., an extra pair of two-strings appears. We determine it exactly and also such regions where $m$ two-string solutions collapse for any positive integers $m$. We illustrate the extra and standard two-string solutions numerically. In the sector we show that the string deviations are exponentially small with respect to $N$ if $N$ is large. We argue that for any finite solution of the spin-1/2 XXX chain there is such a solution of the spin-1/2 XXZ chain that has the same quantum numbers in common with the XXX solution.

arXiv comment38 pages, 9 figures

References in corpus (19)

• Finite temperature Drude weight of the one dimensional spin 1/2 Heisenberg model}
• Stringing Spins and Spinning Strings
• Bethe's equation is incomplete for the XXZ model at roots of unity
• On the completeness of solutions of Bethe's equations
• Deformed strings in the Heisenberg model
• Algebraic Bethe ansatz for singular solutions
• Violation of the string hypothesis and Heisenberg XXZ spin chain
• Singular solutions, repeated roots and completeness for higher-spin chains
• Large N behavior of string solutions in the Heisenberg model
• Violations of the String Hypothesis in the Solutions of the Bethe Ansatz Equations in the XXX-Heisenberg
• Singular Solutions to the Bethe Ansatz Equations and Rigged Configurations
• Thermal form factor approach to the ground-state correlation functions of the XXZ chain in the antiferromagnetic massive regime
• Low-temperature spectrum of correlation lengths of the XXZ chain in the antiferromagnetic massive regime
• Non Self-conjugate Strings, Singular Strings and Rigged Configurations in the Heisenberg Model
• Non-regular eigenstate of the XXX model as some limit of the Bethe state
• Exact quantum numbers of collapsed and non-collapsed 2-string solutions in the Heisenberg spin chain
• Singular eigenstates in the even(odd) length Heisenberg spin chain
• Heisenberg model and Rigged Configurations
• Finite temperature transport in integrable quantum many body systems