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arXiv:1912.01967 · Letters in Mathematical Physics

Free energy asymptotics of the quantum Heisenberg spin chain

Marcin Napiórkowski, Robert Seiringer

Published Mar 9, 2021• 3 citations
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Abstract

Abstract We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin $$S\ge 1/2$$ S ≥ 1 / 2 . We give upper and lower bounds on the free energy, proving that at low temperature it is asymptotically equal to the one of an ideal Bose gas of magnons, as predicted by the spin-wave approximation. The trial state used in the upper bound yields an analogous estimate also in the case of two spatial dimensions, which is believed to be sharp at low temperature.

DOI
10.1007/s11005-021-01375-4

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We prove the leading low-temperature free-energy asymptotics of the nearest-neighbour quantum Heisenberg ferromagnet on Z2\mathbb Z^2Z2. For every fixed S∈{12,1,32,…}S\in\{\frac12,1,\frac32,\ldots\}S∈{21​,1,23​,…}, the free energy per site satisfies lim⁡β→∞β2Sf2(β,S)=−π24. \lim_{β\to\infty}β^2S f_2(β,S)=-\fracπ{24}. limβ→∞​β2Sf2​(β,S)=−24π​. This formula was predicted by Takahashi's modified spin-wave theory [15]. Napiórkowski and Seiringer proved the corresponding upper bound but left the matching lower bound in two dimensions open [13], a gap later highlighted again by Seiringer [14]. We prove this missing bound through a new comparison between the magnon exclusion dynamics and free bosons, based on extending wave functions to collision configurations with controlled kinetic cost.
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Spin wave theory plays a central role in the analysis of low-energy properties of the quantum ferromagnetic Heisenberg model. In this work, we present a first step towards a rigorous justification of this approximation in the context of quantum dynamics in one dimension. According to spin wave theory, the evolution of a spin wave state should be well approximated by its free evolution with the appropriate energy of free quasiparticles called magnons. Our main result establishes this idea rigorously for a class of spin wave packets. We focus on spin-12\frac{1}{2}21​ systems, and consider wave packets composed of two magnons. We argue our result is optimal with respect to the timescales.
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Understanding the low-temperature properties of the Quantum Heisenberg Ferromagnet is central to a mathematical understanding of magnetism. The Quantum Heisenberg Ferromagnet can naturally be reformulated in terms of interacting bosons (called spin-waves or magnons) as an expansion in the inverse spin size. We construct a trial state which provides the first order upper bound to the free energy. We also discuss the temperature dependence and conclude that optimal temperature dependence requires cancellations in higher orders of the expansion.