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2 The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin

Michele Correggi, Alessandro Giuliani

Published Jan 1, 2012• 6 citations
OpenAlex

Abstract

We consider the spin-S ferromagnetic Heisenberg model in three dimensions, in the absence of an external field. Spin wave theory suggests that in a suitable temperature regime the system behaves effectively as a system of non-interacting bosons (magnons). We prove this fact at the level of the specific free energy: if S → ∞ and the inverse temperature β → 0 in such a way that βS stays constant, we rigorously show that the free energy per unit volume converges to the one suggested by spin wave theory. The proof is based on the localization of the system in small boxes and on upper and lower bounds on the local free energy, and it also provides explicit error bounds on the remainder. 1

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Validity of spin-wave theory for the quantum Heisenberg model

Michele Correggi, Alessandro Giuliani, Robert Seiringer

Spin-wave theory is a key ingredient in our comprehension of quantum spin systems, and is used successfully for understanding a wide range of magnetic phenomena, including magnon condensation and stability of patterns in dipolar systems. Nevertheless, several decades of research failed to establish the validity of spin-wave theory rigorously, even for the simplest models of quantum spins. A rigorous justification of the method for the three-dimensional quantum Heisenberg ferromagnet at low temperatures is presented here. We derive sharp bounds on its free energy by combining a bosonic formulation of the model introduced by Holstein and Primakoff with probabilistic estimates and operator inequalities.

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We consider the spin-S ferromagnetic Heisenberg model in three dimensions, in the absence of an external field. Spin wave theory suggests that in a suitable temperature regime the system behaves effectively as a system of non-interacting bosons (magnons). We prove this fact at the level of the specific free energy: if S → ∞ and the inverse temperature β → 0 in such a way that βS stays constant, we rigorously show that the free energy per unit volume converges to the one suggested by spin wave theory. The proof is based on the localization of the system in small boxes and on upper and lower bounds on the local free energy, and it also provides explicit error bounds on the remainder. 1
Abstract. We consider the quantum ferromagnetic Heisenberg model in three di-mensions, for all spins S ≥ 1/2. We rigorously prove the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial con-tribution to the free energy at low temperatures. Our proof comes with explicit, constructive upper and lower bounds on the error term. It uses in an essential way the bosonic formulation of the model in terms of the Holstein–Primakoff representa-tion. In this language, the model describes interacting bosons with a hard-core on-site repulsion and a nearest-neighbor attraction. This attractive interaction makes the lower bound on the free energy particularly tricky: the key idea there is to prove a differential inequality for the two-particle density, which is thereby shown to be smaller than the probability density of a suitably weighted two-particle random process on the lattice. 1.
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.
Abstract We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin S≥1/2S\ge 1/2S≥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.
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.
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.
Spin-wave theory is a key ingredient in our comprehension of quantum spin systems, and is used successfully for understanding a wide range of magnetic phenomena, including magnon condensation and stability of patterns in dipolar systems. Nevertheless, several decades of research failed to establish the validity of spin-wave theory rigorously, even for the simplest models of quantum spins. A rigorous justification of the method for the three-dimensional quantum Heisenberg ferromagnet at low temperatures is presented here. We derive sharp bounds on its free energy by combining a bosonic formulation of the model introduced by Holstein and Primakoff with probabilistic estimates and operator inequalities.