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arXiv:1602.00155

Low-Temperature Spin-Wave Approximation for the Heisenberg Ferromagnet

Michele Correggi, Alessandro Giuliani, Robert Seiringer

Published Jan 30, 2016• 0 citations
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Abstract

We study the low temperature thermodynamics of the quantum Heisenberg ferromagnet in three dimension for any value of the spin $ S \geq 1/2 $. We report on a rigorous proof of the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy as the inverse temperature $ β\to \infty $.

DOI
10.48550/arxiv.1602.00155

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We study the low temperature thermodynamics of the quantum Heisenberg ferromagnet in three dimension for any value of the spin S≥1/2S \geq 1/2S≥1/2. We report on a rigorous proof of the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy as the inverse temperature β→∞β\to \inftyβ→∞.
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.
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.
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
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.
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.
Using Bethe ansatz integral equations, we calculate the free energy and susceptibility of spin-12\frac{1}{2}21​ one-dimensional Heisenberg ferromagnet \(\mathcal{H}{=}J\Sigma(\mbi{S}_{\text{i}}\mbi{S}_{\text{i}+1}-\frac{1}{4})\), ( J <0) at T ≥0.004| J |. We find that the free energy and susceptibility are expanded by T/∣J∣\sqrt{T/|J|}T/∣J∣​ at low temperature: \(f&=|J|\left\{-1.042\left(\frac{T}{|J|}\right)^{3/2}+1.00\left(\frac{T}{|J|}\right)^{2}-0.9\left(\frac{T}{|J|}\right)^{5/2}+O\left(\frac{T}{|J|}\right)^{3}\right\}\\ \chi&=|J|^{-1}\left\{0.1667\left(\frac{|J|}{T}\right)^{2}+0.581\left(\frac{|J|}{T}\right)^{3/2}+0.68\left(\frac{|J|}{T}\right)^{1}+O\left(\frac{|J|}{T}\right)^{1/2}\right\}.\) The first term of free energy coincides with the spin wave calculation. The first term of susceptibility coincides with Fisher's solution of classical Heisenberg Ferromagnet. We conclude α=-0.5 and γ=2.