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arXiv:math-ph/9903004 · Journal of Physics A Mathematical and General

Mathematical structure of magnons in quantum ferromagnets

Tom Michoel, André F. Verbeure

Published Jul 29, 1999• 4 citations
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Abstract

We provide the mathematical structure and a simple, transparent and rigorous derivation of the magnons as elementary quasi-particle excitations at low temperatures and in the infinite spin limit for a large class of Heisenberg ferromagnets. The magnon canonical variables are obtained as fluctuation operators in the infinite spin limit. Their quantum character is governed by the size of the magnetization.

DOI
10.1088/0305-4470/32/32/303

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We provide the mathematical structure and a simple, transparent and rigorous derivation of the magnons as elementary quasi-particle excitations at low temperatures and in the infinite spin limit for a large class of Heisenberg ferromagnets. The magnon canonical variables are obtained as fluctuation operators in the infinite spin limit. Their quantum character is governed by the size of the magnetization.
We use the spin functional renormalization group recently developed by two of us [Krieg and Kopietz, Phys. Rev. B 99, 060403(R) (2019)] to calculate the magnetization M(H,T)M(H,T)M(H,T) and the damping of magnons due to classical longitudinal fluctuations of quantum Heisenberg ferromagnets. In order to guarantee that for vanishing magnetic field $H\ensuremath{\rightarrow}0$, the magnon spectrum is gapless when the spin rotational invariance is spontaneously broken, we use a Ward identity to express the magnon self-energy in terms of the magnetization. In two dimensions, our approach correctly predicts the absence of long-range magnetic order for H=0H=0H=0 at finite temperature TTT. The magnon spectrum then exhibits a gap from which we obtain the transverse correlation length. We also calculate the wave-function renormalization factor of the magnons. As a mathematical byproduct, we derive a recursive form of the generalized Wick theorem for spin operators in frequency space which facilitates the calculation of arbitrary time-ordered connected correlation functions of an isolated spin in a magnetic field.
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 a functional renormalization-group approach, we show that at low temperatures and in a certain range of magnetic fields, the longitudinal dynamic structure factor of quantum Heisenberg ferromagnets in dimensions $D\ensuremath{\le}2$ exhibits a well-defined quasiparticle peak with linear dispersion that we identify as zero-magnon sound. In D>2D>2D>2, the larger phase space available for the decay into transverse spin waves leads only to a broad hump centered at zero frequency whose width scales linearly in momentum.
We compute the zero temperature dynamical structure factor S(q,ω)S({\bf q},\omega)S(q,ω) of the triangular lattice Heisenberg model using a Schwinger Boson approach that includes the Gaussian fluctuations (1/N1/N1/N correction) of the saddle point solution. While the ground state of this model exhibits a well-known 120∘^{\circ}∘ magnetic ordering, the excitation spectrum revealed by S(q,ω)S({\bf q},\omega)S(q,ω) has a strong quantum character that is not captured by low-order 1/S1/S1/S expansions. The low-energy magnons consist of two spinon bound states confined by gauge fluctuations of the auxiliary fields, which can produce internal structure of the magnon peaks. In addition, the high-energy continuum of spinon modes extends up to about three times the single-magnon bandwidth.
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.