The Quantum Self-Consistent Harmonic Approximation: A Unified Framework for Quantum Spin System
G. C. Villela, A. R. Moura
TL;DR
The paper develops the Quantum Self-Consistent Harmonic Approximation (QSCHA) to address quantum fluctuations in spin systems beyond semiclassical approximations. By employing spin coherent states and a variational Gibbs-Bogoliubov framework, it introduces a renormalization parameter $\rho$ that self-consistently captures quantum corrections, yielding a quantum-specific self-consistent equation $\rho(T)=\Lambda(T)(1-\langle (S^z)^2\rangle_0/\tilde{S}^2) e^{-Ξ}$. It provides explicit expressions for the average energy $\langle H\rangle_0$, the angle fluctuation term $Ξ$, and the quantum correction factor $\Lambda(T)$, recovering classical SCHA in the appropriate limit while improving accuracy for $S=1/2$. The approach yields thermodynamic quantities from a quadratic Hamiltonian with a rapidly convergent self-consistent procedure and shows enhanced agreement with Monte Carlo data and experiments, highlighting its potential for quantum magnetism and spintronic applications.
Abstract
The Self-Consistent Harmonic Approximation (SCHA) has been utilized to investigate quantum and thermal phase transitions within magnetic models and, more recently, in spintronic applications. The SCHA methodology involves utilizing simple harmonic Hamiltonians, which are augmented with renormalization parameters that incorporate high-order fluctuations typically overlooked by conventional Linear Spin-Wave (LSW) theories. Although this approach exhibits reasonable accuracy for models defined by large spin values, its reliability diminishes when applied to quantum systems with $S=1/2$. The traditional development of SCHA has incorporated semiclassical assumptions that obscure quantum effects. In this study, we introduce a quantum framework for the SCHA that eliminates the need for semiclassical approximations. Our Quantum Self-Consistent Harmonic Approximation (QSCHA) utilizes the spin coherent states formalism within a fully quantum formulation. Consequently, we derive a novel renormalization parameter that accurately integrates quantum corrections. To assess the efficacy of this new approach, we apply the QSCHA to analyze the critical temperature transitions across various well-documented magnetic models. The findings, combined with the simplified operational procedure relative to other conventional interacting spin-wave methodologies, suggest that QSCHA is a promising tool for advancing research in quantum magnetism and spintronics.