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arXiv:2303.03135 · Physical review. B./Physical review. B

Finite-temperature phase transitions in S=12 three-dimensional Heisenberg magnets from high-temperature series expansions

M. G. Gonzalez, B. Bernu, Laurent Pierre, Laura Messio

Published Jun 29, 2023• 7 citations
arXivHTMLPDF

Abstract

Many frustrated spin models on three-dimensional (3D) lattices are currently being investigated, both experimentally and theoretically, and develop new types of long-range orders in their respective phase diagrams. They present finite-temperature phase transitions, most likely in the Heisenberg 3D universality class. However, the combination between the 3D character and frustration makes them hard to study. We present here several methods derived from high-temperature series expansions (HTSEs), which give exact coefficients directly in the thermodynamic limit up to a certain order; for several 3D lattices, supplementary orders than in previous literature are reported for the HTSEs. We introduce an interpolation method able to describe thermodynamic quantities at $T>{T}_{c}$, which we use here to reconstruct the magnetic susceptibility and the specific heat and to extract universal and nonuniversal quantities (for example, critical exponents, temperature, energy, entropy, and other parameters related to the phase transition). While the susceptibility associated with the order parameter is not usually known for more exotic long-range orders, the specific heat is indicative of a phase transition for any kind of symmetry breaking. We present examples of applications on ferromagnetic and antiferromagnetic models on various 3D lattices and benchmark our results whenever possible.

DOI
10.1103/physrevb.107.235151

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In this paper we reanalyze the series for the fluctuation of the order parameter, the fourth order fluctuation in the order parameter, and the specific heat for the spin [Formula: see text] model on three dimensional lattices. We find for the exponents, respectively, γ = 1.333 ± 0.001, γ2 = 4.725 ± 0.01, and α = 0.00 ± 0.04. We find no evidence for a correction to scaling term with exponent [Formula: see text] for the specific heat or for the susceptibility. Precise estimates for critical entropy and critical energy are obtained. From the critical amplitudes we obtain further support for the lattice–lattice scaling hypothesis and the one scale factor universality hypothesis.
Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.
By combining conventional finite-temperature many-body perturbation theory with cluster expansions, we develop a systematic method to carry out high order arbitrary temperature perturbative calculations on a computer. The method is well suited to studying the thermodynamic properties of quantum disordered and quantum critical phases at finite temperatures. As an application, we calculate the magnetic susceptibility, internal energy, and specific heat of the bilayer Heisenberg model. It is shown that for a wide range of coupling constants these expansions show excellent convergence at all temperatures. Comparing the direct series (without extrapolations) for the bulk susceptibility to quantum Monte Carlo simulations we find an almost perfect agreement between the two methods even at the quantum critical coupling separating the dimerized and antiferromagnetic phases. The convergence fails only at very low temperatures, which are also difficult to reach by quantum Monte Carlo simulations.
This work presents an algorithm for calculating high temperature series expansions (HTSE) of Heisenberg spin models with spin S=1/2 S = 1 / 2 in the thermodynamic limit. This algorithm accounts for the presence of a magnetic field. The paper begins with a comprehensive introduction to HTSE and then focuses on identifying the bottlenecks that limit the computation of higher order coefficients. HTSE calculations involve two key steps: graph enumeration on the lattice and trace calculations for each graph. The introduction of a non-zero magnetic field adds complexity to the expansion because previously irrelevant graphs must now be considered: bridged graphs. We present an efficient method to deduce the contribution of these graphs from the contribution of sub-graphs, that drastically reduces the time of calculation for the last order coefficient (in practice increasing by one the order of the series at almost no cost). Previous articles of the authors have utilized HTSE calculations based on this algorithm, but without providing detailed explanations. The complete algorithm is publicly available, as well as the series on many lattice and for various interactions.
Classical and quantum spin systems are widely used models in both experimental and theoretical condensed matter physics. In many materials, the electronic interactions can be difficult to model exactly. However, in some insulators (Mott insulators), the magnetic (spin) interactions can be captured well with spin-only models. Several models are studied in this thesis. First, I report the solution of a long-standing issue in a classical frustrated spin model (i.e., where the quantum effects are neglected and the complexity is due to competing interactions): the nature of the thermal phase transition to a stripe state in the two dimensional (2D) J1-J2 model. Here J1 and J2 are nearest- and next-nearest-neighbor couplings. Monte Carlo simulations with single-spin updates are used for the calculations, and an extended-ensemble method, a generalization of the Wang-Landau algorithm, is also developed and tested. I focus on the study of "weak universality" behavior (continuously varying critical exponents, with one of the exponents staying fixed), for which I show a correspondence with a known class of conformal field theories with charge c = 1. Next, moving to quantum spins, to shed light on magnetic systems studied experimentally and to investigate new types of quantum states of interest in developing theories of quantum magnetism, I study the S = 1/2 Heisenberg model on the 2D square lattice with added six-spin interactions (the so-called J-Q3 model) as well as a set of 3D quantum antiferromagnets on dimerized lattices. Here I use the stochastic series expansion quantum Monte Carlo method. In the study of the J-Q3 model, I report on a similar weak-universality behavior as in the classical J1-J2 model, but with a mapping to a different known class of c = 1 conformal field theories. The critical behavior of the system again shows continuously changing exponents, in a way which corresponds to a gradual weakening of Z4 symmetry-breaking to an emergent U(1) symmetry. In the study of dimerized antiferromagnets, I report on a universal behavior of the Néel temperature TN, which can be related to ground state parameters independently of the microscopic interaction details in several different models.
High-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heinsenberg model are calculated for various lattices and anisotropies through order ${T}^{\ensuremath{-}8}$ (close-packed lattices) and ${T}^{\ensuremath{-}9}$ (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices $\ensuremath{\gamma}$ (susceptibility), $\ensuremath{\nu}$ (correlation range), and $\ensuremath{\alpha}$ (specific heat) as functions of anisotropy. Our results are consistent with the hypothesis that the critical indices change only when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended.