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arXiv:1901.10643 · Physical review. E

Finite-size scaling for a first-order transition where a continuous symmetry is broken: The spin-flop transition in the three-dimensional XXZ Heisenberg antiferromagnet

Jiahao Xu, Shan-Ho Tsai, D. P. Landau, Kurt Binder

Published Feb 13, 2019• 13 citations
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Abstract

Finite-size scaling for a first-order phase transition where a continuous symmetry is broken is developed using an approximation of Gaussian probability distributions with a phenomenological "degeneracy" factor included. Predictions are compared with data from Monte Carlo simulations of the three-dimensional, XXZ Heisenberg antiferromagnet in a field in order to study the finite-size behavior on a L×L×L simple cubic lattice for the first-order "spin-flop" transition between the Ising-like antiferromagnetic state and the canted, XY-like state. Our theory predicts that for large linear dimension L the field dependence of all moments of the order parameters as well as the fourth-order cumulants exhibit universal intersections. Corrections to leading order should scale as the inverse volume. The values of these intersections at the spin-flop transition point can be expressed in terms of a factor q that characterizes the relative degeneracy of the ordered phases. Our theory yields q=π, and we present numerical evidence that is compatible with this prediction. The agreement between the theory and simulation implies a heretofore unknown universality can be invoked for first-order phase transitions.

DOI
10.1103/physreve.99.023309

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Finite-size scaling for a first-order phase transition where a continuous symmetry is broken is developed using an approximation of Gaussian probability distributions with a phenomenological "degeneracy" factor included. Predictions are compared with data from Monte Carlo simulations of the three-dimensional, XXZ Heisenberg antiferromagnet in a field in order to study the finite-size behavior on a L×L×L simple cubic lattice for the first-order "spin-flop" transition between the Ising-like antiferromagnetic state and the canted, XY-like state. Our theory predicts that for large linear dimension L the field dependence of all moments of the order parameters as well as the fourth-order cumulants exhibit universal intersections. Corrections to leading order should scale as the inverse volume. The values of these intersections at the spin-flop transition point can be expressed in terms of a factor q that characterizes the relative degeneracy of the ordered phases. Our theory yields q=π, and we present numerical evidence that is compatible with this prediction. The agreement between the theory and simulation implies a heretofore unknown universality can be invoked for first-order phase transitions.
Abstract A great triumph of statistical physics in the latter part of the 20th century was the understanding of critical behavior and universality at second-order phase transitions. In contrast, first-order transitions were believed to have no common features. However, we argue that the classic, first-order “spin-flop” transition (between the antiferromagnetic and the rotationally degenerate, canted state) in an anisotropic antiferromagnet in a magnetic field exhibits a new kind of universality. We present a finite-size scaling theory for a first-order phase transition where a continuous symmetry is broken using an approximation of Gaussian probability distributions with a phenomenological degeneracy factor “q” included, where “q” characterizes the relative degeneracy of the ordered phases. Predictions are compared with high resolution Monte Carlo simulations of the three-dimensional, XXZ Heisenberg antiferromagnet in a field to study the finite-size behavior for L×L×L simple cubic lattices. The field dependence of all moments of the order parameters exhibit universal intersections at the spin-flop transition. Our Monte Carlo data agree with theoretical predictions for asymptotic large L behavior. Our theory yields q = π, and we present numerical evidence that is compatible with this prediction. The agreement between the theory and simulation implies a heretofore unknown universality.
Finite size scaling for a first order phase transition, where a continuous symmetry is broken, is tested using an approximation of Gaussian probability distributions with a phenomenological "degeneracy" factor. Predictions are compared to the data from Monte Carlo simulations of the Lebwohl-Lasher model on L × L × L simple cubic lattices. The data show that the intersection of the fourth-order cumulant of the order parameter for different lattice sizes can be expressed in terms of the relative degeneracy q = 4π of the ordered and disordered phases. This result further supports the concept of universality at first order transitions developed recently.
A brief review is given of the ‘effective Lagrangian’ approach of Leutwyler, Hasenfratz and others, which describes the behaviour at low energies or temperatures, or large distances, in lattice systems which undergo a first-order transition involving spontaneous breakdown of a continuous symmetry. Universal predictions can be given, based on a continuum field theory of the massless Goldstone bosons generated by the breakdown of the symmetry, which control the behaviour in these regimes. In particular, the finite-size scaling behaviour can be predicted, in a way very similar to the predictions of conformal invariance for a two-dimensional system at a critical point. Conversely, measurements of the finite-size scaling behaviour can give estimates of the parameters of the Goldstone bosons. These points are illustrated using data for the O(2) Heisenberg spin model in (2+1) dimensions, and the XXZ antiferromagnet on a square lattice.
The Ising antiferromagnet on a face-centered cubic (fcc) lattice with nearest-neighbor interaction only is well known to exhibit a macroscopic (exponential in the system size LLL) ground-state degeneracy. With increasing temperature, this degeneracy is expected to be lifted and the model undergoes a first-order phase transition. For a model with an exponential degeneracy in the whole low-temperature phase, it was recently found that the finite-size scaling behavior is governed by leading correction terms $\ensuremath{\sim}{L}^{\ensuremath{-}2}$ instead of $\ensuremath{\sim}{L}^{\ensuremath{-}3}$ as usual. To test the conjecture that such a transmuted behavior may effectively persist also for the fcc antiferromagnet up to some crossover system size, we have performed parallel multicanonical Monte Carlo simulations for lattices of linear size $L\ensuremath{\le}18$ with periodic boundary conditions and determined various inverse pseudo phase transition temperatures, as well as the extremal values of the specific heat and the energetic Binder parameter. We indeed find that, for the simulated lattice sizes, the conjectured transmuted finite-size scaling ansatz fits the data better than the standard ansatz. On this basis, we extrapolate for the transition temperature an estimate of T0=1.735047(46){T}_{0}=1.735047(46)T0​=1.735047(46).
The results of the large-scale Monte Carlo simulations of the continuous phase transitions in the three-dimensional Ashkin-Teller model on a cubic lattice are presented in the two-parameter space. The invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region and the Metropolis algorithm are exploited. Using the universality hypothesis and the finite-size-scaling analysis, the accurate positions of the continuous phase-transition lines are calculated and their Ising character is confirmed. Approximate locations of the tricritical points are also estimated.