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arXiv:1312.5984 · Physical Review Letters

Nonstandard Finite-Size Scaling at First-Order Phase Transitions

Marco Mueller, Wolfhard Janke E, Desmond A. Johnston

Published May 19, 2014• 31 citations
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Abstract

We note that the standard inverse system volume scaling for finite-size corrections at a first-order phase transition (i.e., $1/{L}^{3}$ for an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ lattice in 3D) is transmuted to $1/{L}^{2}$ scaling if there is an exponential low-temperature phase degeneracy. The gonihedric Ising model which has a four-spin interaction, plaquette Hamiltonian provides an exemplar of just such a system. We use multicanonical simulations of this model to generate high-precision data which provide strong confirmation of the nonstandard finite-size scaling law. The dual to the gonihedric model, which is an anisotropically coupled Ashkin-Teller model, has a similar degeneracy and also displays the nonstandard scaling.

DOI
10.1103/physrevlett.112.200601

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We note that the standard inverse system volume scaling for finite-size corrections at a first-order phase transition (i.e., 1/L31/{L}^{3}1/L3 for an L\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fiL lattice in 3D) is transmuted to 1/L21/{L}^{2}1/L2 scaling if there is an exponential low-temperature phase degeneracy. The gonihedric Ising model which has a four-spin interaction, plaquette Hamiltonian provides an exemplar of just such a system. We use multicanonical simulations of this model to generate high-precision data which provide strong confirmation of the nonstandard finite-size scaling law. The dual to the gonihedric model, which is an anisotropically coupled Ashkin-Teller model, has a similar degeneracy and also displays the nonstandard scaling.
It is known that fixed boundary conditions modify the leading finite-size corrections for an L3 lattice in 3d at a first-order phase transition from 1/L3 to 1/L. We note that an exponential low-temperature phase degeneracy of the form 23L will lead to a different leading correction of order 1/L2. A 3d gonihedric Ising model with a four-spin interaction, plaquette Hamiltonian displays such a degeneracy and we confirm the modified scaling behaviour using high-precision multicanonical simulations. We remark that other models such as the Ising antiferromagnet on the FCC lattice, in which the number of “true” low-temperature phases is not macroscopically large but which possess an exponentially degenerate number of low lying states may display an ef- fective version of the modified scaling law for the range of lattice sizes accessible in simulations.
It is known that fixed boundary conditions modify the leading finite-size corrections for an L 3 lattice in 3d at a first-order phase transition from 1/L 3 to 1/L. We note that an exponential low-temperature phase degeneracy of the form 2 3L will lead to a different leading correction of order 1/L 2 .A 3 d gonihedric Ising model with a four-spin interaction, plaquette Hamiltonian displays such a de generacy and we confirm the modified scaling behaviour using high-precision multicanonical simulations. We remark that other models such as the Ising antiferromagnet on the FCC lattice, in which the number of “true” low-temperature phases is not macroscopically large but which possess an exponentially degenerate number of low lying states may display an effective version of the modified scaling law for the range of lattice sizes accessible in simulations.
A well-known feature of first-order phase transitions is that fixed boundary conditions can strongly influence finite-size corrections, modifying the leading corrections for an L 3 lattice in 3 d from order 1/ L 3 under periodic boundary conditions to 1/ L . A rather similar effect, albeit of completely different origin, occurs when the system possesses an exponential low- temperature phase degeneracy of the form 2 3 L which causes for periodic boundary conditions a leading correction of order 1/ L 2 in 3 d . We discuss a 3 d plaquette Hamiltonian ("gonihedric") Ising model, which displays such a degeneracy and manifests the modified scaling behaviour. We also investigate an apparent discrepancy between the fixed and periodic boundary condition latent heats for the model when extrapolating to the thermodynamic limit.
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).
We review some recent investigations of the 3d plaquette Ising model. This displays a strong first-order phase transition with unusual scaling properties due to the size-dependent degeneracy of the low-temperature phase. In particular, the leading scaling correction is modified from the usual inverse volume behaviour ∝ 1/L 3 to 1/L 2. The degeneracy also has implications for the magnetic order in the model which has an intermediate nature between local and global order and gives rise to novel fracton topological defects in a related quantum Hamiltonian.
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.