Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1810.05006 · Physical review. B./Physical review. B

Finite-size scaling of Monte Carlo simulations for the fcc Ising antiferromagnet: Effects of the low-temperature phase degeneracy

Ronja Stübel, Wolfhard Janke E

Published Nov 9, 2018• 7 citations
arXivHTMLPDF

Abstract

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 $L$) 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 ${T}_{0}=1.735047(46)$.

DOI
10.1103/physrevb.98.174413

Related papers

Transmuted Finite-size Scaling at First-order Phase Transitions

Marco Mueller, Wolfhard Janke E, Desmond A. Johnston

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.

arXiv:1410.7928•Physics Procedia•Jan 1, 2014•6 citations

27th Annual CSP Workshops on "Recent Developments in Computer Simulation Studies in Condensed Matter Physics", CSP 2014 Transmuted finite-size scaling at first-order phase transitions

Marco Mueller, Wolfhard Janke E, Desmond A. Johnston

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.

•Jan 1, 2014•0 citations

Monte Carlo simulation of type-1 fcc Ising antiferromagnets

Alto Stemmer, Alfred Hüller

Using Monte Carlo methods we investigate the phase transition of an Ising system on the fcc lattice, with antiferromagnetic nearest-neighbor interaction $J$ and ferromagnetic next-nearest-neighbor interaction $\ensuremath{\alpha}J$ for \ensuremath{\alpha} values up to 2. The scaling of the microcanonical entropy ${s}_{L}$ with system size $L$ agrees with the assumption of a first-order transition. Since the asymptotic behavior has not yet been reached, even for the largest systems with $L=40,$ our calculations cannot determine the order of the transition unambiguously.

Physical review. B, Condensed matter•Jul 1, 1998•8 citations

Nonstandard Finite-Size Scaling at First-Order Phase Transitions

Marco Mueller, Wolfhard Janke E, Desmond A. Johnston

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.

arXiv:1312.5984•Physical Review Letters•May 19, 2014•31 citations

Finite-size scaling and Monte Carlo simulations of first-order phase transitions

Jooyoung Lee, J. Michael Kosterlitz

We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)=${\mathit{U}}_{0}$(${\mathit{tL}}^{\mathit{d}}$)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$${\mathit{U}}_{1}$(${\mathit{tL}}^{\mathit{d}}$) with t=1-${\mathit{T}}_{\mathit{c}}$/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(${\mathit{e}}_{1}$/${\mathit{e}}_{2}$-${\mathit{e}}_{2}$/${\mathit{e}}_{1}$${)}^{2}$/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature ${\mathit{T}}_{\mathit{c}}^{(2)}$(L)-${\mathit{T}}_{\mathit{c}}$=O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at ${\mathit{T}}_{\mathit{c}}$ using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.

Physical review. B, Condensed matter•Feb 1, 1991•316 citations

First-order phase transition in the fcc Heisenberg antiferromagnet

H. T. Diep, Hikaru Kawamura

The nature of the phase transition of the nearest-neighbor Heisenberg antiferromagnet on a fcc lattice is studied by means of Monte Carlo simulations. Using a finite-size scaling, the transition is shown to be unambiguously of first order. Evidence for a first-order transition in the corresponding XY model is also presented.

Physical review. B, Condensed matter•Oct 1, 1989•78 citations
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∼L\ensuremath−2\ensuremath{\sim}{L}^{\ensuremath{-}2}\ensuremath∼L\ensuremath−2 instead of \ensuremath∼L\ensuremath−3\ensuremath{\sim}{L}^{\ensuremath{-}3}\ensuremath∼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≤18L\ensuremath{\le}18L\ensuremath≤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).
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.
Using Monte Carlo methods we investigate the phase transition of an Ising system on the fcc lattice, with antiferromagnetic nearest-neighbor interaction JJJ and ferromagnetic next-nearest-neighbor interaction $\ensuremath{\alpha}J$ for \ensuremath{\alpha} values up to 2. The scaling of the microcanonical entropy sL{s}_{L}sL​ with system size LLL agrees with the assumption of a first-order transition. Since the asymptotic behavior has not yet been reached, even for the largest systems with L=40,L=40,L=40, our calculations cannot determine the order of the transition unambiguously.
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 $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ 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.
We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)=U0{\mathit{U}}_{0}U0​(tLd{\mathit{tL}}^{\mathit{d}}tLd)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$U1{\mathit{U}}_{1}U1​(tLd{\mathit{tL}}^{\mathit{d}}tLd) with t=1-Tc{\mathit{T}}_{\mathit{c}}Tc​/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(e1{\mathit{e}}_{1}e1​/e2{\mathit{e}}_{2}e2​-e2{\mathit{e}}_{2}e2​/e1{\mathit{e}}_{1}e1​)2{)}^{2})2/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature Tc(2){\mathit{T}}_{\mathit{c}}^{(2)}Tc(2)​(L)-Tc{\mathit{T}}_{\mathit{c}}Tc​=O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at Tc{\mathit{T}}_{\mathit{c}}Tc​ using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.