Physical review. B, Condensed matter

Monte Carlo simulation of type-1 fcc Ising antiferromagnets

Alto Stemmer, Alfred Hüller

Published Jul 1, 1998• 8 citations

Abstract

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.

DOI: 10.1103/physrevb.58.887

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

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Phase transitions and critical characteristics in the layered antiferromagnetic Ising model with next-nearest-neighbor intralayer interactions

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Journal of Experimental and Theoretical Physics LettersMay 1, 201525 citations

Physical review. B, Condensed matter

Monte Carlo simulation of type-1 fcc Ising antiferromagnets

Alto Stemmer, Alfred Hüller

Published Jul 1, 1998• 8 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevb.58.887

Related papers

First-order phase transition in the fcc Heisenberg antiferromagnet

H. T. Diep, Hikaru Kawamura

Physical review. B, Condensed matterOct 1, 198978 citations

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

Ronja Stübel, Wolfhard Janke E

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

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Monte Carlo simulations are used to study the location and nature of phase boundaries for Ising square lattices with antiferromagnetic coupling ${J}_{\mathrm{NN}}$ between nearest neighbors and additional interactions ${J}_{\mathrm{NNN}}$ between next-nearest neighbors and ${J}_{3\mathrm{N}\mathrm{N}}$ between third-nearest neighbors. Results in zero magnetic field are obtained for a wide range of R= ${J}_{\mathrm{NNN}/{J}_{\mathrm{NN}}}$ and R'= ${J}_{3\mathrm{N}\mathrm{N}/{J}_{\mathrm{NN}}}$ . In addition to the c(2\ifmmode\times\else\texttimes\fi{}2) and (2\ifmmode\times\else\texttimes\fi{}1) phases, which also occur for R'=0, we find new (4\ifmmode\times\else\texttimes\fi{}4) and (4\ifmmode\times\else\texttimes\fi{}2) ordered states, for R'\ensuremath{\ne}0, which are separated from the disordered state by lines of first-order transitions. The nonuniversal critical behavior of the (2\ifmmode\times\else\texttimes\fi{}1) phase is studied using the block-distribution method and finite-size scaling. The possible existence of incommensurate phases is also explored.Monte Carlo simulations are used to study the location and nature of phase boundaries for Ising square lattices with antiferromagnetic coupling ${J}_{\mathrm{NN}}$ between nearest neighbors and additional interactions ${J}_{\mathrm{NNN}}$ between next-nearest neighbors and ${J}_{3\mathrm{N}\mathrm{N}}$ between third-nearest neighbors. Results in zero magnetic field are obtained for a wide range of R=${J}_{\mathrm{NNN}/{J}_{\mathrm{NN}}}$ and R'=${J}_{3\mathrm{N}\mathrm{N}/{J}_{\mathrm{NN}}}$. In addition to the c(2\ifmmode\times\else\texttimes\fi{}2) and (2\ifmmode\times\else\texttimes\fi{}1) phases, which also occur for R'=0, we find new (4\ifmmode\times\else\texttimes\fi{}4) and (4\ifmmode\times\else\texttimes\fi{}2) ordered states, for R'\ensuremath{\ne}0, which are separated from the disordered state by lines of first-order transitions. The nonuniversal critical behavior of the (2\ifmmode\times\else\texttimes\fi{}1) phase is studied using the block-distribution method and finite-size scaling. The possible existence of incommensurate phases is also explored.

Physical review. B, Condensed matterMay 1, 1985202 citations

Unusual finite size effects on critical temperature in fcc Ising antiferromagnets

J. Pommier, H. T. Diep, Abdelwahab Ghazali +1 more

A new multispin coding technique is presented for Monte Carlo simulation of antiferromagnetic Ising spin systems on an fcc lattice. The nearest- and next-nearest-neighbor interactions J1 and J2 are included. This technique allows a considerable gain in CPU time and computer memory. As a first application, we have studied samples of 4L3 spins with L up to 48. An unusual behavior of the critical temperature with increasing L is found in the case of nearest-neighbor interaction in zero field. Finite size effects on the locations of tricrical points in the (T,J2/J1) plane are discussed.A new multispin coding technique is presented for Monte Carlo simulation of antiferromagnetic Ising spin systems on an fcc lattice. The nearest- and next-nearest-neighbor interactions J1 and J2 are included. This technique allows a considerable gain in CPU time and computer memory. As a first application, we have studied samples of 4L3 spins with L up to 48. An unusual behavior of the critical temperature with increasing L is found in the case of nearest-neighbor interaction in zero field. Finite size effects on the locations of tricrical points in the (T,J2/J1) plane are discussed.

Journal of Applied PhysicsApr 15, 19886 citations

Computer simulation study of hysteresis and free energy in the fcc Ising antiferromagnet

Hagai Meirovitch

A method for estimating the entropy $S$ from a Monte Carlo simulation, suggested previously by the author, is applied to the fcc Ising antiferromagnet with nearest-neighbor interactions and external magnetic field $h$ . The estimated accuracy for $S$ is 0.01% to 0.5%, which is, at least, 1 order of magnitude higher than has been obtained with thermodynamic integration by Binder. Comparison of the free-energy results for the coexisting ordered and disordered phases enables us to determine with high accuracy the first-order transition points and the discontinuities in the thermodynamic functions. For the residual ground-state entropies per spin $\frac{S(0)}{{k}_{B}}$ at the critical fields $\frac{{h}_{c}}{J}=12$ and 4, we obtain 0.249 89(2) [=0.369 51(3)ln2] and 0.239(1) [=0.345(1)ln2], respectively. The first result is significantly larger than the theoretical upper bound, ${\ensuremath{\sigma}}_{u}^{\ensuremath{'}}=0.2398$ (=0.3460ln2), conjectured by Hadjukovi\ifmmode \acute{c}\else \'{c}\fi{} and Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{}. Our results are also significantly larger than Binder's result, ~⅓ ln2, obtained for both critical fields.A method for estimating the entropy $S$ from a Monte Carlo simulation, suggested previously by the author, is applied to the fcc Ising antiferromagnet with nearest-neighbor interactions and external magnetic field $h$. The estimated accuracy for $S$ is 0.01% to 0.5%, which is, at least, 1 order of magnitude higher than has been obtained with thermodynamic integration by Binder. Comparison of the free-energy results for the coexisting ordered and disordered phases enables us to determine with high accuracy the first-order transition points and the discontinuities in the thermodynamic functions. For the residual ground-state entropies per spin $\frac{S(0)}{{k}_{B}}$ at the critical fields $\frac{{h}_{c}}{J}=12$ and 4, we obtain 0.249 89(2) [=0.369 51(3)ln2] and 0.239(1) [=0.345(1)ln2], respectively. The first result is significantly larger than the theoretical upper bound, ${\ensuremath{\sigma}}_{u}^{\ensuremath{'}}=0.2398$ (=0.3460ln2), conjectured by Hadjukovi\ifmmode \acute{c}\else \'{c}\fi{} and Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{}. Our results are also significantly larger than Binder's result, ~⅓ ln2, obtained for both critical fields.

Physical review. B, Condensed matterSep 1, 198448 citations

Phase transitions and critical characteristics in the layered antiferromagnetic Ising model with next-nearest-neighbor intralayer interactions

Рамазанов Магомедшейх Курбанович, Akai Kurbanovich Murtazaev

Phase transitions and critical characteristics of the layered antiferromagnetic Ising model in the case of a cubic lattice with next-nearest-neighbor intralayer interactions are studied in the framework of the Monte Carlo method implementing the replica algorithm. The characteristics of the phase transitions are analyzed within the histogram method and with the Binder cumulants. For the model under study, it is found that the transition from the superantiferromagnetic phase to the paramagnetic one is a second order phase transition. The static critical exponents for the specific heat α, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent η are calculated using the finite-size scaling theory. It is shown that the three-dimensional Ising model for the cubic lattice with next-nearest-neighbor interactions belongs to the same universality class of critical behavior as the completely frustrated three-dimensional Ising model.Phase transitions and critical characteristics of the layered antiferromagnetic Ising model in the case of a cubic lattice with next-nearest-neighbor intralayer interactions are studied in the framework of the Monte Carlo method implementing the replica algorithm. The characteristics of the phase transitions are analyzed within the histogram method and with the Binder cumulants. For the model under study, it is found that the transition from the superantiferromagnetic phase to the paramagnetic one is a second order phase transition. The static critical exponents for the specific heat α, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent η are calculated using the finite-size scaling theory. It is shown that the three-dimensional Ising model for the cubic lattice with next-nearest-neighbor interactions belongs to the same universality class of critical behavior as the completely frustrated three-dimensional Ising model.

Journal of Experimental and Theoretical Physics LettersMay 1, 201525 citations

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.

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)

Monte Carlo simulations are used to study the location and nature of phase boundaries for Ising square lattices with antiferromagnetic coupling

{J}_{\mathrm{NN}}

between nearest neighbors and additional interactions

{J}_{\mathrm{NNN}}

between next-nearest neighbors and

{J}_{3\mathrm{N}\mathrm{N}}

between third-nearest neighbors. Results in zero magnetic field are obtained for a wide range of R=

{J}_{\mathrm{NNN}/{J}_{\mathrm{NN}}}

and R'=

{J}_{3\mathrm{N}\mathrm{N}/{J}_{\mathrm{NN}}}

. In addition to the c(2\ifmmode\times\else\texttimes\fi{}2) and (2\ifmmode\times\else\texttimes\fi{}1) phases, which also occur for R'=0, we find new (4\ifmmode\times\else\texttimes\fi{}4) and (4\ifmmode\times\else\texttimes\fi{}2) ordered states, for R'\ensuremath{\ne}0, which are separated from the disordered state by lines of first-order transitions. The nonuniversal critical behavior of the (2\ifmmode\times\else\texttimes\fi{}1) phase is studied using the block-distribution method and finite-size scaling. The possible existence of incommensurate phases is also explored.

A method for estimating the entropy

S

from a Monte Carlo simulation, suggested previously by the author, is applied to the fcc Ising antiferromagnet with nearest-neighbor interactions and external magnetic field

h

. The estimated accuracy for

S

is 0.01% to 0.5%, which is, at least, 1 order of magnitude higher than has been obtained with thermodynamic integration by Binder. Comparison of the free-energy results for the coexisting ordered and disordered phases enables us to determine with high accuracy the first-order transition points and the discontinuities in the thermodynamic functions. For the residual ground-state entropies per spin

\frac{S(0)}{{k}_{B}}

at the critical fields

\frac{{h}_{c}}{J}=12

and 4, we obtain 0.249 89(2) [=0.369 51(3)ln2] and 0.239(1) [=0.345(1)ln2], respectively. The first result is significantly larger than the theoretical upper bound, ${\ensuremath{\sigma}}_{u}^{\ensuremath{'}}=0.2398$ (=0.3460ln2), conjectured by Hadjukovi\ifmmode \acute{c}\else \'{c}\fi{} and Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{}. Our results are also significantly larger than Binder's result, ~⅓ ln2, obtained for both critical fields.