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:cond-mat/0407284 · Physical Review E

Multicriticality of the three-dimensional Ising model with plaquette interactions: An extension of Novotny’s transfer-matrix formalism

Yoshihiro Nishiyama

Published Aug 31, 2004• 12 citations
arXivHTMLPDF

Abstract

A three-dimensional Ising model with the plaquette-type (next-nearest-neighbor and four-spin) interactions is investigated numerically. This extended Ising model, the so-called gonihedric model, was introduced by Savvidy and Wegner as a discretized version of the interacting (closed) surfaces without surface tension. The gonihedric model is notorious for its slow relaxation to the thermal equilibrium (glassy behavior), which deteriorates the efficiency of the Monte Carlo sampling. We employ the transfer-matrix (TM) method, implementing Novotny's idea, which enables us to treat an arbitrary number of spins N for one TM slice even in three dimensions. This arbitrariness admits systematic finite-size-scaling analyses. Accepting the extended parameter space by Cirillo et al., we analyzed the (multi-) criticality of the gonihedric model for N</=13. Thereby, we found that, as first noted by Cirillo et al. analytically (cluster-variation method), the data are well described by the multicritical (crossover) scaling theory. That is, the previously reported nonstandard criticality for the gonihedric model is reconciled with a crossover exponent and the ordinary three-dimensional-Ising universality class. We estimate the crossover exponent and the correlation-length critical exponent at the multicritical point as phi=0.6(2) and nu; =0.45(15), respectively.

DOI
10.1103/physreve.70.026120

Related papers

Critical behavior of the three-dimensional Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions

Emilio N. M. Cirillo, Giuseppe Gonnella, Alessandro Pelizzola

The critical and multicritical behavior of the simple cubic Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions is studied using the cube and star-cube approximations of the cluster variation method and the recently proposed cluster variation--Pad\'eapproximant method. Particular attention is paid to the line of critical end points of the ferromagnetic-paramagnetic phase transition: its (multi)critical exponents are calculated, and their values suggest that the transition belongs to a different universality class. A rough estimate of the crossover exponent is also given.

arXiv:cond-mat/9612001•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jan 1, 1997•22 citations

Finite-size-scaling analysis of the XY universality class between two and three dimensions: An application of Novotny’s transfer-matrix method

Yoshihiro Nishiyama

Based on Novotny's transfer-matrix method, we simulated the (stacked) triangular Ising anti-ferromagnet embedded in the space with the dimensions variable in the range 2 < or = d < or = 3. Our aim is to investigate the criticality of the XY universality class for 2 < or = d < or = 3. For that purpose, we employed an extended version of the finite-size-scaling analysis developed by Novotny, who utilized this scheme to survey the Ising criticality (ferromagnet) for 1 < or = d < or = 3. Diagonalizing the transfer matrix for the system sizes N up to N = 17 , we calculated the d -dependent correlation-length critical exponent nu(d). Our simulation result nu(d) appears to interpolate smoothly the known two limiting cases, namely, the Kosterlitz-Thouless (KT) and d = 3 XY universality classes, and the intermediate behavior bears close resemblance to that of the analytical formula via the 1/N-expansion technique. Methodological details including the modifications specific to the present model are reported.

arXiv:cond-mat/0502084•Physical Review E•Apr 12, 2005•11 citations

The Phase Diagram of the Gonihedric 3d Ising Model via CVM

Emilio N. M. Cirillo, G. Gonnella, Desmond A. Johnston +1 more

We use the cluster variation method (CVM) to investigate the phase structure of the 3d gonihedric Ising actions defined by Savvidy and Wegner. The geometrical spin cluster boundaries in these systems serve as models for the string worldsheets of the gonihedric string embedded in Z 3. The models are interesting from the statistical mechanical point of view because they have a vanishing bare surface tension. As a result the action depends only on the angles of the discrete surface and not on the area, which is the antithesis of the standard 3d Ising model. The results obtained with the CVM are in good agreement with Monte Carlo simulations for the critical temperatures and the order of the transition as the selfavoidance coupling κ is varied. The value of the magnetization critical exponent β = 0.062 ±0.003, calculated with the cluster variation–Padè approximant method, is also close to the simulation results. 1 The Model In this paper we discuss the use of the cluster variation method (CVM) in mapping out the phase diagram of the gonihedric 3d Ising model. The gonihedric 3d Ising model is a generalization of the usual 3d Ising model where planar Peierls boundaries between + and − spins can be created at zero energy cost. It has been introduced in [1] in relations with string theory. The CVM method in other contexts has shown itself to be an accurate and economical way of describing the phase diagram both for first order and continuous transitions. As both the model and the method may be unfamiliar, we outline both in turn before going on to describe our results. The genesis of the model is in a novel discretized random surface theory, the so-called gonihedric string [1],

•Jan 1, 1996•18 citations

Slow dynamics in the three-dimensional gonihedric model

P. Dimopoulos, Domènec Espriu, E. Jané +1 more

We study dynamical aspects of three-dimensional gonihedric spins by using Monte-Carlo methods. These models have a purely geometrical motivation, deriving from string and random surface theory. Here, however, we shall analyze this family of models just from a statistical point of view. In particular, we shall be concerned with their ability to exhibit remarkably slow dynamics and seemingly glassy behavior below a certain temperature T(g), without the need of introducing disorder of any kind. We consider first a Hamiltonian that takes into account only a four-spin term (kappa=0), where a first-order phase transition is well established. By studying the relaxation properties at low temperatures, we confirm that the model exhibits two distinct regimes. For T(g)<T<T(c), with long lived metastability and a supercooled phase, the approach to equilibrium is well described by a stretched exponential. For T<T(g), the dynamics appears to be logarithmic. We provide an accurate determination of T(g). We also determine the evolution of particularly long lived configurations. Next, we consider the case kappa=1, where the plaquette term is absent and the gonihedric action consists in a ferromagnetic Ising with fine-tuned next-to-nearest neighbor interactions. This model exhibits a second order phase transition. The consideration of the relaxation time for configurations in the cold phase reveals the presence of slow dynamics and glassy behavior for any T<T(c). Type-II aging features are exhibited by this model.

arXiv:cond-mat/0204403•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Nov 19, 2002•23 citations

Transfer-matrix approach to three-dimensional bond percolation: An application of Novotny’s formalism

Yoshihiro Nishiyama

A transfer-matrix simulation scheme for the three-dimensional (d=3) bond percolation is presented. Our scheme is based on Novotny's transfer-matrix formalism, which enables us to consider arbitrary (integral) number of sites N constituting a unit of the transfer-matrix slice even for d=3. Such an arbitrariness allows us to perform systematic finite-size-scaling analysis of the criticality at the percolation threshold. Diagonalizing the transfer matrix for N=4, 5,..., we obtain an estimate for the correlation-length critical exponent v=0.81(5).

arXiv:cond-mat/0512421•Physical Review E•Jan 12, 2006•2 citations

Crossover Scaling and Critical Behavior at the "Surface-Bulk" Multicritical Point

Kurt Binder, D. P. Landau

With use of a novel Monte Carlo method suited for the study of surface phenomena, the authors have investigated a three-dimensional Ising model with modified surface exchange. The results verify the scaling behavior of the magnetization profile and confirm and extend renormalization-group predictions. The surface-layer magnetization exponent is ${\ensuremath{\beta}}_{1}=0.78\ifmmode\pm\else\textpm\fi{}0.02$ except at the "special" (multicritical) transition where ${{\ensuremath{\beta}}_{1}}^{m}=0.175\ifmmode\pm\else\textpm\fi{}0.025$ and the crossover exponent is $\ensuremath{\phi}=0.56\ifmmode\pm\else\textpm\fi{}0.04$.

Physical Review Letters•Jan 30, 1984•288 citations
A three-dimensional Ising model with the plaquette-type (next-nearest-neighbor and four-spin) interactions is investigated numerically. This extended Ising model, the so-called gonihedric model, was introduced by Savvidy and Wegner as a discretized version of the interacting (closed) surfaces without surface tension. The gonihedric model is notorious for its slow relaxation to the thermal equilibrium (glassy behavior), which deteriorates the efficiency of the Monte Carlo sampling. We employ the transfer-matrix (TM) method, implementing Novotny's idea, which enables us to treat an arbitrary number of spins N for one TM slice even in three dimensions. This arbitrariness admits systematic finite-size-scaling analyses. Accepting the extended parameter space by Cirillo et al., we analyzed the (multi-) criticality of the gonihedric model for N</=13. Thereby, we found that, as first noted by Cirillo et al. analytically (cluster-variation method), the data are well described by the multicritical (crossover) scaling theory. That is, the previously reported nonstandard criticality for the gonihedric model is reconciled with a crossover exponent and the ordinary three-dimensional-Ising universality class. We estimate the crossover exponent and the correlation-length critical exponent at the multicritical point as phi=0.6(2) and nu; =0.45(15), respectively.
The critical and multicritical behavior of the simple cubic Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions is studied using the cube and star-cube approximations of the cluster variation method and the recently proposed cluster variation--Pad\'eapproximant method. Particular attention is paid to the line of critical end points of the ferromagnetic-paramagnetic phase transition: its (multi)critical exponents are calculated, and their values suggest that the transition belongs to a different universality class. A rough estimate of the crossover exponent is also given.
Based on Novotny's transfer-matrix method, we simulated the (stacked) triangular Ising anti-ferromagnet embedded in the space with the dimensions variable in the range 2 < or = d < or = 3. Our aim is to investigate the criticality of the XY universality class for 2 < or = d < or = 3. For that purpose, we employed an extended version of the finite-size-scaling analysis developed by Novotny, who utilized this scheme to survey the Ising criticality (ferromagnet) for 1 < or = d < or = 3. Diagonalizing the transfer matrix for the system sizes N up to N = 17 , we calculated the d -dependent correlation-length critical exponent nu(d). Our simulation result nu(d) appears to interpolate smoothly the known two limiting cases, namely, the Kosterlitz-Thouless (KT) and d = 3 XY universality classes, and the intermediate behavior bears close resemblance to that of the analytical formula via the 1/N-expansion technique. Methodological details including the modifications specific to the present model are reported.
We use the cluster variation method (CVM) to investigate the phase structure of the 3d gonihedric Ising actions defined by Savvidy and Wegner. The geometrical spin cluster boundaries in these systems serve as models for the string worldsheets of the gonihedric string embedded in Z 3. The models are interesting from the statistical mechanical point of view because they have a vanishing bare surface tension. As a result the action depends only on the angles of the discrete surface and not on the area, which is the antithesis of the standard 3d Ising model. The results obtained with the CVM are in good agreement with Monte Carlo simulations for the critical temperatures and the order of the transition as the selfavoidance coupling κ is varied. The value of the magnetization critical exponent β = 0.062 ±0.003, calculated with the cluster variation–Padè approximant method, is also close to the simulation results. 1 The Model In this paper we discuss the use of the cluster variation method (CVM) in mapping out the phase diagram of the gonihedric 3d Ising model. The gonihedric 3d Ising model is a generalization of the usual 3d Ising model where planar Peierls boundaries between + and − spins can be created at zero energy cost. It has been introduced in [1] in relations with string theory. The CVM method in other contexts has shown itself to be an accurate and economical way of describing the phase diagram both for first order and continuous transitions. As both the model and the method may be unfamiliar, we outline both in turn before going on to describe our results. The genesis of the model is in a novel discretized random surface theory, the so-called gonihedric string [1],
We study dynamical aspects of three-dimensional gonihedric spins by using Monte-Carlo methods. These models have a purely geometrical motivation, deriving from string and random surface theory. Here, however, we shall analyze this family of models just from a statistical point of view. In particular, we shall be concerned with their ability to exhibit remarkably slow dynamics and seemingly glassy behavior below a certain temperature T(g), without the need of introducing disorder of any kind. We consider first a Hamiltonian that takes into account only a four-spin term (kappa=0), where a first-order phase transition is well established. By studying the relaxation properties at low temperatures, we confirm that the model exhibits two distinct regimes. For T(g)<T<T(c), with long lived metastability and a supercooled phase, the approach to equilibrium is well described by a stretched exponential. For T<T(g), the dynamics appears to be logarithmic. We provide an accurate determination of T(g). We also determine the evolution of particularly long lived configurations. Next, we consider the case kappa=1, where the plaquette term is absent and the gonihedric action consists in a ferromagnetic Ising with fine-tuned next-to-nearest neighbor interactions. This model exhibits a second order phase transition. The consideration of the relaxation time for configurations in the cold phase reveals the presence of slow dynamics and glassy behavior for any T<T(c). Type-II aging features are exhibited by this model.
A transfer-matrix simulation scheme for the three-dimensional (d=3) bond percolation is presented. Our scheme is based on Novotny's transfer-matrix formalism, which enables us to consider arbitrary (integral) number of sites N constituting a unit of the transfer-matrix slice even for d=3. Such an arbitrariness allows us to perform systematic finite-size-scaling analysis of the criticality at the percolation threshold. Diagonalizing the transfer matrix for N=4, 5,..., we obtain an estimate for the correlation-length critical exponent v=0.81(5).
With use of a novel Monte Carlo method suited for the study of surface phenomena, the authors have investigated a three-dimensional Ising model with modified surface exchange. The results verify the scaling behavior of the magnetization profile and confirm and extend renormalization-group predictions. The surface-layer magnetization exponent is ${\ensuremath{\beta}}_{1}=0.78\ifmmode\pm\else\textpm\fi{}0.02$ except at the "special" (multicritical) transition where ${{\ensuremath{\beta}}_{1}}^{m}=0.175\ifmmode\pm\else\textpm\fi{}0.025$ and the crossover exponent is $\ensuremath{\phi}=0.56\ifmmode\pm\else\textpm\fi{}0.04$.