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

Physical review. B, Condensed matter

Phase transitions in the Ising square lattice with next-nearest-neighbor interactions

David Paul Landau

Published Feb 1, 1980• 107 citations
Publisher

Abstract

An importance sampling Monte Carlo method has been used to study $L\ifmmode\times\else\texttimes\fi{}L$ Ising square lattices with periodic boundary conditions. The behavior of the thermal and magnetic properties was determined over a wide range of $R$, the ratio of next-nearest- to nearest-neighbor coupling. From these data we have extracted the $R$ dependence of the critical temperature and a range of other critical parameters. Within our experimental error the critical exponents are independent of $R$ but the critical amplitudes are not. The data do however suggest a change in $\ensuremath{\alpha}$ for $R$ slightly greater than \textonehalf{}. Variations in critical entropy and internal energy may be as large as those resulting from changes in lattice or spin dimensionalities. Our results are compared and contrasted with those obtained by a range of theoretical methods.

DOI
10.1103/physrevb.21.1285

Related papers

Finite-size behavior of the Ising square lattice

David Paul Landau

An importance-sampling Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N$ Ising square lattices with nearest-neighbor interactions and either free edges or periodic boundary conditions. The internal energy, specific heat, order parameter, susceptibility, and near-neighbor spin-spin correlation functions of the finite lattices are determined as a function of $N$ and extrapolated to the corresponding infinite-system values. The effect of finite size is greater for free edges in all cases. The results agree well with predictions of finite size scaling theory and the shape functions as well as amplitudes of surface contribution terms are determined.

Physical review. B, Solid state•Apr 1, 1976•411 citations

Phase diagrams and critical behavior of Ising square lattices with nearest-, next-nearest-, and third-nearest-neighbor couplings

D. P. Landau, Kurt Binder

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 matter•May 1, 1985•202 citations

Phase transitions in the antiferromagnetic ising model on a square lattice with next-nearest-neighbor interactions

Akai Kurbanovich Murtazaev, Рамазанов Магомедшейх Курбанович, Felix A. Kassan-Ogly +1 more

The phase transitions in the two-dimensional Ising model on a square lattice are studied using a replica algorithm, the Monte Carlo method, and histogram analysis with allowance for the next-nearest-neighbor interactions in the range 0.1 ≤ r < 1.0. A phase diagram is constructed for the dependence of the critical temperature on the next-nearest-neighbor interaction. A second-order phase transition is detected in this range and the model under study.

Journal of Experimental and Theoretical Physics•Dec 1, 2013•28 citations

Critical properties of an antiferromagnetic Ising model on a square lattice with interactions of the next-to-nearest neighbors

Akai Kurbanovich Murtazaev, Рамазанов Магомедшейх Курбанович, М. К. Бадиев

The critical properties of an antiferromagnetic Ising model on a square lattice with interactions of the next-to-nearest neighbors are investigated by a replica Monte-Carlo method. Using the finite-size scaling theory the static critical exponents of specific heat, ordering parameter, susceptibility, correlation radius as well as the Fisher exponent are calculated. An analysis of data is performed both with and without taking into account a correction to the finite-size scaling. It was found that in the model under consideration the second order phase transition is observed. It is shown that this model belongs to the new class of universality of critical behavior.

Low Temperature Physics•Dec 1, 2011•24 citations

The 2D Ising square lattice with nearest- and next-nearest-neighbor interactions

Harold J. W. Zandvliet

An analytical expression for the boundary free energy of the Ising square lattice with nearest- and next-nearest-neighbor interactions is derived. The Ising square lattice with anisotropic nearest-neighbor ( J x and J y ) and isotropic next-nearest-neighbor ( J d ) interactions has an order-disorder phase transition at a temperature T = T c given by the condition e −2 J x / k b T c + e − 2 J y / k b T c + e −2( J x + J y )/ k b T c (2 − e −4 J d / k b T c ) = e 4 J d / k b T c . The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase is in excellent agreement with series expansion, finite scaling of transfer matrix and Monte Carlo results. For a vanishing next-nearest-neighbor interaction Onsager's famous result, i.e. sinh (2 J x / k b T c )sinh (2 J y / k b T c ) = 1, is recaptured.

Europhysics Letters (EPL)•Jan 19, 2006•36 citations

Ising Antiferromagnet with Nearest-Neighbor and Next-Nearest-Neighbor Interactions on a Square Lattice

Akai Kurbanovich Murtazaev, Рамазанов Магомедшейх Курбанович, М. К. Бадиев

The critical properties of two-dimensional antiferromagnetic Ising model in square lattice are investigated using the replica Monte-Carlo method with account of interactions of second nearest neighbors. The diagram of critical temperature dependence on an interaction value of second nearest neighbors is plotted. Static critical exponents of the heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated for this model using the finite-size scaling theory.

Diffusion and defect data, solid state data. Part B, Solid state phenomena/Solid state phenomena•Apr 1, 2014•0 citations
An importance sampling Monte Carlo method has been used to study L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL Ising square lattices with periodic boundary conditions. The behavior of the thermal and magnetic properties was determined over a wide range of RRR, the ratio of next-nearest- to nearest-neighbor coupling. From these data we have extracted the RRR dependence of the critical temperature and a range of other critical parameters. Within our experimental error the critical exponents are independent of RRR but the critical amplitudes are not. The data do however suggest a change in \ensuremathα\ensuremath{\alpha}\ensuremathα for RRR slightly greater than \textonehalf{}. Variations in critical entropy and internal energy may be as large as those resulting from changes in lattice or spin dimensionalities. Our results are compared and contrasted with those obtained by a range of theoretical methods.
An importance-sampling Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N$ Ising square lattices with nearest-neighbor interactions and either free edges or periodic boundary conditions. The internal energy, specific heat, order parameter, susceptibility, and near-neighbor spin-spin correlation functions of the finite lattices are determined as a function of NNN and extrapolated to the corresponding infinite-system values. The effect of finite size is greater for free edges in all cases. The results agree well with predictions of finite size scaling theory and the shape functions as well as amplitudes of surface contribution terms are determined.
Monte Carlo simulations are used to study the location and nature of phase boundaries for Ising square lattices with antiferromagnetic coupling JNN{J}_{\mathrm{NN}}JNN​ between nearest neighbors and additional interactions JNNN{J}_{\mathrm{NNN}}JNNN​ between next-nearest neighbors and J3NN{J}_{3\mathrm{N}\mathrm{N}}J3NN​ between third-nearest neighbors. Results in zero magnetic field are obtained for a wide range of R=JNNN/JNN{J}_{\mathrm{NNN}/{J}_{\mathrm{NN}}}JNNN/JNN​​ and R'=J3NN/JNN{J}_{3\mathrm{N}\mathrm{N}/{J}_{\mathrm{NN}}}J3NN/JNN​​. 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.
The critical properties of an antiferromagnetic Ising model on a square lattice with interactions of the next-to-nearest neighbors are investigated by a replica Monte-Carlo method. Using the finite-size scaling theory the static critical exponents of specific heat, ordering parameter, susceptibility, correlation radius as well as the Fisher exponent are calculated. An analysis of data is performed both with and without taking into account a correction to the finite-size scaling. It was found that in the model under consideration the second order phase transition is observed. It is shown that this model belongs to the new class of universality of critical behavior.
An analytical expression for the boundary free energy of the Ising square lattice with nearest- and next-nearest-neighbor interactions is derived. The Ising square lattice with anisotropic nearest-neighbor ( J x and J y ) and isotropic next-nearest-neighbor ( J d ) interactions has an order-disorder phase transition at a temperature T = T c given by the condition e −2 J x / k b T c + e − 2 J y / k b T c + e −2( J x + J y )/ k b T c (2 − e −4 J d / k b T c ) = e 4 J d / k b T c . The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase is in excellent agreement with series expansion, finite scaling of transfer matrix and Monte Carlo results. For a vanishing next-nearest-neighbor interaction Onsager's famous result, i.e. sinh (2 J x / k b T c )sinh (2 J y / k b T c ) = 1, is recaptured.