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Europhysics Letters (EPL)

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

Harold J. W. Zandvliet

Published Jan 19, 2006• 36 citations
Publisher

Abstract

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.

DOI
10.1209/epl/i2005-10451-1

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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.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
Using the parallel tempering algorithm and graphics processing unit accelerated techniques, we have performed large-scale Monte Carlo simulations of the Ising model on a square lattice with antiferromagnetic (repulsive) nearest-neighbor and next-nearest-neighbor interactions of the same strength and subject to a uniform magnetic field. Both transitions from the (2x1) and row-shifted (2x2) ordered phases to the paramagnetic phase are continuous. From our data analysis, re-entrance behavior of the (2x1) critical line and a bicritical point which separates the two ordered phases at T=0 are confirmed. Based on the critical exponents we obtained along the phase boundary, Suzuki's weak universality seems to hold.
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.
The critical behavior at the phase transition of the ferromagnetic two-dimensional anisotropic Ising model with next-nearest neighbor (NNN) couplings in the presence of the field is determined using mainly Monte Carlo (MC) method. This method is used to investigate the phase diagram of the model and to verify the existence of a divergence at null temperature which often appears in two-dimensional systems. We analyze also the influence of the report of the NNN interactions [Formula: see text] and the magnetic field [Formula: see text] on the critical temperature of the system, and we show that the critical temperature depends on the magnetic field for positive values of the interaction. Finally, we have investigated other thermodynamical qualities such as the magnetic susceptibility [Formula: see text]. It has been shown that their thermal behavior depends qualitatively and quantitatively on the strength of NNN interactions and the magnetic field.