Free Fermion Approximation Research Articles

The dual model for an Ising model with nearest and next-nearest neighbors

We construct and analyze a dual model to the next-nearest-neighbor Ising (NNNI) model on the rectangular lattice. The Hamiltonian of the dual model turns out to contain two- and four-spin interactions. The free fermion approximation suggests that an increase in the critical temperature of the dual model caused by the four-spin interactions is limited to a finite range.

Equilibrium properties of an axial next-nearest-neighbor Ising model in two dimensions

We study thermodynamic properties of an axial next-nearest-neighbor Ising model in two dimensions using a cluster heat bath Monte Carlo method. The method drastically reduces the relaxation time, and enables us to study equilibrium properties of the model. We show that the model really exhibits two phase transitions at ${T}_{c1}$ and ${T}_{c2}(<{T}_{c1}),$ as predicted by a free fermion approximation. The one at ${T}_{c1}$ is shown to be of the Kosterlitz-Thouless type, and the other to be of the Pokrovsky-Talapov type. The higher transition temperature ${T}_{c1}$ is considerably lower than those estimated by previous authors. Low-temperature properties as well as the lower transition temperature ${T}_{c2}$ are found to be well described by the fermion approximation.

The Critical Behavior of the Ising Model on Union Jack Lattice11The project supported by Science Foundation of Shanghai Jiaotong University

In this paper, we have investigated the critical behavior of the ferromagnetic Ising model on union jack lattice. The model is equivalent to the eight-vertex model which can be solved as a free fermion model with the free fermion approximation. The critical exponents have been obtained as and .

The Critical Points of a Generalized Mixed Spin Model with Free Fermion Approximation

A generalized mixed spin model with anisotropic interactions is considered on the union jack lattice. We find that the model is equivalent to a free fermion model. The critical temperature is obtained. In special cases, we get the same results as exact solutions.

FREE-FERMION APPROXIMATE SOLUSION OF THE ISING MODEL ON UNION JACK LATTICE

In this paper,the Ising model with two-body anisotropic interactions and three-body interactions on union Jack lattice is solved by free-fermion approximation. We have obtained its free energy, spontaneous magnetisation and the equation of critical points. When the interaction coefficients are simplified as in a square lattice Ising model, the same solusion as Onsager's is obtained.

Free fermion approximation for the Ising model with further-neighbor interactions on a triangular lattice

The free fermion solution/approximation for the Ising model on a triangular lattice with further-neighbor interactions is derived, using Vdovichenko's method. For isotropic first- and second-neighbor interactionsK,L⩾0, the approximation is a strict lower bound for the partition sum. We have also obtained the approximate critical surface, where the critical behavior is Ising-like, and the exact zero-temperature phase diagram when the interactions are isotropic. A recent extension of the method of Vdovichenko due to Calheiroset al. makes it easy to give the surface free energy and the equilibrium crystal shape as well, in the ferromagnetic regime and for a regime where the phase is ordered in layers.

On the BNNNI model in the free-fermion approximation

In the free-fermion approximation, the authors map the BNNNI model onto the nearest-neighbour Ising model for the general anisotropic case. The resulting phase diagram does not contain floating phases but the transition lines between other phases agree with earlier studies, which were restricted to the isotropic case.

On the 2D ANNNI model in free fermion approximation

Assuming a restricted form of the free fermion approximation, we map the two-dimensional ANNNI model onto the nearest neighbour Ising model. The resulting phase diagram does not contain floating phases but the transition lines between other phases agree with earlier studies.

Critical couplings of mixed spin-1/2-spin-S Ising model: a free-fermion approximation

A free-fermion approximation procedure is applied to the mixed spin-1/2-spin-S Ising models. The author's results for the critical coupling Kc are in very satisfactory agreement with the high-temperature series results of Yousif and Bowers (1984).

A square-lattice spin-1 model in a magnetic field: Free-fermion approximation

A spin-1 model on a square lattice with − βH= Σ ij ( JS i S j + KS 2 i S 2 j ) + Σ i ( i λS i − ΔS 2 i ) is solved when e K cosh J=1and λ= π or zero by a free-fermion approximation procedure and an Ising-type critical line is found.

Free-fermion approach to the commensurate-incommensurate transition in a model of Si/W(110).

The existence and stability of the incommensurate phase in a lattice-gas model of Si adsorbed on W(110) is studied using domain-wall arguments and the free-fermion approximation. The incommensurate phase observed in a previous Monte Carlo study is shown to be stable above both the (5\ifmmode\times\else\texttimes\fi{}1) and (6\ifmmode\times\else\texttimes\fi{}1) phases in agreement with the general prediction for (p\ifmmode\times\else\texttimes\fi{}1) phases. The commensurate-incommensurate transition lines from the ordered (5\ifmmode\times\else\texttimes\fi{}1) and (6\ifmmode\times\else\texttimes\fi{}1) phases to the incommensurate phase are calculated using the low-temperature free-fermion approximation. The results are shown to be in good qualitative agreement with results from previous Monte Carlo simulations.

Nearest-neighbor Ising model with a uniaxial incommensurate phase and a Lifshitz point

The anisotropic triangular nearest-neighbor Ising model, with antiferromagnetic interactions, is studied. The phase diagram, as a function of temperature and field, has a complex structure. Two commensurate ordered phases are found. The transition to one is Ising-type. The second is reached from the disordered phase in one of two ways: either via an intermediate uniaxial incommensurate phase---which is reached at high temperatures by a Kosterlitz-Thouless transition and at low temperatures undergoes an incommensurate-commensurate transition---or by a single continuous transition. The latter transition is in the same universality class as the chiral three-state Potts model, with thermal exponent within 4% of that of the three-state Potts model. These two transition regimes are found to be separated by a Lifshitz point. The phase diagram and critical behavior were determined by analytical (symmetry analysis and free-fermion approximation) as well as numerical (phenomenological renormalization-group and Monte Carlo) methods.

Rectangular lattice gas model with competing interactions and the two-dimensional ANNNI model in a field

A lattice gas model with short range competing interactions for adsorption on (110) surfaces of fcc crystals, in particular for O/Pd(110), as well as its Ising analog, the two-dimensional ANNNI model with antiferromagnetic axial nearest and next-nearest neighbour interactions in a field, are studied using the free fermion approximation and Monte Carlo techniques. The phase diagrams display different commensurate phases and incommensurate regions. Static and dynamic aspects of topological defects (walls and dislocations) characterising the incommensurate structures are investigated.

Nearest-neighbor Ising model with an incommensurate phase and a Lifshitz point

An anisotropic triangular nearest-neighbor Ising model, with antiferromagnetic interactions, was studied. The phase diagram as a function of temperature and magnetic field was determined with the use of symmetry analysis, free-fermion approximation, phenomenological renormalization group, and Monte Carlo simulation. A massless phase, a Lifshitz point, and transition lines to commensurate phases were identified.

A Monte Carlo study of the asymmetric clock or chiral Potts model in two dimensions

The phase diagram of the two-dimensional, three-state chiral Potts or asymmetric clock model is studied using Monte Carlo techniques. The phase boundaries are compared to those obtained using the finite-size renormalization group and the free fermion approximation. The incommensurate phase is described in detail and crossover effects near the Lifshitz point are discussed.