Semi-infinite Ising model

For the semi-infinite Ising model in two or more dimensions, we prove analyticity properties of the surface free energy and map out the phase diagram in the absence of an external magnetic field. We prove that this phase diagram contains critical lines where the parallel and/or the transverse correlation lengths diverge. The critical exponent,v _⊥, of the transverse correlation length is shown to be equal to the exponentv of the Ising model on an infinite lattice. In a second paper, these results will be used to analyze the wetting transition.     

The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

We define a new percolation model by generalising the FK representation of the Ising model, and show that on the triangular lattice and at high temperatures, the critical point in the new model corresponds to the Ising model. Since the new model can be viewed as Bernoulli percolation on a random graph, our result makes an explicit connection between Ising percolation and critical Bernoulli percolation, and gives a new justification of the conjecture that the high temperature Ising model on the triangular lattice is in the same universality class as Bernoulli percolation.     

Deconfinement transition and dimensional cross-over in the 3D gauge Ising model

We present a high-precision Monte Carlo study of the finite-temperature gauge theory in 2 + 1 dimensions. The duality with the 3D Ising spin model allows us to use powerful cluster algorithms for the simulations. For temporal extensions of up to N_t = 16 we obtain the inverse critical temperature with a statistical accuracy comparable with the most accurate results for the bulk phase transition of the 3D Ising model. We discuss the predictions of T.W. Capehart and M.E. Fisher for the dimensional cross-over from 2 to 3 dimensions. Our precise data for the critical exponents and critical amplitudes confirm the Svetitsky-Yaffe conjecture. We find deviations from Olesen's prediction for the critical temperature of about 20%.     

A Solvable Decorated Ising Lattice Model

A decorated lattice is suggested and the Ising model on it with three kinds of interactions K₁, K₂, and K₃ is studied. Using an equivalent transformation, the square decorated Ising lattice is transformed into a regular square Ising lattice with nearest-neighbor, next-nearest-neighbor, and four-spin interactions, and the critical fixed point is found at K₁=0.5769, K₂=-0.0671, and K₃=0.3428, which determines the critical temperature of the system. It is also found that this system and the regular square Ising lattice, and the eight-vertex model belong to the same universality class.     

Union Jack晶格上伊辛模型的自由费密近似解

运用自由费密近似对Union Jack晶格上具有各向异性二体耦合作用及三体相互作用的伊辛模型进行了求解,得到了模型的自由能、自发磁矩和临界点方程。在耦合常数简化为正方晶格上的伊辛模型时,得到了与Onsager一致的解。     

On the Ising model for the simple cubic lattice

The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.     

Condensation and metastability in the 2D Potts model

For the first order transition of the Ising model below , Isakov has proven that the free energy possesses an essential singularity in the applied field. Such a singularity in the control parameter, anticipated by condensation theory, is believed to be a generic feature of first order transitions, but too weak to be observable. We study these issues for the temperature driven transition of the q states 2D Potts model at . Adapting the droplet model to this case, we relate its parameters to the critical properties at and confront the free energy to the many informations brought by previous works. The essential singularity predicted at the transition temperature leads to observable effects in numerical data. On a finite lattice, a metastability domain of temperatures is identified, which shrinks to zero in the thermodynamical limit. Received 30 March 1999     

On the properties of the Ising antiferromagnets in the maximum critical field

We demonstrate that all Ising antiferromagnets with general spin S and arbitrary many-neighbour interactions in the maximum critical field have highly degenerate ground states accompanied with nonzero residual entropies. For finite S, the residual entropies vanish when the range of interaction tends to infinity. The proof is realised by establishing bounds for residual entropies in the case of an Ising system situated on a lattice with arbitrary number of dimensions. In addition we estimate the ground-state entropies for a few two-dimensional lattices.     

Critical temperature for random planar model with frustration

The critical temperature for a quenched Ising model on the square lattice, with vertical random interactions is found exactly. For the usual mixed ferro- and antiferromagnetic δ-distribution of the bonds, the calculated phase transition temperature versus the concentration of antiferromagnetic impurities is in agreement with the results of various molecular field approximations and of the renormalization group calculations on random planar Ising models.     

The Phase Transition of the Quantum Ising Model is Sharp

An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d+1 dimensions. A so-called ‘random-parity’ representation is developed for the latter model, similar to the random-current representation for the classical Ising model on a discrete lattice. Certain differential inequalities are proved. Integration of these inequalities yields the sharpness of the phase transition, and also a number of other facts concerning the critical and near-critical behaviour of the model under study.     

Ising deconfinement transition between Feshbach-resonant superfluids

We investigate the phase diagram of bosons interacting via Feshbach-resonant pairing interactions in a one-dimensional lattice. Using large scale density matrix renormalization group and field theory techniques we explore the atomic and molecular correlations in this low-dimensional setting. We provide compelling evidence for an Ising deconfinement transition occurring between distinct superfluids and extract the Ising order parameter and correlation length of this unusual superfluid transition. This is supported by results for the entanglement entropy which reveal both the location of the transition and critical Ising degrees of freedom on the phase boundary.     

Critical and multicritical behavior in the Ising–Heisenberg universality class

Critical behavior of three-dimensional classical frustrated antiferromagnets with a collinear spin ordering and with an additional twofold degeneracy of the ground state is studied. We consider two lattice models, whose continuous limit describes a single phase transition with a symmetry class differing from the class of non-frustrated magnets as well as from the classes of magnets with non-collinear spin ordering. A symmetry breaking is described by a pair of independent order parameters, which are similar to order parameters of the Ising and O(N) models correspondingly. Using the renormalization group method, it is shown that a transition is of first order for non-Ising spins. For Ising spins, a second order phase transition from the universality class of the O(2) model may be observed. The lattice models are considered by Monte Carlo simulations based on the Wang–Landau algorithm. The models are a ferromagnet on a body-centered cubic lattice with the additional antiferromagnetic exchange interaction between next-nearest-neighbor spins and an antiferromagnet on a simple cubic lattice with the additional interaction in layers. We consider the cases N = 1, 2, 3 and in all of them find a first-order transition. For the N = 1 case we exclude possibilities of the second order or pseudo-first order of a transition. An almost second order transition for large N is also discussed.     

Percolation transition in an irreversible-surface reaction model

Monte Carlo simulation studies of percolation transition in a surface reaction model describing the oxidation of carbon mono-oxide on a catalytic surface are presented. The percolation transition for adsorbed oxygen atoms occurs below the poisoning transition where carbon mono-oxide completely covers the surface of the catalyst and takes place for an oxygen coverage of about 0.525 which is close to the percolation transition in an Ising lattice gas with nearest-neighbor attractive interactions. In several respects the oxygen clusters near the percolation threshold resemble those of the Ising lattice gas near its critical point.     

Finite-size scaling approach for critical wetting: rationalization in terms of a bulk transition with an order parameter exponent equal to zero

Clarification of critical wetting with short-range forces by simulations has been hampered by the lack of accurate methods to locate where the transition occurs. We solve this problem by developing an anisotropic finite-size scaling approach and show that then the wetting transition is a "bulk" critical phenomenon with order parameter exponent equal to zero. For the Ising model in two dimensions, known exact results are straightforwardly reproduced. In three dimensions, it is shown that previous estimates for the location of the transition need revision, but the conclusions about a slow crossover away from mean-field behavior remain unaltered.     

Ising Model on an Infinite Ladder Lattice

In this paper we propose an Ising model on an infinite ladder lattice, which is made of two infinite Ising spin chains with interactions. It is essentially a quasi-one-dimessional Ising model because the length of the ladder lattice is infinite, while its width is finite. We investigate the phase transition and dynamic behavior of Ising model on this quasi-one-dimessional system. We use the generalized transfer matrix method to investigate the phase transition of the system. It is found that there is no nonzero temperature phase transition in this system. At the same time, we are interested in Glauber dynamics. Based on that, we obtain the time evolution of the local spin magnetization by exactly solving a set of master equations.     

Ising Model on an Infinite Ladder Lattice

In this paper we propose an Ising model on an infinite ladder lattice, which is made of two infinite Ising spin chains with interactions. It is essentially a quasi-one-dimessional Ising model because the length of the ladder lattice is infinite, while its width is finite. We investigate the phase transition and dynamic behavior of Ising model on this quasi-one-dimessional system. We use the generalized transfer matrix method to investigate the phase transition of the system. It is found that there is no nonzero temperature phase transition in this system. At the same time, we are interested in Glauber dynamics. Based on that, we obtain the time evolution of the local spin magnetization by exactly solving a set of master equations.     

The Ising model and percolation on trees and tree-like graphs

We calculate the exact temperature of phase transition for the Ising model on an arbitrary infinite tree with arbitrary interaction strengths and no external field. In the same setting, we calculate the critical temperature for spin percolation. The same problems are solved for the diluted models and for more general random interaction strengths. In the case of no interaction, we generalize to percolation on certain tree-like graphs. This last calculation supports a general conjecture on the coincidence of two critical probabilities in percolation theory.Research partially supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship     

Criticality in one dimension with inverse square-law potentials

We demonstrate that the scaled order parameter for ferromagnetic Ising and three-state Potts chains with inverse square interactions exhibits a universal critical jump, in analogy with the superfluid density in helium films. Renormalization-group arguments are combined with numerical simulations of systems containing up to 10(6) lattice sites to accurately determine the critical properties of these models. In strong contrast with earlier work, compelling quantitative evidence for the Kosterlitz-Thouless-like character of the phase transition is provided.     

Ferromagnetic phase transition for the spanning-forest model (q-->0 limit of the Potts model) in three or more dimensions

We present Monte Carlo simulations of the spanning-forest model (q-->0 limit of the ferromagnetic Potts model) in spatial dimensions d=3, 4, 5. We show that, in contrast to the two-dimensional case, the model has a ferromagnetic second-order phase transition at a finite positive value w(c). We present numerical estimates of w(c) and of the thermal and magnetic critical exponents. We conjecture that the upper critical dimension is 6.     

Gauge-invariant lattice gas for the microcanonical Ising model

We introduce a lattice gas for particles with discrete momenta (1, 0, –1) and local deterministic microdynamics, which exactly reproduces Creutz's microcanonical algorithm for the ferromagnetic Ising model. However, because of the manifest gauge invariance of our variables, both the Ising ferromagnetic and spin-glass systems share precisely the same dynamics with different initial conditions. Additional conservation laws in the 1D Ising case result in a completely integrable system in the limit of zero or unbounded demon energy cutoff. Numerical investigations of ergodicity are presented for the pure Ising lattice gas in one and two dimensions.