Correlated correlation functions in random-bond ferromagnets

The two-dimensional random-bond Q-state Potts model is studied for Q near 2 via the perturbative renormalisation group to one loop. It is shown that weak disorder induces cross-correlations between the quenched-averages of moments of the two-point spin/spin and energy/energy correlation functions, which should be observable numerically in specific linear combinations of various quenched correlation functions. The random-bond Ising model in (2+ε) dimensions is similarly treated. As a byproduct, a simple method for deriving the scaling dimensions of all moments of the local energy operator is presented.     

Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line

We study the critical two-dimensional Ising model with a defect line (altered bond strength along a line) in the continuum limit. By folding the system at the defect line, the problem is mapped to a special case of the critical Ashkin-Teller model, the continuum limit of which is the Z₂ orbifold of the free boson, with a boundary. Possible boundary states on the Z₂ orbifold theory are explored, and a special case is applied to the Ising defect problem. We find the complete spectrum of boundary operators, exact two-point correlation functions and the universal term in the free energy of the defect line for arbitrary strength of the defect. We also find a new universality class of defect lines. It is conjectured that we have found all the possible universality classes of defect lines in the Ising model. Relative stabilities among the defect universality classes are discussed.     

Stochastic Quantization Approach for the Ising Model

The Ising model is studied in the fermionicformulation of the stochastic quantization. An exactstochastic equation is given for D = 2 and 3 and in aHartree approximation a method is developed for treating the two-point correlation functions.     

Perturbation calculation for the eight-vertex model around the Ising point for T → Tc−

The dispersion expansion for the two-point electric correlation function in the eight-vertex model is calculated to first order in the four-spin coupling in the scaling limit. The dispersion expansion for the spin-spin-energy density correlation function in the Ising model is used. The result is not a simple extension of the Fredholm structure in the Ising model.     

Phase structure of the FCC quartet Ising model

The phase structure of an Ising model with four-spin interactions defined on an FCC lattice is determined by Monte Carlo techniques. Our results show that this system undergoes a single, first order transition at the self-dual point. This resolves previous mutually contradictory conclusions which have been drawn about this model.     

Shot noise in the self-dual interacting resonant level model

By using two independent and complementary approaches, we compute exactly the shot noise in an out-of-equilibrium interacting impurity model, the interacting resonant level model at its self-dual point. An analytical approach based on the thermodynamical Bethe ansatz allows us to obtain the density matrix in the presence of a bias voltage, which in turn allows for the computation of any observable. A time-dependent density matrix renormalization group technique that has proven to yield the correct result for a free model (the resonant level model) is shown to be in perfect agreement with the former method.     

Dynamic critical behavior of the worm algorithm for the Ising model

We study the dynamic critical behavior of the worm algorithm for the two- and three-dimensional Ising models, by Monte Carlo simulation. The autocorrelation functions exhibit an unusual three-time-scale behavior. As a practical matter, the worm algorithm is slightly more efficient than the Swendsen-Wang algorithm for simulating the two-point function of the three-dimensional Ising model.     

Variational principle for regular and random Ising models on the cactus tree or on the usual lattice in the “cactus approximation”

The distribution functions and the free energy are expressed in terms of the effective fields for the regular and random Ising models of an arbitrary spin S on the generalized cactus tree. The same expressions apply to systems on the usual lattice in the “cactus approximation” in the cluster variation method. For an ensemble of random Ising models of an arbitrary spin S on the generalized cactus tree, the equation for the probability distribution function of the effective fields is set up and the averaged free energy is expressed in terms of the probability distribution. The same expressions apply to the system on the usual lattice in the “cactus approximation”. We discuss the quantities on the usual lattice when the system or the ensemble of random systems has the translational symmetry. Variational properties of the free energy for a system and of the averaged free energy for an ensemble of random systems are noted. The “cactus approximations” are applicable to the Heisenberg model as well as to the Ising model of an arbitrary spin, and to ensembles of random systems of these models.     

On the nature of the nearest singularities of the free energy in the neighborhood of a critical point

Algebraic functions are used to provide an example of multiple-valued functions which coincide with a model (single-valued) free energy on one sheet of the Riemann surface in the neighborhood of a critical point. For the case of homogeneous free energies andα=α′=0, there are enough conditions to determine the behavior of the nearest singularities (branch points) to the critical point of the algebraic function. If no other singularities are present these branch points would represent the spinodal line. The particular exponents of the two-dimensional Ising model are used to provide a specific example.     

Improved phenomenological renormalization schemes

An analysis is made of various methods of phenomenological renormalization based on finite-dimensional scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-dimensional scaling amplitude of the internal energy in the three-dimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-dimensional corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-dimensional equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical index.     

Gauge supersymmetric Ising model on a cubic lattice near the critical point in the free field representation

The nature of the set of free fields that represent the system at the critical point has been revealed by studying the correlation functions of the degrees of freedom of the gauge supersymmetric Ising model on the cubic lattice. The same set of free fields represents the continuous supersymmetric Abelian gauge theory. Thus, the name of the lattice system is appropriate. Comparison with the two-dimensional Ising model is given.     

On the mean-field Ising model in a random external field

We use a method developed by van Hemmen to obtain the free energy of the mean-field Ising model in a random external magnetic field. Some results of previous mean-field calculations are confirmed and generalized. The tricritical point in the global phase diagram is discussed in detail. We also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point.     

Nonanalytic features of the first order phase transition in the Ising model

The absence of the analytic continuation for the free energy near the point of the first order phase transition in thed-dimensional Ising model is proved. It is shown that thermodynamic functions in the metastable phase do not have certain values and can be derived only with an uncertainty. The asymptotic expansion near the point of the phase transition yields the values of thermodynamic functions with the same uncertainty.     

Interface free energy and critical line for the ising model with nearest and next-nearest-neighbor interactions

A method due to Müller-Hartmann and Zittartz is used to calculate the free energy of a diagonal interface in an Ising model on a square lattice with interactions between nearest and next-nearest neighbors. Instead of solving the full bulk problem, one only takes into account a simple subset of interface configurations. The subset, which is somewhat different from that considered by Müller-Hartmann and Zittartz, is chosen in such a way that exact results for the interface free energy are reproduced if the system has only nearest or only next-nearest-neighbor interactions. When the critical line of ferromagnetic transitions is obtained by demanding that the interface free energy as a function of both couplings vanish, a surprisingly accurate approximation is obtained.     

The random field method for the Ising model

The random field method is used for investigation of the Ising model. The generalization of the variational principle and a new representation for the free energy of the Ising model are proposed.     

Kac type models of semi-finite spin systems

The Ising model for systems with free surfaces and Kac type interaction is exactly solved. The free energy and the transition point are similar to those obtained for Valenta's model of thin films. The spherical model for the same system is also investigated, and has the same critical temperature.     

Theory of monomer-dimer systems

We investigate the general monomer-dimer partition function,P(x), which is a polynomial in the monomer activity,x, with coefficients depending on the dimer activities. Our main result is thatP(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e.x=0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away fromx=0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.Work supported by National Science Foundation Grant GP-26526.     

On the application of the critical minimum energy subspace method to disordered systems

We discuss the recent application to strongly disordered systems of the Critical Minimum Energy Subspace (CrMES) method, used to limit the energy subspace of the Wang-Landau sampling. We compare with our results on the 3D random field Ising model obtained by a multi-range Wang-Landau simulation over the whole energy range. We point out some problems that may arise when applying the CrMES scheme to models having a complex free energy landscape.     

Simulations on infinite-size lattices

We introduce a Monte Carlo method, as a modification of existing cluster algorithms, which allows simulations directly on systems of infinite size, and for quantum models also at beta = infinity. All two-point functions can be obtained, including dynamical information. When the number of iterations is increased, correlation functions at larger distances become available. Limits q-->0 and omega-->0 can be approached directly. As examples we calculate spectra for the d = 2 Ising model and for Heisenberg quantum spin ladders with two and four legs.     

Non-linear partial difference equations for the two-spin correlation function of the two-dimensional Ising model

For all temperatures, the two-point function of the two-dimensional Ising model is shown to be expressible in terms of the solution of a non-linear partial difference equation on the lattice. From this difference equation, the known results for the two-point function of the Ising field theory are regained by taking the scaling limit.