The Martinelli-Parisi expansion for the non-linear sigma model

The Martinelli-Parisi expansion around the Migdal-Kadanoff transformation is computed to second order in the shift parameter for the non-linear sigma model on a triangular lattice. The computation is performed in a completely analytical way in the region of weak coupling which is the most crucial region for checking the goodness of the expansion. The value of the beta function improves definitely with respect to the zero- and first-order results, reducing to 13% the error with regard to the exact value of continuum perturbation theory.     

Phase Transition of Ising Model on a Kind of Periodic Fractal Lattices

In this paper we introduced a periodic fractal structure (PFS) and studied the critical behavior of the nearest-neighbor-interacting Ising model within Migdal-Kadanoff bond-moving approximation. The critical curve equation and the formulas of the critical exponents were derived approximately in a special parameter region.     

Interfacial free energy of the two-dimensional Ising model from the renormalization group

The interfacial free energy of a two-dimensional Ising model is calculated by using various renormalization group schemes. The results obtained are quantitatively consistent with known exact results. In addition, a general discussion of various drawbacks within different renormalization group approximations is given. The best result are obtained with the 4×4 finite cluster approximation, while the Migdal-Kadanoff approximation seems to be inherently unsuitable for calculation of interfacial properties.     

Renormalization group (RG) study of the Ising model on higher-dimensional (d>1) quasi-lattices

We suggest two model quasi-lattices in two and three dimensions and use the Migdal-Kadanoff renormalization group scheme to calculate the fixed points and the correlation length exponent ν. The results strongly support the conjecture that Ising models on periodic and quasi-periodic lattices belong to the same universality class.     

Linked-Cluster Expansion of the Ising Model

The linked-cluster expansion technique for the high-temperature expansion of spin modes is reviewed. A new algorithm for the computation of three-point and higher Green's functions is presented. Series are computed for all components of two-point Green's functions for a generalized 3D Ising model, to 25th order on the bcc lattice and to 23rd order on the sc lattice. Series for zero-momentum four-, six-, and eight-point functions are computed to 21st, 19th, and 17th order respectively on the bcc lattice.     

Why temperature chaos in spin glasses is hard to observe

The overlap length of a three-dimensional Ising spin glass on a cubic lattice with Gaussian interactions has been estimated numerically by transfer matrix methods and within a Migdal-Kadanoff renormalization group scheme. We find that the overlap length is large, explaining why it has been difficult to observe spin glass chaos in numerical simulations and experiment.     

Julia sets and complex singularities in hierarchical Ising models

We study the analytical continuation in the complex plane of free energy of the Ising model on diamond-like hierarchical lattices. It is known [12, 13] that the singularities of free energy of this model lie on the Julia set of some rational endomorphismf related to the action of the Migdal-Kadanoff renorm-group. We study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin off. We prove that for almost all (with respect to the harmonic measure) geodesics the complex critical exponent is common, and compute it.     

A Trial of Variational-Cumulant Expansion on Simple Ising Model With Second Order Phase Transition

The standard variational-cumulant expansion (VC) combined with a scheme for the enumeration of connected diagrams is employed to study the simple Ising model with the second-order phase transition. This model is evaluated to the 5th order approximations. A comparison with the 2-dimensional exact results shows that thermodynamic quantities converge rather slowly to the exact values in the intermediate region. A strict analytical calculation of the critical temperature with the 5th order approxinlation both for J_i and cumulant expansion suggests that VC chn achieve rather high standard accuracy if the variational parameters J_i could be determined up to higher degrees of corrections rather than the first one. Anomalies are found for U, C and M at extremely low temperature. A detailed study of M implies that thermodynamic formulae should be carefully chosen in VC approach so as to avoid these anomalies.     

Expansion Around the Mean Field in Quantum Magnetic Systems

We introduce a new definition of ordered phase in a magnetic system based on properties of the local spin state probability. A statistical functional associated to this quantity depends both on amplitude and direction of the local magnetization. We show that it is possible to introduce an expansion at fixed magnetization amplitude in the inverse of lattice coordination number if the direction is selected by an extremum condition. In the limit of infinite coordination number we recover the mean field results. First order corrections are derived for the Ising model in the presence of a transverse field and for the XY model. Our results concerning critical temperature and order parameter compare favorably with other approaches.     

Temperature chaos, rejuvenation, and memory in Migdal-Kadanoff spin glasses

We use simulations within the Migdal-Kadanoff approach to probe the scales relevant for rejuvenation and memory in Ising spin glasses. First we investigate scaling laws for domain wall free energies and extract the chaos overlap length l(T,T'). Then we perform out of equilibrium simulations that follow experimental protocols. We find that (1) a rejuvenation signal arises at a length scale significantly smaller than l(T,T'), and (2) memory survives even if equilibration goes out to length scales larger than l(T,T'). Theoretical justifications of these phenomena are then considered.     

Onsager's reaction field for the Potts model from the path integral

A new path integral formulation for theq-state Potts model is proposed. This formulation reproduces known results for the Ising model (q=2) and naturally extends these results for arbitraryq. The mean field results for both the Ising and the Potts models are obtained as a leading saddle point contribution to the corresponding functional integrals, while the systematic computation of corrections to the saddle point contribution produces the Onsager reaction field terms, which forq=2 coincide with results already known for the Ising model.     

Flame capturing with an advection–reaction–diffusion model

We conduct several verification tests of the advection–reaction–diffusion flame-capturing model, developed by Khokhlov in 1995 for subsonic nuclear burning fronts in supernova simulations. We find that energy conservation is satisfied, but there is systematic error in the computed flame speed due to thermal expansion, which was neglected in the original model. We decouple the model from the full system, determine the necessary corrections for thermal expansion, and then demonstrate that these corrections produce the correct flame speed. The flame-capturing model is an alternative to other popular interface tracking techniques, and might be useful for applications beyond astrophysics.     

Spin-spin correlation function in the two-dimensional Ising model with linear defects. II.T>T c

The dispersion expansion for the spin correlation function in the two-dimensional Ising model with linear defects aboveT _c is derived. The asymptotic behavior is computed by a steepest descent analysis. The lattice is divided into four domains with different asymptotic behaviors. In particular, the correlation length inside certain domains is a function of the defect.     

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.     

Spin-spin correlation function in the two-dimensional Ising model with linear defects. I.T<T c

The dispersion expansion for the spin correlation function in the two-dimensional Ising model with linear defects belowT _c is derived. The asymptotic behavior is computed by a steepest descent analysis. The lattice is divided into four domains with different asymptotic behaviors. In particular, the correlation length inside certain domains is a function of the defect.     

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.     

Finite size effects for the Ising model on random graphs with varying dilution

We investigate the finite size corrections to the equilibrium magnetization of an Ising model on a random graph with N nodes and N^γ edges, with 1<γ≤2. By conveniently rescaling the coupling constant, the free energy is made extensive. As expected, the system displays a phase transition of the mean-field type for all the considered values of γ at the transition temperature of the fully connected Curie-Weiss model. Finite size corrections are investigated for different values of the parameter γ, using two different approaches: a replica based finite N expansion, and a cavity method. Numerical simulations are compared with theoretical predictions. The cavity based analysis is shown to agree better with numerics.     

Critical Behavior of Ising Model with Long Range Correlated Quenched Impurities

The theoretic renormalization-group approach is applied to the study of the critical behavior of the ddimensional Ising model with long-range correlated quenched impurities, which has a power-like correlations r^-(d-p). The asymptotic scaling law is studied in the framework of the expansion in e = 4 - d. In d ~ 4, the dynamic exponent z .is calculated up to the second order in p with ρ= O(ε^1/2). The shape function is obtained in one-loop calculation. When d = 4, the logarithmic corrections to the critical behavior are found. The finite size effect on the order parameter relaxation rate is also studied.     

Calculation of structure factors for a simple model of adsorption on a surface by real-space renormalization-group method

The structure factor S(q) of a simple model describing adsorption on a two-dimensional surface is computed by real-space renormalization-group technique for T ? T_c. The method used is based on first-order cumulant expansion implemented with an ad hoc recursion relation for the coupling constants. The model is equivalent to an antiferromagnetic Ising model in an external field.     

The four-dimensional site-diluted Ising model: A finite-size scaling study

Using finite-size scaling techniques, we study the critical properties of the site-diluted Ising model in four dimensions. We carry out a high-statistics Monte Carlo simulation for several values of the dilution. The results support the perturbative scenario: there is only the Ising fixed point with large logarithmic scaling corrections. We obtain, using the Perturbative Renormalization Group, functional forms for the scaling of several observables that are in agreement with the numerical data.