Renormalization Group Study of Ising Transition in Double－Frequency sine－Gordon Model

We utilize the renormallzation group (RG) technique to analyze the Ising critical behavior in the double frequency Sine-Gordon model. The one-loop RG equations obtained show unambiguously that there exist two Ising critical points besides the trivial Gaussian fixed point. The topology of the RG flows is obtained as well.     

A Renormalization Group Study of the Ising Model on a Triangular Lattice with Long-Range Interactions

The critical exponents of the triangular lattice Ising model with long-range interactions γ^-s are calculated by the real space renormalization group. Using the simplest Kadanoff blocks and the lowest approximation of cumulant expansion, it is shown that there exists a finite critical temperature when 4(1 - ㏑2/㏑3) < s < 4.     

The Renormalization Group Treatment of the Hubbard Model

The article presents the renormalization group treatment to the Hubbard model. To begin with, the bosonization of Hubbard model Hamiltonian is performed. We have obtained the sine-Gordon Hamiltonian. We have further approximated this Hamiltonian by the Hamiltonian of ⁴-theory. Then we utilized Wilson's results of the renormalization group method and obtained the recursion formula for the Hubbard model. Having solved these formulas we have obtained the critical indices for the Hubbard model.     

Renormalization of one-dimensional avalanche models

We investigate a renormalization group (RG) scheme for avalanche automata introduced recently by Pietroneroet al. to explain universality in self-organized criticality models. Using a modified approach, we construct exact RG equations for a one-dimensional model whose detailed dynamics is exactly solvable. We then investigate in detail the effect of approximations inherent in a practical implementation of the RG transformation where exact dynamical information is unavailable.     

Ising Transition in Dimerized XY Quantum Spin Chain

We proposed a simple spin-1/2 model which provides an exactly solvable example to study the Ising criticality with central charge c=1/2.By mapping it onto the real Majorana fermions,the Ising critical behavior in explored explicitly,although its bosonized form is not the double frequency sine-Gordon model.     

采用正方元胞的Ising模型实空间重正化群解

在二维正方形晶格上,将元胞取为4格点正方形,采用3种不同的规则定义块自旋状态,进行了重正化群计算,得出了更为精确的结果;解决了元胞内格点数为偶数的重正化群计算问题.     

Critical Exponents of the Diluted Ising Model between Dimensions 2 and 4

Within the massive field-theoretic renormalization-group approach the expressions for the and functions of the anisotropic mn-vector model are obtained for general space dimension d in three-loop approximation. Resumming corresponding asymptotic series, critical exponents for the case of the weakly diluted quenched Ising model (m = 1, n = 0), as well as estimates for the marginal order parameter component number m _c of the weakly diluted quenched m-vector model, are calculated as functions of d in the region 2 d < 4. Conclusions concerning the effectiveness of different resummation techniques are drawn.     

Improvement of Renormalization Group for Barenblatt Equation

We construct a Hartree-Fock (self-consistent)-like algorithm with renormalization group (RG) approach to calculate the anomalous dimension in a nonlinear diffusion equation. We find that our result improves the original RG work because we include the effect of Heaviside function.     

Exact Renormalization Group for the Brazovskii Model of Striped Patterns

We consider the coarse graining of the generalized Brazovskii free energy functional for striped patterns. The technique developed by Shankar for the Fermi liquids is combined with the irreducible version of the exact renormalization group to calculate the recursion relations for interaction vertices. We perform the one-loop calculations from this method taking the eight-point vertex into account.     

Improvement of Renormalization Group for Barenblatt Equation

We construct a Hartree-Fock (seff-consistent)-like algorithm with renormalization group (RG) approach to calculate the anomalous dimension in a nonlinear diffusion equation. We find that our result improves the original RG work because we include the effect of Heaviside function.     

Universality and Conformal Invariance for the Ising Model in Domains with Boundary

The partition function with boundary conditions for various two-dimensional Ising models is examined and previously unobserved properties of nonformal invariance and universality are established numerically.     

Non-Universality in Ising Models with Four Spin Interaction

We consider two bidimensional classical Ising models, coupled by a weak interaction bilinear in the energy densities of the two systems; the model contains, as limiting cases, the Ashkin–Teller and the Eight-vertex models for certain values of their parameters. We write the energy–energy correlations and the specific heat as Grassman integrals formally describing Dirac 1+1 dimensional interacting massive fermions on a lattice, and an expansion based on Renormalization Group is written for them, convergent up to temperatures very close to the critical temperature for small coupling. The asymptotic behaviour is determined by critical indices which are continuous functions of the coupling.     

Renormalization Group Approach to Multiscale Modelling in Materials Science

Dendritic growth, and the formation of material microstructure in general, necessarily involves a wide range of length scales from the atomic up to sample dimensions. The phase field approach of Langer, enhanced by optimal asymptotic methods and adaptive mesh refinement, copes with this range of scales, and provides an effective way to move phase boundaries. However, it fails to preserve memory of the underlying crystallographic anisotropy, and thus is ill-suited for problems involving defects or elasticity. The phase field crystal (PFC) equation—a conserving analogue of the Swift-Hohenberg equation—is a phase field equation with periodic solutions that represent the atomic density. It can natively model elasticity, the formation of solid phases, and accurately reproduces the nonequilibrium dynamics of phase transitions in real materials. However, the PFC models matter at the atomic scale, rendering it unsuitable for coping with the range of length scales in problems of serious interest. Here, we show that a computationally-efficient multiscale approach to the PFC can be developed systematically by using the renormalization group or equivalent techniques to derive appropriate coarse-grained coupled phase and amplitude equations, which are suitable for solution by adaptive mesh refinement algorithms. PACS numbers: 81.16.Rf, 05.10.Cc, 61.72.Cc, 81.15.Aa     

Renormalization Group for Strongly Coupled Maps

Systems of strongly coupled chaotic maps generically exhibit collective behavior emerging out of extensive chaos. We show how the well-known renormalization group (RG) of unimodal maps can be extended to the coupled systems, and in particular to coupled map lattices (CMLs) with local diffusive coupling. The RG relation derived for CMLs is nonperturbative, i.e., not restricted to a particular class of configurations nor to some vanishingly small region of parameter space. After defining the strong-coupling limit in which the RG applies to almost all asymptotic solutions, we first present the simple case of coupled tent maps. We then turn to the general case of unimodal maps coupled by diffusive coupling operators satisfying basic properties, extending the formal approach developed by Collet and Eckmann for single maps. We finally discuss and illustrate the general consequences of the RG: CMLs are shown to share universal properties in the space-continuous limit which emerges naturally as the group is iterated. We prove that the scaling properly ties of the local map carry to the coupled systems, with an additional scaling factor of length scales implied by the synchronous updating of these dynamical systems. This explains various scaling laws and self-similar features previously observed numerically.     

The Renormalization Group and Optimization of Entropy

We illustrate the possible connection that exists between the extremal properties of entropy expressions and the renormalization group (RG) approach when applied to systems with scaling symmetry. We consider three examples: (1) Gaussian fixed-point criticality in a fluid or in the capillary-wave model of an interface; (2) Lévy-like random walks with self-similar cluster formation; and (3) long-ranged bond percolation. In all cases we find a decreasing entropy function that becomes minimum under an appropriate constraint at the fixed point. We use an equivalence between random-walk distributions and order-parameter pair correlations in a simple fluid or magnet to study how the dimensional anomaly at criticality relates to walks with long-tailed distributions.     

Replicon Modes in the Renormalization Group with Replica Symmetry Breaking

Abstract The renormalization group with replica symmetry (RSB) breaking is studied for the disordered ferromagnets. The ε-expansion (ε= 4-D, D is the spatial dimension) is used. For fixed points with step-like RSB structufe replicon eigqnvalpes are defined to describe the stability of the fixed points with respect to the continuous lESB perturbation.It is shown that this metliod can be generalized to the cases in which the number of the steps in the step-like RSB structure ahd the order of expansion are arbitrary.     

Block Renormalization Group Approach for Correlation Functions of Interacting Fermions

Inspired by a decomposition of the lattice Laplacian operator into massive terms (coming from the use of the block renormalization group transformation for bosonic systems), we establish a telescopic decomposition of the Dirac operator into massive terms, with a property named orthogonality between scales. Making a change of Grassmann variables and writing the initial fields in terms of the eigenfunctions of the operators related to this decomposition, we propose a multiscale structure for the generating function of interacting fermions. Due to the orthogonality property we obtain simple formulas, establishing a trivial link between the correlation functions and the effective potential theories. In particular, for the infrared analysis of some asymptotically free models, the two point correlation function is written as a dominant term (decaying at large distances as the free propagator) plus a correction with faster decay, and the study of both terms is straightforward once the effective potential theory is controlled.     

O(N)模型破缺相的红外重正化群计算

利用一个与质量有关的重正化方案,研究了O(N)模型破缺相的红外重正化群性质.得到了与质量有关的一圈重正化群系数以及在红外极限下的二圈重正化群系数.利用标准的微扰场论计算方法,发现特殊的抵消效应,该效应使得二圈的重正化群系数与一圈重正化群系数在红外极限下相同.     

Renormalization Group Flow of the Two-Dimensional O(N) Spin Model

We develop a new block spin transformation and apply it to the 2D O(N) spin model. The transformation does not yield complicated non-local terms and then the transformation recursion formula seems to be controllable for any initial inverse temperature > 0. The main part of the block spin transformation of the model with large N converges to a massive state, no matter how low the initial temperature 1/ is, and is close to the flow of the hierarchical model advocated by Dyson and Wilson several decades ago.