Application of the real space dynamic renormalization group method to the one-dimensional kinetic ising model

We apply the recently developed real space dynamic renormalization group method to the one-dimensional kinetic Ising model. We show how one can develop block spin methods that lead to recursion relations for the space and time dependent correlation functions that correspond to the observables for this system. We point out the importance of carefully choosing the appropriate parameters governing the behavior of individual blocks of spins and the necessity of worrying about the high temperature properties of the temperature recursion relations if one is to obtain the proper exponential decay of correlation functions at large distances away from the critical point at zero temperature. We systematically investigate the accuracy of our approximate recursion relations for various correlation functions by checking them against the known exact results. Our simple methods work surprisingly well over a wide range of temperatures, wavenumbers and frequencies.     

Solvable real space renormalization group for the kinetic Ising chain

A real-space renormalization group for the one-dimensional kinetic Ising model is established. The parameter space of the model must be enlarged to include non-Markovian kernels in the equation of motion. The recursion relations for these kernels can be iterated analytically so that the global flow under the renormalization group can be traced exactly. The resulting fixed-point equation is non-Markovian.     

The irrelevance of detailed balance for the dynamical critical exponent

We construct the most general real space renormalization for the two-dimensional kinetic Ising model on the triangular lattice which, to second order in the high-temperature expansion, conserves detailed balance and avoids fast transients for the cell spins. We show the corresponding dynamical recursion relations (as well as the exponentz) to be unaltered with respect to the ones found, in a previous paper, for a completely different class of transformations. This finding resolves long-standing confusions and controversies.     

Irrelevant operators and momentum-shell recursion relations ind=2 + ∈ dimensions

The momentum-shell recursion relations of Nelson and Pelcovits for ann-vector model near two dimensions are reexamined. The renormalization of the infinite set of relevant and marginal operators present in the system is studied. Ambiguities obtained in the ensuing recursion relations are shown to involve irrelevant operators only, thus justifying the procedure of Nelson and Pelcovits. The cases of finite external fieldh and finite spin anisotropyg are both considered.     

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.     

嵌套正方格子上反铁磁高斯模型的相变

采用等效变化的方法,把嵌套正方晶格转化为可求解的正方晶格。利用重整化群变换,我们求得了正方系统的临界点。结合本文中给出的两个变换关系,得到了嵌套正方晶格上反铁磁高斯模型的临界点为 K=-0.707b。     

An introduction to the fundamentals of the renormalization group in critical phenomena

The fundamental concepts underlying the application of the renormalization group and related techniques to critical phenomena are reviewed at an elementary level. Topics discussed include: the definition of the renormalization group as a functional integral over high momentum components of the spin field, the behaviour of the renormalization group near the fixed point and the derivation of scaling, Wilson's approximate recursion relation, trivial and non-trivial fixed points of isotropic spin systems near d = 4, Feynman graph expansions for critical exponents, ? = 4 ? d and 1/n-expansions, the derivation of exact recursion relations and co-ordinate space transformations for d = 2 Ising systems     

嵌套正方格子上反铁磁高斯模型的相变

采用等效变换的方法,把嵌套正方晶格转化为可求解的正方晶格.利用重整化群变换,我们求得了正方系统的临界点.结合本文中给出的两个变换关系,得到了嵌套正方晶格上反铁磁高斯模型的临界点为K*=-0.707b.     

格点规范理论(Ⅰ)

本文介绍了由Wilson等人发展起来的处理粒子间强相互作用的格点规范理论。由于这个理论是建立在点阵上的规范理论,故首先讨论了点阵上体系的场论性质和统计物理性质之间的联系,介绍了处理粒子禁闭问题的Wilson判据,点阵的哈密顿形式。然后讨论了各种具体模型的计算方法,如规范场的点阵模型、紧致QED模型、费米子模型、阿贝尔Higgs模型等。在此基础上,总结出Wilson定理。本文也讨论了格点规范理论中的实空间重正化群方法,介绍了Heisenberg平面模型的重正化群分析,一维的二维的复现关系及Migdal近似。最后评介了近年来对于Wilson回路算子的一些研究,内容包括’t Hooft代数和Wilson回路算子方程等。     

Field Diffeomorphisms and the Algebraic Structure of Perturbative Expansions

We consider field diffeomorphisms in the context of real scalar field theories. Starting from free field theories we apply non-linear field diffeomorphisms to the fields and study the perturbative expansion for the transformed theories. We find that tree-level amplitudes for the transformed fields must satisfy BCFW type recursion relations for the S-matrix to remain trivial. For the massless field theory these relations continue to hold in loop computations. In the massive field theory the situation is more subtle. A necessary condition for the Feynman rules to respect the maximal ideal and co-ideal defined by the core Hopf algebra of the transformed theory is that upon renormalization all massive tadpole integrals (defined as all integrals independent of the kinematics of external momenta) are mapped to zero.     

Phase transitions and renormalized structure of lattice gauge theories with fermions

Using recursion relations of the type proposed by Migdal and Kadanoff we discuss the fixed points relevant to the renormalization of lattice gauge theories in four dimensions. The role of the topological excitation for the U(1) case is evaluated.     

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.     

Generalized renormalization group treatment of the growth kinetics of unstable systems

We discuss the generalization of a renormalization group technique developed previously for the study of ordering in unstable systems in the context of the ferromagnetic Ising model with spin-flip dynamics. Difficulties encountered in earlier work are eliminated through the use of new recursion relations dependent on a continuous spatial rescaling factorb 1. A more careful analysis and implementation of the approximation scheme is carried out. Our improved method allows the study of the anisotropy of the time-dependent structure factor and the pre-scaling behavior of the shape function.     

Phase diagram of the two-dimensional ferromagnetic three-color Ashkin-Teller model

In the present work we study the critical properties of the ferromagnetic three-color Ashkin-Teller model (3AT) by means of a Migdal-Kadanoff renormalization group approach on a diamond-like hierarchical lattice. The analysis of the fixed points and flux diagram of the recursion relations is used to determine the corresponding phase diagram (including its symmetry properties) and critical exponents. Our numerical results show the presence of four universality classes, three of them are associated to the Potts model with q=2, 4 and 6 states. Finally, a connection between our findings and some known results from the literature is presented.     

Distillation protocols for mixed states of multilevel qubits and the quantum renormalization group

We study several properties of distillation protocols to purify multilevel qubit states (qudits) when applied to a certain family of initial mixed bipartite states. We find that it is possible to use qudits states to increase the stability region obtained with the flow equations to distill qubits. In particular, for qutrits we get the phase diagram of the distillation process with a rich structure of fixed points. We investigate the large-D limit of qudits protocols and find an analytical solution in the continuum limit. The general solution of the distillation recursion relations is presented in an appendix. We stress the notion of weight amplification for distillation protocols as opposed to the quantum amplitude amplification that appears in the Grover algorithm. Likewise, we investigate the relations between quantum distillation and quantum renormalization processes.Received: 23 April 2003, Published online: 12 August 2003PACS: 03.67.-a Quantum information - 03.67.Lx Quantum computation     

Migdal-Kadanoff study of Z4 symmetric systems with generalized action

For the Z₄ lattice gauge theory in four dimensions or the related spin system in two dimensions, we consider real space renormalization group transformations on the most general single plaquette or nearest neighbor action. The Migdal-Kadanoff recursion relations in differential form have several fixed points and predict a rich phase structure.     

Mass generations in two-dimensional hierarchical Heisenberg model of Migdal-Kadanoff type

Finiteness of correlation length in the 2D Heisenberg model is established within the Migdal-Kadanoff approximate renormalization recursion formulas.     

Non-perturbative effects in two-dimensional lattice O(N) models

Non-abelian analogues of Kosterlitz-Thouless vortices may have important effects in two-dimensional lattice spin systems with O(N) symmetries. Renormalization group equations which include these effects are developed in two ways. The first set of equations extends the renormalization group equations of Kosterlitz to O(N) spin systems, in a form suggested by Cardy and Hamber. The second is derived from a Villain-type O(N) model using Migdal's recursion relations. Using these equations, the part played by topological excitations in the crossover from weak to strong coupling behavior is studied. Another effect which influences crossover behavior is also discussed: irrelevant operators which occur naturally in lattice theories can make important contributions to the renormalization group flow in the crossover region. When combined with conventional perturbative results, these two effects may explain the observed crossover behavior of these models.     

On the large-N limit of 3D and 4D hermitian matrix models

The large-N limit of the hermitian matrix model in three and four euclidean space-time dimensions is studied with the help of the approximate Renormalization Group recursion formula. The planar graphs contributing to wave-function, mass and coupling-constant renormalization are identified and summed in this approximation. In four dimensions the model fails to have an interacting continuum limit, but in three dimensions there is a non-trivial fixed point for the approximate RG relations. The critical exponents of the three-dimensional model at this fixed point are ν = 0.67 and η = 0.20. The existence (or non-existence) of the fixed point and the critical exponents display a fairly high degree of universality since they do not seem to depend on the specific (non-universal) assumptions made in the approximation.     

Monte Carlo renormalization group study of the percolation problem of discs with a distribution of radii

A real space renormalization group is formulated for continuum (off-lattice) percolation problems. It is applied to the system of overlapping discs with a variety of distributions of disc radii. Monte Carlo method is used for obtaining recursion relations. The results support universality: The Harris criterion seems to work for percolation. The position of the critical point shows stability against introducing a distribution in the disc radii.Supported in part by SFB 125 Aachen-Jülich-Köln