Analytic renormalization using many space-time dimensions

A renormalization method originally proposed by Ashmore is reformulated and shown to be, in fact, a renormalization. The method involves use of different complex dimensions associated with various subgraphs of a graph, and appears to combine the best features of complex dimensional and analytic renormalization.     

Inverse Monte Carlo renormalization group transformations for critical phenomena

We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.     

量子多体系统的线性张量重正化群方法

文章简述了数值重正化群方法的历史发展,包括威耳逊（Wilson）的数值重正化群算法,S.R.White的密度矩阵重正化群方法,以及近期迅速发展的处理强关联量子系统的几种张量网络态与张量网络算法.在此基础上,文章重点介绍了作者最近提出的用于研究量子多体系统热力学性质的线性张量重正化群方法,以及该方法在一维和二维量子系统中的应用.     

Tensor renormalization group approach to two-dimensional classical lattice models

We describe a simple real space renormalization group technique for two-dimensional classical lattice models. The approach is similar in spirit to block spin methods, but at the same time it is fundamentally based on the theory of quantum entanglement. In this sense, the technique can be thought of as a classical analogue of the density matrix renormalization group method. We demonstrate the method - which we call the tensor renormalization group method - by computing the magnetization of the triangular lattice Ising model.     

Dynamically driven renormalization group

We present a detailed discussion of a novel dynamical renormalization group scheme: the dynamically driven renormalization group (DDRG). This is a general renormalization method developed for dynamical systems with non-equilibrium critical steady state. The method is based on a real-space renormalization scheme driven by a dynamical steady-state condition which acts as a feedback on the transformation equations. This approach has been applied to open nonlinear systems such as self-organized critical phenomena, and it allows the analytical evaluation of scalling dimensions and critical exponents. Equilibrium models at the critical point can also be considered. The explicit application to some models and the corresponding results are discussed.     

Real space renormalization group theory of the percolation model

A cluster expansion renormalization group method in real space is-developed to determine the critical properties of the percolation model. In contrast to previous renormalization group approaches, this method considers the cluster size distribution (free energy) rather than the site or bond probability distribution (coupling constants) and satisfies the basic renormalization group requirement of free energy conservation. In the construction of the renormalization group transformation, new couplings are generated which alter the topological structure of the clusters and which must be introduced in the original system. Predicted values of the critical exponents appear to converge to presumed exact values as higher orders in the expansion are considered. The method can in principle be extended to different lattice structures, as well as to different dimensions of space.This paper is dedicated to Prof. Philippe Choquard.     

Complex networks renormalization: flows and fixed points

Recently, it has been claimed that some complex networks are self-similar under a convenient renormalization procedure. We present a general method to study renormalization flows in graphs. We find that the behavior of some variables under renormalization, such as the maximum number of connections of a node, obeys simple scaling laws, characterized by critical exponents. This is true for any class of graphs, from random to scale-free networks, from lattices to hierarchical graphs. Therefore, renormalization flows for graphs are similar as in the renormalization of spin systems. An analysis of classic renormalization for percolation and the Ising model on the lattice confirms this analogy. Critical exponents and scaling functions can be used to classify graphs in universality classes, and to uncover similarities between graphs that are inaccessible to a standard analysis.     

A renormalization group approach to a Yang–Mills two matrix model

A Yang–Mills type two matrix model with mass terms is studied by use of a matrix renormalization group approach proposed by Brezin and Zinn-Justin. The renormalization group method indicates that the model exhibits a critical behavior similar to that of two-dimensional Euclidean gravity. A massless limit and the generation of quadratic terms along the renormalization group flow are discussed.     

Summed analytic renormalization

With a small modification, analytic renormalization is shown to be multiplicative and the renormalization group equation is derived. For quantum electrodynamics the method is compatible with gauge invariance at the one-loop level.     

Differential real space renormalization: The linear Ising chain

The differential real space renormalization method, recently introduced by Hillhorst et al., is applied to the linear Ising chain. It is shown that chains with spatially homogeneous as well as inhomogeneous or quenched random interactions can be treated. For the first two cases the free energy is computed by renormalization. The discussion includes also the case with a magnetic field, higher order interactions and the behavior of correlation functions under renormalization.     

Quasiparticle renormalization of the memory function approach for response functions

The problem of renormalization of the double-time Green function method for response functions is treated in the framework of the Mori-theory for the special case of Fermi liquids and finite Fermi systems. It is shown that the quasiparticle-quasihole renormalization of the response function can be carried out under conditions which are physically equivalent to the conditions under which the quasiparticle-quasihole renormalization is performed in the many-time Green function theory of Fermi liquids.     

Linear response within the projector-based renormalization method: many-body corrections beyond the random phase approximation

The explicit evaluation of linear response coefficients for interacting many-particle systems still poses a considerable challenge to theoreticians. In this work we use a novel many-particle renormalization technique, the so-called projector-based renormalization method, to show how such coefficients can systematically be evaluated. To demonstrate the prospects and power of our approach we consider the dynamical wave-vector dependent spin susceptibility of the two-dimensional Hubbard model and also determine the subsequent magnetic phase diagram close to half-filling. We show that the superior treatment of (Coulomb) correlation and fluctuation effects within the projector-based renormalization method significantly improves the standard random phase approximation results.     

Monte Carlo Renormalization Group Study of Φ4 Theory on 4-Dimensional Random Triangle Laticte

A Monte Carlo renormalization group investigation of 4-dimensional Φ⁴ theory is performed. Using special block method, the Ising renormalization flows and the Gaussian renormalization flows have been obtained, and the critical exponent v has been measured. The critical behavior supports the prediction of the triviality of the 4-dimensional iP4 theory.     

无碰撞等离子体的相干重整化理论

本文提出对Власов-Poisson方程进行微扰处理的一种重整化方案。利用图形展开方法证明了该理论到任意阶微扰的可重整化性质。给出了重整化传播量的一般形式。分析了相干项和绝对非相干项的物理意义。给出了重整化介电函数的正确表示,并对它的意义做了讨论。通过和以往重整化理论的比较,指出这种重整化方案是一种真正的完全重整化。 关键词：     

Lectures on the functional renormalization group method

These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scaler theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.     

Momentum space renormalization for the sine-Gordon model

A momentum space renormalization is presented for the sine-Gordon model with an arbitrary cut-off function. In contrast to previous work the present method reproduces the slope of the critical line as found from the Kosterlitz renormalization of the equivalent Coulomb gas and is also applicable to the case of a sharp cut-off function.     

The renormalization constants of QED in unitary approximation

A unitary approximation for the time-evolution operator given through the exponential representation is used to calculate the renormalization constants of QED. The results obtained by this method are the same as the renormalization group improvement of the usual perturbation series taking into account terms up to the second order ine.