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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Operator renormalization group

D. Horn, W. G. J. Langeveld, Helen R. Quinn, Marvin Weinstein

Published Nov 15, 1988• 6 citations
Publisher

Abstract

We introduce a novel operator renormalization-group method. This is a new and more powerful variant of the t expansion combining that method with the real-space renormalization-group approach. The aim is to extract infinite-volume physics at t\ensuremath{\rightarrow}\ensuremath{\infty} from calculations of only a few powers of t. Good results are obtained for the (1+1)-dimensional Ising model. The method readily generalizes to higher-dimensional spin theories and to a gauge-invariant treatment of gauge theories. These theories will require greater computational power.

DOI
10.1103/physrevd.38.3238

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We introduce a novel operator renormalization-group method. This is a new and more powerful variant of the t expansion combining that method with the real-space renormalization-group approach. The aim is to extract infinite-volume physics at t\ensuremath{\rightarrow}\ensuremath{\infty} from calculations of only a few powers of t. Good results are obtained for the (1+1)-dimensional Ising model. The method readily generalizes to higher-dimensional spin theories and to a gauge-invariant treatment of gauge theories. These theories will require greater computational power.
The results of a new hybrid method for calculating ground-state energies for lattice Hamiltonians are presented. This method combines the ttt expansion with the real-space renormalization-group approach, with the hope of extracting infinite-volume physics at $t\ensuremath{\rightarrow}\ensuremath{\infty}$ from calculations of only a few powers of ttt. We calculate the ground-state energy for the (1+1)-dimensional anisotropic Heisenberg model and the (1+1)-dimensional Ising model. Using a blocking that treats the sites and links symmetrically, we were able to determine the exact critical point of the Ising system. In both of these systems we see that the operator renormalization-group method substantially improves upon the results of either the ttt expansion or real-space renormalization-group methods used separately.
We present a real-space renormalization-group scheme for both spin and gauge systems within a Hamiltonian formalism. The approximation, in particular, preserves gauge invariance at every step of the calculation. We apply this scheme to the (1 + 1)-dimensional Ising model in a transverse field and to the (2 + 1)-dimensional Ising gauge theory. We find reasonable results for the critical coupling and for those critical exponents which are related to energy gaps. We also obtain the correct qualitative behavior for order and disorder parameters and correlation functions. In particular, the calculation yields exponential decay for correlation functions in the disordered phase. However, the critical indices we find for spacelike quantities are not good. This defect of the approximation is related to the asymmetric scaling of space and time under the renormalization group.
A general method for optimizing real-space renormalization-group transformations is presented. It is applicable to both spin systems and lattice gauge theories, with use of Monte Carlo computer simulations. It provides substantial improvement in the convergence of critical exponents for the d=3d=3d=3 Ising model.
Self-Similarity and Scale Invariance. Renormalization Group Approach to Chaos. Renormalization Approach to Percolation. Renormalization Group and Critical Phenomena. The Ising Model. Renormalization Group for the Ising Model. Other Real-Space Renormalization Group Methods for the Ising Model. Mean Field Theory and the Gaussian Fixed Point. The epsis Expansion. The Spherical Model and the 1/n Expansion. The Two-Dimensional X-Y Model and the Kosterlitz-Thouless Transition. Appendices. Author Index. Subject Index.
The analysis of the error of perturbative series by comparing it to the exact solution is an important tool to understand the non-perturbative physics of statistical models. For some toy models, a new method can be used to calculate higher order weak coupling expansion and modified perturbation theory can be constructed. However, it is nontrivial to generalize the new method to understand the critical behavior of high dimensional spin and gauge models. Actually, it is a big challenge in both high energy physics and condensed matter physics to develop accurate and efficient numerical algorithms to solve these problems. In this thesis, one systematic way named tensor renormalization group method is discussed. The applications of the method to several spin and gauge models on a lattice are investigated. theoretically, the new method allows one to write an exact representation of the partition function of models with local interactions. E.g. O(N) models, Z2 gauge models and U(1) gauge models. Practically, by using controllable approximations, results in both finite volume and the thermodynamic limit can be obtained. Another advantage of the new method is that it is insensitive to sign problems for models with complex coupling and chemical potential. Through the new approach, the Fisher's zeros of the 2D O(2) model in the complex coupling plane can be calculated and the finite size scaling of the results agrees well with the Kosterlitz-Thouless assumption. Applying the method to the O(2) model with a chemical potential, new phase diagram of the models can be obtained. The structure of the tensor language may provide a new tool to understand phase transition properties in general.
Abstract 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.