Tensor renormalization group methods for spin and gauge models

Abstract

In this thesis, we study the physics and critical behavior of several statistical models, e.g. O(N) model and lattice gauge models by using different approach, which including the conventional perturbative expansion and later and mostly, tensor renormalization group methods. We first start from the exact solvable model, 1-d O(2) model with open boundary conditions (OBC) and periodic boundary conditions (PBC) on a finite lattice and take the perturbative approach. We discuss the error of perturbative series by comparing it to the exact solution of the partition function and the average energy. The error (nonperturbative part) for both boundary conditions can be parametrized as Aβ−Be−Cβ but with different coefficients. For PBC, the error comes from the vortices solutions. We calculate the weak coupling expansion for finite PBC systems up to term 12 by using a new method and modify the ordinary perturbative series of the 1-link model. We also compare the small E expansion of the density of states with numerical values. We search for the Fisher’s zeros for a system with PBC and construct Migdal-Kadanoff flows for the OBC system. We conclude that the Fisher’s zeros control the geometrical properties of the flows. Motivated by recent attempts to find nontrivial infrared fixed points in 4dimensional lattice gauge theories, we discuss 2D nonlinear O(N) sigma models on a finite lattice, in the large-N limit. We explain the Riemann sheet structure and singular points of the finite L mappings between the mass gap and the ’t Hooft