Renormalization group and phase transitions in spin, gauge, and QCD like theories

In this thesis, we study several different renormalization group (RG) methods, including the conventional Wilson renormalization group, Monte Carlo renormalization group (MCRG), exact renormalization group (ERG, or sometimes called functional RG), and tensor renormalization group (TRG).

We use the two dimensional nearest neighbor Ising model to introduce many conventional yet important concepts. We then generalize the model to Dyson's hierarchical model (HM), which has rich phase properties depending on the strength of the interaction. The partition function zeros (Fisher zeros) of the HM model in the complex temperature plane is calculated and their connection with the complex RG flows is discussed. The two lattice matching method is used to construct both the complex RG flows and calculate the discrete β functions. The motivation of calculating the discrete β functions for various HM models is to test the matching method and to show how physically relevant fixed points emerge from the complex domain.

We notice that the critical exponents calculated from the HM depend on the blocking parameter b. This motivated us to analyze the connection between the discrete and continuous RG transformation. We demonstrate numerical calculations of the ERG equations. We discuss the relation between Litim and Wilson-Polchinski equation and the effect of the cut-off functions in the ERG calculation.

We then apply methods developed in the spin models to more complicated and more physically relevant lattice gauge theories and lattice quantum chromodynamics (QCD) like theories. Finite size scaling (FSS) technique is used to analyze the Binder cumulant of the SU(2) lattice gauge model. We calculate the critical exponent nu and omega of the model and show that it is in the same universality class as the three dimensional Ising model. Motivated by the walking technicolor theory, we study the strongly coupled gauge theories with conformal or near conformal properties. We compare the distribution of Fisher zeros for lattice gauge models with four and twelve light fermion flavors. We also briefly discuss the scaling of the zeros and its connection with the infrared fixed point (IRFP) and the mass anomalous dimension.

Conventional numerical simulations suffer from the critical slowing down at the critical region, which prevents one from simulating large system. In order to reach the continuum limit in the lattice gauge theories, one needs either large volume or clever extrapolations. TRG is a new computational method that may calculate exponentially large system and works well even at the critical region. We formulate the TRG blocking procedure for the two dimensional O(2) (or XY ) and O(3) spin models and discuss possible applications and generalizations of the method to other spin and lattice gauge models.

We start the thesis with the introduction and historical background of the RG in general.