Haiyuan Zou

Exact blocking formulas for spin and gauge models

Using the example of the two-dimensional (2D) Ising model, we show that in contrast to what can be done in configuration space, the tensor renormalization group formulation allows one to write exact, compact, and manifestly local blocking formulas and exact coarse-grained expressions for the partition function. We argue that similar results should hold for most models studied by lattice gauge theorists. We provide exact blocking formulas for several 2D spin models [the O(2) and O(3) sigma models and the SU(2) principal chiral model] and for the three-dimensional gauge theories with groups Z(2), U(1) and SU(2). We briefly discuss generalizations to other groups, higher dimensions and practical implementations.

Tensor renormalization group study of classical XY model on the square lattice

Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the thermodynamic properties of the continuous XY model on the square lattice. The temperature dependence of the free energy, the internal energy, and the specific heat agree with the Monte Carlo calculations. From the field dependence of the magnetic susceptibility, we find the Kosterlitz-Thouless transition temperature to be 0.8921(19), consistent with the Monte Carlo as well as the high temperature series expansion results. At the transition temperature, the critical exponent δ is estimated as 14.5, close to the analytic value by Kosterlitz.

Controlling sign problems in spin models using tensor renormalization

We consider the sign problem for classical spin models at complex beta = 1/g(0)(2) on L x L lattices. We show that the tensor renormalization group method allows reliable calculations for larger Im beta than the reweighting Monte Carlo method. For the Ising model with complex beta we compare our results with the exact Onsager-Kaufman solution at finite volume. The Fisher zeros can be determined precisely with the tensor renormalization group method. We check the convergence of the tensor renormalization group method for the O(2) model on L x L lattices when the number of states D-s increases. We show that the finite size scaling of the calculated Fisher zeros agrees very well with the Kosterlitz-Thouless transition assumption and predict the locations for larger volume. The location of these zeros agree with Monte Carlo reweighting calculation for small volume. The application of the method for the O(2) model with a chemical potential is briefly discussed.

Tensor renormalization group study of the 2d O(3) model

We present our progress on a study of the

Progress towards quantum simulating the classical O(2) model

O(3)

Fine structure of the entanglement entropy in the O(2) model

model in two-dimensions using the Tensor Renormalization Group method. We first construct the theory in terms of tensors, and show how to construct

Comparing Tensor Renormalization Group and Monte Carlo calculations for spin and gauge models

n

Fisher's zeros, complex RG flows and confinement in LGT models.

-point correlation functions. We then give results for thermodynamic quantities at finite and infinite volume, as well as 2-point correlation function data. We discuss some of the advantages and challenges of tensor renormalization and future directions in which to work.

Approaching conformality with the Tensor Renormalization Group method

Department of Energy under DOE [DE-FG02-05ER41368, DE-SC0010114, DE-FG02-91ER40664]; Army Research Office of the Department of Defense [W911NF-13-1-0119]; NSF [DMR-1411345]

Volume Effects in Discrete Beta Functions

We compare two calculations of the particle density in the superfluid phase of the O(2) model with a chemical potential μ in 1+1 dimensions. The first relies on exact blocking formulas from the Tensor Renormalization Group (TRG) formulation of the transfer matrix. The second is a worm algorithm. We show that the particle number distributions obtained with the two methods agree well. We use the TRG method to calculate the thermal entropy and the entanglement entropy. We describe the particle density, the two entropies and the topology of the world lines as we increase μ to go across the superfluid phase between the first two Mott insulating phases. For a sufficiently large temporal size, this process reveals an interesting fine structure: the average particle number and the winding number of most of the world lines in the Euclidean time direction increase by one unit at a time. At each step, the thermal entropy develops a peak and the entanglement entropy increases until we reach half-filling and then decreases in a way that approximately mirrors the ascent. This suggests an approximate fermionic picture.