Tensor Network Studies on Partition Function Zeros of two-dimensional Classical Spin Models exhibiting BKT transition

Abstract

We employ tensor network methods to investigate the finite-size-scaling behavior of the partition function zeros for the classical spin models. We first review the tensor network representation of the partition function on the square lattice, and definition of the local tensor on the lattice sites with two-dimensional Ising model as an example. We review three tensor network methods employed in this thesis: the tensor renormalization group (TRG) algorithm, the baseline of the tensor contraction methods, and the two improved algorithms, higher-order tensor renormalization group (HOTRG) and loop optimization of tensor network renormalization (LoopTNR). We show that LoopTNR achieves high accuracy in the partition function calculation and scaling dimensions with the transfer matrix. And we show that the entanglement filtering (EF) technique can also improve the loop optimization step. We investigate the finite-size-scaling (FSS) behavior of the leading Fisher zero of the partition function in the complex temperature plane in the p-state clock models of p = 5 and 6. We derive the logarithmic finite-size corrections to the scaling of the leading zeros which we numerically verify by performing the HOTRG calculations in the square lattices of a size up to 128×128 sites. The necessity of the deterministic HOTRG method in the clock models is noted by the extreme vulnerability of the numerical i leading zero identification against stochastic noises that are hard to be avoided in the Monte-Carlo approaches. We characterize the system-size dependence of the numerical vulnerability of the zero identification by the type of phase transition, suggesting that the two transitions in the clock models are not of an ordinary first- or second-order type. In the direct FSS analysis of the leading zeros in the clock models, we find that their FSS behaviors show excellent agreement with our predictions of the logarithmic corrections to the Berezinskii-Kosterlitz-Thouless ansatz at both of the high- and low-temperature transitions. We study the logarithmic correction to the scaling of the first Lee–Yang zero at the critical point in the classical XY model on square lattices by using tensor renormalization group methods. In comparing the HOTRG and the LoopTNR, we find that the entanglement filtering in LoopTNR is crucial to gaining high accuracy for the characterization of the logarithmic correction, while HOTRG still proposes approximate upper and lower bounds for the zero location associated with two different bond merging algorithms of the higher-order singular value decomposition and the oblique projectors. Using the LoopTNR data computed up to the system size of L = 1024 in the L × L lattices, we estimate the logarithmic correction exponent r = −0.0643(9) from the extrapolation of the finite-size effective exponent, which is comparable to the renormalization group prediction of r = −1/16.