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arXiv:2406.10081 · Physical review. D/Physical review. D.

S U ( 2 ) principal chiral model with tensor renormalization group on a cubic lattice

Shinichiro Akiyama, Raghav G. Jha, Judah Unmuth-Yockey

Published Aug 30, 2024• 8 citations
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Abstract

We study the continuous phase transition and thermodynamic observables in the three-dimensional Euclidean S U ( 2 ) principal chiral field model with the triad tensor renormalization group and the anisotropic tensor renormalization group methods. Using these methods, we find results that are consistent with previous Monte Carlo estimates and the predicted renormalization group scaling of the magnetization close to criticality. These results bring us one step closer to studying finite-density quantum chromodynamics in four dimensions using tensor network methods. Published by the American Physical Society 2024

DOI
10.1103/physrevd.110.034519

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We study the three-dimensional SU(2)SU(2)SU(2) principal chiral model (PCM) using different tensor renormalization group methods based on the triad and anisotropic decomposition of the tensor. The tensor network representation is formulated based on the character expansion of the Boltzmann weight. We compare the average action obtained using these two tensor network algorithms and confirm that the resulting critical coupling and exponent are comparable with the recent estimations from the Monte Carlo methods.
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The tensor-network renormalization group (TNRG) is an accurate numerical real-space renormalization group method for studying phase transitions in both quantum and classical systems. Continuous phase transitions, as an important class of phase transitions, are usually accompanied by spontaneous breaking of various symmetries. However, the understanding of symmetries in the TNRG is well-established mainly for global on-site symmetries like U(1) and SU(2). In this paper, we demonstrate how to incorporate lattice symmetries (including reflection and rotation) and the PT symmetry in the TNRG in two dimensions (2D) through a case study of the hard-square lattice gas with nearest-neighbor exclusion. This model is chosen because it is well-understood and has two continuous phase transitions whose spontaneously-broken symmetries are lattice and PT symmetries. Specifically, we write down proper definitions of these symmetries in a coarse-grained tensor network and propose a TNRG scheme that incorporates these symmetries. We demonstrate the validity of the proposed method by estimating the critical parameters and the scaling dimensions of the two phase transitions of the model. The technical development in this paper has made the 2D TNRG a more well-rounded numerical method.
We investigate the phase transition of the four-dimensional Ising model with two types of tensor network scheme, one is the higher-order tensor renormalization group and the other is the anisotropic tensor renormalization group. The results for the internal energy and magnetization obtained by the former algorithm with the impure tensor method, enlarging the lattice volume up to 102441024^410244, are consistent with the weak first-order phase transition. For the later algorithm, our implementation successfully reduces the execution time thanks to the parallel computation and the results provided by ATRG seems comparable to those with HOTRG.
We show that the Tensor Renormalization Group (TRG) method can be applied to O(N) spin models, principal chiral models and pure gauge theories (Z2, U(1) and SU(2)) on (hyper) cubic lattices.We explain that contrarily to some common belief, it is very difficult to write compact formulas expressing the blockspinning of lattice models.We show that in contrast to other approaches, the TRG formulation allows us to write exact blocking formulas with numerically controllable truncations.The basic reason is that the TRG blocking separates neatly the degrees of freedom inside the block and which are integrated over, from those kept to communicate with the neighboring blocks.We argue that the TRG is a method that can handle large volumes, which is crucial to approach quasi-conformal systems.The method can also get rid of some sign problems.We discuss recent results regarding the critical properties of the 2D O(2) nonlinear sigma model with complex β and chemical potential.As some of these results appeared in a recently published paper (PRD 88, 056005) and two recent preprints (