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Lattice and PT symmetries in tensor-network renormalization group: a case study of a hard-square lattice gas model

Xinliang Lyu

Abstract

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.

Lattice and PT symmetries in tensor-network renormalization group: a case study of a hard-square lattice gas model

Abstract

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.

Paper Structure

This paper contains 20 sections, 1 theorem, 48 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The model and its tensor-network representation
  3. The known phase transitions of the model
  4. The relevant symmetries of the two phase transitions
  5. Tensor-network representation of the model and its symmetries
  6. Stability of the spontaneous-symmetry-breaking phase under RG maps
  7. Exploit the lattice and $\mathcal{PT}$ symmetries in TNRG with loop optimization
  8. Definition of the lattice and $\mathcal{PT}$ symmetries for the coarse-grained tensor network
  9. Weak form of the lattice symmetries
  10. Strong form of the lattice symmetries
  11. Symmetric SVD splitting
  12. TRG with the lattice and $\mathcal{PT}$ symmetries
  13. Incorporating the loop optimization
  14. Relationship with the symmetric loop-TNR
  15. Numerical results
  16. ...and 5 more sections

Key Result

Theorem 1

When the tensor $A$ satisfies the diagonal reflection symmetry along both the $l_1$ and $l_2$ axes in Eq. eq:scheme:Arefll1l2sym, the tensor $v_{\Lambda}$ in its symmetric SVD splitting along the $l_1$ axis in Eq. eq:scheme:symSVD can be chosen to satisfy the following symmetry property: where the first equal sign comes from the definition of the arrow in the diagrammatic representation of $v$: a

Figures (9)

  • Figure 1: (a) Black dots represent sites of the square lattice, and they are divided into two sublattices like a chessboard. If a particle is in the central shaded site, it excludes other particles from its four neighbor unshaded sites. (b) A large-length-scale picture of the 1NN hard-square model at large activity $z$. "S" denotes the area where particles occupy the shaded sublattice, while "U" for the unshaded sublattice.
  • Figure 2: SSB of lattice symmetries for positive-$z$ transition. Number $0, 1$ on the site means number of particle. When the activity $z=+\infty$, all particles occupy either the shaded (left) or unshaded (right) sites. These two configurations can be mapped to each other through any of the following symmetry transformations: 1) translation by one lattice constant in any direction, 2) $90^\circ$ lattice rotation around the central cross point, and 3) lattice reflection along either the $l_1$ or the $l_2$ axis.
  • Figure 3: The four largest eigenvalues of the transfer matrix of the 1NN hard-square model with size $L=12$ for activity near the repulsive-core singularity $z_c^{-}$. The largest eigenvalues are normalized to have unit norm. Black solid lines are real parts, and blue dashed lines are imaginary parts. When the activity $z$ decreases and goes over $z_c^{-}$, the first two eigenvalues develop imaginary parts and become a complex-conjugate pair, which is SSB of the $\mathcal{PT}$ symmetry.
  • Figure 4: The RG flows of the degeneracy index $X$ near the positive-$z$ transition point $z_c^+$ generated by (a) the HOTRG and (b) the TRG at bond dimension $\chi=10$. The solid line in the middle is the RG flow closest to the transition point $z_c^{+} \approx 3.80$. Two lines below have $z < z_c^{+}$ and belong to the trivial phase, while two lines above have $z > z_c^+$ and belong to the SSB phase. For the flows generated by the HOTRG starting with $z>z_c^+$, the tensor stays at the SSB fixed point ($X=2$) until the RG step $n\approx 11$. The corresponding flows generated by the TRG are more stable, and the tensor starts to move away from the SSB fixed point after $n\approx 26$ RG steps.
  • Figure 5: The RG flows of the degeneracy index $X$ near the negative-$z$ transition point $z_c^-$ generated by the TRG at bond dimension $\chi=10$. The solid line in the middle is the RG flow closest to the repulsive-core singularity $z_c^- \approx -0.119$. Two lines below have $z>z_c^-$ and belong to the trivial phase, while two lines above have $z<z_c^-$ and belong to the SSB phase. (a) The RG flow starts with a complex tensor-network representation of the model. Perturbation due to machine-precision error breaks the $\mathcal{PT}$ symmetry and thus makes the SSB fixed point ($X=2$) unstable after $n\approx 26$ RG steps. (b) The RG flow starts with a real tensor-network representation of the model. Since the SVD splittings in the TRG make sure all tensors remain real, the $\mathcal{PT}$ symmetry is exactly preserved. Therefore, the SSB fixed point ($X=2$) should be strictly stable. Although the figure only shows up to the RG step $n=51$, we have checked that this stability persists even after $n=100$ RG steps.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Remark
  • Remark
  • Remark
  • Theorem 1: Symmetry of the 3-leg tensor $v_{\Lambda}$ in symmetric SVD splitting
  • proof
  • Remark
  • Remark
  • Remark
  • Remark
  • Remark
  • ...and 2 more