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Rotations, Negative Eigenvalues, and Newton Method in Tensor Network Renormalization Group

Nikolay Ebel, Tom Kennedy, Slava Rychkov

TL;DR

This work introduces a Newton-method approach to directly solve the tensor RG fixed-point equation at criticality, overcoming marginal deformations by embedding a π/2 rotation into the RG map. The rotated map yields an invertible Jacobian near the fixed point, enabling rapid convergence to isotropic fixed-point tensors for the 2D Ising and 3-state Potts models with Gilt-TNR at χ=30 and accuracy ~10^{-9}. By examining the Jacobian spectrum, the authors connect RG perturbations to conformal field theory operators, observing universal quasiprimary eigenvalues and nonuniversal derivative modes, with rotation modifying spin-dependent phases. The results establish that the Newton method with rotation not only improves fixed-point accuracy compared to shooting but also provides a robust framework for extracting CFT data from tensor RG, while revealing anisotropy manifolds and period-2 RG behavior in anisotropic cases. The paper outlines open questions on gauge-fixing, higher bond dimensions, and extending the approach to other models and fixed points, marking a significant step toward rigorous fixed-point construction in tensor RG.

Abstract

In the tensor network approach to statistical physics, properties of the critical point of a 2D lattice model are encoded by a four-legged tensor which is a fixed point of an RG map. The traditional way to find the fixed point tensor consists in iterating the RG map after having tuned the temperature to criticality. Here we develop a different and more direct technique, which solves the fixed point equation via the Newton method. This is challenging due to the existence of marginal deformations -- linear transformations of the coordinate frame, which parametrize a two-dimensional family of fixed points. We address this challenge by including a 90 degree rotation into the RG map. This flips the sign of the problematic marginal eigenvalues, rendering the fixed point isolated and accessible via the Newton method. We demonstrate the power of this technique via explicit computations for the 2D Ising and 3-state Potts models. Using the Gilt-TNR algorithm at bond dimension $χ=30$, we find the fixed point tensors with $10^{-9}$ accuracy, much higher than what was previously achieved.

Rotations, Negative Eigenvalues, and Newton Method in Tensor Network Renormalization Group

TL;DR

This work introduces a Newton-method approach to directly solve the tensor RG fixed-point equation at criticality, overcoming marginal deformations by embedding a π/2 rotation into the RG map. The rotated map yields an invertible Jacobian near the fixed point, enabling rapid convergence to isotropic fixed-point tensors for the 2D Ising and 3-state Potts models with Gilt-TNR at χ=30 and accuracy ~10^{-9}. By examining the Jacobian spectrum, the authors connect RG perturbations to conformal field theory operators, observing universal quasiprimary eigenvalues and nonuniversal derivative modes, with rotation modifying spin-dependent phases. The results establish that the Newton method with rotation not only improves fixed-point accuracy compared to shooting but also provides a robust framework for extracting CFT data from tensor RG, while revealing anisotropy manifolds and period-2 RG behavior in anisotropic cases. The paper outlines open questions on gauge-fixing, higher bond dimensions, and extending the approach to other models and fixed points, marking a significant step toward rigorous fixed-point construction in tensor RG.

Abstract

In the tensor network approach to statistical physics, properties of the critical point of a 2D lattice model are encoded by a four-legged tensor which is a fixed point of an RG map. The traditional way to find the fixed point tensor consists in iterating the RG map after having tuned the temperature to criticality. Here we develop a different and more direct technique, which solves the fixed point equation via the Newton method. This is challenging due to the existence of marginal deformations -- linear transformations of the coordinate frame, which parametrize a two-dimensional family of fixed points. We address this challenge by including a 90 degree rotation into the RG map. This flips the sign of the problematic marginal eigenvalues, rendering the fixed point isolated and accessible via the Newton method. We demonstrate the power of this technique via explicit computations for the 2D Ising and 3-state Potts models. Using the Gilt-TNR algorithm at bond dimension , we find the fixed point tensors with accuracy, much higher than what was previously achieved.
Paper Structure (41 sections, 1 theorem, 77 equations, 16 figures, 7 tables)

This paper contains 41 sections, 1 theorem, 77 equations, 16 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Intuition for RG eigenvalues
  3. Tensor RG for the 2D Ising
  4. CFT expectations for RG eigenvalues
  5. Symmetry preserved by the RG map
  6. Rotation symmetry breaking and eigenvalues 1
  7. Adding a rotation to the tensor RG map
  8. Newton method
  9. Results for the non-rotating Gilt-TNR
  10. Review of the non-rotating Gilt-TNR algorithm
  11. Isotropic fixed point by the "shooting" method
  12. Jacobian eigenvalues
  13. Comparison to prior shooting method results
  14. Locating the fixed point
  15. Jacobian eigenvalues
  16. ...and 26 more sections

Key Result

Lemma 4.1

Assume $M$ is diagonalizable with the eigenvalues $\lambda_1$, $\lambda_2$,… arranged in order of decreasing absolute value. Suppose none of the first $s$ eigenvalues are 1. Then $M_s$ is diagonalizable with eigenvalues $0, \cdots, 0, \lambda_{s+1}$, $\lambda_{s+2}$ …, where eigenvalue $0$ has multi

Figures (16)

  • Figure 1: Expected pattern of tensor RG evolution for the 2D Ising model. The dashed line is the curve of initial tensors corresponding to varying $T$ in $A^{(0)}(T)$.
  • Figure 2: Anisotropy ellipse parametrizing translationally invariant stress-tensor perturbations of the CFT.
  • Figure 3: This figure corrects \ref{['fig:exp']} to show that we expect a two-dimensional manifold ${\cal M}_2$ of fixed point tensors. (Since two of the three dimensions in the figure represent directions orthogonal to the fixed point manifold, we had to represent ${\cal M}_2$ by a one-dimensional thick red line.) Microscopically, we may consider the anisotropic 2D Ising model with varying $J_y/J_x$. Tensor RG flows starting from the critical point of this model (dotted line) will be attracted to a one-dimensional submanifold of ${\cal M}_2$. Adding the next to nearest neighbor interaction we may cover the whole ${\cal M}_2$.
  • Figure 4: The RG flow of tensors starting from $A^{(0)}$ corresponding to the Ising model at the reduced temperature $t=1.0000110043$ and with anisotropy $J_x/J_y=1$ (i.e. the isotropic tensor), in terms of the 20 largest singular values along the diagonal, \ref{['fig:singval']}, normalized by the first one. Gilt-TNR parameters (see \ref{['tab:Giltparams']}): $\chi=30$, $\epsilon_{\rm gilt}=6\times 10^{-6}$.
  • Figure 5: The Hilbert-Schmidt distance between two subsequent gauge-fixed tensors as a function of the RG step. Gilt-TNR parameters are the same as in \ref{['fig:sval_traj_nr']}.
  • ...and 11 more figures

Theorems & Definitions (7)

  • Remark 3.1
  • Remark 3.2
  • Remark 4.1
  • Lemma 4.1
  • proof : Proof of Lemma \ref{['lem:eigs']}
  • proof
  • Remark D.1