Lattice-reflection symmetry in tensor-network renormalization group with entanglement filtering in two and three dimensions

This paper proposes a method to incorporate lattice-reflection symmetry into the tensor-network renormalization group (TNRG) with entanglement filtering for both two and three dimensions by defining the symmetry in tensor-network language and introducing a transposition trick to preserve it during key operations, thereby enabling the separate extraction of scaling dimensions in specific symmetry sectors.

The Gist

Imagine you are trying to understand a massive, intricate tapestry woven from billions of threads. This tapestry represents a physical system, like a magnet made of tiny atomic spins. To understand how the whole thing behaves, you can't look at every single thread; it's too much data. Instead, you need a way to zoom out, grouping threads together to see the bigger picture without losing the essential patterns.

This is what Tensor Network Renormalization Group (TNRG) does. It's a mathematical "zoom-out" tool used by physicists to study how materials behave, especially at the "critical point"—the exact moment a magnet loses its magnetism or water turns to steam.

However, zooming out is tricky. If you just squint and guess, you might lose important details or introduce errors. This paper introduces a clever new set of rules to make that zooming process smarter, faster, and more accurate, specifically by respecting the symmetry of the pattern.

Here is the breakdown of the paper's ideas using everyday analogies:

1. The Problem: The "Blurry" Zoom

Think of the tapestry as a giant grid of tiles. When you zoom out, you combine four small tiles into one big tile.

• The Issue: Sometimes, when you combine them, you accidentally throw away a tiny bit of information that looks like "noise" but is actually a crucial pattern.
• The Solution (Entanglement Filtering): The authors use a technique called "Entanglement Filtering." Imagine you have a sieve. You pour the combined tiles through the sieve to filter out the "redundant" threads (the ones that don't add new information) while keeping the "essential" threads. This keeps the picture sharp even when you zoom out.

2. The New Trick: The "Mirror" Move (Lattice-Reflection Symmetry)

The tapestry isn't just a random mess; it has rules. If you look at a magnet, flipping it upside down or looking at it in a mirror often reveals the same pattern. This is called Symmetry.

• The Old Way: Previously, when the computer tried to zoom out, it treated every direction as unique. It didn't realize that the pattern on the left was just a mirror image of the pattern on the right. This meant the computer had to do double the work, calculating things it already knew.
• The New Trick (Transposition Trick): The authors propose a "Mirror Move." Before they zoom out, they flip certain parts of the tapestry over (like turning a page in a book).
  • Analogy: Imagine you are folding a piece of paper. If you know the left side is a mirror image of the right, you only need to draw half the picture. The "Transposition Trick" is the instruction to the computer: "Hey, this side is just a mirror of that side. Don't calculate both; just calculate one and flip it!"
  • Why it matters: This drastically reduces the number of calculations needed. In 3D, it cuts the number of variables the computer has to juggle from 24 down to just 3. It's like going from solving a Sudoku puzzle with 81 clues to one with only 3.

3. The "SWAP-Gauge" Matrix: The Magic Label

When you flip a tile in a mirror, sometimes the colors or labels on the tile need to be swapped to match the original perfectly.

• The paper introduces a concept called the SWAP-gauge matrix. Think of this as a "magic label" or a "translator" that tells the computer how to rearrange the threads after the mirror flip so that the pattern still makes sense.
• Without this label, the mirror image would look broken. With it, the symmetry is perfectly preserved, ensuring the computer doesn't make mistakes when it zooms out.

4. Sorting the "Charges": The Library Analogy

Once the computer has zoomed out and cleaned up the image, it wants to find the "scaling dimensions." These are like the "DNA" of the critical point—they tell us exactly how the material behaves (e.g., how fast heat spreads).

• The Challenge: In the past, all these "DNA" strands were mixed together in a big pile, making it hard to find the specific ones you wanted.
• The Solution: The authors show how to use the mirror symmetry to sort these strands into different "bins" or "sectors" (like a library sorting books by genre).
  • Some strands belong to the "Mirror-Symmetric" bin.
  • Some belong to the "Mirror-Anti-Symmetric" bin.
  • By sorting them first, the computer can analyze each group separately. This is much more efficient and prevents errors where one type of pattern gets confused with another.

5. The Results: A Sharper Picture

The authors tested their new method on the Ising Model (a classic model for magnets) in both 2D (flat) and 3D (cubes).

• In 2D: They successfully identified the "DNA" of the magnet with high precision, matching theoretical predictions perfectly.
• In 3D: This is where it gets really exciting. 3D calculations are notoriously difficult and slow. By using their mirror trick, they were able to get reliable results for 3D magnets that were previously very hard to calculate. They found that the "DNA" of the 3D magnet follows the same beautiful patterns predicted by theory, just organized into those neat "bins" they created.

Summary

This paper is like giving a cartographer a new set of rules for drawing a map of a complex city.

1. Old Way: Draw every street, every building, and every tree, even if the city is perfectly symmetrical. It takes forever and you might make mistakes.
2. New Way: Realize the city is symmetrical. Draw half the city, use a "mirror trick" to copy the other half, and use a "magic label" to ensure the streets connect correctly. Then, sort the map into clear sections.
3. Result: You get a perfect, detailed map of the city in a fraction of the time, with fewer errors.

This breakthrough allows physicists to study complex 3D materials with a level of accuracy and efficiency that was previously out of reach, paving the way for understanding new phases of matter in the future.

Technical Summary

1. Problem Statement

Tensor-Network Renormalization Group (TNRG) is a powerful real-space method for studying criticality in classical and quantum lattice systems. While incorporating global on-site symmetries (like $Z_2$ spin-flip) is well-established, incorporating lattice symmetries (specifically reflection and rotation) remains a challenge.

• The Gap: Existing TNRG algorithms often break lattice symmetries due to arbitrary choices in contraction orders or numerical approximations. This leads to unstable fixed points (e.g., the low-temperature fixed point of the Ising model flowing to the high-temperature one) and prevents the isolation of specific critical operators.
• The Specific Challenge: While practitioners often intuitively exploit lattice symmetries to reduce computational costs, there has been no systematic, general framework to define, preserve, and impose lattice-reflection symmetry within TNRG, particularly when combined with Entanglement Filtering (EF) in both 2D and 3D.
• Goal: To develop a general framework for exploiting lattice-reflection symmetry in TNRG with EF, enabling the calculation of scaling dimensions in distinct symmetry sectors and simplifying algorithm implementation.

2. Methodology

The authors propose a systematic framework based on projective truncations and a novel "transposition trick."

A. Definition of Lattice-Reflection Symmetry

The paper formalizes the definition of lattice-reflection symmetry for tensor networks in 1D, 2D, and 3D.

• SWAP-Gauge Matrices: Unlike simple matrix transposition, the symmetry definition in TNRG requires the introduction of SWAP-gauge matrices ($g$). For a tensor $A$, reflection symmetry is defined as $A = A^{\text{transposed}}$ with the action of $g$ on specific legs. These matrices satisfy $g^2 = 1$ (reflecting the $Z_2$ nature of reflection).
• Origin: The paper proves that these $g$ matrices naturally arise when relating the coarse-grained tensor to the original tensor under reflection, specifically due to the reversal of arrow directions in isometric tensors during coarse-graining.

B. The Transposition Trick

To impose and preserve this symmetry, the authors introduce a transposition trick:

• Concept: Before performing the block-tensor transformation (coarse-graining), the algorithm transposes specific legs of tensors in a checkerboard pattern (every other row/column in 2D, every other layer in 3D).
• Effect: This transforms the RG map from $A' = AA$ to $A' = AA^\top$ (in 1D analogy).
• Benefit: This construction ensures that the resulting coarse-grained tensor is symmetric by construction, regardless of numerical errors in the input. It effectively "imposes" the symmetry rather than just hoping it is preserved.

C. Application to Entanglement Filtering (EF) and Projective Truncations

The framework is applied to the two core operations of modern TNRG:

1. Entanglement Filtering (EF):
   • Without the transposition trick, exploiting reflection symmetry in 3D EF would require complex handling of SWAP-gauge matrices for 24 independent filtering matrices.
   • With the trick: The number of independent filtering matrices in 3D reduces dramatically from 24 to 3 (one per spatial direction). The symmetry properties of the filtering matrices become straightforward (e.g., $W_+ = E_+$), and the SWAP-gauge matrices largely disappear from the algorithmic implementation.
2. Projective Truncations:
   • The isometric tensors ($p$) used for truncation are shown to inherit the same symmetry properties as the exact coarse-graining case.
   • The SWAP-gauge matrices for the coarse-grained tensor can be determined directly from the isometric tensors using a specific diagrammatic relation.

D. Linearization of the RG Map

The authors develop a method to linearize the RG map around a fixed point within specific lattice-reflection charge sectors.

• Chain Rule: They apply a chain rule to decompose the linearization of the full RG map (which involves EF and multiple collapse directions) into linearized maps for each step.
• Sector Separation: By defining charge sectors $(c_x, c_y, c_z)$ where $c_i \in \{0, 1\}$ (symmetric/antisymmetric), they derive explicit formulas for the linearized map $R^{(c)}$. This allows the extraction of scaling dimensions for operators belonging to specific symmetry sectors separately.

3. Key Contributions

1. General Framework: Established the first systematic definition and proof of lattice-reflection symmetry preservation in TNRG with EF for both 2D and 3D.
2. Transposition Trick: Introduced a simple yet powerful technique to impose reflection symmetry, simplifying the algorithm and removing the need to track SWAP-gauge matrices in many steps.
3. Computational Efficiency: Demonstrated that exploiting symmetry in 3D EF reduces the number of variational parameters (filtering matrices) from 24 to 3, significantly lowering computational costs.
4. Linearization Formalism: Provided a rigorous method to linearize the RG map in separate symmetry sectors, enabling the isolation of scaling dimensions for specific conformal field theory (CFT) operators.
5. Diagrammatic Proofs: Developed a "dragging tensors" technique to prove symmetry properties, which is applicable to other lattice symmetries (e.g., rotation).

4. Numerical Results

The authors validated their framework using the Ising model in 2D and 3D.

• 2D Ising Model:

  • Calculated scaling dimensions for the spin-flip even and odd sectors.
  • Successfully resolved scaling dimensions up to $x \approx 3.125$.
  • The results matched CFT predictions, correctly identifying the degeneracy structure of operators (e.g., distinguishing $\partial_x \epsilon$ from $\partial_y \epsilon$ based on their reflection charges).
  • Note: While the method improved symmetry handling, it still struggled to resolve very high scaling dimensions ($x > 2$) accurately compared to transfer matrix methods, a known limitation of current TNRG linearization.

• 3D Ising Model:

  • Applied the 3D algorithm with bond dimension $\chi=6$.
  • Successfully extracted scaling dimensions for primary operators ($\sigma, \epsilon$) and their descendants.
  • Key Achievement: The numerical results correctly reproduced the degeneracy structure of the scaling dimensions across different lattice-reflection sectors (e.g., the 3-fold degeneracy of first descendants $\partial_k \epsilon$). This provided strong numerical justification for the proposed linearization scheme in 3D.
  • Estimates for primary fields were within ~5-6% of bootstrap estimates.

5. Significance and Future Outlook

• Theoretical Foundation: This work bridges the gap between intuitive symmetry usage and rigorous mathematical formulation in TNRG. It clarifies the role of SWAP-gauge matrices and provides a proof mechanism for symmetry preservation.
• Algorithmic Improvement: The reduction of filtering matrices in 3D makes high-precision TNRG simulations more feasible.
• Path to Rotation Symmetry: The techniques developed (especially the diagrammatic proofs and transposition trick) lay the groundwork for incorporating lattice-rotation symmetry in TNRG, which is currently an open problem, particularly in 3D where the rotation group is non-abelian.
• Complex Tensors: The current work focuses on real-valued tensors. The authors note that extending this to complex-valued tensors (where SWAP-gauge matrices might be unitary) is a natural next step.

In summary, this paper provides a robust toolkit for enforcing lattice-reflection symmetry in TNRG, significantly enhancing the accuracy and efficiency of critical exponent calculations in both 2D and 3D lattice models.