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

This paper extends the tensor-network renormalization group (TNRG) method to incorporate lattice and PT symmetries in two dimensions by applying it to the hard-square lattice gas model, thereby enabling accurate estimation of critical parameters and scaling dimensions for its continuous phase transitions.

The Gist

Imagine you are trying to understand a massive, complex city by looking at it from a helicopter. As you fly higher, the individual buildings blur together, and you start to see neighborhoods, districts, and the overall layout of the city. This is essentially what physicists do when they study phase transitions (like water turning to ice or a magnet losing its magnetism). They use a mathematical tool called the Tensor Network Renormalization Group (TNRG) to "zoom out" and see the big picture of how a system changes.

However, there's a catch. When you zoom out, you have to throw away a lot of tiny details to keep the math manageable. If you throw away the wrong details, your map of the city becomes distorted, and you might miss the most important features.

This paper, by Xinliang Lyu, is about building a better, more careful zoom-out tool that respects the specific "rules of symmetry" of the city it's studying.

Here is the breakdown using everyday analogies:

1. The Problem: The "Broken Mirror" and the "Ghost City"

The author is studying a specific model called the Hard-Square Lattice Gas. Imagine a chessboard where you are placing coins (particles) on the squares. The rule is simple: No two coins can touch. They must have at least one empty square between them.

This system has two very different "moods" (phase transitions):

• Mood A (The Solid Crowd): When you have a lot of coins, they eventually crowd onto just the black squares or just the white squares of the chessboard. They break the symmetry of the board. This is like a crowd of people suddenly deciding to all stand on the left side of a room.
• Mood B (The Ghostly Math): There is a second, more abstract transition involving "negative" coins (mathematically speaking). This transition breaks a different rule called PT Symmetry. Think of this as a "ghost" rule where the system behaves differently if you look at it in a mirror and reverse time.

The Issue: Previous zoom-out tools (TNRG methods) were great at handling simple rules (like "spin up" or "spin down"), but they were clumsy with these specific "lattice" and "ghost" rules. When they tried to zoom out, the tiny errors in the math would accidentally break these symmetries, causing the map to drift off course and give wrong answers.

2. The Solution: The "Symmetry-Aware" Zoom Lens

The author proposes a new way to zoom out that acts like a smart lens. Instead of just blurring the image, this lens is programmed to know exactly what the rules of the city are.

• The Lattice Symmetry (The Chessboard Rules): The lens knows that the board has rotational symmetry (it looks the same if you turn it 90 degrees) and reflection symmetry (it looks the same in a mirror). The new method forces the math to respect these rules at every step. It's like telling the camera, "No matter how much you zoom out, the image must always look like a perfect chessboard."
• The PT Symmetry (The Ghost Rule): For the "negative coin" transition, the lens ensures that all the numbers used in the calculation remain real numbers (no imaginary ghosts). This keeps the "ghost" symmetry intact, preventing the math from collapsing into nonsense.

3. The Secret Sauce: "Loop Optimization" (The Noise Canceller)

Even with a smart lens, zooming out creates "noise" (truncation errors). To fix this, the author adds a step called Entanglement Filtering (specifically, Loop Optimization).

• The Analogy: Imagine you are trying to listen to a conversation in a noisy room. You have a microphone (the zoom lens), but the room is loud.
• The Old Way: You just turn up the volume, but the noise gets louder too.
• The New Way: The author adds a "noise-cancelling" feature. Before you zoom out, the algorithm looks at the small loops of connections in the network and filters out the redundant "static." It cleans the signal so that when you zoom out, you get a crystal-clear picture of the critical moment where the phase change happens.

4. The Results: A Sharper Map

The author tested this new method on the Hard-Square model.

• Without the new method: The map of the city started to wobble and drift after a few zoom steps. The estimated "tipping point" (where the phase change happens) was slightly off.
• With the new method: The map stayed perfectly stable. The author could pinpoint the exact moment the coins decided to crowd onto one side of the board, and they could calculate the "scaling dimensions" (the mathematical fingerprints of the transition) with incredible precision—far better than previous methods.

Why Does This Matter?

Think of this paper as upgrading the GPS in a self-driving car.

• Before: The car could drive on a straight highway (simple symmetries like the Ising model), but if the road had complex curves or weird intersections (lattice symmetries), the car would get confused and take a wrong turn.
• Now: The car has a new navigation system that understands complex road rules. It can handle the twists and turns of "hard-core" particle systems, allowing physicists to study more complex materials and phase transitions that were previously too difficult to simulate accurately.

In short: The author built a smarter, symmetry-respecting zoom tool that cleans up the math noise, allowing scientists to see the hidden patterns of how matter organizes itself with unprecedented clarity.

Technical Summary

1. Problem Statement

The Tensor-Network Renormalization Group (TNRG) is a powerful numerical method for studying phase transitions in classical and quantum systems. While TNRG has been well-established for systems with global on-site symmetries (like $U(1)$ or $SU(2)$), its application to lattice symmetries (reflection, rotation) and $\mathcal{PT}$ symmetry (parity-time reversal) in two dimensions (2D) remains underdeveloped.

The core problem addressed is that standard TNRG schemes (like TRG or HOTRG) often fail to explicitly preserve these symmetries during coarse-graining. This leads to:

• Instability: Fixed points corresponding to spontaneously symmetry-broken (SSB) phases become unstable under Renormalization Group (RG) flow due to numerical artifacts (e.g., machine precision errors or algorithmic asymmetry).
• Inaccuracy: The lack of symmetry constraints increases the degrees of freedom, leading to larger truncation errors and less accurate estimates of critical parameters and scaling dimensions.
• Gap in Methodology: Existing Entanglement Filtering (EF) enhanced schemes (like Loop-TNR) often break lattice symmetries because they update tensors in a specific order, which is an artifact of the algorithm.

The paper uses the 1NN hard-square lattice gas model as a benchmark because it possesses two distinct continuous phase transitions:

1. Positive activity ($z > 0$): A transition to an ordered phase breaking lattice symmetries (belonging to the 2D Ising universality class).
2. Negative activity ($z < 0$): A non-physical singularity (Yang-Lee edge) breaking $\mathcal{PT}$ symmetry (belonging to the non-unitary minimal model $M(2,5)$).

2. Methodology

The author proposes a novel Symmetric TNRG scheme enhanced by Loop Optimization (Entanglement Filtering). The methodology consists of three main pillars:

A. Symmetric Tensor Network Representation

The partition function is represented as a tensor network where the initial tensors explicitly encode the symmetries:

• Lattice Symmetries: The initial 4-leg tensor $A_0$ satisfies specific reflection and rotation properties. The network is constructed using copies of $A_0$ and its rotations ($A_{\pi/2}, A_{\pi}, A_{3\pi/2}$) to ensure the network structure respects the lattice geometry.
• $\mathcal{PT}$ Symmetry: For negative activity, the model requires real-valued tensors. The $\mathcal{PT}$ symmetry is enforced by ensuring all tensors remain real-valued throughout the RG flow, preventing the eigenvalues of the transfer matrix from acquiring imaginary parts prematurely.

B. Symmetric Singular Value Decomposition (SVD)

Standard TRG uses SVD to split tensors. The author introduces a Symmetric SVD based on the Eigendecomposition (ED) for tensors satisfying diagonal reflection symmetry.

• Instead of standard SVD, the tensor is treated as a symmetric matrix along the reflection axis.
• The decomposition yields a diagonal matrix of eigenvalues ($\Lambda$) and eigenvectors ($v$).
• The sign of the eigenvalues is extracted into a new bond matrix $\sigma'$, while the absolute values are distributed to the tensors.
• Key Theorem: This splitting naturally preserves the "weak form" of lattice symmetries and ensures the resulting 3-leg tensors satisfy a specific "SWAP-gauge" symmetry property.

C. Loop Optimization with Symmetry Constraints

To reduce truncation errors (Entanglement Filtering), the author integrates Loop Optimization (from Loop-TNR) into the symmetric TRG framework.

• Challenge: Standard loop optimization updates tensors in a series, which breaks lattice rotation/reflection symmetries.
• Solution: The optimization is performed on a "redundant entanglement" loop (Corner-Double Line, CDL) using a specific update rule derived from the Full Environment Truncation (FET) framework.
• Initialization: The symmetric SVD provides a high-quality initialization for the loop optimization, ensuring the CDL structure is filtered out efficiently without breaking the underlying symmetries.
• Algorithm: The RG step involves: (1) Rotation trick to align tensors, (2) Symmetric SVD splitting, (3) Loop optimization of the 3-leg tensor, and (4) Recombination.

3. Key Contributions

1. Formal Definition of Symmetries in Coarse-Grained Tensors: The paper defines "Weak" and "Strong" forms of lattice symmetries for coarse-grained tensors and proves that the proposed RG transformation preserves and imposes the Strong form.
2. Symmetric SVD Splitting: A new decomposition technique that respects lattice reflection/rotation symmetries and naturally generates the necessary bond matrices ($\sigma$) without introducing complex numbers for real-valued systems.
3. Symmetry-Preserving Loop Optimization: The first EF-enhanced TNRG scheme that explicitly incorporates lattice and $\mathcal{PT}$ symmetries. It resolves the issue where standard EF methods destabilize SSB fixed points.
4. Handling Non-Unitary CFTs: The method is successfully applied to the Yang-Lee edge singularity (a non-unitary Conformal Field Theory with $c = -22/5$), demonstrating that EF can effectively control RG errors even when the standard entanglement area law arguments fail.

4. Results

The method was tested on the 1NN hard-square model at bond dimensions $\chi$ ranging from 6 to 50.

• Critical Activity ($z_c$) Estimation:
  • Positive-$z$ (Ising): The proposed scheme achieved a relative error of $\sim 10^{-7}$ at $\chi=12$, whereas the standard TRG required $\chi=50$ to reach similar accuracy.
  • Negative-$z$ (Yang-Lee): The proposed scheme achieved a relative error of $\sim 10^{-8}$ at $\chi=20$, significantly outperforming standard TRG.
• Stability of RG Flows:
  • In standard TRG/HOTRG, the RG flow near the SSB fixed point drifts away due to symmetry breaking (measured by the degeneracy index $X$).
  • In the proposed scheme, the SSB fixed points remained strictly stable (degeneracy index $X$ remained constant) for hundreds of RG steps, even in the presence of machine precision limits.
• Scaling Dimensions:
  • The extracted scaling dimensions for both transitions converged rapidly to exact theoretical values.
  • For the non-unitary Yang-Lee transition, the proposed method produced stable estimates at $\chi=10$ that were more accurate than standard TRG at $\chi=50$.
• Error Suppression: The loop optimization successfully tamed the growth of RG truncation errors, reducing them from $\sim 10^{-2}$ (standard TRG) to $\sim 10^{-7}$ (proposed scheme) after a few RG steps.

5. Significance

• Generalization of TNRG: This work transforms 2D TNRG from a method primarily suited for on-site symmetries into a versatile tool capable of handling spatial lattice symmetries and $\mathcal{PT}$ symmetry.
• Improved Accuracy: By reducing the search space of the RG flow through symmetry constraints, the method drastically improves the precision of critical parameter estimation and reduces the computational cost (lower bond dimensions required).
• Stability in SSB Phases: It solves the long-standing issue of fixed-point instability in ordered phases, ensuring that numerical artifacts do not destroy the physical degeneracy of the ground state.
• Non-Unitary Systems: It demonstrates the viability of TNRG for non-unitary theories (like Yang-Lee edge singularities), expanding the scope of tensor network applications beyond standard unitary quantum systems.
• Future Applications: The framework is applicable to any hard-core lattice gas model with larger exclusion ranges, offering a path to study complex phase transitions in statistical mechanics that were previously difficult to simulate accurately.