Extracting conformal data from finite-size tensor-network flow in critical two-dimensional classical models

This paper presents a general framework for extracting conformal data, such as the central charge and scaling dimensions, from critical two-dimensional classical lattice models by identifying a self-consistent finite-size window in tensor-network flow that separates finite-size and finite-entanglement scaling regimes, thereby enabling accurate results without prior knowledge of the underlying conformal field theory.

The Gist

Imagine you are trying to understand the rules of a massive, complex city (a physical system) by looking at a tiny, blurry photograph of it. This is the challenge physicists face when studying materials at their "critical point"—the exact moment a material changes state, like ice melting into water or a magnet losing its magnetism. At this moment, the material behaves in a wild, chaotic way that follows deep, universal mathematical rules called Conformal Field Theory (CFT).

The problem is, these rules are usually hidden. To see them, you need to look at an infinitely large system, which is impossible to simulate on a computer.

This paper presents a clever new "lens" to see these hidden rules using a technique called Tensor Networks. Here is the story of how they did it, explained simply.

1. The Problem: The "Pixelated" View

Think of a computer simulation of a material as a digital image. To make the image clear, you need high resolution (more pixels). In physics, this resolution is called the bond dimension.

• Low resolution: The image is blurry. You can see the general shape, but the details are lost.
• High resolution: The image is crisp, but it takes a supercomputer years to render.

The authors realized that if you try to simulate a system that is too big, the computer runs out of memory (resolution), and the image distorts. This creates a "blurry zone" where the data is useless.

2. The Solution: Finding the "Sweet Spot"

The authors developed a method to find a Goldilocks Zone—a specific size of the system that is:

1. Big enough to show the universal rules of the city (the CFT data).
2. Small enough that the computer's "blur" (finite bond dimension) hasn't ruined the picture yet.

They call this the Self-Consistent Window.

The Analogy: Imagine listening to a song on a radio.

• If the station is too far away, you hear static (the computer's error).
• If the station is too close, the signal is too weak to hear the melody (the system is too small to show the physics).
• The authors found the exact distance where the music is crystal clear.

3. How They Did It: The "Flow" and the "Spin"

To find this sweet spot, they used two main tricks:

A. The Flow of Time (Tensor Network Flow)
They didn't just look at one size; they watched the system "grow" step-by-step. As they increased the size of the system, they watched how the data changed.

• At first, the data looks chaotic.
• Then, it settles into a smooth, predictable pattern (the "flow").
• Finally, the computer's limits kick in, and the pattern breaks down again.
  They simply stopped the analysis right before the pattern broke.

B. The Compass (Conformal Spin)
How do they know the data is "real" and not just computer noise? They used a property called Conformal Spin.

• The Metaphor: Think of the particles in the material as dancers. In a perfect, rule-following city, these dancers must spin in perfect integer numbers (0, 1, 2 spins).
• If the computer simulation is accurate, the dancers spin in perfect integers.
• If the simulation is getting too blurry (too large for the computer), the dancers start spinning in weird, fractional numbers (like 1.43 spins).
• The Rule: As soon as the dancers stop spinning in whole numbers, the authors know they have left the "Sweet Spot" and thrown away the data.

4. The Results: Cracking the Code

They tested this method on two famous models: the Ising Model (like a grid of tiny magnets) and the 3-State Clock Model (a more complex version).

• The Outcome: They were able to extract the "DNA" of these materials (the central charge, scaling dimensions, and spins) with incredible accuracy.
• The Surprise: They found that one specific method, called HOTRG (Higher-Order Tensor Renormalization Group), was the best "camera" for this job. It gave the clearest pictures with the least amount of computer power.

5. Why This Matters

Before this, physicists often needed to know the answer before they could find it, or they had to find a perfect, theoretical "fixed point" which is mathematically very hard to define.

This paper says: "You don't need to know the answer in advance."
You just need to watch the system grow, look for the moment the "dancers" start spinning weirdly, and take your measurement right before that happens.

Summary

The authors built a smart filter for computer simulations. Instead of trying to simulate the entire universe (which is impossible), they found a way to look at a small, manageable slice of it, verify that the slice is still "real" using a spin-check, and then extract the universal laws of nature from that slice.

It's like being able to predict the weather of an entire continent just by looking at a single, perfectly clear cloud, provided you know exactly how to measure the wind before the cloud gets too foggy.

Technical Summary

1. Problem Statement

Critical two-dimensional (2D) classical lattice models are described in the continuum limit by Conformal Field Theory (CFT). A major challenge in computational physics is extracting universal conformal data—specifically the central charge ($c$), scaling dimensions ($\Delta$), and conformal spins ($S$)—directly from lattice calculations without relying on prior knowledge of the underlying CFT.

Traditional tensor-network renormalization (TNR) approaches often rely on finding a unique "critical fixed-point tensor." However, this strategy is technically delicate due to issues like gauge redundancies, short-range entanglement structures (e.g., corner double-line contributions), and the lack of a rigorous, general definition of tensor-network fixed points. Furthermore, standard finite-size scaling in tensor networks is complicated by finite-entanglement scaling effects induced by bond-dimension truncation, which can obscure the true critical behavior.

2. Methodology

The authors propose a general framework based on finite-size tensor-network flow that avoids the need for a unique fixed-point tensor. The core methodology involves:

• Transfer Matrix Spectra: The partition function of a system on an $L_x \times L_y$ torus is represented via a transfer matrix $T(L_y)$. The eigenvalues of $T$ correspond to the energy spectrum of an effective Hamiltonian $H(L_y)$.
• Finite-Size Estimators:
  • Scaling Dimensions: Extracted from the excitation gaps: $\Delta_i \approx \frac{L_y}{2\pi}(E_i - E_0)$.
  • Conformal Spins: Extracted from the lattice momentum via a shift operator $T_P$, where the phase of the eigenvalue of $T T_P$ yields the spin $S_i$.
  • Central Charge: Estimated via two complementary methods:
    1. Ground-state energy ($c_{E0}$): Based on the finite-size scaling of the ground state energy $E_0(L_y) \sim -\frac{\pi c}{12 L_y}$.
    2. Entanglement Entropy ($c_S$): Based on the bipartite entanglement entropy of the dominant eigenvector, scaling as $S_E \sim \frac{c}{3} \ln L_y$.
• Self-Consistent Window Identification:
  • The framework identifies a crossover scale $L_y^*$ that separates the universal finite-size scaling regime from the non-universal finite-entanglement regime caused by bond-dimension truncation.
  • Criterion: $L_y^*$ is defined as the system size where the derivative of the scaling dimension (or central charge) with respect to $\ln L_y$ is minimized ($|\frac{d}{d \ln L_y} X| = \text{min}$).
  • Spin Consistency: Conformal spins are required to be integers (within a threshold). Data points where spins deviate significantly from integers are discarded, defining the upper bound of the self-consistent window.
• Tensor Network Schemes: The method is benchmarked using three distinct renormalization schemes to ensure robustness:
  1. HOTRG (Higher-Order Tensor Renormalization Group).
  2. PTMRG (Periodic Transfer-Matrix Renormalization Group).
  3. CTRG (Core-Tensor Renormalization Group).

3. Key Contributions

• Fixed-Point Independent Framework: The approach extracts conformal data without requiring the existence or identification of a unique critical fixed-point tensor, relying instead on general structural properties of CFT (e.g., integer spins, charge sectors).
• Operational Definition of Entanglement Scaling: The paper provides a natural operational definition for entanglement scaling in classical tensor-network calculations, offering a complementary estimator for the central charge ($c_S$) alongside the traditional energy-based estimator ($c_{E0}$).
• Crossover Scale Criterion: It establishes a robust, data-driven criterion ($L_y^*$) for selecting the optimal system size for estimation, effectively filtering out truncation errors.
• Spin-Resolved Subgrouping: By utilizing conformal spin to split degenerate energy levels, the method can resolve complex operator manifolds and identify specific CFT operators even in the presence of near-degeneracies.

4. Results

The framework was benchmarked on the critical 2D Ising model ($c=1/2$) and the 3-state clock model ($c=4/5$), both of which have known exact CFT data.

• Accuracy and Range:
  • Ising Model: Accurate extraction of conformal data up to scaling dimensions $\Delta \approx 7$ (in the $Q=0$ sector) and $\Delta \approx 7.125$ (in the $Q=1$ sector). Relative errors for scaling dimensions range from $10^{-6}$ (low levels) to $10^{-3}$ (high levels) using HOTRG with bond dimension $D=100$.
  • 3-State Clock Model: Accurate extraction up to $\Delta \approx 4.8$ ($Q=0$) and $\Delta \approx 4.33$ ($Q=\pm 1$). The usable window is narrower than the Ising model due to the higher density of states, but relative errors remain within $10^{-5}$ to $10^{-2}$.
• Scheme Comparison:
  • HOTRG consistently provided the best overall accuracy and trade-off between computational cost and precision across all observables.
  • PTMRG and CTRG served as valuable consistency checks but generally required larger bond dimensions to match HOTRG's accuracy.
• Central Charge Estimators:
  • Both $c_{E0}$ and $c_S$ converged to the exact values within the identified self-consistent window.
  • The entropy-based estimator ($c_S$) was found to be slightly more accurate than the energy-based estimator ($c_{E0}$) for the tested parameters.
• Operator Identification: The method successfully identified primary operators and their descendants, correctly reproducing degeneracies and conformal spins, even for groups containing multiple operators with identical scaling dimensions.

5. Significance

This work significantly advances the application of tensor networks to critical phenomena by:

1. Democratizing CFT Extraction: It allows for the extraction of high-level conformal data without the technical bottleneck of finding fixed-point tensors, making the method applicable to a broader class of models, including non-unitary theories or those with complex spectra where fixed-point approaches fail.
2. Validating Finite-Size Scaling: It rigorously demonstrates that finite-size scaling in tensor networks is a viable route to universal information, provided a careful analysis of the crossover scale is performed.
3. Methodological Robustness: By cross-checking three different renormalization schemes, the authors confirm that the extracted conformal data is a universal property of the flow, not an artifact of a specific algorithm.
4. Future Directions: The framework opens pathways for studying more complex classical models, systematic comparisons across tensor-network schemes, and extensions to systems where traditional renormalization-group fixed points are difficult to control.

In conclusion, the paper establishes a robust, general, and highly accurate protocol for extracting conformal field theory data from finite-size tensor-network simulations, bridging the gap between lattice calculations and continuum CFT predictions.