Note on searching for critical lattice models as entropy critical points from strange correlator

1. Problem Statement

The identification of critical lattice models (systems undergoing second-order phase transitions) and the extraction of their Conformal Field Theory (CFT) data, such as the central charge ($c$), are fundamental challenges in statistical mechanics and condensed matter physics.

• Traditional Limitations: Standard methods often rely on computing entanglement entropy ($S_{EE}$) and looking for logarithmic scaling ($S_{EE} \sim \frac{c}{3} \ln l$). This requires relatively large system sizes (long spin chains) to observe the asymptotic behavior, making computations expensive and difficult for complex models.
• The Specific Challenge: In the context of the Strange Correlator formalism, which constructs 2D critical models from 3D topological quantum field theories (TQFTs) by tuning boundary conditions, the search space for critical parameters is vast. Finding the specific boundary conditions that yield a gapless (critical) phase is computationally intensive.
• The Goal: The authors aim to test a recently proposed method (Ref. [1]) that uses an entropy function as a diagnostic tool for criticality. They investigate whether this method can efficiently identify critical points and estimate central charges using extremely small lattice systems (as few as 4 spins) within the Strange Correlator framework.

2. Methodology

The paper combines two distinct theoretical frameworks:

A. The Entropy Function Criterion (Ref. [1])

The authors utilize an identity derived from the modular Hamiltonian and entanglement entropy of 1+1D CFT ground states.

• The Entropic Function ($S_\Delta$): Defined for a state $|\psi\rangle$ on an infinite line or circle with intervals $A, B, C$:
  $$S_\Delta(|\psi\rangle) = (S_{AB} + S_{BC}) - \eta(S_A + S_C) - (1-\eta)(S_B + S_{ABC})$$
  where $\eta$ is the cross-ratio of the intervals.
• The Critical Condition: For a CFT ground state, the variation of this entropy function vanishes ($dS_\Delta = 0$). Thus, CFT ground states correspond to critical points (extrema) of the entropy function in the parameter space of the Hamiltonian or boundary conditions.
• Vector Fixed Point Equation (VFPE): A related condition $K_\Delta |\psi\rangle = \frac{c}{3}h(\eta)|\psi\rangle$ is also discussed, where $K_\Delta$ is an operator constructed from entanglement Hamiltonians.

B. Strange Correlator Formalism

• Topological Ground State ($|\Psi\rangle$): Constructed from a 3D TQFT (specifically a string-net model on a square-octagon lattice) represented as a tensor network with quantum 6j-symbols.
• Boundary Condition ($\langle \Omega|$): Constructed via competing condensates. The boundary state is a superposition of different Frobenius algebra objects ($A_i$) sharing a common module ($M$).
  $$\langle \Omega(\{r_i\})| = \bigotimes_{\text{cells}} \left( \sum_i r_i \langle \frac{M}{A_i} | \right)$$
• The Search Strategy:
  1. Construct the transfer matrix for the inner product $\langle \Omega | \Psi \rangle$ on a small cylinder (e.g., 4 sites).
  2. Find the ground state (dominant eigenvector) of this transfer matrix for a given set of coupling parameters $\{r_i\}$.
  3. Calculate the entropy function $S_\Delta$ for this ground state.
  4. Scan the parameter space $\{r_i\}$ to find the values where $S_\Delta$ is maximized (or its derivative is zero). These points are hypothesized to be the critical points.

3. Key Contributions

1. Validation of Small-System Diagnostics: The paper demonstrates that the entropy function criterion is highly effective even on 4-site lattices. This is a significant reduction in computational cost compared to traditional scaling analysis.
2. Efficient Phase Diagram Mapping: The method allows for the rapid plotting of entire phase diagrams in multi-dimensional parameter spaces (e.g., 2D or 3D coupling spaces) to identify phase boundaries and tricritical points.
3. Central Charge Estimation: The value of the entropy function at the critical point is directly related to the central charge ($S_\Delta \approx c/3$). The authors show this allows for the estimation of $c$ on tiny systems, though with some finite-size error at higher bond dimensions.
4. Distinction of Phase Transitions: The method is tested on both second-order (continuous) and weak first-order transitions, showing its ability to locate critical points even in complex scenarios like the 5-state Potts model.

4. Results

The authors applied this method to several lattice models derived from the $A_k$ and $Z_N$ categories:

• A-Series Minimal Models ($A_{k+1}$):

  • Setup: Competition between condensates $0$ and $0 \oplus 2$ with module $1$.
  • Result: The entropy function showed clear peaks near the theoretical critical couplings ($r^*_k$) for $k=2$ to $6$.
  • Accuracy: For $k=3,4$, the estimated central charges ($c \approx 0.5, 0.7$) matched theory well. For higher $k$ ($A_5, A_6, A_7$), finite-size effects caused deviations in $c$, but the location of the critical coupling remained accurate.
  • Efficiency: Each data point was generated in ~1 second on a standard laptop.

• Ashkin-Teller Like Model ($A_4$):

  • Setup: Competition between non-simple algebras $0 \oplus 2$ and $0 \oplus 3$.
  • Result: A peak was found at $r \approx 1.17$, close to the theoretical $r^* \approx 1.14$. The estimated $c \approx 1.5$ was higher than the theoretical $c=1.2$ (Ising + Tri-Ising), attributed to finite-size effects.

• Tricritical Point ($A_5$):

  • Setup: Competition between three algebras ($0, 0 \oplus 4, 0 \oplus 2 \oplus 4$).
  • Result: The method successfully mapped a 2D phase diagram, revealing three phase boundaries intersecting at a tricritical point. The entropy values along the boundaries matched $c/3$ for the respective gapless theories.

• Potts Models ($Z_N$):

  • Setup: Competition between $0$ and the sum of all objects in $Z_N$.
  • Result: Correctly identified second-order critical points for $N \le 4$. For $N=5$ (first-order transition), the method still found a maximum in the entropy function, suggesting a connection to a nearby complex CFT, though it could not distinguish the order of the transition solely by the peak existence.

5. Significance and Conclusion

• Computational Efficiency: The primary significance is the drastic reduction in computational resources required to scan for criticality. By reducing the system size to 4 spins, the method bypasses the need for large-scale simulations and finite-size scaling analysis.
• Robustness: The method proves robust across different topological orders and symmetry groups, successfully recovering known critical points and phase diagrams.
• Limitations: While the location of critical points is highly accurate, the estimation of the central charge suffers from finite-size effects, particularly for models with high bond dimensions (large $k$). The authors suggest combining this entropy-based screening with Tensor Network Renormalization (TNR) for larger systems to refine central charge estimates.
• Future Outlook: This approach provides a "first screen" for criticality in the vast landscape of Strange Correlator models, enabling the systematic discovery of new CFTs and the exploration of complex phase diagrams that were previously computationally prohibitive.

In summary, Jin and Hung successfully demonstrate that the entropy function critical point is a powerful, cost-effective, and efficient tool for identifying critical boundary conditions in topological holographic models, effectively turning the search for CFTs into a local optimization problem on very small lattices.