Exploring the limit of the Lattice-Bisognano-Wichmann form describing the Entanglement Hamiltonian: A quantum Monte Carlo study

This paper presents a general framework combining a lattice-Bisognano-Wichmann ansatz with multi-replica-trick quantum Monte Carlo methods to accurately reconstruct entanglement Hamiltonians in diverse two-dimensional quantum systems, demonstrating that the ansatz remains a valid approximation even in the absence of Lorentz invariance and translational symmetry.

The Gist

Imagine you have a giant, complex puzzle made of billions of tiny, interconnected pieces. In the world of quantum physics, these pieces are particles, and they are all "entangled," meaning they are mysteriously linked no matter how far apart they are.

Physicists want to understand how these pieces are linked. To do this, they use a mathematical tool called the Entanglement Hamiltonian (EH). Think of the EH as a master recipe that tells you exactly how the pieces in one half of the puzzle are connected to the pieces in the other half. If you have this recipe, you can predict everything about the puzzle's hidden connections.

The Problem: The Recipe is Missing

For a long time, scientists only knew this "recipe" for very special, perfect puzzles where the rules of the universe (specifically, Einstein's relativity) applied perfectly. This known recipe is called the Bisognano-Wichmann (BW) theorem.

However, most real-world quantum systems are messy. They aren't perfect, they don't follow those strict relativistic rules, and they often have broken patterns. Scientists didn't know if the "BW recipe" would still work for these messy systems. They needed a way to test it.

The Solution: A Digital Simulator

The authors of this paper, a team of researchers, built a powerful digital simulator using a method called Quantum Monte Carlo. Think of this as a super-advanced video game engine that can simulate the behavior of these quantum puzzles with incredible precision.

They used a clever trick called the "multi-replica trick." Imagine you have one copy of the puzzle, but to understand the deep connections, you make several identical copies (replicas) and stack them on top of each other. By studying how these stacked copies interact, they could reverse-engineer the "master recipe" (the EH) without knowing it beforehand.

The Experiment: Testing the Recipe

The researchers took two types of puzzles to test their new method:

1. The Perfect Puzzle (Transverse-Field Ising Model): This is a system where the pattern is uniform and regular.

   • Result: As expected, the "BW recipe" worked perfectly. It described the connections accurately.

2. The Messy Puzzle (Dimerized Heisenberg Model): This is a system where the pattern is broken. Some connections are strong, and some are weak, creating a "striped" or "patchy" look. This is where things got interesting.

The Big Discovery: It Depends on How You Cut the Cake

When dealing with the messy puzzle, the researchers had to decide how to split it into two halves (System A and System B). They found that how they made the cut mattered immensely.

• Cutting the Strong Links (The "Anomalous" Cut): Imagine the puzzle has some very thick, strong ropes holding it together. If you cut right through these strong ropes, you create a jagged, unstable edge.

  • Result: The "BW recipe" failed. It couldn't describe the connections accurately. The cut introduced a "glitch" or an anomaly that the simple recipe couldn't handle.

• Cutting the Weak Links (The "Ordinary" Cut): Now, imagine cutting through the thin, weak strings instead. This leaves the strong structure of the puzzle intact on both sides.

  • Result: The "BW recipe" worked perfectly, even though the puzzle was messy and didn't follow the strict relativistic rules!

The Analogy: The "Ordinary" vs. "Special" Edge

The authors realized that the success of the recipe depends on the nature of the edge where the cut is made.

• Ordinary Edge: Like cutting a loaf of bread cleanly through the soft crumb. The inside of the bread (the bulk physics) is perfectly reflected on the cut surface. The recipe works.
• Special/Anomalous Edge: Like cutting through a hard crust or a layer of jam. The cut surface is weird and distorted. It doesn't just reflect the inside; it adds its own weird properties. The recipe fails.

Why This Matters

This discovery is huge for two reasons:

1. It breaks the rules: It shows that you don't need the universe to be perfectly "relativistic" (following Einstein's rules) for this mathematical recipe to work. You just need to make sure you aren't cutting through the "strong ropes" that create anomalies.
2. A New Tool: The researchers have created a general toolkit. They can now take almost any complex quantum system, simulate it, and figure out its "entanglement recipe." This helps us understand everything from new materials to the nature of quantum computers.

In a Nutshell

The paper is like finding out that a specific map (the BW recipe) works for almost any terrain, as long as you don't try to draw the map right over a cliff edge. If you cut the terrain gently (ordinary cut), the map is perfect. If you cut it through a cliff (strong bond/anomaly), the map breaks. This gives scientists a powerful new way to navigate the complex landscape of quantum entanglement.

Technical Summary

1. Problem Statement

The Entanglement Hamiltonian (EH), defined via the reduced density matrix $\rho_A = e^{-H_A}$, encodes the full entanglement structure of a quantum many-body system. While the Bisognano-Wichmann (BW) theorem provides an exact analytical form for the EH in Lorentz-invariant quantum field theories (where $H_A$ is proportional to the energy density weighted by the distance from the entanglement boundary), its validity on lattice systems is less understood.

Key challenges addressed in this work include:

• Lack of General Analytical Form: For general lattice models, especially those without Lorentz invariance or translational symmetry, the functional form of the EH is unknown.
• Dependence on Sound Velocity: The lattice-BW (LBW) ansatz requires a prefactor $\epsilon_{EH} = 2\pi/v$ (where $v$ is the sound velocity). In many systems, $v$ is not known a priori, making the ansatz difficult to apply.
• Validity Beyond Lorentz Invariance: It is unclear whether the LBW form remains a good approximation for systems that lack Lorentz invariance or translational symmetry (e.g., dimerized systems).
• Boundary Effects: The impact of how the system is partitioned (cutting strong vs. weak bonds) on the validity of the LBW ansatz has not been systematically explored.

2. Methodology

The authors propose a comprehensive numerical framework combining Multi-Replica Quantum Monte Carlo (QMC) techniques with the LBW ansatz to reconstruct and validate the EH.

A. The LBW Ansatz

For a lattice system with Hamiltonian $H$, the LBW ansatz for the entanglement Hamiltonian $\tilde{H}_A$ is constructed by modulating the coupling constants based on the distance from the entanglement boundary:
$$\tilde{H}_A = \epsilon_{EH} \left[ \sum_{\text{bonds}} \Gamma(x) h_{\text{bond}} + \sum_{\text{sites}} \Theta(x) l_{\text{site}} \right]$$
where the coupling strengths $\Gamma(x)$ and $\Theta(x)$ scale linearly with the distance $x$ from the boundary. The parameter $\epsilon_{EH}$ is the unknown energy scale to be determined.

B. Multi-Replica QMC Simulation

To verify the ansatz without knowing the exact analytical form of the true EH ($H_A$), the authors utilize the multi-replica trick:

1. Exact EH Simulation: They simulate the ensemble of the exact EH at integer effective inverse temperatures $\beta_A = n$ ($n=1, 2, \dots$) by constructing $n$ replicas of the system. The reduced density matrix is effectively $\rho_A^n \propto \text{Tr}_B(e^{-\beta H})^n$.
2. LBW EH Simulation: They simulate the ansatz Hamiltonian $\tilde{H}_A$ using standard finite-temperature QMC (Stochastic Series Expansion) at various effective temperatures $\beta_A$.

C. Parameter Fitting via Imaginary-Time Correlations

To determine the unknown parameter $\epsilon_{EH}$ (and thus the sound velocity $v$):

• They compute momentum-resolved imaginary-time correlation functions $C_k(\tau)$ for both the exact-EH and the LBW-EH.
• In the large-$\tau$ limit, $C_k(\tau) \sim e^{-\tau \Delta E}$.
• By comparing the decay slopes of the logarithmic correlations between the exact and ansatz systems, they extract the ratio of energy gaps, allowing them to fit the sound velocity $v$ and fix $\epsilon_{EH}$.

D. Validation

Once $\epsilon_{EH}$ is fixed, they compare equal-time correlation functions (specifically along the entanglement boundary) between the LBW-EH and the exact-EH across different phases (gapped, gapless, critical) and temperatures.

3. Key Contributions

1. General Numerical Scheme: Developed a robust method to numerically reconstruct the EH and determine its scaling parameters without prior knowledge of the sound velocity.
2. Extension to Non-Lorentzian Systems: Demonstrated that the LBW ansatz is applicable beyond Lorentz-invariant field theories, extending to lattice models with broken translational symmetry.
3. Identification of "Ordinary" vs. "Anomalous" Cuts: Discovered that the validity of the LBW ansatz depends critically on the geometry of the entanglement cut (specifically, whether the cut introduces surface anomalies).
4. Connection to Surface Criticality: Linked the success of the LBW ansatz to the concept of "ordinary" surface criticality, where the boundary does not host extra gapless modes.

4. Results

A. Transverse-Field Ising Model (TFIM)

• System: 2D TFIM (Translationally invariant, Lorentz-invariant at criticality).
• Findings: The LBW ansatz provides an excellent approximation across the ferromagnetic phase, paramagnetic phase, and the quantum critical point (QCP).
• Validation: The fitted sound velocities matched independent estimates (MCRG) within a few percent. Equal-time correlation functions for the LBW-EH and exact-EH showed near-perfect overlap.

B. Dimerized Heisenberg Model (Broken Translational Symmetry)

• System: 2D Columnar Dimerized Heisenberg model (Strong bonds $J_1$, Weak bonds $J_2$).
• Three Partitioning Scenarios:
  1. Strong-Bond Cut (Anomalous): The entanglement boundary cuts through strong bonds, creating a dangling spin chain with a Lieb-Schultz-Mattis (LSM) anomaly.
     • Result: The LBW ansatz fails to accurately describe the EH (except at the isotropic limit $J_r=1$). Significant discrepancies appear in correlation functions, even as the system approaches the ground state.
  2. Weak-Bond Cut (Ordinary - Horizontal): The boundary cuts through weak bonds.
     • Result: The LBW ansatz provides a near-perfect description of the exact EH across all phases (Néel, QCP, Dimer) and system sizes.
  3. Weak-Bond Cut (Ordinary - Vertical): Similar to the horizontal case, cutting weak bonds vertically.
     • Result: The LBW ansatz remains highly accurate.

C. System Size Dependence

Finite-size scaling checks (up to $L=32$) confirmed that the discrepancy in the strong-bond cut is a systematic feature, not a finite-size artifact, while the agreement in weak-bond cuts persists at larger sizes.

5. Significance and Conclusion

• Theoretical Insight: The study reveals that the Lorentz invariance of the bulk is not a strict requirement for the LBW ansatz to hold. Instead, the crucial condition is that the entanglement boundary must be "ordinary" (free of surface anomalies like dangling spin chains).
• Physical Interpretation: When a cut introduces an anomaly (e.g., cutting strong bonds in a dimerized system), it induces extra gapless edge modes that modify the entanglement structure, rendering the simple LBW form insufficient. Conversely, "ordinary" cuts purely reflect bulk criticality, which the LBW form captures accurately.
• Methodological Impact: The proposed multi-replica QMC framework offers a powerful, general tool for investigating the analytical structure of entanglement in complex quantum many-body systems, potentially applicable to other EH ansatzes and geometries.
• Resolution of Contradictions: The findings help explain recent contradictions in entanglement entropy behaviors observed under different splitting schemes, attributing them to the presence or absence of boundary anomalies.

In summary, this work establishes a general framework for validating entanglement Hamiltonians and identifies the nature of the entanglement boundary as the primary determinant of the LBW ansatz's validity, extending its utility to a broad class of non-Lorentzian, non-translationally invariant quantum systems.