Worldline deconfinement and emergent long-range interaction in the entanglement Hamiltonian and in the entanglement spectrum

Using quantum Monte Carlo simulations on a square-octagon lattice Heisenberg model, this study reveals that gapless modes induce a sublinear magnon dispersion in the entanglement spectrum, signaling emergent long-range interactions in the entanglement Hamiltonian that can be explained by the deconfinement of worldlines in the path integral formulation.

The Gist

Imagine you are trying to understand a massive, complex city (a quantum material) by only looking at a single neighborhood (a subsystem). In physics, this "neighborhood view" is called entanglement. Usually, when we look at this neighborhood, we expect to see a quiet, orderly street where interactions only happen between immediate neighbors (like neighbors chatting over a fence). This is the standard rule for most materials.

However, this paper discovers something strange and exciting: When the city enters a specific "critical" or "gapless" state, the rules of the neighborhood change completely. Suddenly, people in the neighborhood start having loud, meaningful conversations with people on the other side of the city, even though they can't physically walk there.

Here is a breakdown of the paper's discoveries using simple analogies:

1. The Setup: The "Entanglement Map"

Think of the Entanglement Spectrum as a "map of connections" for our neighborhood.

• Normal State (Gapped): In a normal, stable material, this map looks like a standard city grid. You only talk to your immediate neighbors. The "energy" of these conversations follows a straight, predictable line (like a car driving at a constant speed).
• The Discovery: The researchers looked at a specific type of magnetic material (a square-octagon lattice) and found that when it hits a critical point, the map changes. The "conversations" (interactions) suddenly become long-range.

2. The "M-Shape" Mystery

In a normal material, if you plot the energy of these interactions, it looks like a smooth, straight ramp (linear).

• What they found: In this special state, the graph looks like an "M" shape. It's flat in the middle and then shoots up.
• The Analogy: Imagine a roller coaster. Normally, it's a smooth hill. But here, the track suddenly flattens out and then curves sharply. This "sublinear" shape is a fingerprint of long-range interactions. It means the "neighbors" are somehow influencing each other across vast distances, defying the usual rules of the neighborhood.

3. The "Wormhole" vs. The "Deconfinement"

Why does this happen? The paper offers a fascinating explanation using Worldlines (which are like the paths particles take through time).

• The "Wormhole" (Normal State): Imagine the neighborhood is surrounded by a thick, expensive fog. To talk to someone far away, you have to walk through the fog. It costs so much energy that you just don't do it. You stay "confined" to your block. The researchers call this Worldline Confinement. It's like a wormhole that is too small to use.
• The "Deconfinement" (Critical State): When the material hits that critical point, the fog suddenly evaporates. Now, a particle can "jump" from one side of the neighborhood to the other almost instantly, as if a wormhole opened up in the fabric of space-time.
• The Result: Because these "jumps" (wormholes) are now cheap and easy, the neighborhood effectively becomes a long-distance communication hub. This is called Worldline Deconfinement. The "entanglement Hamiltonian" (the rulebook for the neighborhood) suddenly gains long-range powers it didn't have before.

4. The Comparison: A One-Way Street vs. A Superhighway

To prove this, the researchers compared their 2D material to a simpler 1D chain of magnets that already had long-range rules.

• They found that the "M-shape" and the weird "M-roller-coaster" in their complex 2D material looked exactly like the behavior of the 1D chain with long-range rules.
• The Takeaway: Even though the original material only had short-range rules (neighbors only), the act of "zooming in" on the entanglement created a new, emergent reality where long-range rules suddenly appeared.

5. Why This Matters

This is a big deal because it changes how we understand quantum materials.

• Old Belief: We thought that if a material is "gapless" (fluid and active), its entanglement rules would stay simple and short-range.
• New Reality: The paper shows that in these fluid states, the "virtual" rules governing the entanglement can become wildly complex and long-range.

In a nutshell:
The paper reveals that when a quantum system is on the edge of a phase transition, it's like the "fog" of the universe lifts. Suddenly, distant parts of the system can "talk" to each other instantly. This creates a new kind of "virtual physics" inside the entanglement, where the rules of the game change from "talk to your neighbor" to "talk to anyone, anywhere." This discovery helps us better understand the hidden, complex structures of quantum matter.

Technical Summary

1. Problem Statement

The entanglement spectrum (ES) is a powerful diagnostic tool for topological phases, where the low-lying ES often mirrors the edge energy spectrum of the physical system (Li-Haldane conjecture). In gapped systems, the entanglement Hamiltonian ($H_E$) is typically short-ranged, and the ES resembles the edge spectrum of a gapped bulk with a gapless edge.

However, the behavior of the ES in gapless systems (specifically at quantum critical points or in gapless ordered phases) remains unclear. A central open question is whether the entanglement Hamiltonian in gapless regimes retains its short-range nature or if relevant long-range interactions emerge. If such interactions exist, they could fundamentally alter the dispersion relations and the nature of the ES, potentially violating standard assumptions about the correspondence between the ES and physical edge states.

2. Methodology

The authors employ a combination of advanced numerical techniques to investigate the 2D square-octagon lattice (SOL) Heisenberg model ($S=1/2$):

• Quantum Monte Carlo (QMC) with Replica Trick: They utilize a recently developed QMC method based on path integral sampling of the reduced density matrix ($\rho_A$). By introducing a replica number $n$, they simulate an effective imaginary-time evolution $\tau_A = n$ for the entanglement Hamiltonian ($Z_A^{(n)} \propto \text{Tr}[\rho_A^n] = \text{Tr}[e^{-n H_E}]$). This allows access to the ES of large-scale 2D systems, which is intractable for Exact Diagonalization (ED) or DMRG.
• Stochastic Analytic Continuation (SAC): To extract the real-frequency entanglement spectrum from the imaginary-time correlation functions measured in QMC, the authors use SAC. This technique reconstructs the spectral function $G(\omega, q)$ from the noisy imaginary-time data $G(\tau_A, q)$.
• Model System: The study focuses on the SOL Heisenberg model, which hosts a rich phase diagram including the Affleck-Kennedy-Lieb-Tasaki (AKLT) phase, the $S=1/2$ Néel phase, and a plaquette valence bond crystal (PVBC). The system is tuned across quantum critical points (QCPs) by varying the ratio of inter-unit-cell to intra-unit-cell coupling ($g = J_2/J_1$).
• Benchmarking: To interpret the results, the authors compare the ES of the 2D model against the known excitation spectrum of a 1D $S=1/2$ long-range Heisenberg chain with power-law interactions ($J_r \propto 1/r^\alpha$).

3. Key Contributions and Results

A. Emergence of M-Shape Magnon and Sublinear Dispersion

The authors observe a dramatic evolution in the ES as the system transitions from the gapped AKLT phase to the gapless Néel phase:

• AKLT Phase ($g < g_c$): The ES exhibits a gapless two-spinon continuum, consistent with the Li-Haldane conjecture where the boundary behaves like a 1D Luttinger liquid.
• Néel Phase ($g > g_c$): The ES transforms into sharp magnon modes. Crucially, the dispersion near the momentum $q = \pi$ is not linear. Instead, it exhibits an M-shape with a sublinear dispersion relation:
  $$\omega(q) \propto |q - \pi|^s$$
  where the exponent $s \approx 0.2$. This deviates significantly from the linear dispersion ($s=1$) expected for conventional short-range magnons.

B. Evidence of Relevant Long-Range Interactions

The sublinear dispersion ($s < 1$) and the "M-shape" are signatures of relevant long-range interactions within the entanglement Hamiltonian.

• By comparing the 2D ES with the 1D long-range Heisenberg chain, the authors map the observed exponent $s$ to an effective interaction power law $\alpha$ in the 1D chain, finding $\alpha < 2.23$.
• Finite-size scaling analysis confirms that the dispersion exponent $s$ remains significantly less than 1 in the thermodynamic limit for the Néel phase, indicating that the long-range interactions are not merely irrelevant perturbations but fundamentally shape the ES.
• The ES gap at $q_n = \pi - 2\pi/L$ appears to open in finite systems but closes in the thermodynamic limit, consistent with a gapless spectrum modified by long-range forces.

C. Mechanism: Worldline Deconfinement

The paper proposes a physical mechanism to explain the emergence of these long-range interactions: Worldline Deconfinement.

• Wormhole Picture (Gapped Phase): In gapped phases, worldlines in the path integral formulation of the reduced density matrix are "confined" near the entanglement boundary. They can jump between replicas via a "wormhole" effect (virtual path through the environment trace) with low cost, but crossing the bulk is energetically prohibitive. This leads to a short-range $H_E$.
• Deconfinement (Gapless Phase): In the gapless Néel phase, the bulk gap vanishes ($\Delta_b \to 0$). Consequently, worldlines can propagate freely across the bulk and connect spatially distant spins with low energy cost. This deconfinement of worldlines allows for non-local correlations to develop in the entanglement Hamiltonian, effectively generating long-range interactions.
• Numerical Verification: The authors measure the imaginary-time correlation function $|G(\tau_A, r)|$ on the entanglement boundary. In the AKLT phase, correlations decay rapidly (confinement). In the Néel phase, correlations saturate to a constant even at large distances and imaginary times, confirming the deconfinement of worldlines.

4. Significance

• Fundamental Insight into Entanglement: The work demonstrates that the nature of the entanglement Hamiltonian is not static; it can undergo a qualitative change from short-range to long-range when the underlying system enters a gapless phase. This challenges the assumption that $H_E$ is always a local, short-range operator.
• New Diagnostic for Gapless Phases: The specific signature of an M-shaped magnon with sublinear dispersion in the ES serves as a robust numerical indicator for the presence of emergent long-range interactions in gapless quantum phases.
• Unification of Concepts: The paper successfully bridges the concepts of path integral worldlines, entanglement Hamiltonians, and long-range physics, providing a unified framework (worldline deconfinement) to understand how gapless bulk modes induce non-locality in the entanglement structure.
• Methodological Advancement: It showcases the power of QMC-based replica methods combined with SAC to probe the ES of 2D systems, a task previously limited to small sizes or 1D systems.

In conclusion, the paper establishes that in gapless quantum phases, the entanglement Hamiltonian acquires relevant long-range interactions driven by the deconfinement of worldlines, fundamentally altering the entanglement spectrum from a standard edge-like structure to one characterized by sublinear dispersion.