Eigenstate entanglement entropy in Bose-Hubbard models

This paper investigates the eigenstate entanglement entropy of mid-spectrum states in Bose-Hubbard models, deriving a volume-law coefficient via a generalized mean-field approach and revealing that while disorder does not alter the volume-law term, the subleading O(1) contribution exhibits distinct behaviors depending on particle-number conservation and local bosonic cutoffs.

The Gist

Imagine you have a giant, chaotic party. The guests are quantum particles (bosons), and the room is a grid of spots where they can sit. Sometimes, the guests follow strict rules (like "we must have exactly 100 people in the room"), and sometimes, people can magically appear or disappear. Sometimes the room is perfectly symmetrical, and sometimes it's cluttered with random obstacles.

This paper is about measuring how "mixed up" or "entangled" these guests get with each other when the party is in full swing (a highly excited state). Specifically, the authors are looking at the "middle" of the party's energy spectrum—the most chaotic, lively part of the night.

Here is the breakdown of their findings, translated into everyday language:

1. The Main Question: How Entangled is the Party?

In quantum physics, entanglement is like a secret handshake between two groups of people. If Group A (the left side of the room) is highly entangled with Group B (the right side), it means the state of a person in Group A is deeply connected to the state of someone in Group B, even if they are far apart.

The authors wanted to know: Does the way the room is set up (perfectly ordered vs. messy/disordered) change how much the guests mix?

2. The "Volume Law" (The Big Picture)

First, they looked at the main trend: how entanglement grows as the room gets bigger.

• The Analogy: Imagine the amount of gossip spreading in a room. If the room doubles in size, the gossip (entanglement) roughly doubles too. This is called a "volume law."
• The Finding: They discovered that it doesn't matter if the room is perfectly symmetrical (like a ballroom) or messy and disordered (like a cluttered garage). As long as the guests are interacting and the party is chaotic, the amount of gossip spreading is exactly the same.
• Why it's cool: In simpler systems (like non-interacting particles), the room's layout does change the gossip. But in these interacting Bose-Hubbard models, the chaos is so strong that the layout doesn't matter. The "volume law" is robust.

3. The "O(1)" Term (The Tiny, Tricky Detail)

After accounting for the main volume of gossip, there is a tiny, leftover amount of entanglement. This is the O(1) term. Think of this as the "fine print" or the "secret sauce" that random chance alone can't explain.

The authors found that this "fine print" behaves very differently depending on the rules of the party:

Case A: The Strict Party (Particle-Number Conservation)

The Rule: The total number of guests is fixed. No one enters or leaves.

• The Finding: The "fine print" is complicated. It depends heavily on how crowded the room is (particle density) and how many seats each spot can hold (the local cutoff).
• The Metaphor: Imagine a dance floor where you have exactly 50 dancers. If you pack them in tightly, the way they bump into each other is different than if they are spread out. The "secret sauce" of their entanglement changes based on exactly how many people are there and how much space they have. It's not a universal constant; it's specific to the crowd size.

Case B: The Wild Party (No Particle-Number Conservation)

The Rule: Guests can magically appear and vanish. The total number isn't fixed.

• The Finding: Here, the "fine print" seems to settle on a universal constant.
• The Metaphor: Even though people are popping in and out, the underlying "vibe" of the party settles into a specific, predictable pattern that doesn't change based on the crowd size. This pattern matches a prediction made for other types of quantum systems (like fermions), suggesting there is a fundamental "quantum rule" for how chaos settles down when the rules are loose.

4. The "Random Pure State" Prediction

The authors compared their real-world quantum simulations to a theoretical "Random Party."

• The Theory: If you just threw people into a room randomly without any specific rules, you could predict exactly how much they would mix.
• The Reality:
  • For the Strict Party, the real quantum system sometimes deviates from the random prediction in a way that depends on the crowd density.
  • For the Wild Party, the real system seems to have an extra layer of entanglement (the universal O(1) term) that the simple random prediction missed. It's as if the quantum particles have a hidden instinct to mix slightly more than pure randomness would suggest.

Summary

• Big Picture: Whether the room is ordered or messy, the main amount of quantum mixing (entanglement) is the same.
• The Twist: The tiny, leftover details of that mixing tell a different story.
  • If the number of particles is fixed, the details are messy and depend on the crowd size.
  • If particles can appear/disappear, the details settle into a neat, universal constant that might be a fundamental law of nature.

In short: The authors found that while the "loud" part of the quantum noise is the same everywhere, the "whispers" (the tiny corrections) reveal deep secrets about whether the system follows strict rules or allows for chaos.

Technical Summary

1. Problem Statement

While the entanglement entropy of highly excited eigenstates (mid-spectrum) has been extensively studied in fermionic systems and spin-1/2 models, bosonic systems remain less understood. Specifically, there is a need to characterize:

• The volume-law coefficient of entanglement entropy in bosonic systems with a tunable local Hilbert space dimension (soft-core bosons).
• The impact of translational invariance versus disorder on these coefficients.
• The nature of the subleading $O(1)$ contributions to the entanglement entropy, particularly in the presence of particle-number conservation (global $U(1)$ symmetry) versus systems where this conservation is broken.
• Whether universal $O(1)$ corrections exist beyond the predictions of random pure state ensembles (Page curves) for bosonic models.

2. Methodology

The authors employ a combination of analytical derivations and numerical simulations using Exact Diagonalization (ED) and Polynomially Filtered Exact Diagonalization (POLFED).

Models Studied

Three variations of the Bose-Hubbard model on a 1D lattice with $L$ sites and local bosonic cutoff $n_{\text{max}}$ (limiting occupation per site to $n_j \le n_{\text{max}}$):

1. Translationally-Invariant (TI) Model: Standard Bose-Hubbard with nearest-neighbor hopping ($t$) and on-site repulsion ($U$). Conserves total particle number $N$.
2. Disordered Model: Adds an on-site random potential ($W$) breaking translational symmetry but conserving particle number.
3. Generalized Model: Adds particle creation/annihilation terms ($g$) breaking the global $U(1)$ symmetry and particle-number conservation.

Analytical Approach

• Mean-Field Derivation: The authors generalize the mean-field approach (previously applied to hard-core bosons) to soft-core bosons. They construct random grandcanonical pure states to derive the volume-law coefficient for the entanglement entropy.
• Random Pure State Predictions: They utilize generalized Page formulas for random states with fixed particle number (Eq. 3) and without (Eq. 1) to serve as baselines for comparison.

Numerical Implementation

• System Sizes: Up to $L=14$ for particle-number conserving models and $L=12$ for non-conserving models, depending on $n_{\text{max}}$.
• Statistics: Averages are taken over mid-spectrum eigenstates within symmetry sectors (for TI) or over disorder realizations (for disordered models).
• Metrics: The study analyzes the mean entanglement entropy $\langle S_A \rangle$, the distribution $P(S_A)$, and the statistical mode $Mo(S_A)$. They specifically look at the difference between numerical results and theoretical random-state predictions to isolate $O(1)$ terms.

3. Key Contributions and Results

A. Volume-Law Coefficient Derivation

The authors derive the volume-law coefficient $F(n)$ for the entanglement entropy $S_A \approx F(n)L_A$ for random grandcanonical pure states with arbitrary local cutoff $n_{\text{max}}$.

• Result: The coefficient is given by the Legendre transform of the logarithm of the generating function: $F(n) = \ln \zeta(\mu) - \mu n$.
• Validation: This analytical result matches previous numerical findings for translationally invariant systems and generalizes the hard-core boson result to soft-core bosons and true bosons ($n_{\text{max}} \to \infty$).

B. Role of Translational Invariance

The authors compare the TI model with the weakly disordered model ($W=0.4$).

• Finding: Unlike non-interacting fermionic systems (where translational invariance significantly alters the volume-law coefficient), translational invariance does not modify the volume-law term in interacting Bose-Hubbard models.
• Evidence: The difference in mean entanglement entropy between TI and disordered models vanishes as $1/L$. The $O(1)$ terms also appear identical within numerical precision.

C. $O(1)$ Contributions with Particle-Number Conservation

For the disordered Bose-Hubbard model with conserved particle number:

• Dependence: The subleading $O(1)$ term exhibits a non-trivial dependence on the particle-number density $n$ and the local cutoff $n_{\text{max}}$.
• Dominant Sector: The deviation from the random pure state prediction is most pronounced when the system is in the dominant particle-number sector ($n = n^* = n_{\text{max}}/2$).
• Thermodynamic Limit: It remains inconclusive whether a universal non-zero $O(1)$ term persists as $L \to \infty$ for all densities. However, for specific cases (e.g., $n=1$ with no cutoff), the data suggests the deviation may vanish, implying the random pure state prediction is exact even at the $O(1)$ level.

D. $O(1)$ Contributions without Particle-Number Conservation

For the generalized model where particle number is not conserved:

• Universal Term: The numerical results strongly suggest the emergence of a universal $O(1)$ contribution that is not captured by the standard random pure state prediction (Eq. 1).
• Magnitude: The difference between the numerical mean and the theoretical prediction converges to a value close to $c_1(f=1/2) = 1/2 + \ln(1/2)/2 \approx -0.153$ (or related constants discussed in the text regarding Eq. 4).
• Significance: This indicates that even without particle-number conservation, the eigenstate entanglement entropy in chaotic bosonic systems possesses a universal correction, likely related to total energy conservation, similar to conjectures for fermionic systems.

4. Significance

• Generalization of Theory: The paper successfully extends the mean-field theory of entanglement entropy from hard-core bosons/spin-1/2 systems to general soft-core bosons with arbitrary local Hilbert space dimensions.
• Universality in Bosons: It establishes that the volume-law coefficient for thermalizing bosonic systems is robust against the breaking of translational invariance, distinguishing them from free fermion systems.
• Symmetry Effects: The work highlights the complex interplay between particle-number conservation, local cutoffs, and entanglement. It suggests that while particle-number conservation introduces density-dependent subtleties, the breaking of this conservation leads to a potentially universal $O(1)$ correction, providing a clearer numerical playground for detecting such universal terms compared to spin-1/2 models.
• Experimental Relevance: Since Bose-Hubbard models are experimentally realizable in optical lattices and entanglement can be measured, these theoretical predictions provide a benchmark for future experimental verification of eigenstate thermalization and entanglement scaling in bosonic quantum simulators.