Chaos, thermalization and breakdown of quantum-classical correspondence in a collective many-body system

This paper investigates the collective Bose-Hubbard model on a four-site lattice to reveal that quantum-classical correspondence breaks down in an intermediate energy regime due to quantum dynamics remaining trapped in symmetry-breaking sectors despite classical chaos, leading to unexpectedly slow convergence to the classical limit even for large particle numbers.

The Gist

Imagine you have a giant, magical dance floor with four corners. On this floor, there are thousands of tiny dancers (particles) who can move between the corners. They can hop around freely, but they also push and pull on each other depending on how crowded a corner gets. This is the Bose-Hubbard model, a simplified way physicists study how huge groups of atoms behave.

The big question this paper asks is: If we watch these dancers for a long time, will they eventually spread out evenly across all four corners, or will they get stuck in just one or two?

Usually, physicists expect that if you have enough dancers, the quantum world (where things are fuzzy and wave-like) will eventually look exactly like the classical world (where things are solid and predictable). This is called Quantum-Classical Correspondence. It's like expecting a blurry, spinning top to eventually look exactly like a sharp, stationary top once it slows down.

However, this paper discovers a surprising glitch in that expectation. Here is the story of what they found, broken down into three acts:

Act 1: The Locked Rooms (Low Energy)

Imagine the dance floor is divided into four separate rooms, and the doors between them are locked.

• The Classical View: If you put a dancer in the top-left room, they are stuck there forever. They can't get out.
• The Quantum View: The quantum dancers are also stuck in that room.
• The Result: Everyone stays in their corner. The system is "symmetry-breaking" because the corners aren't equal; one is full, the others are empty. This part works exactly as expected.

Act 2: The Narrow Bridges (The "Middle" Energy)

Now, imagine we give the dancers a little more energy. The doors between the rooms unlock, but they are replaced by tiny, fragile bridges.

• The Classical View: A classical dancer (a solid ball) can eventually cross these narrow bridges. It might take a very long time—maybe a million years—but eventually, they will wander into every room and spread out evenly. The system becomes "ergodic" (everything mixes).
• The Quantum View: Here is the surprise. The quantum dancers are like ghosts. Even though the bridges exist, the quantum dancers refuse to cross them. They get "trapped" in their original corner, just like in the locked rooms.
• The Glitch: The classical dancers say, "We are mixing!" while the quantum dancers say, "No, we are still stuck!"
• The Analogy: Imagine a crowd of people trying to cross a bridge that is so narrow it feels like a tightrope. A classical person (a solid human) might eventually wobble across. But a quantum person is like a wave of water; the wave hits the tightrope and bounces back, never making it to the other side.

This mismatch is the paper's main discovery. Even when the system is chaotic and should be mixing, the quantum world stays "stuck" in a specific pattern, while the classical world moves on.

Act 3: The Wide Open Floor (High Energy)

Finally, we give the dancers even more energy. The narrow bridges disappear, and the whole floor becomes one giant, open space.

• The Result: Both the classical dancers and the quantum dancers now run around freely and mix perfectly. The quantum world finally catches up to the classical world. The correspondence is restored.

The Big Surprise: "Finite-Size" Stubbornness

Usually, scientists think that if you just add more dancers (make the system bigger), the quantum behavior will eventually smooth out and look exactly like the classical behavior.

This paper found that this isn't true here. Even when they increased the number of dancers from 35 to 75 (which is a lot in this context), the quantum dancers still refused to cross the narrow bridges. They stayed stuck in their corners, while the classical dancers mixed.

It's as if you have a rule that says, "The more people you add to the party, the more they will mix." But in this specific quantum party, adding more people just made the "stuck" behavior stronger, not weaker.

Why Does This Matter?

This is a big deal because it challenges our understanding of how the universe works.

1. Thermalization: It shows that some quantum systems might never truly "thermalize" (reach a balanced, hot state) in the way we expect, even if they look like they should.
2. The Classical Limit: It proves that the transition from the weird quantum world to the normal classical world is much slower and more complicated than we thought. You can't just assume "big enough = classical."
3. Symmetry Breaking: It reveals that quantum systems have a "memory" of their starting position that is incredibly hard to erase, even when the laws of physics say they should move on.

In a nutshell: The researchers found a "trap" in the quantum world that doesn't exist in the classical world. Even when the path to freedom is open, quantum particles seem to get stuck in their lanes, and adding more particles doesn't help them escape. It's a reminder that the quantum world is stubborn, weird, and full of surprises that don't always follow the rules of everyday life.

Technical Summary

1. Problem Statement

The paper investigates the mechanisms of thermalization and the validity of the quantum-classical correspondence in isolated collective many-body systems. Specifically, it addresses a gap in the standard understanding of the Eigenstate Thermalization Hypothesis (ETH). While ETH generally predicts that chaotic quantum systems thermalize and that their long-time behavior converges to classical predictions as the system size ($N$) increases (the semiclassical limit), this work explores scenarios where this correspondence breaks down.

The authors focus on the collective Bose-Hubbard (BH) model with periodic boundary conditions. They aim to understand how Excited-State Quantum Phase Transitions (ESQPTs) and symmetry-breaking affect the equilibration process, particularly in the intermediate energy regime where classical phase space is topologically connected but quantum dynamics may remain trapped in symmetry-breaking sectors.

2. Methodology

The study employs a multi-faceted approach combining classical phase-space analysis, spectral statistics, and real-time dynamical simulations:

• Model: The $N$-site Bose-Hubbard model with $N$ bosons. The Hamiltonian includes hopping ($J$) and on-site interaction ($U$). The study primarily focuses on the 4-site case ($N_{sites}=4$) with varying particle numbers ($N$) to approach the thermodynamic limit.
• Classical Limit: The classical limit is defined by $N \to \infty$ (equivalent to an effective Planck constant $\hbar_{eff} \propto 1/N \to 0$). The authors derive a classical Hamiltonian in terms of canonical coordinates $(q, p)$ and analyze the phase space structure $\Omega$.
• Symmetry Analysis: The system possesses a discrete dihedral symmetry group $D_N$. The authors utilize a generalized imbalance operator ($\hat{I}$) as an order parameter to detect symmetry-breaking (where particle populations on sites are unequal, $\langle \hat{n}_i \rangle \neq 1/N$).
• Chaos Diagnostics:
  • Classical: Calculation of Lyapunov exponents to determine the fraction of regular vs. chaotic trajectories ($f_{reg}$).
  • Quantum: Analysis of level spacing distributions ($P(s)$) using the Berry-Robnik distribution to quantify the degree of quantum chaos and compare it with classical regularity.
• Dynamical Simulations:
  • Classical: Time evolution of single trajectories and ensembles of trajectories (averaged over a Gaussian distribution).
  • Quantum: Time evolution of coherent states (initial states mimicking classical points) under the Schrödinger equation.
  • Comparison: Direct comparison of the long-time average of the occupation number $\langle \hat{n}_{max}(t) \rangle$ between classical and quantum dynamics for different energy regimes.

3. Key Contributions and Results

The authors identify three distinct dynamical regimes based on energy relative to the ESQPT critical energies ($\epsilon_c$):

A. Low-Energy Regime ($\epsilon < \epsilon_{c1}$)

• Classical: The phase space consists of disconnected wells. Trajectories initialized in one well remain trapped, leading to symmetry-breaking equilibrium ($\langle \hat{I} \rangle \neq 0$).
• Quantum: Eigenstates are symmetry-breaking. The quantum dynamics matches the classical prediction: the system remains trapped in a symmetry-broken state.
• Correspondence: Restored. Quantum and classical behaviors agree.

B. Intermediate-Energy Regime ($\epsilon_{c1} < \epsilon < \epsilon_{c2}$)

This is the core finding of the paper, where the correspondence breaks down.

• Classical: The phase space becomes topologically connected via narrow "bridges" between the wells. While the system is technically ergodic (chaotic), the bridges are so narrow that classical trajectories exhibit intermittency. They get effectively trapped in a well for extremely long times before hopping, leading to very slow convergence to the symmetric equilibrium ($\langle \hat{I} \rangle = 0$).
• Quantum: Despite the classical phase space being connected, the quantum coherent state fails to delocalize. It remains trapped in a symmetry-breaking sector.
  • Mechanism: The breakdown is driven by the population of imbalance-carrying eigenstates (eigenstates with non-zero eigenvalues of the imbalance operator). Due to the quasi-degeneracy of energy levels in symmetry-broken sectors (gaps closing exponentially with $N$), the quantum wavepacket does not explore the full phase space.
  • Scaling: Crucially, this mismatch persists and even grows as the particle number $N$ increases (tested up to $N=75$). The quantum expectation value of the occupation number remains far from the symmetric value ($0.25$), while the classical average (over many trajectories) eventually approaches it.
• Correspondence: Broken. The quantum system does not thermalize to the classical equilibrium value, even for large $N$.

C. High-Energy Regime ($\epsilon > \epsilon_{c2}$)

• Classical: The phase space is fully connected and chaotic; the "bridges" disappear, and the system is ergodic.
• Quantum: The system thermalizes, and eigenstates are symmetric.
• Correspondence: Restored. Quantum and classical values coincide rapidly.

4. Significance

• Breakdown of Semiclassical Expectations: The paper challenges the standard semiclassical view that quantum systems in the collective limit ($N \to \infty$) will inevitably follow classical trajectories. It demonstrates that symmetry-breaking effects and ESQPTs can create robust finite-size effects that survive even at large particle numbers.
• Role of ESQPTs: It highlights that ESQPTs are not just static spectral features but fundamentally alter dynamical timescales, creating "bottlenecks" in phase space that affect thermalization differently in quantum vs. classical systems.
• Thermalization Mechanism: The work suggests that in systems with long-range interactions (collective models), thermalization is not guaranteed solely by chaos. The structure of the eigenstate spectrum (specifically the presence of symmetry-breaking eigenstates with non-vanishing imbalance) can prevent the system from reaching the classical equilibrium state.
• Finite-Size Effects: The finding that the quantum-classical mismatch increases with $N$ in the intermediate regime implies that "large" systems in collective models may still exhibit strong non-ergodic behavior, complicating the prediction of thermodynamic limits in such systems.

In conclusion, the authors provide a concrete example where the Eigenstate Thermalization Hypothesis holds in the sense of chaotic spectral statistics, yet the dynamical thermalization to the classical equilibrium state fails due to the specific interplay between symmetry, ESQPTs, and the population of specific eigenstates.