Suppressed correlation-spreading in a one-dimensional Bose-Hubbard model with strong interactions

This paper demonstrates that in a strongly interacting one-dimensional Bose-Hubbard model, doublon-holon exchange drives slow, non-ergodic correlation spreading via domain-wall excitations, a phenomenon further suppressed by parabolic traps and accurately described by mapping the system to an antiferromagnetic transverse-field Ising model.

The Gist

Imagine a long, narrow hallway lined with lockers. Inside this hallway, we have a group of energetic dancers (the atoms) who love to pair up. In this specific experiment, the dancers start in a very strict formation: every other locker is packed with a dancing pair, and the lockers in between are completely empty. It looks like this: [Pair] [Empty] [Pair] [Empty]...

This is the "doubly occupied density-wave state" mentioned in the paper.

The researchers wanted to see what happens when they let these dancers move around freely. Usually, in a chaotic system, if you start with an ordered pattern, it quickly dissolves into a messy, random crowd. This is called "thermalization" or "ergodicity"—the system forgets its past and settles into a uniform average.

However, this paper discovers that when the dancers are very strongly attracted to staying in their pairs (strong interactions), the system behaves differently. It refuses to forget its past. Here is a breakdown of what happens, using simple analogies:

1. The "Rebound" Effect (Strong Interactions)

Imagine the dancers are so tightly bound in their pairs that they can't easily break apart. If a pair tries to split up to fill an empty locker, it costs a huge amount of energy (like trying to pull apart two magnets glued together).

Instead of breaking apart, the pairs start "jumping" as a unit. A pair in one locker jumps over to the next empty spot, effectively swapping places with the empty space.

• The Analogy: Think of a row of people holding hands in pairs, with empty chairs between them. Instead of letting go and sitting randomly, the pairs slide over together.
• The Result: This creates a "domain wall." Imagine a wave of "swapped" pairs moving down the hallway. The left side is still in the original order, but the right side has flipped. This wave travels across the system, but it moves very slowly.

2. The "Frozen Edges" (The Trap)

Now, imagine the hallway isn't flat; it's slightly curved like a shallow bowl (this is the "parabolic trap"). The lockers at the very ends of the hallway are slightly higher up, making it harder for the dancers to move there.

The researchers found that even a very slight curve in the hallway is enough to freeze the dancers at the edges.

• The Analogy: It's like trying to roll a marble down a very gentle hill. If the marble is heavy enough (strong interaction), it gets stuck at the very top edge and refuses to roll down.
• The Surprise: Usually, you need a huge wall to stop something from moving. Here, a tiny, almost invisible slope was enough to stop the "correlation wave" from spreading to the edges. The edges become "frozen," and the chaos never reaches them.

3. Different Types of "Memories" (Correlations)

The paper looked at how the dancers "remember" each other. They checked three different ways of looking at the crowd:

• The "Solo" Memory (Single-particle correlation): If you ask, "Is there a dancer at locker #5?", the answer is very localized. The dancers don't seem to know what's happening far away. It's like a shy person who only talks to their immediate neighbor.
• The "Pair" Memory (Pair correlation): If you ask, "Are there two dancers together?", they still mostly stick to their immediate neighbors. They are a bit more social than the solo dancers, but they still don't travel far.
• The "Pattern" Memory (Density-density correlation): This is the most interesting one. If you look at the pattern of pairs vs. empties, you can see a wave moving down the hallway. Even though the individual dancers haven't moved far, the information about the pattern has traveled. However, this wave moves slowly and gets stuck at the edges if the "bowl" trap is present.

4. The "Spin" Shortcut (The Magic Mapping)

To understand why this happens so clearly, the authors used a clever mathematical trick. They realized that this complex system of dancing pairs could be perfectly described by a much simpler system: a line of tiny magnets (spins) that can point Up or Down.

• The Analogy: Instead of tracking hundreds of complex dancers, they realized the whole system acts like a row of light switches.
  • Switch Up = A pair is here.
  • Switch Down = The pair has moved to the next spot.
• Why it helps: In this "magnet" world, the movement of the wave is just a "spin flip" traveling down the line. The speed of this wave is determined by how hard it is to flip a switch (which depends on the interaction strength). The stronger the interaction, the slower the flip, and the slower the wave travels.

The Big Takeaway

In a world of quantum particles, we often expect chaos to win quickly. Everything should mix up and become random.

This paper shows that if you have strong interactions (particles that really want to stay in pairs) and even a tiny bit of a trap (a slight slope), the system can get "stuck." It creates a slow-moving wave of change that refuses to spread out fully. The system retains a memory of its initial order for a very long time, effectively breaking the rules of normal thermal equilibrium.

It's like a line of dominoes that, instead of falling over quickly, only tips over one by one at a glacial pace, and the ones at the very end never fall at all.

Technical Summary

1. Problem Statement

The paper investigates the breakdown of ergodicity and the mechanisms of thermalization in isolated quantum many-body systems. Specifically, it focuses on the one-dimensional (1D) Bose-Hubbard model starting from a doubly occupied density-wave state ($|2, 0, 2, 0, \dots\rangle$).

While previous studies have shown that singly occupied density-wave states thermalize quickly and that many-body localization (MBL) occurs in the presence of strong disorder or tight traps, the dynamics of the doubly occupied state under strong interactions ($U \gg J$) remain less understood. The authors aim to determine:

• Whether correlation spreading is suppressed in this specific regime.
• How the interplay between strong on-site repulsion ($U$) and a weak parabolic trapping potential affects the relaxation dynamics.
• The underlying physical mechanisms (excitations) driving these dynamics.

2. Methodology

The authors employ quasi-exact numerical simulations using Matrix Product States (MPS) and the Time-Evolving Block Decimation (TEBD) algorithm.

• Model: A 1D Bose-Hubbard chain with open boundary conditions, subject to a parabolic trapping potential $V_i \propto (i - (M+1)/2)^2$.
• Parameters:
  • Lattice size: $M=16$ (for dynamics) and $M=12$ (for long-time averages).
  • Interaction strength: Strong regime ($U/J \gg 1$, specifically up to $U/J = 40$).
  • Trap strength: Weak trap ($\Omega/J = 0.02$) to minimize direct localization effects while isolating interaction-driven phenomena.
  • Initial State: $|\psi_0\rangle = |2, 0, 2, 0, \dots\rangle$.
• Observables:
  • Site occupation dynamics ($n_i$).
  • Correlation functions: Single-particle ($C_{sp}$), Pair ($C_{pair}$), and Connected Density-Density ($C_{dd}$).
  • Correlation width ($\sigma(t)$) to quantify the spatial spread of correlations.
• Analytical Approach: Derivation of an effective Hamiltonian using the Schrieffer-Wolff transformation to map the bosonic system to a spin model in the limit $U \gg J$.

3. Key Contributions and Results

A. Dominance of Doublon-Holon Exchange

In the strong interaction limit ($U \gg J$), single-particle hopping is energetically suppressed. Instead, the dynamics are dominated by second-order pair tunneling (doublon-holon exchange).

• Mechanism: A pair of bosons on one site tunnels to a neighboring empty site, effectively flipping a $|2, 0\rangle$ configuration to $|0, 2\rangle$.
• Domain Wall Excitation: This process creates a "domain wall" separating regions of $|2, 0\rangle$ and $|0, 2\rangle$ pairs.
• Propagation: In the absence of a trap, this domain wall excites at the edge (where an empty site exists) and propagates across the chain, eventually inverting the system configuration.

B. Suppression of Correlation Spreading by Traps

The presence of even a weak parabolic trap significantly alters the dynamics:

• Edge Freezing: The trap creates an energy gradient that freezes the occupation at the edges. The pair-hopping process at the edges becomes energetically costly, preventing the excitation of the domain wall from the boundaries.
• Correlation Localization: While single-particle and pair correlations are already highly localized (decaying rapidly beyond nearest neighbors), the density-density correlations ($C_{dd}$) show a distinct behavior.
  • Without a trap: $C_{dd}$ shows a "negated staggered pattern" propagating outward, indicating domain wall motion.
  • With a trap: The propagation of these correlations is suppressed. The correlation width $\sigma(t)$ does not grow indefinitely but decays or saturates at smaller values as interaction strength increases.
• Timescale: The relaxation timescale is dictated by the inverse of the interaction strength ($\sim U/J$), leading to slow dynamics.

C. Mapping to an Effective Transverse-Field Ising (TFI) Model

To provide an intuitive understanding, the authors map the effective Bose-Hubbard Hamiltonian (restricted to $\nu_i \in \{0, 2\}$) to a Transverse-Field Ising (TFI) model:

• Mapping: $|2, 0\rangle \to |\uparrow\rangle$ and $|0, 2\rangle \to |\downarrow\rangle$.
• Hamiltonian: The resulting effective Hamiltonian includes a transverse field term (spin flip) and an Ising interaction term, with an effective coupling $\tilde{J} = 2J^2/U$.
• Validation:
  • The TFI model successfully reproduces the site occupation dynamics and density-density correlations of the original Bose-Hubbard model.
  • The domain wall excitation in the bosonic model corresponds to a spin-flip excitation in the TFI model.
  • The propagation velocity of correlations in the original model matches the group velocity of spin-wave excitations in the TFI model ($v_g \propto \tilde{J}$).

4. Significance and Implications

• Non-Ergodic Signatures: The paper identifies specific dynamical signatures of non-ergodicity (Hilbert Space Fragmentation) in a clean system without disorder. The suppression of correlation spreading is a robust indicator of this behavior.
• Role of Traps: It demonstrates that weak trapping potentials, often considered negligible in bulk studies, can fundamentally alter relaxation dynamics by freezing edge modes and suppressing the propagation of excitations.
• Theoretical Insight: The successful mapping to the TFI model provides a powerful analytical tool for understanding complex bosonic dynamics in the strong-coupling regime, linking them to well-known spin physics concepts like spin waves and domain walls.
• Experimental Relevance: The findings are directly relevant to cold atom experiments in optical lattices, suggesting that correlation spreading can be controlled and suppressed by tuning interaction strengths and trap geometries, even in regimes previously thought to be thermalizing.

Conclusion

The study reveals that in a 1D Bose-Hubbard model with strong interactions starting from a doubly occupied density-wave state, correlation spreading is dominated by doublon-holon exchange. While the system eventually relaxes in the absence of a trap, the introduction of a weak trap freezes edge dynamics, leading to a significant suppression of correlation propagation. This behavior is accurately captured by an effective Transverse-Field Ising model, where the slow relaxation is governed by the group velocity of spin-wave excitations.