Non-equilibirum physics of density-difference dependent Hamiltonian: Quantum Scarring from Emergent Chiral Symmetry

This paper demonstrates that a density-difference-dependent Hamiltonian, characterized by an emergent chiral symmetry, hosts two distinct classes of quantum many-body scars—a charge density wave ordered scar and an edge-mode scar—that exhibit robust thermalization breaking dynamics.

The Gist

The Big Picture: A Quantum Puzzle

Imagine a crowded dance floor where everyone is supposed to eventually mix with everyone else, forgetting who they started with. In physics, this is called "thermalization" or "ergodicity." Usually, if you start a quantum system (like a group of atoms) in a specific pattern, it quickly gets messy, scrambles, and forgets its original shape.

However, this paper discovers a special "glitch" in the rules. The authors found a way to build a system where the dancers refuse to mix. Instead of forgetting their starting position, they keep dancing in a loop, remembering exactly where they began. In physics, these stubborn, non-mixing states are called Quantum Many-Body Scars.

The researchers studied a specific set of rules (a Hamiltonian) for how particles move. They found that this system has two different "superpowers" that create these scars, depending on how the rules are tweaked.

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Mechanism 1: The "Perfect Cancellation" Dance (Charge Density Wave)

The Setup: Imagine a line of dancers. The rules say they can hop to the next spot, but there's a catch: if a neighbor is already there, the hop changes.

The Analogy: Think of this like a game of musical chairs where the chairs are moving.

• The Problem: Usually, if a dancer tries to move left, they might get stuck or bounce back randomly.
• The Solution: The authors found a specific setting (using "imaginary" numbers in the math) where two forces cancel each other out perfectly.
  • Imagine a dancer trying to hop forward.
  • Simultaneously, a "correlated" force tries to pull them backward.
  • If the timing is perfect, these two forces are like two people pushing a car from opposite sides with equal strength. The car doesn't move.
• The Result: This "destructive interference" locks the particles into a specific pattern called a Charge Density Wave (like an alternating pattern of occupied and empty spots: Occupied-Empty-Occupied-Empty).
• The Catch: This "glitch" is a bit fragile. If you make the line of dancers infinitely long (the "thermodynamic limit"), the perfect cancellation starts to fail, and the pattern eventually breaks down. It's a "weak" scar—it works for a while, but it's not permanent in an infinite system.

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Mechanism 2: The "Trapped Edge" Ghosts (Many-Body Edge Modes)

The Setup: Now, imagine the same line of dancers, but this time the rules are slightly different (using "real" numbers).

The Analogy: Think of a long hallway with a very thick, sticky carpet in the middle, but the very ends of the hallway are smooth, slippery ice.

• The Middle: In the middle of the system, the particles are "bound" together in tight clusters. They act like a single heavy unit that can't easily move around.
• The Edges: At the very ends of the line, the rules change. Because the line stops, the particles at the edge get "trapped" in a special state.
• The "Fock-Space Lattice": The authors used a clever trick to visualize this. Instead of thinking of particles moving on a physical line, they imagined the particles moving on a map of all possible arrangements. On this map, the edge particles look like they are stuck in a small, isolated room at the end of a long corridor.
• The Result: These edge particles bounce back and forth between the very end of the line and the spot next to it, never venturing into the messy middle. Because they are stuck at the edge, they don't mix with the rest of the system.
• Why it's special: This is a "strong" scar. Even if the system is large, these edge ghosts stay put. They are protected by a symmetry in the math (called "chiral symmetry") that pins them to a specific energy level, making them immune to the chaos happening in the middle.

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How They Proved It

The researchers didn't just guess; they ran simulations to prove these patterns exist:

1. Entanglement Check: In a normal chaotic system, particles become deeply "entangled" (connected) with everything else, creating a huge mess of information. In their "scar" systems, the entanglement stayed very low. It was like the dancers at the edge were wearing noise-canceling headphones, ignoring the chaos around them.
2. The "Revival" Test: They started the system in a specific pattern and watched it evolve. In a normal system, the pattern would disappear instantly. In their system, the pattern would fade, then suddenly snap back to its original shape, over and over again. This "revival" is the signature of a quantum scar.

Summary

The paper shows that by tweaking how particles interact based on their neighbors, you can create two types of "memory" in a quantum system:

1. The Wave Scar: A pattern that survives because opposing forces cancel each other out (works well for a while, but fades in huge systems).
2. The Edge Scar: Particles that get trapped at the ends of the line, protected by the geometry of the system and the rules of the game, refusing to ever mix with the crowd.

This helps physicists understand how the orderly, predictable world we see in daily life might emerge from the chaotic, scrambling world of quantum mechanics.

Technical Summary

Technical Summary: Non-equilibrium physics of the density-difference-dependent Hamiltonian

Problem and Context
Quantum many-body scars (QMBS) represent a form of weak ergodicity breaking where a small subset of eigenstates in a chaotic system retains memory of specific initial conditions, preventing thermalization. While the Eigenstate Thermalization Hypothesis (ETH) generally explains the emergence of classical statistical mechanics from the Schrödinger equation, QMBS challenge this by exhibiting low entanglement entropy and persistent oscillations in time-dependent fidelity. Previous mechanisms for QMBS formation include kinetic constraints (e.g., the PXP model), geometric frustration, and Hilbert space fragmentation. However, the mechanisms governing scar formation in systems with density-dependent interactions and dynamical gauge fields remain less explored. This work investigates the density-difference-dependent Hamiltonian, a model previously studied in Floquet engineering and non-Hermitian few-body limits, to identify new mechanisms for stabilizing QMBS.

Methodology
The authors analyze a one-dimensional bosonic chain of length $L$ with $N$ particles at half-filling ($\nu = 1/2$) governed by the Hamiltonian:
$$H = \sum_{j} a^\dagger_{j+1}[-J + \gamma(n_{j+1} - n_j)]a_j + a^\dagger_j[-J + \gamma^*(n_{j+1} - n_j)]a_{j+1}$$
where $J$ is the hopping parameter and $\gamma$ is a complex coupling to the density-difference-dependent hopping. The study employs both periodic (PBC) and open boundary conditions (OBC).

The methodology involves:

1. Spectral Analysis: Calculating the average level spacing ratio ($r_n$) to confirm the chaotic nature of the system (matching Gaussian Orthogonal or Unitary Ensemble predictions) and identifying deviations indicative of scars.
2. Entanglement Entropy: Computing the bipartite entanglement entropy ($S$) to identify eigenstates with anomalously low entropy, signaling ETH violation.
3. Time Dynamics: Simulating the time evolution of fidelity and entanglement entropy for specific initial states (Charge Density Wave and edge-mode configurations) to observe revivals and non-ergodic behavior.
4. Analytical Modeling: Constructing effective models, including a Fock-space lattice description for clustered states and a Krylov subspace analysis, to derive the mechanisms stabilizing the scars.
5. Numerical Techniques: Utilizing exact diagonalization for small systems and the Time-Dependent Variational Principle (TDVP) for larger systems to verify stability in the thermodynamic limit.

Key Contributions and Results

The paper identifies two distinct mechanisms for QMBS formation in this Hamiltonian, dependent on the nature of the coupling $\gamma$:

1. Charge Density Wave (CDW) Scar (Imaginary $\gamma$)

• Mechanism: When $\gamma$ is purely imaginary, the system exhibits a CDW-like scar stabilized by the destructive interference between bare hopping ($-J$) and correlated hopping (gauge field hopping). The authors propose an ansatz where the scar state is a superposition of the CDW state and configurations with "doublons" (two particles on a site) created by unbinding processes.
• Symmetry: This mechanism relies on the system's chiral symmetry and parity.
• Nature of the Scar: The authors characterize this as a "weak" scar. While it exhibits non-ergodic fidelity oscillations and low entanglement for finite systems, the weight of the scarred configuration ($|c_0|^2$) scales as $1/L$. Consequently, the non-ergodic behavior vanishes in the thermodynamic limit.
• Dynamics: In OBC, the system exhibits oscillations between two CDW orders ($|CDW\rangle$ and $|CDW'\rangle$) driven by the unoccupied edge, with frequencies proportional to $J$ and inversely proportional to chain length $L$.

2. Many-Body Edge-Mode Scar (Real $\gamma$)

• Mechanism: When $\gamma$ is purely real, the system supports scars associated with many-body edge modes. These states arise from the interplay between energy detuning of edge states (relative to the bulk spectrum) and the system's chiral symmetry.
• Effective Model: The authors map the many-body problem to an effective Fock-space lattice. In this picture, the system behaves like a Su-Schrieffer-Heeger (SSH) model where "sites" represent clusters of particles. The edge modes are identified as "truncation-induced" states resulting from the termination of this effective lattice.
• Nature of the Scar: Unlike the CDW case, these edge-mode scars are robust. They appear as tensor products of edge states localized on the boundaries (e.g., $N/2$ particles on the left edge and $N/2$ on the right). They exhibit near-zero entanglement entropy and persistent fidelity revivals even in the thermodynamic limit (demonstrated via TDVP for $N=10$).
• Stability: The stability of these scars is sensitive to the ratio $J/|\gamma|$. If the hopping $J$ becomes too large relative to $\gamma$, the edge states merge with the extended bulk spectrum, destroying the scar. The critical value $J_c$ depends on the particle number.

Significance and Claims
The authors claim that this work demonstrates the existence of QMBS in a density-difference-dependent Hamiltonian without relying on previously known mechanisms like strict kinetic constraints or Hilbert space fragmentation.

• New Mechanisms: The paper proposes two novel stabilization mechanisms: destructive interference of hopping terms for CDW scars and the interplay of chiral symmetry with truncation-induced edge states in an effective Fock-space lattice for edge-mode scars.
• Role of Symmetry: The study highlights the critical role of chiral symmetry in pinning states to zero energy and stabilizing these non-ergodic phases.
• Weak vs. Strong Scarring: The work distinguishes between "weak" scarring (CDW), which is unstable in the thermodynamic limit, and "strong" scarring (edge modes), which remains robust.
• Broader Implications: The authors suggest that these findings may provide insight into the general conditions under which QMBS appear and how they are disrupted in the classical limit. They note that the use of effective single-particle models (via Krylov Hamiltonians or cluster models) to explain many-body scars could bridge the gap between QMBS and topological states, as evidenced by the topologically nontrivial nature of the effective Hamiltonians derived in their analysis.

The paper concludes that while the ETH generally holds for non-integrable systems, specific symmetries and interaction structures can lead to robust violations, offering a deeper understanding of the connection between classical statistical mechanics and quantum unitary evolution.