Geometric signatures of the onset of many-body ergodicity

Technical Summary: Geometric Signatures of the Onset of Many-Body Ergodicity

Problem Statement
Identifying universal, robust, and interpretable signatures for the onset of ergodicity in quantum many-body systems remains a significant challenge. While existing diagnostics—such as entanglement entropy scaling, inverse participation ratios, eigenvalue statistics (gap ratios, spectral form factors), and dynamical features (operator growth)—provide precise characterizations deep within thermalizing or non-thermalizing regimes, they often exhibit ill-behaved or ambiguous behavior near the boundary between ergodic and integrable phases. Furthermore, previous approaches utilizing the Adiabatic Gauge Potential (AGP) as a probe of quantum chaos have been limited by their dependence on the specific nature of the perturbation; without prior knowledge of the Hamiltonian's symmetries, a single-parameter AGP analysis may mischaracterize the thermalizing regime.

Methodology
The authors generalize the AGP framework to multi-parameter perturbations to construct an emergent quantum geometry. They define a Hilbert-Killing (HK) metric, $G_{\mu\nu}$, which quantifies the susceptibility of eigenstates to infinitesimal changes in Hamiltonian parameters $x_\mu$.

1. Construction of the Metric:

   • Starting with a Hamiltonian $\hat{H}[x]$ parametrized by $n$ parameters, the authors consider the transformation of eigenstates $|\psi_k(x)\rangle$ under parameter variations.
   • They define the infidelity between states at $x$ and $x+dx$ as a metric on the manifold of parameterized quantum states.
   • The single-eigenstate metric components are defined as $g^{(k)}_{\mu\nu} = \text{Re}[\langle\psi_k|A_\mu A_\nu|\psi_k\rangle - \langle\psi_k|A_\mu|\psi_k\rangle\langle\psi_k|A_\nu|\psi_k\rangle]$, where $A_\mu$ is the adiabatic gauge potential (generator of parallel transport).
   • To characterize the entire spectrum, they construct the HK metric by averaging over the spectrum: $G_{\mu\nu} = \frac{1}{d} \sum_k g^{(k)}_{\mu\nu}$.
   • The authors demonstrate that this metric is equivalent to the entanglement fidelity of the unitary transformation and maps directly to the Killing form of the Lie algebra $\mathfrak{gl}(d)$, hence the name "Hilbert-Killing."

2. Numerical Investigation:

   • The HK metric is applied to two distinct models:
     • The 1D mixed-field Ising model with transverse ($h_x$) and longitudinal ($h_z$) fields.
     • The constrained PXP model with disorder, which hosts localized (MBL), constrained MBL, and ergodic phases.
   • The study involves computing various components of the metric tensor (diagonal $G_{\mu\mu}$ and off-diagonal $G_{\mu\nu}$) and performing finite-size scaling analysis to determine the growth rate of the metric with system size $L$.

Key Contributions

• Generalization of AGP: The work extends the AGP from a single-parameter probe to a multi-parameter geometric structure, providing a symmetry-agnostic tool for diagnosing ergodicity.
• The Hilbert-Killing Metric: The introduction of the HK metric as a unified geometric quantity that encodes eigenstate deformations across the entire spectrum.
• Identification of a Universal Scaling Signature: The paper identifies that the boundary between integrable and ergodic regimes is uniquely characterized by the fastest growth of the HK metric with system size.
• Directional Sensitivity: The authors demonstrate that different components of the HK metric exhibit disparate behaviors within the same regime, highlighting that the choice of perturbation direction is critical for accurate diagnosis.

Results

• Ising Model: In the phase diagram defined by $(h_x/J, h_z/J)$, the HK metric components (e.g., $G_{h_x h_x}$) show a significantly larger response precisely at the boundary between the integrable/free and ergodic phases.
  • Scaling Behavior: Deep in the ergodic phase, the metric exhibits exponential scaling with system size (log-derivative plateaus). In the integrable phase, it shows sub-exponential scaling (log-derivative decays). Crucially, at the ergodic-integrable boundary, the log-derivative of the metric increases with system size, indicating the strongest scaling growth.
  • Perturbation Dependence: While the boundary signature is robust, the specific magnitude of the response depends on the perturbation direction. For instance, the free phase is highly susceptible to perturbations breaking its integrability ($G_{h_z h_z}$), whereas the ergodic phase shows strong responses across directions.
• PXP Model: Similar results are observed in the disordered PXP model. The HK metric is maximized at the transition boundaries (both the critical boundary between localized and ergodic phases and the constrained boundary).
  • The scaling analysis confirms that the strongest system-size dependence is pinned specifically to the ergodic-integrable boundary, distinguishing it from the bulk of the ergodic or integrable phases.
  • The metric successfully distinguishes between different types of non-ergodic phases (e.g., diagonal MBL vs. constrained MBL) and the onset of thermalization.

Significance
The paper claims that the Hilbert-Killing metric provides a robust and interpretable geometric signature for the onset of many-body ergodicity. By capturing the fastest scaling growth at the phase boundary, the HK metric overcomes limitations of previous metrics that are often ill-behaved near critical transitions. The work suggests that quantum geometry, as endowed by the HK metric, offers a natural bridge connecting ergodicity, information theory (via entanglement fidelity), and differential geometry (via the Killing form). The authors emphasize that this approach allows for a more comprehensive understanding of the specific directions and perturbations that drive or inhibit thermalization in complex quantum systems.