Tracking Adiabaticity in Non-Equilibrium Many-Body Systems: The Hard Case of the X-ray Photoemission in Metals

Technical Summary: Tracking Adiabaticity in Non-Equilibrium Many-Body Systems

Problem Statement
The paper addresses the challenge of reliably tracking adiabaticity in complex, non-equilibrium many-body quantum systems. While the Quantum Adiabatic Criterion (QAC) is the standard tool for assessing adiabatic evolution, it faces significant limitations: it relies on perturbative assumptions, struggles with continuous energy spectra (common in metals), and fails to account for memory effects or non-Markovian dynamics. The authors propose testing recently developed metric-based methods—specifically those utilizing the Bures distance, trace distance, and Natural Local Density (NLD) distance—on a "hard case" scenario: X-ray photoemission in metals. This phenomenon is characterized by a strongly out-of-equilibrium character, a continuum energy spectrum, and the Anderson orthogonality catastrophe, where the ground state of the system becomes orthogonal to the initial state due to a localized scattering potential.

Methodology
The authors investigate the time evolution of a Fermi gas subject to a time-dependent localized scattering potential $K(t)$, modeling the interaction of an X-ray photon with inner-shell electrons in a metal. The system is described by a 1D tight-binding model with a half-filled band.

1. Theoretical Framework:

   • The authors utilize the metric-space approach to quantum mechanics. They define the adiabaticity of a system evolving from an initial ground state $|\phi_0(0)\rangle$ to a time-dependent state $|\Psi(t)\rangle$ by measuring the distance between $|\Psi(t)\rangle$ and the instantaneous ground state $|\phi_0(t)\rangle$.
   • They derive an explicit analytical upper bound for the NLD distance ($D_n$) in terms of the trace distance ($D_T$) and the overlap probability $|c_0(t)|^2 = |\langle \Psi(t) | \phi_0(t) \rangle|^2$. Specifically, they prove $D_n \leq 2 D_T$, establishing a rigorous connection between the computationally cheaper local density metric and the quantum state metrics.
   • A new analytical solution is derived for the time evolution of the system coefficients. This solution assumes that for continuous $K(t)$, the dynamics are dominated by single particle-hole excitations, neglecting higher-order excitations. This allows for a computationally efficient calculation of $|c_0(t)|^2$ across a wide parameter range.

2. Numerical Implementation:

   • To validate the analytical approach and handle the full many-body complexity, the authors employ the Real-Space Numerical Renormalization Group (eNRG) method. This involves discretizing the metallic band and using a smoothing procedure (integrating over the offset parameter $\theta$) to eliminate non-physical oscillations.
   • The time evolution is computed using the Crank-Nicolson method.

3. Parameters:

   • The study varies the maximum scattering potential $\bar{K}$, the ramp-up time $T$, and the energy gap $\Delta\varepsilon$ (controlled by system size $N$).
   • The authors compare results against the standard QAC, which is approximated using only the ground and first excited states.

Key Results

• Validity of Metric-Based Methods: The study confirms that metric-based methods (Bures, trace, and NLD distances) remain valid and effective for tracking adiabaticity even in the presence of the Anderson orthogonality catastrophe and continuum spectra.
• Analytical Solution Accuracy: The derived analytical solution, which considers only single particle-hole excitations, shows excellent agreement with the eNRG numerical results. It remains accurate even when the system is far from adiabaticity (down to $|c_0(t)|^2 \approx 0.6$). Deviations occur only when two-particle-hole excitations become significant.
• Universal Behavior: The authors identify a universal behavior for the ground-state occupation $|c_0(T)|^2$ at the end of the potential ramp. This behavior depends on the ratio of the ramp time to the gap time scale ($T/T_{\Delta\varepsilon}$) and the scattering potential strength.
• Failure of the QAC: The standard Quantum Adiabatic Criterion fails to predict or track adiabaticity for this system. The QAC incorrectly suggests that adiabaticity improves with time or depends linearly on the product $\bar{K} \cdot T$, whereas the metric-based analysis reveals a more complex dependence on $\bar{K}$ and the ratio $T/T_{\Delta\varepsilon}$. The QAC underestimates the non-adiabatic region for fast ramps and overestimates adiabaticity for slow ramps in the continuum limit.
• Superiority of Local Density Distance: The NLD distance not only tracks adiabaticity but also captures dynamical information missed by the Bures and trace distances. Specifically, for $t > T$ (after the potential stops changing), the Bures distance remains constant (as the projection on the ground state is conserved in a closed system), while the NLD distance decreases. This decrease reflects the system's relaxation via Friedel oscillations and charge redistribution toward the instantaneous ground state configuration, a non-equilibrium dynamic invisible to state-projection metrics.

Significance and Claims
The paper claims that the Natural Local Density distance is a robust, computationally efficient, and experimentally accessible tool for characterizing adiabaticity in complex many-body systems. By establishing an explicit upper bound relating the NLD distance to the trace distance, the authors provide a theoretical justification for using local density (which is easier to compute, e.g., via Density Functional Theory) as a proxy for full quantum state evolution.

The study demonstrates that for X-ray photoemission, the local density metric provides a more complete picture of the system's dynamics than traditional quantum state distances, particularly in capturing post-quench relaxation processes. Furthermore, the work highlights the inadequacy of the standard QAC for systems with continuous spectra and strong correlations, advocating for metric-based approaches that do not rely on perturbative expansions or discrete gap conditions. The derived analytical solution offers a fast and reliable method for determining adiabatic regimes in Fermi gases with localized potentials, valid well beyond the strict adiabatic limit.