Many-body localization edge in the random-field Heisenberg chain (1411.0660v2)

Abstract: We present a large scale exact diagonalization study of the one dimensional spin $1/2$ Heisenberg model in a random magnetic field. In order to access properties at varying energy densities across the entire spectrum for system sizes up to $L=22$ spins, we use a spectral transformation which can be applied in a massively parallel fashion. Our results allow for an energy-resolved interpretation of the many body localization transition including the existence of an extensive many-body mobility edge. The ergodic phase is well characterized by Gaussian orthogonal ensemble statistics, volume-law entanglement, and a full delocalization in the Hilbert space. Conversely, the localized regime displays Poisson statistics, area-law entanglement and non ergodicity in the Hilbert space where a true localization never occurs. We perform finite size scaling to extract the critical edge and exponent of the localization length divergence.

• The paper employs large-scale exact diagonalization to identify a pronounced many-body mobility edge across the energy spectrum.
• It differentiates ergodic and localized phases using level statistics, entanglement entropy scaling, and multifractal analysis.
• Finite-size scaling uncovers a critical exponent that challenges conventional predictions and prompts further experimental investigation.

Many-Body Localization in Random-Field Heisenberg Chains

The paper details a comprehensive paper of many-body localization (MBL) phenomena in the one-dimensional random-field Heisenberg spin-$1/2$ chain. Employing large-scale exact diagonalization, the authors explore properties across the energy spectrum for system sizes up to 22 spins. The research rigorously investigates transitions between ergodic and localized phases within such quantum systems, contributing to the broader understanding of disorder and interaction-induced phenomena.

Methodology and Findings

The investigation employs a spectral transformation technique, allowing extensive numerical analyses via exact diagonalization to access properties at varying energy densities. This approach efficiently tackles challenges posed by the size of Hilbert spaces, scaling to dimensions around $705,432$ for $L=22$ systems. The paper reveals a pronounced many-body mobility edge, delineating regions on the disorder-energy plane where transitions occur.

Characterization of Phases

The research identifies distinct characteristics of ergodic and localized regimes, as follows:

• Ergodic Phase: Characterized by Gaussian orthogonal ensemble statistics, volume-law scaling of entanglement entropy (EE), and delocalization in the Hilbert space.
• Localized Phase: Marked by Poisson statistics, area-law EE, and a multifractal state with reduced but non-zero localization properties within the Hilbert space.

Numerical Tools and Scaling Analyses

The paper benefits from several numerical tools and scaling analyses, including:

• Level Statistics and Kullback-Leibler Divergence (KLd): Employing level spacing statistics and KLd to discern regime transitions. The transition from Gaussian to Poisson distributions is explored via spectral features and eigenstates correlations.
• Entanglement Entropy: EE serves as a primary tool to differentiate phases, with scaling behaviors confirming volume-to-area law transitions.
• Bipartite Fluctuations and Participation Entropy: These metrics offer additional insights into system properties and the nature of localization, particularly emphasizing multifractal analyses of participation entropy to reinforce phase characterizations.

Critical Exponents and Phase Diagram

The paper reports a critical exponent $\nu \approx 0.8(3)$ through finite-size scaling analyses across different energy densities, pinpointing regions in the phase diagram corresponding to transitions. However, this violates the Harris-Chayes criterion ($\nu \geq 2/d$), raising questions about finite-size effects or potential deviations from standard scaling behavior at the MBL transition.

Implications and Future Directions

The implications are significant for understanding quantum statistical mechanics and phase transitions in disordered systems. The findings are pivotal in advancing potential experimental measurements of MBL, particularly in cold atom setups and spin chain systems. The methodological advances, especially regarding energy-resolved exact diagonalization and measurement of entropy scaling, are poised to influence future research in many-body quantum systems.

Avenues for future work include tackling the questions raised by the critical exponent findings and exploring the efficacy of representing localized states via matrix product states as inspired by these insights. The extension of such numerical approaches to broader classes of disordered models could further illuminate the robustness and universality of MBL phenomena.