Characterizing the Many Body Localization Crossover as a Metal-Insulator Transition: Localization length from Polarization and Quantum Metric

TL;DR

This work links the geometry of many-body quantum states to many-body localization by relating a localization length from the modern theory of polarization to the many-body quantum metric under twist boundary conditions. By applying these concepts to both noninteracting Anderson insulators and the disordered Bose-Hubbard chain, the authors show that the MBQM $g_N$ and localization parameter $D_N$ agree in insulating regimes and diverge in ergodic regimes for finite systems, enabling a finite-size diagnostic of the MBL crossover. They introduce a dimensionless diagnostic $\Delta$ to quantify the agreement and extract a natural localization length $\ell_N = \sqrt{D_N}/(2\pi n) = \sqrt{g_N}/n$ that remains meaningful across insulating phases. The results provide a geometrical, experimentally accessible route to characterize MBL and illuminate finite-size effects and possible thermodynamic-limit behavior in 1D. The study also highlights potential experimental pathways to measure MBQM and to infer localization lengths, with implications for understanding the fate of MBL in the thermodynamic limit.

Abstract

Many body localization (MBL) represents a unique physical phenomenon, providing a testing ground for exploring thermalization, or more precisely its failure. Here we characterize the MBL regime geometrically by the many-body quantum metric (MBQM), defined in the parameter space of twist boundary, and the localization parameter as defined in the modern theory of polarization and insulators. First, we demonstrate that the quantum metric can be used to characterize disordered insulating states by applying this theoretical framework to excited states of the 1D Anderson insulator. There we observe that the MBQM and localization parameter are related in finite realizations despite the states being gapless in the thermodynamic limit. Then, we consider a disordered 1D Bose-Hubbard model and find that we can characterize the ergodic-MBL crossover by comparing the MBQM and localization parameter. We find that we can extract a natural localization length in the MBL regime that characterizes the real space spread of the wave function and can be measured by extracting the quantum metric. Our analysis provides complementary insight into the MBL regime focusing on its insulating properties and providing a localization length whose definition is consistent across a range of insulating phases.

Figures (7)

• Figure 1: MBQM $L^2g$ (solid), variance (dashed) and $(L/2\pi)^2 D_1$ (dotted) as a function of disorder strength $W$ for a single particle Anderson Insulator system. The line color corresponds to different system lengths. These quantities are all equivalent in a localized state. The discrepancy at low disorder arises from the localization length becoming comparable to the system size.
• Figure 2: Quantum metric, $g_N$, and localization length parameter, $D_N$, as a function of disorder strength $W$ for a half-filled Anderson insulator. The colors correspond to different lengths of the system, $L$, as labeled in the legend. The key feature to notice is that for large enough disorder the two quantities agree and become independent of system size. Disagreements at low disorder are attributed to finite size effects.
• Figure 3: The parameter $\Delta$ for the half-filled non interacting disordered chain. The point size and color correspond to different system lengths, $L$, as labeled in the legend with larger points representing longer chains. For large disorder we see that the parameter drops to zero as we have reached a large enough system size to capture the insulating nature of the state.
• Figure 4: Many-body quantum metric $g_N=4\pi^2Ng$ (solid) and localization parameter $D_N=-N\ln|z_N|^2$ (dashed) versus disorder strength for different fixed interaction strengths $U$. The line color corresponds to different system lengths, $L$, as labeled in the legend. Vertical dashed lines correspond to the contour presented on the phase diagram on the right panel of Fig. \ref{['fig:PDfinite']} where the parameter $\Delta=0.5$ for the system with $L=18$.
• Figure 5: The parameter $\Delta$ as a function of disorder strength $W$ for different interaction strengths $U$. Each plot contains results for system sizes $L=8$ to $18$ with marker size proportional to system length.