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Many-body localization edge in the random-field Heisenberg chain
David J. Luitz, Nicolas Laflorencie, Fabien Alet
TL;DR
The paper studies how the many-body localization transition varies across the spectrum of a random-field Heisenberg chain. Using finite-size scaling and dynamical-spin-fraction data, it extracts transition parameters while finding stability limits from system size and spectral-edge sampling.
Problem
The study examines how dynamical memory distinguishes ergodic and localized phases across disorder and system size.
Source: Dynamical spin fraction · p. 6Method
The authors use finite-size scaling of dynamical spin fraction and other observables, with stability fits and bootstrap error estimates.
Source: Supplementary material · p. 6, Dynamical spin fraction · p. 7Results
The dynamical spin fraction shows a crossing at ϵ = 0.5, and scaling collapse yields hc and ν values compatible with other estimates.
Source: Dynamical spin fraction · p. 7Takeaways & Limitations
The dynamical spin fraction provides a transition estimate consistent with the paper’s other finite-size-scaling analyses.
Source: Dynamical spin fraction · p. 7Fits restricted to L ≥16 are difficult, entanglement scaling is problematic with only even sizes, and gap-ratio analysis suffers at spectral edges.
Source: Supplementary material · p. 6
Abstract
from arXiv · showhide
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.
Supplementary material
The supplementary analysis tests the stability of finite-size scaling estimates for hc and ν across fit windows and system-size cutoffs. It identifies a stable fitting choice while quantifying statistical and systematic uncertainty and documenting spectral and size-related limitations.
Supplementary material
- The scaling analysis fits r, f, SE/L, and F/L to g[(h−hc)L^(1/ν)] while varying fit-window size and minimum system size.The universal function is modeled by a cubic polynomial, and drift terms were unnecessary for good fit quality.Source: Supplementary material · p. 6
- ν = 0.8(3) is the estimated critical exponent, with Figure 7 showing fit-window results against the mean and error bounds.Targets from 0.1 to 0.9 are included in the displayed stability analysis.Source: Figure 7 · p. 6
- Fits restricted to L ≥16 generate most outliers because the reduced size range makes scaling difficult, especially for SE/L using only even system sizes.The entanglement-entropy analysis is particularly problematic under this restriction.Source: Supplementary material · p. 6
- At low and high energy densities, the low density of states makes gap-ratio analysis particularly problematic.This limits the reliability of level-statistics scaling near the spectrum edges.Source: Supplementary material · p. 6
- The most stable fits use Lmin = 14, or Lmin = 15 for gap ratios, with fit-window half-width w = 0.8.These settings are used for the results presented in the rest of the Letter.Source: Supplementary material · p. 6
- The stability spread represents systematic uncertainty, while bootstrap estimates provide statistical errors for hc and ν.The authors caution that the systematic error is of the order of the fit-result spread.Source: Supplementary material · p. 6
Dynamical spin fraction
The dynamical spin fraction measures memory of an initial spin density after long-time evolution. At the spectrum center, its disorder- and size-dependent data show a crossing and can be collapsed with finite-size scaling.
Dynamical spin fraction
- The dynamical spin fraction f equals 1 in the ergodic phase and decays to zero in the localized phase as initial spin-density memory is lost.It is defined from the long-time remainder of an initially prepared spin density.Source: Dynamical spin fraction · pp. 6–7
- At ϵ = 0.5, f versus disorder strength shows a crossing for different system sizes L.The crossing is used to analyze the transition at the spectrum center.Source: Dynamical spin fraction · p. 7, Figure 8 · p. 7
- The scaling form g[(h−hc)L^(1/ν)] collapses the dynamical-spin-fraction data and yields hc and ν values compatible with other estimates.The best-fit parameters are reported in the Figure 8 inset.Source: Dynamical spin fraction · p. 7