Inner structure of the many-body localization transition and the fulfillment of the Harris criterion

TL;DR

This paper reveals an inner structure of the many-body localization transition by introducing two symmetry-resolved entanglement measures derived from half-chain entropy: the probability distribution deviation $|d(p_n)|$ and the symmetry-block entropy $S_{ extrm{vN}}^{n}(D_n= ext{max})$. The measures are shown to be independent, yielding two distinct transition points in interacting 1D disordered Heisenberg chains, with the localized and thermalization transitions separated in the interacting case while coinciding in the noninteracting Aubry–André limit. Finite-size scaling produces universal exponents $ u oughly2$ for both transitions in random disorder, satisfying the Harris-CCFS bound, and $ u oughly1.5$ for the noninteracting quasiperiodic case, aligning with the Harris–Luck bound. The results provide an experimentalizable framework linking symmetry and entanglement to the organization of eigenstate phase transitions and resolve longstanding Harris-criterion tensions in ED studies by exposing the two-transition structure.

Abstract

We treat the disordered Heisenberg model in 1D as the standard model of many-body localization (MBL). Two new and independent order parameters stemming solely from the half-chain von Neumann entanglement entropy $S_{\textrm{vN}}$ are introduced to probe the eigenstate phase transition in this model. From the symmetry-endowed entropy decomposition, they are the probability distribution deviation $|d(p_n)|$ and the von Neumann entropy $S_{\textrm{vN}}^{n}(D_n\!=\!\mbox{max})$ of the maximally dimensional symmetry subdivision. The finite-size scaling reveals that $\{p_n\}$ drives the localization transition, preceded by a thermalization breakdown transition governed by $\{S_{\textrm{vN}}^{n}\}$. For the noninteracting case, these transitions coincide, but in the interacting circumstance they separate. Such separability creates an intermediate phase regime and discriminates between the Anderson and MBL transitions. One obstacle whose solution eludes the community to date concerns the violation of the Harris criterion in most numerical investigations of MBL. Upon elucidating the mutually independent measures comprising $S_{\textrm{vN}}$, it becomes clear that the previous studies may lack the resolution to pinpoint thus potentially overlook the crucial internal structure of the transition. We show that after this necessary decomposition, the universal critical exponents for both transitions of $|d(p_n)|$ and $S_{\textrm{vN}}^{n}(D_n\!=\!\mbox{max})$ fulfill the Harris criterion: $ν\approx2\ (ν\approx1.5)$ for quench (quasirandom) disorder. Our work puts forth symmetry combined with entanglement as an organization principle for the generic eigenstate matter and phase transition.

Figures (7)

• Figure 1: In the three prototype scenarios, the Heisenberg chain (\ref{['ham']}) randomized in the diagonal field strength displays two disentangled transition points $h^{1,2}_c$, each characterized by their own critical exponents $\nu^{1,2}$.
• Figure 2: The finite-size scaling of $S_{\textrm{vN}}$, $|d(p_n)|$, and $S^n_{\textrm{vN}}(D_n=\max)$ for the noninteracting Aubry-André model $(J_z=0)$.
• Figure 3: Upper (lower) row: The finite-size scaling of $S_{\textrm{vN}}$, $|d(p_n)|$, and $S^n_{\textrm{vN}}(D_n=\max)$ for the quasiperiodically (quench) disordered Heisenberg spin chain model, where the interaction strength $J_z=1$ is uniform and finite.
• Figure 4: The finite-size scaling of the correlator $C_n(D_n=\textrm{max})$ [see definition (\ref{['cnmax']})] as the experimentally more accessible alternative to detect the transition of $S^n_{\textrm{vN}}(D_n=\textrm{max})$ presented in Fig. \ref{['main_pic3']}(f).
• Figure 5: (a),(b) The finite-size scaling of $S_\textrm{vN}^{n}(D_n\!=\!\max)$ and $|d(p_n)|$ for the quench disordered Heisenberg spin chain model, where the interaction strength $J_z=1$ is uniform and finite. Here we use the shift-invert diagonalization method to reach the maximal system size $L=20$, and we focus exclusively on the eigenstate that is closest to the middle of the eigenspectrum. (c) displays the collected crossing points of the neighboring data lines in (b) as a function of the inverse system size $1/L$. The red solid line in (c) is a best linear fit for these available crossing points whose intercept $b$ represents an estimate for the critical disorder strength at the transition point in (b) in the infinite-$L$ limit.