Many-body localization edge in the random-field Heisenberg chain

We present a large-scale exact diagonalization study of the one-dimensional $\text{spin}-1/2$ 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 nonergodicity 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.

Figure 1

Disorder ($h$)—Energy density ($\epsilon$) phase diagram of the disordered Heisenberg chain, Eq. (1). The ergodic phase (dark region with a participation entropy volume law coefficient $a_1\simeq 1$) is separated from the localized regime (bright region with $a_1\ll 1$). Various symbols (see legend) show the energy-resolved MBL transition points extracted from finite-size scaling performed over system sizes $L\in \{14,15,16,17,18,19,20,22\}$. Red squares correspond to a visual estimate of the boundary between volume and area-law scaling of entanglement entropy $S_E$.

Figure 2

Adjacent gap ratio (top) and Kullback Leibler divergence (bottom) as a function of disorder strength in the spectrum center $\epsilon =0.5$. Insets: (top) data collapse used to extract the critical disorder strength $h_c$ and exponent $\nu$. The $h$ axis is transformed by $(h-h_c)L^{1/\nu }$; (bottom) distribution of KLd in both phases.

Figure 3

Entanglement entropy per site $S^{\mathrm{E}}/L$ and its variance ${\sigma }_E$, as a function of system size $L$ for different disorder strengths in the middle of the spectrum (left) and in the upper part (right). The volume-law scaling leading to a constant $S^{\mathrm{E}}/L$ for weak disorder contrasts with the area law (signaled by a decreasing $S^{\mathrm{E}}/L$) at larger disorder. Black line: $S^{\mathrm{E}}/L$ for a random state [58]. Close to the transition, the prefactor of the volume law is expected to converge only for larger system sizes.

Figure 4

Bipartite fluctuations of half-chain magnetization as a function of disorder strength at $\epsilon =0.3$. Inset: Data collapse using the best estimates for the critical disorder strength $h_c=3.09\left(7\right)$ and $\nu =0.77\left(4\right)$.

Figure 5

Participation entropy as a function of $S_0^{\mathrm{P}}=ln\left(\text{dim}\mathcal{H}\right)$ for $q=1,2$ and $\epsilon =0.4$. In the ergodic phase ($h=1.8$), $S_q^{\mathrm{P}}$ grows linearly with $S_0^{\mathrm{P}}$ while the linear scaling term vanishes within our error bars in the localized regime ($h=4.8$). Our fits (solid lines; see text) constrain $a_q\in [0,1]$ and yield a logarithmic scaling prefactor $l_q\approx 2\left(1\right)$ at $h=4.8$, consistent with a (slow) growth of $S_q^{\mathrm{P}}$ with system size in the localized phase.