Many-Body Localization in a Disordered Quantum Ising Chain

Many-body localization occurs in isolated quantum systems when Anderson localization persists in the presence of finite interactions. Despite strong evidence for the existence of a many-body localization transition, a reliable extraction of the critical disorder strength is difficult due to a large drift with system size in the studied quantities. In this Letter, we explore two entanglement properties that are promising for the study of the many-body localization transition: the variance of the half-chain entanglement entropy of exact eigenstates and the long time change in entanglement after a local quench from an exact eigenstate. We investigate these quantities in a disordered quantum Ising chain and use them to estimate the critical disorder strength and its energy dependence. In addition, we analyze a spin-glass transition at large disorder strength and provide evidence for it being a separate transition. We, thereby, give numerical support for a recently proposed phase diagram of many-body localization with localization protected quantum order [Huse et al., Phys. Rev. B 88, 014206 (2013)].

Figure 1

Phase diagram of the Ising model Eq. (1) in the $h/2=J_2=0.3$ plane with $\epsilon =2(E-E_{\min })/(E_{\max }-E_{\min })$ being the energy density relative to the total bandwidth. The axes give the energy density above the ground state and the disorder strength. The colored areas are guides to the eye. The data are obtained from finite size scaling of entanglement difference Eq. (2) after a local quench and the spin-glass order parameter Eq. (4); only statistical error bars are given, see text. The schematic on top of the phase diagram shows a caricature of the spatial domain wall probability distribution in the different phases. The thermal phase is characterized by extended domain walls, the MBL paramagnetic phase by localized domain walls which are created and removed in pairs (dashed), and the MBL spin glass by localized nonoverlapping domain walls.

Figure 2

(a) Standard deviation of entanglement over the disorder ensemble as a function of disorder strength $\delta J$ for different system sizes $L$ and $D$ independent disorder realizations, at a fixed energy density in the middle of the spectrum ($\epsilon =59/60$). The left inset shows the mean entanglement entropy with dashed lines giving the values $S=(L\ln 2-1)/2$ and $S=\ln 2$. The right inset gives the scaling collapse of the data in the main panel. (b) Entanglement difference as a function of disorder strength for a local quench from an eigenstate in the middle of the spectrum. The inset gives the scaling collapse of the data.

Figure 3

Spin-glass order parameter (4) as function of disorder strength $\delta J$ for different system sizes $L$ and $D$ independent disorder realization, at a fixed energy in the middle of the energy spectrum ($\epsilon =59/60$). The dashed line gives the expected value $⟨{\chi }^{\mathrm{SG}}⟩=1$, determined by normalization, in the absence of spin glass order. In the spin glass phase $⟨{\chi }^{\mathrm{SG}}⟩$ is proportional to $L$.The inset shows the scaling collapse of the data.