Many-Body Critical Phase: Extended and Nonthermal

The transition between ergodic and many-body localization (MBL) phases lies at the heart of understanding quantum thermalization of many-body systems. Here, we predict a many-body critical (MBC) phase with finite-size scaling analysis in the one-dimensional extended Aubry-André-Harper-Hubbard model, which is different from both the ergodic phase and MBL phase, implying that the quantum system hosts three different fundamental phases in the thermodynamic limit. The level statistics in the MBC phase are well characterized by the so-called critical statistics, and the wave functions exhibit deep multifractal behavior only in the critical region. We further study the half-chain entanglement entropy and thermalization properties and show that the former, in the MBC phase, manifest a volume law scaling, while the many-body states violate the eigenstate thermalization hypothesis. The results are confirmed by the state-of-the-art numerical calculations with system size up to $L=22$. This work unveils a novel many-body phase which is extended but nonthermal.

Figure 1

Main results. (a) Noninteracting phase diagram. The regions I, II, and III correspond to extended, critical, and localized phases, respectively. (b) Phase diagram for the extended interacting Harper model with $U=1$, which contains three phases: the ergodic phase (region I), the MBL phase (region III) and the MBC phase (region II). The transition points (marked by gray dots) can be obtained with finite-size scaling analyses for EE and level statistics, and the phase boundaries are fitted by corresponding curves. Only in the MBC phase of region II, the level statistics follow the critical statistics and the many-body wave functions are multifractal.

Figure 2

Number variance ${\mathrm{\Sigma }}^2(M)$ for different incommensurate potential strengths $V$ with fixed $\mu =0.5$, and for different $\mu$ with fixed $V=1$. In the ergodic phase and when $M$ is small, ${\mathrm{\Sigma }}^2(M)$ shows a slow logarithmic increase (red solid curve). In the MBL phase, ${\mathrm{\Sigma }}^2(M)$ is linear with slope one (black solid curve). Inset: Number variance in the critical regime and near the transition point. They are linear with slopes $1/2<\chi <1$, which is a signature of critical statistics. Here, we fix the number of sites $L=16$.

Figure 3

(a) Fractal dimension $a$ as a function of $V$ and $\mu$. (b) $-\ln ⟨\mathrm{IPR}⟩$ as a function of $\ln D$ with system size varying from $L=12$ to $L=22$. The fitting coefficients are $a=1.11\pm 0.15$, $b=-3.48\pm 1.02$ for $V=2$, $\mu =0.5$ (in the ergodic phase), $a=0.0514\pm 0.0668$, $b=0.21\pm 0.04$ for $\mu =0.5$, $V=5$ (in the MBL phase, the range of $a$ covers 0) and $a=0.51\pm 0.08$, $b=-0.77\pm 0.28$ for $V=1$, $\mu =1.2$ (in the MBC phase).

Figure 4

(a) The sample averaged EE $⟨S⟩$ as a function of $V$ with fixed $\mu =0.5$ and (b) $⟨S⟩$ as a function of $\mu$ with fixed $V=1$. Finite-size scaling analysis of $⟨S/S_T⟩$ as a function of $(V-V_c)L^{1/\nu }$ with fixed $\mu =0.5$ in (c) and $(\mu -{\mu }_c)L^{1/\nu }$ with fixed $V=1$ in (d). Rescaled $⟨S/S_T⟩$ so that all curves for different sizes converge to a single curve. Here, the number of the samples is (50,10,100,50) for $L=(16,18,20,22)$.

Figure 5

The observable $O(E)$ versus eigenvalues for (a) $V=1$, $\mu =0.5$ in the ergodic phase, (b) $V=4$, $\mu =0.5$ in the MBL phase and (c) $V=1$, $\mu =1.2$ in the MBC phase. Here, the system size has been fixed as $L=2N=16$ and the initial phase $\delta =0$. $⟨T⟩$ as a function of (d) $V$ with fixed $\mu =0.5$ and (e) $\mu$ with fixed $V=1$ for different sizes.