Many-body localization and level repulsion

Insertion of disorder in thermal interacting quantum systems decreases the amount of level repulsion and can lead to many-body localization. In this paper we use the many-body picture to perturbatively study the effect of level repulsion in the localized phase. We find that most eigenstates can be described accurately in an approximate way, including many with rare resonances. A classification of the rare resonances shows that most types are exponentially rare and requires exponential fine tuning in an approximate description. The classification confirms that no rare thermal eigenstates exist in a fully localized phase and we argue that all types of resonances need to become common if a continuous transition into a thermal phase should occur.

Figure 1

(a) Sorted weights $|c_j{|}^2$ for random eigenstates in the middle of the spectra for thermal states (dashed lines), from random matrices [31], and MBL states (solid lines), from Eq. (1) with $\delta J=6.0$, for system sizes $L=8,10,12,14$ (in descending order), using ED. (b) Energy errors as more states are added to ${\tilde{H}}_N$ with decreasing $C$ [Eq. (6)] at $\delta J=8.0$. Four typical states (color) and a $N_{\text{dis}}=100$ random realization average (black) for $L=14$ (solid lines). One typical $L=24$ state (dashed line).

Figure 2

ED: (a) Lower half of the energy spectrum in one sector in a $L=6$ system. Two levels with a rare resonance (inset from box) are highlighted (red and green). (b) The entanglement entropy for the 16 states in (a). Same $\delta J$ range as the inset in (a).

Figure 3

(a) The number of $n_s$ eigenstates with rare resonances labeled after their $p$ (black) and $q$ (red) values, with entanglement entropy $1.5\text{log}\left(2\right)<S<2.5\text{log}\left(2\right)$. $N_{rr}=1000$ rare resonances detected in a random state in the middle of the $L=12$ spectrum at $\delta J=8$ from $N_{\text{dis}}=16256$ disorder realizations with ED. (b) Number of eigenstates $n_s$ with entanglement entropy $(w-1/2)\text{log}\left(2\right)<S_E<(w+1/2)\text{log}\left(2\right)$ for $L=10$ (green), $L=12$ (blue), $L=14$ (blue), and $L=16$ (black) from $N_{\text{dis}}={10}^5$ random states ($N_{\text{dis}}={10}^4$ for $L=16$) at $\delta J=10$ with ED.

Figure 4

Sorted energy errors $|E_N-{\tilde{E}}_N|$ (a) and entanglement entropy errors $|S_N-{\tilde{S}}_N|$ (b) at $\delta J=8$ from $N_{\text{dis}}=2000$ random disorder realizations $\lambda$ of ${\tilde{H}}_N$ with $n_{\text{ps}}$ product states for $L=12$ (solid lines) and $L=10$ (dashed lines). The average level spacing $\delta E$ in the middle of the spectrum of a $L=12$ system (black dotted line). Insets highlight the largest parts.

Figure 5

(a) Disorder averaged entanglement entropy $S_E$ and level statistics $r$ (inset) for the thermal Hamiltonian $H_{\text{th}}$ [Eq. (13)] as a function of the strength of the level repulsion $t/\langle \delta e\rangle \propto {10}^{-\alpha }$ for system sizes $L=8,10,12$ with standard error (not visible) from ED. Dashed lines indicate expected values in the thermal [Eq. (12) and ${\langle r\rangle }_{\text{th}}=0.531$] and MBL (${\langle r\rangle }_{\text{MBL}}=0.386$) phases [4]. (b) The resonance length scale $\langle \tilde{\xi }\rangle$ [see Eq. (11)] as a function of $\delta J$ in the middle of the energy spectrum (solid lines) and at $E_{1/5}=\frac{1}{2}(E_{\text{GS}}-E_{\text{max}})/5$ (dashed lines). The dashed line indicates the phase transition ${\xi }_c$. Data from ED averaged over $N_{\text{dis}}={10}^4$ disorder realizations for $L=8,10,12$. (Inset) $\langle p\rangle$ (dashed lines) and $\langle q\rangle$ (dashed dotted lines) from single states at $E_{1/5}$ averaged over $N_{rr}=1000$ rare resonances for $L=10$ (blue), 12 (green), and $N_{rr}=100$ for $L=14$ (black), with standard error from ED.