Many-body localization and transition by density matrix renormalization group and exact diagonalization studies

A many-body localized (MBL) state is a new state of matter emerging in a disordered interacting system at high-energy densities through a disorder-driven dynamic phase transition. The nature of the phase transition and the evolution of the MBL phase near the transition are the focus of intense theoretical studies with open issues in the field. We develop an entanglement density matrix renormalization group (En-DMRG) algorithm to accurately target highly excited states for MBL systems. By studying the one-dimensional Heisenberg spin chain in a random field, we demonstrate the accuracy of the method in obtaining energy eigenstates and the corresponding statistical results of quantum states in the MBL phase. Based on large system simulations by En-DMRG for excited states, we demonstrate some interesting features in the entanglement entropy distribution function, which is characterized by two peaks: one at zero and another one at the quantized entropy $S=ln2$ with an exponential decay tail on the $S>ln2$ side. Combining En-DMRG with exact diagonalization simulations, we demonstrate that the transition from the MBL phase to the delocalized ergodic phase is driven by rare events where the locally entangled spin pairs develop power-law correlations. The corresponding phase diagram contains an intermediate or crossover regime, which has power-law spin-$z$ correlations resulting from contributions of the rare events. We discuss the physical picture for the numerical observations in this regime, where various distribution functions are distinctly different from results deep in the ergodic and MBL phases for finite-size systems. Our results may provide new insights for understanding the phase transition in such systems.

Figure 1

(a) The logarithm of the energy variance ${\sigma }^2$ as a function of the number of En-DMRG sweeps for eight randomly selected En-DMRG processes for system size $N=72$ and $h=8$ targeting the state at the middle of the energy spectrum. (b) The entropy $S$ evolution for the same processes. (c) For smaller system $N=18$ at $h=8$, we show the energy eigenvalues for $i\text{th}$ eigenstate found in the energy interval $(-2.272,-2.257)$ near the energy spectrum center, where an excellent agreement between the En-DMRG and ED results is found. The absolute energy difference $\left|\mathrm{\Delta }E\right|$ between En-DMRG and ED eigenstates are shown in the inset of (c), which typically is around ${10}^{-9}$. (d) For all 36 states we found in (c), we demonstrate their one-to-one correspondence for entanglement entropy $S$ between ED and En-DMRG results.

Figure 2

(a) The logarithm of the energy variance ${\sigma }^2$ as a function of the number of the maximum bond dimension $m$ used in En-DMRG averaged over 10000 configurations for bond dimensions $m\le 48$ and more than 1000 configurations for larger $m$. (b) The typical wave-function overlap $O$ between the current lowest-energy state (for $H_2)$ and the state at the previous step for the En-DMRG with and without the proper optimization for each DMRG sweep measured when the En-DMRG sweeps to the middle of the system. (c) The expectation value of spin-$z\langle S_{N/2}^z\rangle$ during the En-DMRG sweeping for the cases described in (b).

Figure 3

(a) The spin-$z$ expectation values $\langle S_{N/2}^z\rangle$ for $N/2$-th site for eigenstates obtained by En-DMRG for 1000 random disorder configurations at $h=8$ and $N=72$. (b) The probability density distributions $P\left(S^z\right)$ for systems with weaker disorder $h=2$ and 3 obtained by ED at $N=18$ using an ensemble of 2000 disorder configurations with 50 states from each configuration. $P\left(S^z\right)$ has the Gaussian distribution for $h=2$ while there are two peaks at $\langle S^z\rangle \sim \pm 0.5$ for $h=3$. (c) $P\left(S^z\right)$ for different system sizes from $N=18$ to $N=72$ for strong disorder $h=8$ using an ensemble of 20 000 disorder configurations. The sharp peaks at $\langle S^z\rangle \sim \pm 0.5$ demonstrate the violation of the ETH for this system. The results obtained by En-DMRG and ED are near identical at $N=18$ establishing the unbiased sampling for our En-DMRG. Shown in the inset is $P\left(0\right)$ as a function of $h$. For (b)–(c), the typical standard error bar (obtained by dividing ensembles into 10 groups) is about the size of the symbols.

Figure 4

(a) The entropy probability density distribution function $P\left(S\right)$ for $N=18$, 30, and 72 at $h=8$. For $N=18$, ED and En-DMRG results find excellent agreement. (b) The evolution of $P\left(S\right)$ from the MBL phase to the delocalized phase by reducing $h$ from 8 to 3 for $N=18$ from both ED and En-DMRG calculations. All curves have a peak at or near $S=0$ and a second peak near $S=0.69\sim ln2$. $P\left(S\right)$ at larger $S>ln2$ side can be fitted by a power-law behavior $P\left(S\right)\sim 1/S^x$ for $h\le 4$. (c) The evolution of the probability density $P\left(F\right)$ for the variance of the half-system magnetization $F$ for $h=8,5,4$, and 3. The second peak is located at a quantized value $F=1/4$. For results shown in Fig. 4, we use at least $50000$ energy eigenstates for each system size. The typical standard error bar for (a)–(c) is about the size of symbols and it is larger near the tail of distributions.

Figure 5

(a) The disorder averaged spin-$z$ $C^{zz}\left(\right|i-j\left|\right)$ and spin-$xy$ $C^{xy}\left(\right|i-j\left|\right)$ correlations obtained from En-DMRG at $h=6$ for $N=18$ and $N=30$. The exponential decays can be best fit by the correlation lengths ${\xi }_{xy}=1.1$ and ${\xi }_z=2.78$, respectively. (b) The $C^{zz}\left(\right|i-j\left|\right)$ develops power-law correlations (see solid line fitting) for $h=3$ and 4 obtained by ED. The dashed lines represent exponential fittings for longer distance data with ${\xi }_z=18.0$ and ${\xi }_z=8.0$ for $h=3$ and 4 respectively, which show less overall agreement from the raw data comparing to the power-law fittings. (c) The hopping correlations $C^{xy}\left(\right|i-j\left|\right)$ decay exponentially at $h=3$ and 4 obtained by ED. The fitting correlation lengths are ${\xi }_{xy}=2.7$ and $1.8$ for $h=3$ and 4, respectively.

Figure 6

(a) For system size $N=20$, if we fit all correlations using exponential decay functions shown in Figs. 5–5, then the correlation length for spin-$z$ is much larger than the one for spin-$xy$. (b) A phase diagram with a delocalized ergodic phase, an intermediate critical Griffiths regime, and a MBL phase. See more discussions on the finite-size effect of the critical regime in the main text.