Long tail distributions near the many-body localization transition

The random field $S=\frac{1}{2}$ Heisenberg chain exhibits a dynamical many body localization transition at a critical disorder strength, which depends on the energy density. At weak disorder, the eigenstate thermalization hypothesis (ETH) is fulfilled on average, making local observables smooth functions of energy, whose eigenstate-to-eigenstate fluctuations decrease exponentially with system size. We demonstrate the validity of ETH in the thermal phase as well as its breakdown in the localized phase and show that rare states exist which do not strictly follow ETH, becoming more frequent closer to the transition. Similarly, the probability distribution of the entanglement entropy at intermediate disorder develops long tails all the way down to zero entanglement. We propose that these low entanglement tails stem from localized regions at the subsystem boundaries which were recently discussed as a possible mechanism for subdiffusive transport in the ergodic phase.

Figure 1

Local magnetization for site 5 in the chain as a function of energy. Here, we have generated one disorder configuration at $h=1.0$ for the largest system size $L_{\text{max}}=24$, smaller systems are cut out of this configuration for comparable results. The spread of the results in the middle of the spectrum reduces significantly with system size, thus leading to a smooth function of energy in the thermodynamic limit. Clearly, at the high end of the spectrum, this is no longer true and the variance of the local magnetization does not reduce with system size. This is a signal for many body localization, which is expected at $\epsilon \gtrsim 0.9$ at $h=1$ in accord with Ref. [15] and thus shows the many body mobility edge.

Figure 2

Disorder averaged difference of local magnetizations as a function of disorder strength in the middle (left) and upper part (right) of the spectrum. For weak disorder, eigenstates thermalize and the local magnetization of adjacent states becomes identical in the thermodynamic limit, thus yielding a smooth function of energy. Here, the mobility edge for the many body localization transition is also visible as the region, in which adjacent states yield very different results—the MBL phase—is much larger in the upper part of the spectrum at energy density $\epsilon =0.8$. The location of the previously estimated [15] critical disorder strength $h_c\approx 3.7$ at $\epsilon =0.5$ and $h_c\approx 2.3$ at $\epsilon =0.8$ is given by the dashed line.

Figure 3

Finite size scaling of the disorder averaged difference of local magnetizations $\langle n\left|S_z^i\right|n\rangle$ of adjacent eigenstates as a function of the dimension of the Hilbert space $\text{dim}\left(\mathcal{H}\right)$. The inset displays the (discrete) logarithmic derivative, yielding the exponent of the power law in the thermodynamic limit, which slowly approaches $0.5$. The data presented here includes only states from the middle of the spectrum ($\epsilon =0.5$).

Figure 4

Disorder averaged Frobenius norm of the difference of reduced density matrices of a subsystem of size $\ell$ as a function of subsystem size.

Figure 5

Disorder averaged norm of the difference of the reduced density matrices for adjacent eigenstates as a function of the dimension of the Hilbert space for two different subsystem sizes $\ell$ and different disorder strengths. The difference of the reduced density matrices decays as a power law in the dimension of the Hilbert space in the same way as the difference of local magnetizations in Fig. 3. The black line corresponds to $\text{dim}{\left(\mathcal{H}\right)}^{-\frac{1}{2}}$ for comparison and the exponent $-0.5$ is in good agreement with the results at weak disorder. In the MBL phase, the difference norm is constant with system size as expected.

Figure 6

Probability density of the difference ${\mathrm{\Delta }}_{n,n+1}{S}_i^z$ of the local magnetization $S_z^i$ of two eigenstates with adjacent energies for different disorder strengths. The histograms are obtained from the central ($\epsilon =0.5$) 50 eigenstates of ${10}^3$ disorder realizations and contain the results for each site $i$ on which the local magnetization is compared. Even at very weak disorder, the distribution is already leptokurtic, showing heavier tails than the normal distribution. In the limit of very strong disorder, the distribution is dominated by the bimodal distribution of the local magnetization itself, which leads to the three dominant differences $-1,0,1$ stemming from random configurations of $S_z^i=-0.5,0.5$ in the MBL phase.

Figure 7

Probability density of the difference ${\mathrm{\Delta }}_{n,n+1}{S}_i^z$ of the local magnetization $S_z^i$ of two eigenstates with adjacent energies for different system sizes and disorder strengths. For weak disorder, the reduction of the variance with system size is evident, however the distributions deviate increasingly from a Gaussian distribution with growing disorder. In the MBL phase, the distributions become independent of system size, only governed by the localization length, which creates the nonzero weight between the three peaks of the distribution.

Figure 8

Probability density of the difference ${\mathrm{\Delta }}_{n,n+1}{S}_i^z$ of the local magnetization $S_i^z$ of two eigenstates with adjacent energies, normalized by the standard deviation $\sigma$ to compare the shape of the distributions for different system sizes. For $h=0.8$, the weight of the tails seems to decrease with system size and approach the Gaussian distribution. This is no longer true for stronger disorder and the weight of the tails increases strongly starting from $h\gtrsim 1.6$, accompanied by a growing kurtosis with system size in the same regime, making the distributions strongly leptokurtic. The kurtosis is defined by the fourth central moment divided by the squared variance, yielding a value of 3 for the gaussian distribution and larger values if the distribution has heavier tails, i.e., if it is leptokurtic.

Figure 9

Probability density of the entanglement entropy $S_E$ of states in the middle of the spectrum ($\epsilon =0.5$) for different disorder strengths. We use the bipartition $L_A=L_B$ for the periodic $L=20$ chain. The evolution from the volume law behavior at weak disorder to the area law at strong disorder is clearly visible. At very weak disorder, the distribution is close to Gaussian and develops exponential tails at intermediate disorder. They correspond to rare regions with low entanglement, responsible for slow (subdiffusive) transport.

Figure 10

Probability density of the entanglement entropy $S_E$ of states in the middle of the spectrum for different system sizes at disorder strengths (a) $h=1$, (b) $h=2$, (c) $h=3$, and (d) $h=4$.

Figure 11

Top panel: Spatial variation of the entanglement entropy $S_E$ for the central 63 eigenstates of one disorder realization $h=2$ in the periodic $L=22$ chain (The disorder configuration $h_i$ as a function of the site index $i$ is shown in the bottom panel.) as a function of the position of the cut between subsystem $A$ and $B$. The subsystem size is $\ell =L/2$. Center panel: Same data as in the top panel as line graph. Depending on the cut position, the entanglement entropy varies and is found to be particularly low for some cuts, corresponding to a weaker link between the subsystems.

Figure 12

Top panel: Spatial variation of the entanglement entropy $S_E$ for the central 67 eigenstates of one disorder realization $h=3$ in the periodic $L=18$ chain (The disorder configuration $h_i$ as a function of the site index $i$ is shown in the bottom panel.) as a function of the position of the cut between subsystem $A$ and $B$. The subsystem size is $\ell =L/2$. Center panel: Same data as in the top panel as line graph. Depending on the cut position, the entanglement entropy varies and is found to be particularly low for some cuts, corresponding to a weaker link between the subsystems.