Many-body localization in one dimensional optical lattice with speckle disorder

With the model defined we study first its spectral properties to verify the presence of the ergodic-MBL transition. Consider a mean gap ratio, $\overline{r}$, calculated as an average of

where $s_{i}=E_{i+1}-E_{i}$ (with $E_{i}$ being eigenvalues of the system) and the average is taken over a region of spectrum and over disorder realizations. The mean gap ratio was proposed as a simple probe of level statistics in Oganesyan and Huse (2007) with $\overline{r}\approx 0.38$ for Poisson statistics (PS) (corresponding to localized, integrable case) and $\overline{r}\approx 0.53$ for Gaussian orthogonal ensemble (GOE) Mehta (1990); Haake (2010) of random matrices well describing statistically an ergodic system.

We determine the mean gap ratio as a function of the disorder amplitude $W$ for several system sizes. Since the total spin projection on the $z$ axis is conserved, we consider the largest nontrivial sector of the Hamiltonian with $\sum S_{i}^{z}=0$ – that restricts system sizes considered to $L$ even. Typically, we consider about $N=300$ eigenenergies from the middle of the spectrum, i.e. $\varepsilon\approx 0.5$ where $\varepsilon=(E-E_{min})/(E_{max}-E_{min})$ with $E_{min}(E_{max})$ being the lowest (highest) eigenvalue for a given disorder realization. The results are averaged over 1000 disorder realizations or twice that number close to the estimated transition point. Data for $L=14$ and $L=16$ are obtained by full exact diagonalization, the ones for $L=18$ and $L=20$ are obtained using POLFED algorithm Sierant et al. (2020a). The results are shown in Fig. 2. Curves corresponding to different system sizes cross typically in the vicinity of $\overline{r}\approx 0.4$ - this crossing point is taken as an estimate of the critical disorder value, $W_{c}$ for different disorders. We refrain from using the procedure of a single parameter finite size scaling Luitz et al. (2015); Mondaini and Rigol (2015); Khemani et al. (2017) as it has became apparent recently Laflorencie et al. (2020); Šuntajs et al. (2020) that the transition may be of Kosterlitz-Thouless type and such a finite size scaling approach may be not valid.

The critical values of disorder for $\varepsilon\approx 0.5$ for all considered models are given in Table I. Note that for UR disorder we get $W_{c}\approx 3.4$ close to $W_{c}\approx 3.3$ obtained with finite size scaling in the system with open boundary conditions Chanda et al. (2020c). The critical disorder values are used to rescale the disorder amplitude $w=W/W_{c}$ to facilitate a comparison between various models of disorder. From now on we shall use, whenever possible, the rescaled disorder, $w$.

Comparison of Fig. 2(a) and Fig. 2(b) shows that the crossover between ergodic and localized phases for UR disorder is sharper than for the uncorrelated speckle disorder ($\sigma=0$). The disorder strength at which the average gap ratio $\overline{r}$ departs from the GOE value reaches $w\approx 0.5$ for the UR disorder and system size $L=20$ and $w\approx 0.25$ for speckle disorder with $\sigma=0$. This can be traced back to the unbounded on-site distribution of the speckle potential: the probability of having a fully ergodic system at e.g. $w=0.3$ is diminished, in comparison to the UR disorder case, by the rare events in which the field $h_{i}$ is large on one of the sites. The effect of broadening of the crossover is further enhanced when the correlations ($\sigma>0$) are introduced in the speckle potential.

This suggests that finite size effects are stronger for the correlated disorder which is seemingly contradicted by the fact that curves for different sizes $L$ are much closer to each other for the correlated speckle disorder. However, a weaker size dependence of $\bar{r}$ is a sign of strong finite size effects – relatively small changes in system sizes $L$ available to exact diagonalization are simply too small to have a visible effect on $\overline{r}$ dependence for the correlated speckle disorder.

A more detailed characteristic of ergodic to MBL transition is obtained when one examines variation of system properties for individual disorder realizations Sierant and Zakrzewski (2019). To that end we average the gap ratios $r_{i}$ obtained from $300$ eigenvalues from the middle of the spectrum ($\varepsilon=0.5$) for a single disorder realization which yields a sample-averaged gap ratio $\bar{r}_{s}$. The distributions of $\bar{r}_{s}$ (obtained from calculating $\bar{r}_{s}$ for many disorder realizations) vastly differ between UR disorder and quasiperiodic potential Sierant and Zakrzewski (2019) pointing out the dominant role of inter-sample randomness in the MBL transition in the UR disorder. The large inter-sample randomness, in turn, may be linked to the existence of rare Griffiths regions. Hence, it was suggested Khemani et al. (2017) that MBL transitions for UR disorder and quasiperiodic potential belong to different universality classes.

The distribution of the sample averaged gap ratio $P(\bar{r}_{s})$ obtained for the speckle disorder are compared with the UR disorder case in Fig. 3. Three values of the rescaled disorder are considered: $w=0.15$ and $w=2.35$ corresponding to delocalized and localized phases, respectively, and intermediate one corresponding to $w=0.5$ (for RU disorder $w=0.6$). Such intermediate values of disorder correspond to the maximal inter-sample randomness in the system with the biggest variance of the $\bar{r}_{s}$ distribution and in the thermodynamic limit they tend to the respective critical values $W_{c}$ provided that MBL persists in the thermodynamic limit (this issue is a topic of the current debate Šuntajs et al. (2019); Panda et al. (2020); Sierant et al. (2020b, a); Šuntajs et al. (2020)).

Observe that, as it could be expected, the distributions for the localized and delocalized cases are quite similar for UR and for the speckle disorders with different correlation lengths. The situation is markedly different in the transition regime. For the system with UR disorder the $P(\bar{r}_{s})$ distribution is unimodal, although broad. On the other hand, for the speckle disorder one can observe a bimodal shape. With the increase of the correlation length $\sigma$ the two peaks observed for speckle potential become more pronounced and overlap more the distributions for delocalized and localized phases.

Apparently, the sample averaged gap ratio distribution catches an important difference between the speckle and UR disorder. The speckle distribution is exponential favoring small values of disorder. Thus, it is quite probable to obtain nearby sites with very small difference in the local potential – that facilitates transport. This observation may be quantified by finding the probability distribution for the difference between consecutive random numbers which, for speckle disorder is also exponential. At the intermediate rescaled disorder, $w=0.5$, the probability of having an ergodic system $(\bar{r}_{s}\approx 0.53)$ is enhanced for the uncorrelated speckle disorder. At the same time, the unbounded exponential distribution may give rise to few sites with a significantly larger values of $h_{i}$ resulting in a localized sample $(\bar{r}_{s}\approx 0.39)$, explaining the bimodal structure of the $P(\bar{r}_{s})$ distribution for the speckle disorder with $\sigma=0$. This mechanism is further reinforced by the presence of correlations in the speckle potential ($\sigma=1,2$).

All in all, our results confirm that the inter-sample randomness plays a significant role in the ergodic-MBL transition both for the UR and speckle disorder. This suggests that both UR and speckle disorder correspond to the same universality class of MBL transition, dominated by strong fluctuations in sample-to-sample properties and interplay of ergodic and localized grains. This is in sharp contrast to the MBL transition in quasiperiodic potential observable in current experimental setups where the system properties are much more uniform Agrawal et al. (2020).

While gap ratio statistics provides statistical information about eigenvalues of the model, additional information may be gained from eigenvector properties. Those may be probed via e.g. participation entropies. Following Macé et al. (2019) we consider participation entropies $S_{q}$ defined via the $q$-th moments of wavefunction $\left|\Psi\right>$ following the exprespages = 036206,sion:

where $c_{i}$ are the coefficients of wavefunction $\left|\Psi\right>$ in the basis state $\left|n\right>$, i.e. $\left|\Psi\right>=\sum_{i=1}^{N}c_{i}\left|n\right>$, $N$ is the dimension of Hilbert space. While providing supplementary information to that hidden in eigenvalues one should remember that the participation entropies are basis dependent (becoming trivial in e.g. the eigenbasis of the Hamiltonian). We shall consider the eigenbasis of $S^{z}_{i}$ operators, equivalent to the basis of Fock states in the language of spinless fermions. On the delocalized side this basis is to a large extent unbiased. On the localized size, since the so called local integrals of motion of the Heisenberg chain Serbyn et al. (2013); Huse et al. (2014); Ros et al. (2015); Imbrie (2016); Wahl et al. (2017); Mierzejewski et al. (2018); Thomson and Schiró (2018) can be thought of as dressed $S^{z}_{i}$ operators this basis is rather close to the eigenbasis (for any reasonable measure of the basis distance Bengtsson and Życzkowski (2006)). While this may be considered as a drawback, this choice assures that participation entropies are sensitive to the localization transition. We considered only the lowest moments $q=1$ and $q=2$: $S_{1}(\Psi)=-\sum_{i=1}^{N}|c_{i}|^{2}\ln|c_{i}|^{2}$ and $S_{2}(\Psi)=-\ln\left(\sum_{i=1}^{N}\left|c_{i}\right|^{4}\right)$ that are equal to to the Shannon entropy of $|c_{i}|^{2}$ distribution and logarithm of the inverse participation ratio (IPR), respectively.

To probe the distributions of participation entropies we again consider $L=16$ where we have accumulated data for $4500$ disorder realizations for all cases studied. We take $300$ eigenfunctions corresponding to the middle of the spectrum around $\varepsilon=0.5$. We show here the results for $S_{2}$ - the logarithm of IPR - the celebrated measure of localization studies in single particle physics Fyodorov and Sommers (1995); Evers and Mirlin (2000) as reviewed in Evers and Mirlin (2008).

The $S_{2}$ distributions are shown in Fig. 4. Top panel compares UR disorder with the speckle for localized and delocalized regimes highlighting differences in the two regimes. In the delocalized case, for UR disorder we observe a narrow, almost symmetric gaussian-like distribution. This is not the case for speckle potential despite the fact that $w=0.15$ lays deeply in the delocalized regime. Distributions of $S_{2}$ show a pronounced asymmetry with a broad tail extending towards smaller values of $S_{2}$. The tail significantly grows with the speckle correlation length. The presence of this tail indicates relatively rare situations where a partial localization occurs within the sample. A reversed trend is observed on the localized side. Here the participation entropy $S_{2}$ for an UR disorder shows a characteristic shape with a tail decaying as $S_{2}^{-\alpha}$, for this value of $w$ we find $\alpha\sim 2$. The uncorrelated speckle disorder leads to a similar distribution but with, again, the tail which decays more slowly. The tail gets heavier when the correlations in speckle disorder are introduced. In the vicinity of the transition (see bottom panel of Fig. 4) the distributions become very broad for both types of disorder, corroborating our claims about the same universality class for MBL transition in UR and speckle disorder, even in presence of a finite range correlations in the latter case. We also note that for the speckle potential an apparent excess of large $S_{2}$ corresponding to delocalized samples occurs.

The distribution shapes differ for UR and speckle disorder. A quantitative analysis is obtained by finding the $S_{q}$ scaling with the system size - here we follow closely the similar analysis performed for RU disorder Macé et al. (2019). It was shown that in that case to a good approximation $S_{q}=D_{q}\ln N+b_{q}$, where $N$ is the Hilbert space dimension of the system studied. The eigenstates are multifractal if the fractal dimensions $D_{q}$ differ among themselves. This is to be contrasted with $D_{q}=0/1$ for fully localized/delocalized case, respectively. It was found that the Heisenberg chain in MBL regime possesses multifractal eigenstates.

We show that this is also the case for the speckle disorder – compare Fig. 5 where the scaling with the system size is studied in the deeply localized regime $w=2.35$. The speckle potential leads to slightly higher multifractal dimensions in comparison to UR case as it is already visible for $\sigma=0$. The increase of speckle correlation length shows a further increase of multifractal dimensions revealing that localization is somehow “weaker” for the speckle potential at same, linearly rescaled disorder. Interestingly, observe, however, that the subleading $b_{q}$ coefficient remains positive for all cases considered (suggesting a localization as noted in Macé et al. (2019)).

While eigenvalues and eigenvector properties provide us with an understanding of the difference between random and speckle potential they are not directly accessible in experiments. Standard MBL experiments Schreiber et al. (2015); Lüschen et al. (2017); Lukin et al. (2019); Rispoli et al. (2019) consider instead the dynamics inferring the information from time evolution of appropriately chosen initial states. The first and simple conceptually approach Schreiber et al. (2015) considers the evolution of the density-wave like state with every second site occupied and every second empty (for spinful fermions). Analogously, for the Heisenberg spin chain one may consider a Néel state $|\psi(0)\rangle$ with spins up/down on even/odd sites, respectively (or vice versa). In time evolution starting from the state $|\psi(0)\rangle$, the spin correlation function defined as

where $D$ is a normalization constant (so that $I(0)=1$) is followed. The spin correlation function $I(t)$ for the Néel state is mapped, via Jordan-Wigner transformation, to the difference of populations of spinless fermions at even and odd sites at time $t$, hence we refer to it as an imbalance. While in the delocalized regime the imbalance rapidly decays to zero in accordance with eigenstate thermalization hypothesis Rigol et al. (2008); D’Alessio et al. (2016), in fully localized case, after an initial transient, it saturates to a certain value dependent on the disorder amplitude. Theoretical and experimental studies Luitz et al. (2016); Lüschen et al. (2017) addressed the time dependence of imbalance also in the transition regime observing typically its power-law decay. This effect has been used to estimate the critical disorder for MBL transition for large system sizes Sierant and Zakrzewski (2018); Doggen and Mirlin (2019); Chanda et al. (2020c, 2020).

The exemplary behavior of the imbalance (4) for the studied disorder types is depicted in Fig. 6. Results are obtained for $L=20$ system using Chebyshev propagation technique Tal-Ezer and Kosloff (1984); Fehske and Schneider (2008). The curves indeed show the algebraic decay with $I(t)\propto t^{-\alpha}$ that persists to long times and disorder strengths. The algebraic decay describes well the time dependence of $I(t)$ for both UR and speckle disorder.

Fig. 7 shows the fitted exponents $\alpha$ as a function of the scaled disorder amplitude for system sizes $L=16,18,20$. Firstly, we observe, that the size dependence is similar for UR disorder and for the speckle potential, there is little difference between exponents obtained for the available system sizes. A question that we leave for further studies is whether the slow increase in the exponent $\alpha$ observed for system sizes $20\leqslant L\leqslant 200$ for UR Doggen and Mirlin (2019); Chanda et al. (2020c, 2020) appears for the speckle disorder as well. Another interesting observation is that the power $\alpha$ changes differently with disorder amplitude for various cases depicted in Fig. 7. The functional form of $\alpha(w)$ for UR disorder is well approximated by an exponential decay (a straight line in the lin-log plot). A similar dependence is apparent for an uncorrelated speckle potential, with one difference: the exponent $\alpha$ decreases significantly more slowly with $w$. The presence of correlations in the speckle disorder further enhances the slow decay of imbalance at large disorder strengths as we observe a non-zero exponent $\alpha$ even for $w>2$. This suggest that a linear rescaling of the disorder (by the critical disorder value) does not fully compensate for correlations in the disorder, the transition for larger correlation length is “broader” with larger transition region. This parallels a similar observation made for participation entropies as well as the dependence of the fractal dimensions on the speckle correlation length $\sigma$ and can be linked to a formation of small ergodic grains deep in the MBL phase as we show in the next section.

Is it possible to reveal in a more pronounced way the different properties of MBL for UR and speckle disorders in time dynamics? After all those differences were quite strikingly visible in the eigenvector spectral properties was well as in participation entropies. The answer is positive. What we need is a measure of local localization properties, as the broad entropic distributions discussed above convincingly revealed the presence, even in the same sample, of regions seemingly localized to a different degree. Such a measure may be constructed as a local imbalance $I_{2}^{k}$, $k\in[1,L/2]$ involving spins on $2k-1,2k$ sites:

The global imbalance is just a sum of $I_{2}^{k}$. The local imbalances provide information about local scrambling of spin degrees of freedom.

We consider a set of local imbalances $\{I_{2}^{k}(t_{i})\}^{k=L/2-2}_{k=2}$ at times $t_{i}=1000-i$ (in the $J^{-1}$ units) for $i=0,1,...,30$. Gathering the sets of local imbalances $\{I_{2}^{k}(t_{i})\}$ for a large number of disorder realizations we plot the resulting distributions of local imbalance, $P(I_{2})$, in Fig. 8. Consider first the distribution for UR disorder shown in panel (a): for large disorder the distribution is sharply peaked near its maximal unit value indicating an almost complete localization and a good memory of the initial state. On the contrary, for small disorder we observe a smooth gaussian-like profile centered at $I_{2}=0$ - a signature of a lost memory of the initial state. With increasing disorder this distribution sharpens, becomes asymmetric (as a total imbalance becomes positive) developing a tail at positive values of $I_{2}$. Around a critical disorder value the distribution of $I_{2}$ is broad reaching the edge at unity. For speckle uncorrelated disorder – Fig. 8(b) – the curves look similar although a careful inspection reveals that for the localized case the tail extending to small $I_{2}$ is higher than for UR disorder. The broadest distribution, extending almost uniformly between zero and unity is obtained at slightly different $w$ value, in comparison to the UR case. The picture changes for the correlated speckle disorder. We observe a spectacular narrowing of distributions in the delocalized case. This may be easily understood, once some disorder value is chosen for a given site, the next correlated site has, with a large probability, a similar disorder value facilitating delocalization. Interestingly the broadest distributions (at values of disorder shifting towards slightly larger values) develop local maxima at $I_{2}=0,1$ showing the abundance of completely delocalized as well as localized local imbalances. Correlated speckle disorder leads thus to the formation of localized and delocalized grains of small size. This behavior becomes even more pronounced at $\sigma=2$ when even for large $w$ corresponding globally to the deep MBL case, a noticeable maximum at $I_{2}=0$ still exists pointing towards the existence of small regions that locally thermalize.

Note the real resemblance between bimodal distributions observed for local imbalance with the similarly shaped characteristics of the sample averaged gap ratio shown in Fig. 3. The local imbalance allows us to get a similar understanding of the system behavior as eigenvalue statistics not only on the level of the global properties reflected by the imbalance but on a deeper, local level. Let us stress that the measurement of local imbalances requires a single site resolution - however in cold atomic systems as well as in spin models such resolution is already achieved experimentally Gross and Bloch (2017).