Many-body dynamics in long-range hopping model in the presence of correlated and uncorrelated disorder

TL;DR

This paper investigates how entanglement entropy and participation ratio scale in long-range hopping models with disorder, revealing new sub-extensive behaviors and the effects of correlated versus uncorrelated disorder on many-body dynamics.

Contribution

It uncovers the scaling behavior of EE and PR in long-range models with disorder, highlighting the impact of correlations and the nature of single-particle states on many-body eigenstates.

Findings

Sub-extensive scaling of EE and PR in algebraic localization.

Universal variation of EE scaling exponent with localization exponent.

Secondary slow growth of EE observed only in correlated disorder cases.

Abstract

Much have been learned about universal properties of entanglement entropy (EE) and participation ration (PR) for Anderson localization. We find a new sub-extensive scaling with system size of the above measures for algebraic localization as noticed in one-dimensional long-range hopping models in the presence of uncorrelated disorder. While the scaling exponent of EE seems to vary universally with the long distance localization exponent of single particle states (SPSs), PR does not show such university as it also depends on the short range correlations of SPSs. On the other hand, in presence of correlated disorder, an admixture of two species of SPSs (ergodic delocalized and non-ergodic multifractal or localized) are observed, which leads to extensive (sub-extensive) scaling of EE (PR). Considering typical many-body eigenstates, we obtain above results that are further corroborated with the asymptotic dynamics. Additionally, a finite time secondary slow growth in EE is witnessed only for correlated case while for uncorrelated case there exists only primary growth followed by the saturation. We believe that our findings from typical many-body eigenstate would remain unaltered even in the weakly interacting limit.

Tables

Table 1. Table 1: Summary of the main differences between different phases in the non-interacting long range systems. We note that β 𝛽 \beta and γ 𝛾 \gamma are model dependent exponents.

SPS	$S_{\infty }$	$P{R}_{\infty }$
Exponential localization	$L^0$	$L^0$
Algebraic localizaion	$L^{\gamma }$,	$L^{\beta }$,
Ergodic (delocalized)	$L$	$L$
Multi-fractal (non-ergodic)	$L$	$L^{\beta }$,
Mobility-edge	$L$	$L^{\beta }$,

Equations

$$H=-\sum _{i,j\ne i}\frac{1}{|i-j{|}^a}({\hat{c}}_i^{†}{\hat{c}}_j+\text{H.c.})+\sum _i{ϵ}_i{\hat{n}}_i$$

Full text

Introduction:

In one and two dimensions, an arbitrarily weak amount of disorder is sufficient to exponentially localize all eigenstates of a system of non-interacting particles, known as Anderson localization Anderson (1958); Abrahams et al. (1979); Lee and Ramakrishnan (1985). However, correlated disorder in one dimensional system can lead to a coexistence of exponentially localized and delocalized states, separated by mobility edge (ME) Das Sarma et al. (1990); Ganeshan et al. (2015); Modak and Mukerjee (2015). Interestingly, in the presence of interactions a transition between delocalized (ergodic) to many-body localized (MBL) phase can be observed  Basko et al. (2006); Gornyi et al. (2017). Algebraic localization is another variety of localization that draws significant attention in recent times Burin and Maksimov (1989); Levitov (1989, 1990); Mirlin et al. (1996); Rodríguez et al. (2000); Malyshev et al. (2004); Mirlin and Evers (2000); Yao et al. (2014); Burin (2015a, b); Gopalakrishnan (2017); Nandkishore and Sondhi (2017); Tikhonov and Mirlin (2018); Luitz and Bar Lev (2019); De Tomasi (2019); Roy and Logan (2019). Entanglement entropy (EE), estimating the bipartite quantum correlations, and participation ratio (PR), quantifying the information about the localization properties of wave-function, happen to be the primary measures of localized and delocalized phases Iyer et al. (2013); Serbyn et al. (2013); Celardo et al. (2016); Vosk and Altman (2013a).

There is an upsurge of studies with disordered models in presence of long range hopping that decays with distance $l$ as a power-law $1/l^{a}$ can show algebraic localization Singh et al. (2017); Cantin et al. (2018); Botzung et al. (2018); Nosov et al. (2018). Interestingly, Levitov’s conjecture Levitov (1989, 1990); Burin and Maksimov (1989); Malyshev et al. (2004) about the absence of localization in $d$ dimensional model with $a<d$, is violated in one of such non-interacting long range model where also single particle states (SPSs) are found to be algebraically localized Deng et al. (2018); Burin and Maksimov (1989); Nosov et al. (2018). Recent advancement in experiments with atomic, molecular and optical systems Kastner (2010); Gong et al. (2014), power-law spin interactions with tunable exponent $0<a<3$ can be realized in laser-driven cold atom setup Richerme et al. (2014); Britton et al. (2012). The dipolar ($a=3$) and van-der-Waals ($a=6$) couplings have also been experimentally observed for the ground-state of neutral atoms and Rydberg atoms Baier et al. (2018); Tang et al. (2018); Weimer et al. (2008); Viteau et al. (2012).

Thanks to the availability of analytical and computational methods, many compelling results have been obtained for EE e.g., it satisfies area (volume) law for exponentially localized (delocalized) phase Iglói et al. (2012); Serbyn et al. (2013); Agarwal et al. (2015); Bardarson et al. (2012); Ghosh et al. (2019); Bera et al. (2015); Iyer et al. (2013); Zhao et al. (2016). Another important diagnostic PR is expected to follow the similar behavior Celardo et al. (2016); Rodríguez et al. (2003). While turning into dynamics, EE for clean (disordered) systems show a faster power law (a slower double log-type) growth with time Calabrese and Cardy (2016, 2005); Vosk and Altman (2013b). Having known all of these, we here pose the question that what would be the nature of EE and PR in an algebraically localized phase and do they scale identically? Additionally, the big underlying quest is to predict the Fock space picture for weakly interacting model by performing a many-body analysis (statics and dynamics) on the non-interacting system.

In particular, we study EE and PR for non-interacting power-law hopping model in the presence of uncorrelated disorder (referred as model I), which supports algebraically localized phase, and correlated disorder (referred as model II), that contains ME and multifractal phases. We do find that algebraic localization leads to new sub-extensive scaling of EE and PR with system size for model I, while EE (PR) satisfies extensive (sub-extensive) scaling for model II. Sub-extensive nature of PR is possibly a manifestation of multifractality of many-body wave-functions in Fock space Mirlin et al. (2006); Evers and Mirlin (2008). Additionally, probing the associated exponents, we can convey that EE (PR) can capture the long-range (short-range) correlation of SPSs.

Model:

We study noninteracting fermions in 1D lattice in the presence of disordered potential. The system is described by the following long-range power-law hopping Hamiltonian,

[TABLE]

where $\hat{c}^{{\dagger}}_{i}$ ($\hat{c}_{i}$) is the fermionic creation (annihilation) operator at site $i$, $\hat{n}_{i}=\hat{c}^{{\dagger}}_{i}\hat{c}_{i}$ is the number operator, and $L$ is the size of the system.

We consider two cases here. 1) Model I (with uncorrelated disorder): $\epsilon_{i}$ are chosen randomly from a uniform distribution between $[-W,W]$. In this paper, we choose $W=20$, for which a very tiny fraction of states are delocalized for $a<3/2$ and all states are localized for $a>3/2$ (see Ref. sup for details). It has been shown that the single-particle wave function $\psi(x)$ of this model displays power-law localization  Burin and Maksimov (1989) (not exponential) $|\psi(x)|^{2}\sim 1/|x-x_{0}|^{\nu}$ in the limit $x>>x_{0}$, where, $x_{0}$ is the localization center and $\nu$ is the localization exponent. $\nu$ shows duality $\nu(a)=\nu(2-a)$ around $a=1$ for $0<a<2$ as investigated numerically Deng et al. (2018) and analytically Nosov et al. (2018). However, we like to point out that near $x_{0}$, in the limit $|x-x_{0}|<<L$, the SPSs are completely different in both sides of $a$, while one finds an exponential decay of wave-function for $a>1$, the decay is still algebraic for $a<1$. $a=1$ point has been shown to be critically localized  Burin and Maksimov (1989). We note that as $a\to\infty$, SPS becomes completely exponentially localized with $L\to\infty$. On the other hand, at $a=0$ the model is exactly solvable and wave-functions in the bulk of the spectrum are critically multifractal  Owusu et al. (2008); Ossipov (2013); Modak et al. (2016); Scaramazza et al. (2016). We note that the algebraic localization is also observed for Hamiltonian residing in the family of power-law random banded matrix model e.g., model III Levitov (1989, 1990) (see Ref. sup for details).

2. Model II (with correlated disorder): $\epsilon_{i}=h\cos(2\pi\sigma i+\phi)$, where $\sigma=(\sqrt{5}-1)/2$ and $\phi$ is an offset chosen from a uniform random distribution [0,1] and closely related to the self-dual quasiperiodic model Gopalakrishnan (2017); Biddle et al. (2011). This seemingly innocent difference has drastic consequences on the physics of this model compared to the previous one. Interestingly, for $a<1$, all SPSs are extended. However, depending on the choice of parameters ($h$ and $a$), there are different phases where ergodic and multifractal (MF) states coexist. We will refers this phase as MF phase. On the other hand, for $a>1$ there is a coexistence of delocalized and localized SPSs, hence, mobility edge (ME) exists (we will refer this phase as ME phase). In either side of $a$, different regimes, denoted by $P_{s}$, are characterized by a fraction $\sigma^{s}<1$ of ergodic SPSs at the bottom of the band and the rest are either localized (for $a>1$) or multifractal (for $a\leq 1$) (see sup for details) Deng et al. (2019).

For all calculations in this paper, we restrict ourself at half-filling. All quantities are obtained after algebraically averaging over $10^{3}$ disorder realizations (see Ref. sup for details). All time evolution calculations are done starting from an initial product state $|\Psi_{0}\rangle=\prod_{i=1}^{L/2}\hat{c}^{{\dagger}}_{2i}|0\rangle$.

I Entanglement entropy(EE)

In this section we will discuss the eigenstate EE and also the non-equilibrium dynamics of EE after a global quench starting from above product state. We note that a typical measure of the entanglement in a quantum system is bipartite von Neumann entanglement entropy $S$ defined as, $S=-\mbox{Tr}\,_{A}[\rho_{A}\ln\rho_{A}],$ where $\rho_{A}=\mbox{Tr}\,_{B}|\Psi\rangle\langle\Psi|$ is the reduced density matrix of a sub-system $A$ after dividing the system into two equal adjacent parts $A$ and $B$, both comprised of $L/2$ sites. $|\Psi\rangle$ is many-body wave function of the composite system.

Eigenstate EE:

For model I, we notice that the typical eigenstate EE Vidmar et al. (2018); Vidmar and Rigol (2017); Vidmar et al. (2017); Hackl et al. (2019) (for details, see Ref. sup ) marked by lines in Fig. 1 (a) shows the absence of data collapse in $S/L$. However, the data collapse appears when we replace $S/L$ by $S/L^{\gamma(a)}$ with $\gamma(a)<1$ as shown in Fig. 1(c). Interestingly, $\gamma(a)$ exhibits the duality around $a=1$: $\gamma(a)\simeq\gamma(2-a)$ as shown in Fig. 1(d) ( see Ref. sup for more details). This duality is the consequence of the duality present in the spatial exponent associated with the algebraically decaying long tail of SPSs for this model in either side of $a<1$ and $a>1$  Deng et al. (2017). Furthermore, the inset of Fig. 1(d) suggests that the expoent $\gamma$ follows a universal behavior with spatial exponent $a$ as long as SPSs are algebraically localized. The sub-extensive $L^{\gamma}$ law can be naively understood from the spatial algebraic structure of SPSs. The total probability of finding a particle at any site $\in B$ while its localization center is at site $x_{0}\in A$ becomes $p\sim\sum_{x\in B}|x-x_{0}|^{-2a}\sim L^{1-2a}f(x_{0},a,L)$ where $f$ is a non-linear function of $x_{0},a,L$. This type of fractional scaling with $L$ is absent for exponentially localized SPSs where $p\sim\xi f(x_{0},\xi,L)$ and $\xi$ being the localization length. Moreover, in the many-body case, EE becomes a complex function of $x_{0}$ as the different SPSs have different localization centres. The absence of the coherent length scale $\xi$ in algebraic localization can lead to non-trivial sub-extensive behavior in many-body EE.

On the other hand, Fig.  1(b) shows the presence of data collapse in $S/L$ for model II. There exists a fraction of delocalized ergodic phase yielding $S$ extensive. Even though, in both MF ($a<1$) and ME phase ($a>1$), $S\sim L$ [as shown in Fig. 1 (b)], one can distinguish them by their corresponding numerical values of $S/L$. This is higher in MF phase compared to ME phase because all SPSs are essentially delocalized in MF phase. Moreover, in both sides of $a=1$, transition between different $P_{s}$ phases are clearly visible. The volume law of eigenstate EE has also been recently observed for 1D short-range noninteracting model in the presence of correlated disorder, where there exists a mobility edge in single particle spectrum Modak et al. (2018); Deng et al. (2017).

Asymptopic EE: We shall now extensively investigate the scaling of asymptotic saturation value $S(L,t\to\infty)\equiv S_{\infty}$ with $a$ as marked by symbols, starting from an initial product state $|\Psi_{0}\rangle$. Figure  1(b) and (a) show the presence of data collapse in $S_{\infty}/L$ for model II and absence of it in model I, respectively, for different values of $L$. Similar to the typical eigenstates, we here in dynamics find sub-extensive (extensive) nature of EE and duality in $\gamma$ for model I (model II) as shown in Fig. 1(c) and (d) (Fig. 1(b)). However, the proportionality factor associated with $S/L$ changes from its eigenstate value. Since, these are noninteracting systems, we expect that $S_{\infty}$ obtained from dynamics should show similar behavior as typical eigenstates. Note that interaction leads to dephasing mechanism via scattering in the system and hence, for interacting system the above expectation may not hold true. MBL systems are one such examples, where we see that EE of many-body eigenstates obey area law, however, $S_{\infty}\sim L$Serbyn et al. (2013); Bardarson et al. (2012). Since, for both models, we do not have a parameter regime where all states are ergodic (delocalized), hence, $S_{\infty}/L$ and also the eigenstates EE density always become less than the Page value Page (1993). The EE associated with the mid-spectrum eigenstates of a generic interacting non-integrable systems obeying ETH, Bera et al. (2015); D’Alessio et al. (2016); Rigol and Srednicki (2012) satisfy this bound.

Finite time rise: Having studied finite size scaling of aymptotic EE $S_{\infty}$, now we analyze the finite time growth of EE. The results are shown for model I in Fig. 2(a), and (b). We observe for $a\leq 1$, a power law rise occurs, $S\sim t^{\alpha}$. This growth exponent $\alpha$ becomes larger near the point $a=1$ (see Ref. sup for details). For $a>1$, growth exponent $\alpha$ decreases and for $a\sim 2$, EE shows a logarithmic rise. Note that in the case of Hamiltonian (1) even without disorder, initial growth of $S(t)$ is sub-linear in $t$ Buyskikh et al. (2016); sup . Since, SPSs behave differently in either side of $a$; the presence of rapid fall of single particle wave function near the localization center causes relatively slow rise of EE for $1<a<2$ compared to $a<1$ regime Singh et al. (2017). We note that our power law growth of EE resembles with the out of time ordered correlator showing a deviation from light-cone like behavior in the context of long range models Luitz and Bar Lev (2019).

Turning into model II as shown in Fig. 2(c), and (d), one can see a fast power law rise $(S\sim t^{\alpha_{1}})$ with $\alpha_{1}<1$ followed by a much slower rise in EE. We observe the value of $\alpha_{1}$ is larger for $a>1$ in comparison to $a<1$ case. We also note that in the MF phase, the growth exponent depends on $P_{s}$ phases, where as, in the ME phase, $\alpha$ remains almost same in different $P_{s}$ phases (see Ref. sup for details). This is presumably the consequence of the fact that the spatial structure of the mutifractal SPS are different in different $P_{s}$ phases. Contrastingly, for $a>1$ even in different $P_{s}$ ME phases, the spatial structure of localized states wave function are different from $a<1$. For $a>1$, the long time slow growth, visible in a reasonably large time window, is found to be logarithmic. On the other hand, for $a<1$, we see a similar secondary slow rise of $S$, since the time window is much smaller we can not comment on it whether this is a power-law with exponent $\alpha_{2}<\alpha_{1}$ or logarithmic. One can naively connect our results with non-interacting central site model Hetterich et al. (2017), where multifractality, appeared due to the coupling of a single central bound state with all Anderson localized states, can give rise to a slow logarithmic growth in entanglement dynamics. Previously, logarithmic growth of $S$ was thought to be a unique feature of MBL systemsBardarson et al. (2012); Žnidarič et al. (2008). Interestingly, our results indicate that the presence of two different types of SPS, there exists a secondary slow rise in the finite time evolution of $S$ for model II; this growth is completely missing for the uncorrelated disordered model I.

Based on our anaysis of finite time EE in Model I and II, we can convey that the sub-linear temporal growth $S\sim t^{\alpha(\alpha_{1})}$ is related to the detail (structure and fraction of delocalized states) of SPSs in these systems: 1) This is apparently evident form the behavior of $S$ for $a<3/2$ in Model I, and 2) for model II, $\alpha_{1}$ remains unaltered with $a$ as long as one stays inside a fixed $P_{s}$ phase. However, the value of $\alpha$ interestingly changes as one varies $a$; similarly, $\alpha_{1}$ changes as one goes from MF side to ME side even within the same $P_{s}$ phase. A fnite fraction of delocalized SPSs can also cause the two-stage growth of EE in model II. On the other hand, this fraction becomes vanishly small for model I originating the single stage growth (see Ref. sup for detailed analysis).

II Participation ratio (PR)

Having examined the scaling of eigenstate EE and $S_{\infty}$ with $L$, in the similar spirit, we now look for the scaling of eigenstate PR and the saturation values of PR (designated by PR*∞*). We shall use the definition of many-body PR as introduced in Ref. Bera et al. (2015). It is defined for half-filling case as, PR=L\bigl{[}2\sum_{\alpha=1}^{L}n_{\alpha}\sum_{j=1}^{L}|\phi_{\alpha}(j)|^{4}\bigr{]}^{-1}, where, $|\phi_{\alpha}\rangle$ and $n_{\alpha}$ are eigenvectors and eigenvalues of one-body density matrix $\rho_{ij}=\langle\hat{c}^{{\dagger}}_{i}\hat{c}_{j}\rangle$ respectively. PR$\sim L$ for delocalized ergodic systems and PR$\sim\xi$ for exponentially localized many-body states Celardo et al. (2016); Rodríguez et al. (2003).

We study the characteristics of PR*∞* and eigenstate PR in Fig. 3. From the data collapse of PR/$L^{\beta}$ as a function of $a$ with different system sizes (see Fig. 3 (a) and (b)), we show for both the models that PR exhibits a sub-extensive scaling with $L$ (see Ref. sup for detail). In order to analyze the exponent $\beta$ more concretely, we show the variation of $\beta$ with $a$ for model I and II in Fig. 3 (c) and (d), respectively. $\beta$ remains fixed at a higher value for $a<1$ while it decreases monotonically for $a>1$ for model I. Very surprisingly, unlike the EE, PR does not exhibit any duality with $a$ around $a=1$. The reason being PR is a local quantity, it is not able to capture the duality of SPSs in the long-distance scale where power law tail is observed in either side of $a=1$. Precisely, PR accounts for the short distance behavior of SPSs where exponential and algebraic decay are present for $a>1$ and $a<1$, respectively, hence, $\beta$ is completely asymmetric around $a=1$.

On the other hand, for model II, $\beta$ shows a kind of symmetric behavior around $a=1$ (see Fig. 3). This result may be counter intuitive in the sense that phases in both sides of $a=1$ are completely different i.e., MF phase for $a<1$ and ME phase for $a>1$. Multifractal SPSs in this model have a form of multiple sharp peak on the top of almost flat background in contrast to the ergodic delocalized SPSs that are extended all over the lattice sup . The structure of SPSs for MF phase is kind of similar to the exponentially localized SPSs having only one peak and the background is suppressed exponentially with distance from that peak. Since, PR is an inappropriate measure to identify the fine tuned long distance structure of SPSs for ME and MF phases, we find similar behavior of $\beta$ in either side of $a$. However, we note that $\beta$ is much closer to $1$ in MF phase compared to ME phase. Moreover, from the variation of $\beta$ with $a$, we can roughly identify different $P_{s}$ phases in either side of $a=1$ [see Fig. 3(d)]. Moreover, we note that in the calculation with typical eigenstates, we discard a few bottom spectrum delocalized states to minimize their effect (see Ref. sup for details). On the other hand, all the energy states come automatically into the non-eignestate dynamics. We believe that this is the origin of the apparent dissimilarities between the predictions from typical eigenstate and long time dynamics as observed in Fig. 1 and Fig. 3.

Conclusion:

We summarize our main results in Table. 1. One of the most intriguing finding is to show the sub-extensive scaling in EE and PR, when SPSs are algebraically localized as observed in model I. This is firmly evident from both the eigenstate and long time dynamics. The absence of length scale thus imprints its’ signature unlike the exponentially localized phase. Moreover, these behaviors are not the artifact of delocalized states present in Model I (at least for $a<3/2$) as the number of such states has measure zero for $L\to\infty$ de Moura et al. (2005); Ossipov (2013); Deng et al. (2018). Turning to model II, asymptotic and eigenstate EE both obey volume law due to the presence of ergodic SPSs; however, interestingly, the proportionality factors change in different $P_{s}$ phases. The adiabatic connectivity allows us to conjecture that algebraically localized quasi-local integrals of motion would survive even in the weakly interacting limit Serbyn et al. (2013); Imbrie (2016); Modak et al. (2016); De Tomasi (2019) and hence, the eigenstate EE scaling should remain unaltered even in the above limit. One might not expect the similar scaling of asymptotic EE, obtained from the long time dynamics, due to the dephasing mechanism caused by the interaction.

Our study further reveals the connection between the exponents ($\gamma$ for EE and $\beta$ for PR), and the spatial structure of the SPSs. The EE is maximally governed by the long distance nature of the SPSs and thus the duality in $\gamma$ is closely connected to the duality of the localization exponent $\nu$ as noticed for model I Deng et al. (2018). Moreover, $\gamma$ follows a universal behavior as far as the algebraically localized SPSs are concerned. In contrary, PR captures the short distance nature of correlation leading to the fact that exponent $\beta$ does not show duality around $a=1$. The short distance behavior of SPSs are very different for $a<1$ and $a>1$ for model I. Surprisingly, model II shows duality like behavior within a small window around $a=1$. This can be related to the peculiar spatial distribution of multifractal SPSs at short distance (see Ref. sup for detail).

Another important contribution of our work is to show how the structure of SPSs can influence the finite time rise of EE. An unprecedented two-stage growth of EE for model II is exclusively observed while model I exhibits single-stage growth. The secondary rise in EE for model II might be related to the fact that there exist finite fraction of two types of SPSs i.e., multifractal and delocalized or localized and delocalized. The initial algebraic temporal growth is common in both the models. Recent studies also find signatures of temporal power law growth of EE in long range interacting models Safavi-Naini et al. (2019); De Tomasi (2019). The connection between the temporal power-law growth of EE and algebraical SPSs (LIOM) for non-interacting (interacting) model is still an open field of research. Given the experimental realizability of spin models Baier et al. (2018); Tang et al. (2018); Weimer et al. (2008); Viteau et al. (2012), we believe that our study would initiate a plethora of work in this direction.

III Acknowledgements

Authors thank P. Calabrese and L. Vidmar for reading the manuscript and for several comments. The authors also thank the anonymous referee for useful comments.