Relaxation mechanisms in a disordered system with the Poisson level statistics

TL;DR

This paper investigates how disorder and spin symmetry influence localization in a disordered anisotropic t-J model, revealing that symmetry-preserving disorder allows localization within sectors but enables certain operators to relax due to symmetry sector mixing.

Contribution

It demonstrates that in the anisotropic t-J model, localization persists within symmetry sectors under symmetry-preserving disorder, while symmetry-breaking disorder leads to full localization, highlighting the role of symmetry in many-body localization.

Findings

Localization occurs within each symmetry sector for symmetry-preserving disorder.

Odd operators relax due to transitions between symmetry sectors.

Level statistics suggest localization within sectors despite operator relaxation.

Abstract

We discuss the interplay between many-body localization and spin-symmetry. To this end, we study the time evolution of several observables in the anisotropic t-J model. Like the Hubbard chain, the studied model contains charge and spin degrees of freedom, yet it has smaller Hilbert space and thus allows for numerical studies of larger systems. We compare the field disorder that breaks the Z_2 spin symmetry and a potential disorder that preserves the latter symmetry. In the former case, sufficiently strong disorder leads to localization of all studied observables, at least for the studied system sizes. However, in the case of symmetry-preserving disorder, we observe that odd operators under the Z_2 spin transformation relax towards the equilibrium value at relatively short time scales that grow only polynomially with the disorder strength. On the other hand, the dynamics of even operators and the level statistics within each symmetry sector are consistent with localization. Our results indicate that localization exists within each symmetry sector for symmetry preserving disorder. Odd operators' apparent relaxation is due to their time evolution between distinct symmetry sectors.

Equations

$$H$$

$$|{\Psi }_0⟩$$

$$|{\bar{\Psi }}_0⟩$$

$${\hat{P}}_c$$

$${\hat{P}}_s$$

$$\left(\begin{array}[]{cc}0&-t_{0}\\ -t_{0}&V^{S/A}\end{array}\right)\left(\begin{array}[]{c}|\uparrow,0,\downarrow\rangle^{S/A}\\ |\uparrow,\downarrow,0\rangle^{S/A}\\ \end{array}\right)=E^{S/A}\left(\begin{array}[]{c}|\uparrow,0,\downarrow\rangle^{S/A}\\ |\uparrow,\downarrow,0\rangle^{S/A}\\ \end{array}\right),$$

$${\hat{G}}_n=\frac{1}{L}\sum _i{n}_i{n}_{i+1},G_n(t)=⟨{\hat{G}}_n{⟩}_t,$$

$${\hat{G}}_{S^z}=\frac{1}{L}\sum _i{S}_i^z{S}_{i+1}^z,G_{S^z}(t)=⟨{\hat{G}}_{S^z}{⟩}_t,$$

$$\hat{\Gamma }=\frac{8}{N_f}\sum _{i=1,3,5,\dots ,L-1}(-1{)}^{1+(i+1)/2}{S}_i^z{S}_{i+2}^z{S}_{i+4}^z,$$

$$S_n$$

$$S_{S^z}$$

$$P_P(r)=\frac{2}{(1+r{)}^2},$$

$$P_{GOE}(r)=\frac{27}{4}\frac{r+r^2}{(1+r+r^2{)}^{5/2}}.$$

Full text

I Introduction

The many–body localization (MBL)gornyi05 ; basko_aleiner_06 ; oganesyan_huse_07 ; AbaninRMP phenomena has been most frequently studied in one–dimensional (1D) disordered systems with either charge or spin degrees of freedom barisic_prelovsek_10 ; luitz15 ; luitz16 ; torres15 ; sirker14 ; pal10 ; bera15 ; Hauschild_2016 ; Devakul2015 ; bertrand_garciagarcia_16 ; Husex2017 ; Doggen2018 . Even though research in this field mainly focused on a few simplest prototype model Hamiltonians for MBL such as the disordered XXZ model, the type of the transition and even the existence of the MBL phase in the thermodynamic limit are still under intense considerationsuntajs_bonca_20a ; suntajs2020 ; sirker_2020 ; Luitz20 ; Vasseur20 ; Polkovnikov20 ; sieran2021a ; vidmar2021 ; Huse21 ; sels2021 ; Sirker21 . One of the specific open problems is the effect of various symmetries on the existence of MBL phaseVasseur2016 ; Chandran2014 . There are reports that non–Abelian symmetry precludes MBL Vasseur2016 ; proto2017 ; vasseur2021 while other investigations claim that the non-Abelian symmetry is protected by MBLfriedman2017 .

Shifting focus to more complex prototype models that contain charge and spin degrees of freedom, such as the 1D Hubbard model with potential disorder, the existence of the full MBL phase is even more elusive. In Ref. prelovsek16 ; sroda19 ; kozarzewski2019 ; protopopov2018 authors investigate the time evolution of spin and charge imbalance as well as transport properties in the Hubbard model. Their results are consistent with localized/nonergodic charge degrees of freedom while due to the preserved SU(2) symmetry the spin degrees of freedom remain delocalized/ergodic up to extremely large values of potential disorder. Similar conclusions were drawn based on the statistics of adjacent energy levels mondaini15 and by counting the maximal number of local integrals of motionmierzejewski2018 . Subdifusive time evolution of charge particles was found in the related $t-J$ modelbonca17 with potential disorder.

The effect of symmetry preserving disorder has been addressed already in non–interacting one-dimensional random hopping systems. In the case of systems with chiral or sublattice symmetry where particles can hop only between even- and odd- lattice sites, there is a diverging mean density of states at zero–energy dyson1953 ; cohen1976 ; reidinger1978 , which can lead to the delocalization transition brouwer1998 ; furusaki_NPB ; furusaki2000 ; evers2008 .

The main goal of this work is to compare the effect of spin-symmetry-preserving and symmetry-breaking disorders on the dynamics of charge and spin degrees of freedom. In the case of potential disorder, the behavior of specific spin degrees of freedom is inconsistent with the full MBL state due to the $\mathbb{Z}_{2}$ spin-symmetry. This observation is based on relatively fast relaxation of spin observables that are odd under the later $\mathbb{Z}_{2}$ spin transformation, i.e., the only non–zero matrix elements of these operators connect two distinct symmetry sectors. While this observation seems to preclude MBL state, the statistics of adjacent energy levels computed within each symmetry sector at large potential disorder approaches Poisson statistics.

We have organized the manuscript as follows: in the introduction, we present the model and the numerical method; we also discuss how the symmetry properties of the model depend on the type of the disorder. Next, we present the time evolution of the charge and spin imbalance and present a simplified model that explains the unusual relaxation of the spin imbalance in the presence of the potential disorder. We further present time evolutions of various charge and spin correlation functions, followed by the analysis of charge and spin entanglement entropies. Based on the spectral level statistics, we discuss the apparent inconsistency between the relaxation of the spin imbalance and the Poisson level statistics, both observed under the influence of strong potential disorder.

II Model and method.

We investigate the anisotropic $t-J$ model on a one–dimensional ring with $L$–sites and $N_{f}=L/2$ fermions in the total $S^{z}=0$ subspace in the presence of either a random external magnetic field $h_{i}\in[-2W_{s},2W_{s}]$ or random potential $\epsilon_{i}\in[-W_{c},W_{c}]$

[TABLE]

The fermion operators, $c_{i,\sigma}$ and $n_{i}=\sum_{\sigma}c_{i,\sigma}^{\dagger}c_{i,\sigma}$, as well as the spin operators, $S^{\pm,z}_{i}$, act in the Hilbert space spanned locally by only three states, $|0\rangle_{i}$, $|\uparrow\rangle_{i}$, $|\downarrow\rangle_{i}$. Absence of doubly-occupied states, $|\uparrow\downarrow\rangle_{i}$, in the $t$-$J$ model allows studying charge and spin dynamics for much larger systems than it would be possible for the Hubbard chain. This property is the main motivation for the choice of Hamiltonian. We perform multiple time-evolutions based on the Lanczos technique, each evolution starting from a different set of either random ${h_{i}}$ or ${\epsilon_{i}}$. The main goal of this work is to compare the time evolution of spin and charge degrees of freedom under the influence of two different types of disorder. For this reason we choose, following Ref.mondaini15 , the initial state that possesses a charge–density–wave order as well as a staggered spin orientation configuration $|\Psi_{0}\rangle$ defined as

[TABLE]

while $|\bar{\Psi}_{0}\rangle$ represents a state with a globally reversed spin projections, $S^{z}_{i}$ , via the unitary transformation $\hat{U}=\prod_{i}(1-n_{i}+S^{+}_{i}+S^{-}_{i})$. In addition, we compute the level statistics of the energy spectra. We typically take $N_{r}=500-1500$ realisations of the disorder. We measure time in units of $[1/t_{0}]$ and set $t_{0}=J_{z}=1$, and $J_{\perp}=1.5$.

There exists a significant difference between the two systems under the influence of potential ($W_{c}\not=0,W_{s}=0$) and field disorder ($W_{s}\not=0,W_{c}=0$). Since we have set $J_{z}\not=J_{\perp}$, the $SU(2)$ symmetry is broken even at $W_{c(s)}=0$. In the case of the potential disorder and for $S^{z}_{\mathrm{tot}}=0$, the Hamiltonian remains invariant under the $\mathbb{Z}_{2}$ spin-transformation, $\hat{U}$, which is closely related to the global $\pi$-rotation around the $x$–axis santos_2013 . Since $\hat{U}\hat{U}=\hat{U}\hat{U}^{\dagger}=1$, each eigenstate consists of either symmetric ($\hat{U}|\Psi_{S}\rangle=|\Psi_{S}\rangle$) or antisymmetric ($\hat{U}|\Psi_{A}\rangle=-|\Psi_{A}\rangle$) combination of states that differ by a global reversal of $S^{z}_{i}$, $|\Psi_{S/A}\rangle=(|\Psi\rangle\pm\hat{U}|\Psi\rangle)/2$. The Hamiltonian thus separates into two blocks with equal number of symmetric and antisymmetric functions.

III Charge and spin imbalance

We start by presenting time propagation of charge and spin imbalance as defined by the following operators

[TABLE]

The initial state is chosen such that $\hat{P}_{c(s)}|\Psi_{0}\rangle=|\Psi_{0}\rangle$. Note also that $\hat{P}_{s}$ is odd under the $\mathbb{Z}_{2}$ spin-transformation, $\hat{U}\hat{P}_{s}\hat{U}=-\hat{P}_{s}$, thus it connects the symmetric with the antisymmetric sectors, while its matrix elements within each symmetry block vanish, $\langle\Psi_{S}|\hat{P}_{s}|\Phi_{S}\rangle=\langle\Psi_{A}|\hat{P}_{s}|\Phi_{A}\rangle=0$. Consequently, in the basis of the eigenstates of the Hamiltonian, $\hat{P}_{s}$ has no diagonal matrix-elements. In contrast $\hat{P}_{c}$ is even, $\hat{U}\hat{P}_{c}\hat{U}=\hat{P}_{c}$ and $\langle\Psi_{S}|\hat{P}_{c}|\Phi_{A}\rangle=0$.

We first show the charge imbalance, presented in Fig. 1(a), $P_{c}(t)=\langle\hat{P}_{c}\rangle_{t}$, where $\langle\dots\rangle_{t}$ indicates multiple time evolutions from the initial $|\psi_{0}\rangle$, averaged over different random realizations of either potential or magnetic field disorder. We observe a similar time evolution for $t\lesssim 10$ under the influence of either potential or field disorder. At larger $t$ and for $W_{c(s)}\leq 6$, $P_{c}(t)$ relaxes towards zero faster in the case of field disorder. At larger $W_{c(s)}\geq 10$ $P_{c}(t)$ shows slow, logarithmic decrease in time. Providing that the time dependence would further follow a logarithmic form $A\log(t)+B$, as displayed with thin dashed lines for $W_{c(s)}=10$ in Fig. 1(a), $P_{c}(t)$ would equilibrate under $W_{c}$ or $W_{s}$ to zero around $\tau_{c}\sim 27.000$ and $\tau_{s}\sim 460.000$, respectively. It is worth stressing that relaxation time becomes longer in the case of potential disorder at larger values of $W_{c(s)}\geq 10$, moreover, relaxation might be prevented by the onset of the many body localization.

In the case of the spin imbalance $P_{s}(t)$, see Fig. 1(b), we find exceedingly different time evolution when comparing systems with potential or field disorder. While $P_{s}(t)$ in the case of the field disorder shows qualitatively similar behaviour as $P_{c}(t)$, the time evolution in the case of the potential disorder shows relaxation for all $W_{s}$ on a time scale accessible by our calculations. Moreover, the corresponding relaxation times $\tau$ show a quadratic $W_{c}$–dependence as shown in the inset of Fig. 1(b). The latter quadratic dependence may be explained by recalling that for large $W_{c},(W_{s}=0)$ a redistribution of charge may be energetically very costly, however reversing the spin orientation does not change the energy. Therefore the charge redistribution that is necessary for the spin dynamics is realized via virtual processes shown in Fig. 1(c). In order to estimate the relevant energy scale (i.e. also the time-scale) one may study a toy model with four local states shown in Fig. 1(c): two initial states $|\uparrow,0,\downarrow\rangle^{S/A}$ and two virtually-generated states $|\uparrow,\downarrow,0\rangle^{S/A}$. The corresponding eigenproblem is up to a constant energy shift given by $2\times 2$ matrices

[TABLE]

where the potentials $V^{S/A}=\epsilon_{2}-\epsilon_{3}-\frac{1}{2}(J_{z}\mp J_{\perp})$. If the charge disorder is strong then the typical values of $|V^{S/A}|\sim|\epsilon_{2}-\epsilon_{3}|$ are large. As a consequence only two eigenstates have large projections on the initial states, $|\uparrow,0,\downarrow\rangle^{S/A}$. Then, the dynamics of odd operators is governed by the difference of corresponding eigenenergies in symmetric and antisymmetric sectors $\tau=|E^{S}-E^{A}|^{-1}\sim W_{c}^{2}/t_{0}^{2}J_{\perp}$.

IV Charge and spin correlation functions

We next explore the neighboring density–density and spin–spin correlation function defined as:

[TABLE]

For the proper analysis of long–time behavior it is important to note that the energy of the initial state after averaging over different random realizations, $E_{\mathrm{ave}}=\langle\psi_{0}|H|\psi_{0}\rangle_{\mathrm{ave}}=0$, is located near the middle of the energy spectrum, which in the microcanonical sense corresponds to infinite–temperature ($T\to\infty$). Based on the eigenstate thermalization hypothesis srednicki_94 ; rigol_dunjko_08 ; deutsch_91 ; dalessio_kafri_16 ; eisert_friesdorf_15 ; deutsch_18 ; mori_ikeda_18 it is expected that for small $W_{c(s)}$ and $t\to\infty$ $G_{n(S^{z})}$ approach their respective $T\to\infty$ limits, as indicated by dashed horizontal lines in Fig. 2. It is indicative that charge and spin correlation functions for short times, $t\lesssim 1$ display qualitatively similar time dependence. Initially, only the hopping part (the first term) of the Hamiltonian in Eq. (1) is active since the exchange interaction can only act between particles on neighbouring sites. The change of either potential energy or field energy after hopping between neighboring lattice sites is in both cases comparable, which leads to a similar time–dependence for times comparable to inverse hopping time $1/t_{0}$. For larger $W_{c(s)}\gtrsim 10$ and $t\gtrsim 50$, $G_{n}$ shows logarithmic increase in time while $G_{S^{z}}$ shows logarithmic decrease. The difference is due to substantially different $T\to\infty$ limits. In contrast to $G_{n}$, $G_{S^{z}}$ shows distinct dependence with regard to the type of disorder. At large $W_{c}\geq 10$, $G_{S^{z}}$ approaches significantly closer the ergodic $T\to\infty$ limit than in the case of $W_{s}\geq 10$. Still, $G_{S^{z}}$ does not show relaxation as is the case of $P_{s}(t)$, shown in Fig. 1(b). The explanation for this seeming discrepancy can be found again in the symmetry argument. While $\hat{P}_{s}$ is odd under the $\mathbb{Z}_{2}$ spin-transformation, $\hat{G}_{S^{z}}$ is even, it has non-zero matrix elements within a fixed symmetry sector, and the matrix elements which are diagonal in the eigenstates of Hamiltonian may be non-zero as well.

To test this assumption, we define a three–site operator

[TABLE]

which is also odd under the $\mathbb{Z}_{2}$ spin-transformation, $\hat{U}\hat{\Gamma}\hat{U}=-\hat{\Gamma}$. As seen in Fig. 3, $\Gamma(t)=\langle\hat{\Gamma}\rangle_{t}$ as well shows relaxation with the respective relaxation times $\tau_{\Gamma}$ scaling with $W_{c}^{2}$ just as in the case of $P_{c}(t)$, shown in Fig. 1(b).

V Spin and charge entanglement entropy

We now turn to comparison of the dynamics of the entanglement properties of charge and spin degrees of freedom. To this end we split the system into two halves. We then compute the charge and spin entanglement entropylukin_rispoli_19 ; sirker_2020 , respectively:

[TABLE]

where $S^{z}_{\mathrm{max}}=N_{f}/4$ and $p_{n}$ and $p_{S^{z}}$ represent probabilities that the subsystem contains either $n-$ fermions or the $z-$component of the total spin equals $S^{z}$, respectively.

The charge entropy $S_{n}$, shown in Fig. 4(a), displays qualitatively similar behavior with respect to either the potential or the field disorder for small $W_{c(s)}\leq 6$. In both cases $S_{n}$ approaches its maximal value, characteristic for a thermal state at $t\to\infty$. For larger $W_{c(s)}\geq 10$ we observe a stronger deviation for different types of disorder in the long–time limit. In both case se observe a slow logarithmic increase, characteristic for MBL systems.

In contrast, the spin entropy $S_{S^{z}}$ quantitatively differs in comparison to potential or field disorder in the whole range of $W_{c(s)}$’s. The most significant difference is observed when comparing results for larger $W_{c(s)}\geq 10$ where $S_{S^{z}}(W_{c})$ grows significantly faster than $S_{S^{z}}(W_{s})$. For example: while at $W_{c(s)}=10$ $S_{S^{z}}(W_{c})$ nearly reaches its maximal value $S_{S^{z}}^{\mathrm{max}}$, $S_{S^{z}}(W_{s})$ displays slow logarithmic growth. For even larger $W_{c}\geq 14$ there seems to be no observable time interval with a logarithmic growth of $S_{S^{z}}(W_{c})$. In contrast, it shows a tendency towards saturation towards a non–thermal value.

To gain a deeper physical picture we first note that while the field disorder affects charge as well as spin degrees of freedom, the potential disorder affects only charge degrees of freedom. For example: when a fermion with spin–up hops between neighboring sites under the influence of field disorder, it feels different Zeeman energy just as when in the presence of the potential disorder it feels different potential energy. There exist connected spin chains separated by empty sites. Spins that form a particular connected spin chain do not feel the potential disorder as long as they remain attached to the chain. Spin excitation can freely propagate along a connected spin chain in the presence of the potential disorder. When such connected spin chains extend between the boundaries of the two subsystems, they contribute to the growth of spin entropy.

VI Spectral level statistics

Motivated by the rather unexpected difference in the time evolution of charge vs. spin imbalance, observed in Fig. 1(b) as well as other observables probing charge or spin degrees of freedom under the influence of potential disorder, we next explore statistical properties of the energy spectra. We compute adjacent energy level spacing ratios, $r_{n}=\mathrm{min}[\Delta_{n},\Delta_{n-1}]/\mathrm{max}[\Delta_{n},\Delta_{n-1}]$ where $\Delta_{n}=E_{n}-E_{n-1}$ and $\{E_{n}\}$ represents the ordered set of energy levels of Hamiltonian in Eq. (1). For each realization of disorder we compute average value of $\bar{r}$ and then instead of computing average over different realizations, we calculate the cumulative distribution function for $\bar{r}$, $F(\bar{r})$. In Fig. 5(a) we first present $F(\bar{r})$ for the case of potential disorder, $W_{c}$, taking into account the full spectrum. Since the Hamiltonian in Eq. (1) is non–integrable, one expects that at small $W_{c}=2$ its spectrum resembles the spectrum of the Gaussian random matrices where $\bar{r}_{\mathrm{ave}}=\bar{r}_{\mathrm{GOE}}\simeq 0.53$mukerjee_2006 ; oganesyan_huse_07 . In contrast, at large $W_{c}=20$, the average value of $\bar{r}$ should not drop below $r_{\mathrm{Poisson}}=0.386$, characteristic for a random distribution of energy levels. Distributions $F(\bar{r})$, shown in Fig. 5(a) are not consistent with either of the above predictions.

For a proper analysis of the spectral level statistics in the case of the potential disorder we have computed $F(\bar{r})$ separately for each symmetry subspace. In Figs. 5(b) and 5(c) we present $F(\bar{r})$ for different values of $W_{c(s)}$. There are two types of nearly overlapping curves (full and dashed lines) in the case of potential disorder, see Fig. 5(b) representing separately $F(\bar{r})$ for each symmetry sector. For small $W_{c(s)}=2$, presented in Figs. 5(b) and (c), $F(\bar{r})$ can be fitted with the Error function positioned at $\bar{r}_{\mathrm{ave}}=0.53$, which agrees with $r_{\mathrm{GOE}}$. At large $W_{c(s)}=20$ $F(\bar{r})$ again resemble Error functions, however in this regime close to $r_{\mathrm{Poisson}}$, which on a finite system indicate localization. For the intermediate values of $W_{c(s)}\in[4,6]$ we observe broad distributions $F(\bar{r})$ that result from a mixture of systems where a part of them exhibit level statistics that resembles ergodic systems while others the one closer to non–ergodic/localized.

We have also calculated the distribution of the gap ratios without averaging $r_{n}$ over multiple energy levels. To this end we have generated a set containing $r_{n}$ for various $n$ as well as for various disorder realizations and calculated the probability density, $P(r)$, from the histogram of $\{r_{n}\}$. While the distribution $F(\bar{r})$ in Fig. 5 contains information about differences between various realizations of disorder, such information is not directly encoded in $P(r)$. Nevertheless, $P(r)$ allows for a more detailed comparison with the random matrix theory. For the Poisson level statistics one gets oganesyan_huse_07

[TABLE]

while an approximate distribution for the Gaussian-Orthogonal- Ensemble (GOE) can be derived from the Wigner surmise atas2013 ; alet2022 ; fremling2022 for 3 energy levels

[TABLE]

Figure 6 shows the distributions, $P(r)$, obtained from the symmetric (S) or the antisymmetric (A) parts of spectra with charge disorder. In order to identify artifacts arising from the presence of the localization edge, the distributions have been generated either from all levels or only from a central part ($1/3$ or $1/9$) of the levels in the middle of the spectrum. Results obtained for all three cases accurately overlap (see Figs. 6 and 7 ) indicating absence of artifacts originating from the localization edge.

As expected, numerical results for weak disorder [Fig. 6(a)] accurately reproduce Eq. (13) whereas for strong disorder shown in Fig. 6(c), the distribution agrees with Eq. (12). In the vicinity of the transition, $P(r)$ can be well approximated by a mixed Wigner surmise discussed very recently in Ref. fremling2022 . More precisely, the dashed curve in Fig. 6(b) shows distribution for $2\times 2$ GOE matrices mixed with two uncorrelated energy levels, see Eq. (20) in Ref. fremling2022 . Such mixture of GOE and Poisson distributions may be interpreted as a coexistence of localized (insulating) and delocalized (metalic) domains whereby the absence of level crossings, $F(r\to 0)=0$ in Fig. 6(b), indicates that localization within the former domains is not perfect. Similar results concerning the spatial separation of conducting and insulating domains have been recently found for the random-field Heisenberg model herbrych2022 .

Figure 7 shows the distribution of the gap ratios obtain for charge disorder from the full spectrum that includes both symmetric and antisymmetric levels. Results obtained for weak disorder can be very accurately approximated by two independent GOE ensembles. In particular, dashed line in Fig. 7(a) shows the distribution derived for mixture of two $2\times 2$ GOE matrices, see Eq. (24) in Ref. fremling2022 . Rather unexpected is the case of strong disorder when $P(r)$ for $r\ll 1$ significantly exceeds the distribution derived for the Poisson statics, as it is shown in Fig. 7(b). Comparing Fig. 6(c) and 7(b) one identifies an attraction between symmetric and antisymmetric energy levels. Such scenario is consistent with results for the average ratio shown in Fig. 5(a).

VII Summary

We have studied spin- and charge-dynamics in a disordered chain such that an unperturbed system has $\mathbb{Z}_{2}$ spin-symmetry. Then we considered two types of disorder: a random magnetic field that breaks the $\mathbb{Z}_{2}$ symmetry and random charge potential, which preserves the spin-symmetry. In the former case with broken spin-symmetry, the dynamics of all studied observable are consistent with localization on finite lattices in that their expectation values do not approach the results for thermal equilibrium. However, for the symmetry-preserving disorder, the observables that are odd under the $\mathbb{Z}_{2}$ spin transformation seem to thermalize while even observables do not. Numerical studies of the time-evolution for the symmetry-preserving disorder have been accompanied by the level statistics. Interestingly, the level statistics obtained separately for odd and even symmetry sectors accurately reproduce a crossover/transition from the GOE for weak charge disorder to the Poisson distribution for the strong disorder. The time evolution and the level statistics suggest that localization exists within each symmetry sector, i.e. for odd or even eigenstates and observables with matrix elements only within a given symmetry sector. The apparent relaxation of odd operators is not inconsistent with the level statistics since such operators evolve between the two sectors. Similar numerical results have been found for the dynamics in the Hubbard model with charge disorder, which, however, has SU(2) symmetryprelovsek16 ; kozarzewski18 ; sroda19 ; kozarzewski2019 ; protopopov2019 . In the latter model, the spin imbalance decays subdiffusively prelovsek16 ; kozarzewski18 ; sroda19 while the spin-energy-density seems not to thermalize kozarzewski2019 ; protopopov2019 . In our studies, we have not considered the stability of the localized phase in the thermodynamic limit, so we do not exclude that localization represents extremely slow dynamics that eventually may lead to a thermal equilibrium of an infinite chain.

Comparison of the entanglement entropies of spin and charge degrees of freedom reveals an essential difference between the field disorder that affects charge and spin degrees of freedom and the potential disorder that influences only charge degrees of freedom. Spins that form a particular connected spin chain do not feel potential disorder. Spin excitations can thus freely propagate along a connected spin chain. This may explain the absence of a logarithmic growth of the spin entanglement entropy even in the regime of strong potential disorder, where in contrast, the charge entanglement entropy displays logarithmic time evolution.

The original motivation for this work stems from experiments on cold atoms schreiber_hodgman_15 where charge imbalance was measured in cold atoms experiment setup. Recent advances in the spin- and density–resolved microscopyBloch2016 ; Bloch2020 ; Bloch2020a might allow measurements of the charge and the spin imbalance under the non–symmetry–breaking disorder.

Acknowledgements.

We acknowledge the support by the National Science Centre, Poland via project 2020/37/B/ST3/00020 (M.M.), the support by the Slovenian Research Agency (ARRS), Research Core Fundings Grants P1-0044 ( J.B.) and the support from the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Los Alamos National Laboratory (Contract 89233218CNA000001) and Sandia National Laboratories (Contract DE-NA-0003525) (J.B.).