Spin subdiffusion in disordered Hubbard chain

TL;DR

This paper investigates the anomalous subdiffusive spin transport in a disordered Hubbard chain, deriving an effective model that links spin dynamics to disorder-induced localization effects.

Contribution

It introduces a new effective spin model explaining subdiffusion in disordered Hubbard systems, connecting spin exchange interactions to Anderson localization.

Findings

Spins are delocalized while charges are localized under strong disorder.

Subdiffusive transport arises from a singular distribution of spin exchange interactions.

The subdiffusion exponent depends on the Anderson localization length and electron density.

Abstract

We derive and study the effective spin model that explains the anomalous spin dynamics in the one-dimensional Hubbard model with strong potential disorder. Assuming that charges are localized, we show that spins are delocalized and their subdiffusive transport originates from a singular random distribution of spin exchange interactions. The exponent relevant for the subdiffusion is determined by the Anderson localization length and the density of electrons. While the analytical derivations are valid for low particle density, numerical results for the full model reveal a qualitative agreement up to half-filling.

Equations

$$H$$

$${\chi }_{a^{\prime }{b}^{\prime }}^{ab}$$

$$H_U=\frac{1}{2}\sum _{a\ne b}{J}_{ab}\left(\frac{1}{4}{n}_a{n}_b-{\vec{S}}_a\cdot {\vec{S}}_b\right),$$

$$J_{ab}=2U{\chi }_{ab}^{ab}=2U\sum _i|{\varphi }_{ia}{|}^2|{\varphi }_{ib}{|}^2.$$

$$J_{ab}\simeq 2U\exp (-x_{ab}/\lambda ),\lambda \sim \xi$$

$$H_H=-\sum _i{J}_i{\vec{S}}_i\cdot {\vec{S}}_{i+1},$$

$${\int }_0^y\mathrm{d}x\frac{1}{d}\exp \left(-\frac{x}{d}\right)={\int }_{2U\exp (-y/\lambda )}^{2U}\mathrm{d}J{f}_J(J).$$

$$f_{\tilde{J}}(\tilde{J})=\tilde{\lambda }{\tilde{J}}^{\tilde{\lambda }-1},\tilde{\lambda }=\lambda /d.$$

$$S_L(t)=⟨S_i^z{S}_i^z(t)⟩=\frac{1}{\mathrm{Tr}1}⟨\mathrm{Tr}\left[S_i^z(t)S_i^z\right]{⟩}_{\mathrm{dis}}$$

$${\left[{\int }_{{\tilde{J}}_0}^1\mathrm{d}\tilde{J}{f}_{\tilde{J}}(\tilde{J})\right]}^M={\int }_{{\tilde{J}}_0}^1\mathrm{d}{\tilde{J}}_m{f}_m({\tilde{J}}_m),$$

$$⟨{\tilde{J}}_m⟩$$

$$\Lambda \sim Md\propto {\left(2Ut\right)}^{\tilde{\lambda }},S_L(t)\propto {\Lambda }^{-1}\propto {\left(2Ut\right)}^{-\tilde{\lambda }}.$$

Full text

Introduction.– The many–body localization (MBL) Basko et al. (2006); Oganesyan and Huse (2007) has recently been intensively studied. Vast amount of numerical data allowed to identify the main properties of the MBL systems: vanishing steady transport, Berkelbach and Reichman (2010); Barišić and Prelovšek (2010); Agarwal et al. (2015); Bar Lev et al. (2015); Steinigeweg et al. (2016); Barišić et al. (2016); Kozarzewski et al. (2016); Prelovšek and Herbrych (2017) absence of thermalization Pal and Huse (2010); Serbyn et al. (2013a); Bar Lev and Reichman (2014); Schreiber et al. (2015); Serbyn et al. (2015); Khemani et al. (2015); Luitz et al. (2015); Gornyi et al. (2005); Altman and Vosk (2015); De Luca and Scardicchio (2013); Gramsch and Rigol (2012); De Luca et al. (2014); Huse et al. (2013); Nandkishore and Huse (2015); Rademaker and Ortuño (2016); Chandran et al. (2015); Ros et al. (2015); Eisert et al. (2015); Serbyn et al. (2014); Nandkishore and Huse (2015); Sierant et al. (2017); Prelovšek et al. (2018) and logarithmic growth of the entanglement entropy Žnidarič et al. (2008); Bardarson et al. (2012); Kjäll et al. (2014); Serbyn et al. (2013b, 2015); Luitz et al. (2015). It has also been found that MBL prevents a driven system from heating Kozarzewski et al. (2016); Abanin et al. (2017, 2015); Ponte et al. (2015); Bordia et al. (2017a); Choi et al. (2017); Zhang et al. (2017). These unusual properties can be explained via the existence of a macroscopic number of local integrals of motion Serbyn et al. (2013a); Huse et al. (2014); Imbrie et al. ; Chandran et al. (2015); Rademaker and Ortuño (2016); O’Brien et al. (2016); Inglis and Pollet (2016); Goihl et al. (2017); Mierzejewski et al. (2018).

While most of theoretical studies so far concentrated on the one-dimensional (1D) disordered model of interacting spinless fermions, the experiments on MBL are performed on cold–fermion lattices Schreiber et al. (2015); Choi et al. (2016); Bordia et al. (2016, 2017b) where the relevant model is the Hubbard model with spin–$1/2$ fermions, whereby the disorder enters only via a random (or quasi-periodic) charge potential. Recent numerical studies of such a model Mondaini and Rigol (2015); Prelovšek et al. (2016); Bonča and Mierzejewski (2017, 2017); Mierzejewski et al. (2018) reveal that even at strong disorder, localization and nonergodicity occurs only in the charge subsystem, implying a partial MBL. Unless one introduces also an additional random magnetic field Lemut et al. (2017); Mierzejewski et al. (2018), the spin remain delocalized, Potter and Vasseur (2016); Chandran et al. (2014); Protopopov et al. (2017); Friedman et al. (2017); Prelovšek et al. (2016); Parameswaran and Gopalakrishnan (2017), although the spin transport is anomalously slow and subdiffusive Prelovšek et al. (2016).

In the present work we focus on the explanation of the slow spin dynamics and subdiffusion within the disordered 1D Hubbard model. We first demonstrate that in the case of potential disorder and low particle density the spin dynamics can be described by a squeezed isotropic Heisenberg model, whereby the distribution of the random exchange interactions is singular. Such an effective model can be studied numerically quite in detail, but also analytically taking into account that the 1D spin dynamics is dominated by weak links. In this manner we show that spin excitations spread over distance $M$ in a characteristic time $t$ such that $M\propto t^{\alpha}$ with ${\alpha\simeq\lambda/(d+\lambda)}$, where $d$ is the average distance between singly–occupied sites and $\lambda$ is determined by the Anderson localization length in the noninteracting system. While the mapping on the Heisenberg model is valid for dilute systems $d\gg 1$, numerical results for strongly disordered Hubbard model reveal that the same qualitative spin dynamics remains valid for all densities even up to half–filling.

Model.– Our aim is to establish the spin dynamics in the disordered Hubbard chain,

[TABLE]

where $n_{i}=n_{i\uparrow}+n_{i\downarrow}$ and $n_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$. We study the model with $L$ sites and $N$ electrons, fixing also total spin projection $S^{z}_{\rm tot}=0$. We assume a uniform distribution of random charge potentials, $\varepsilon_{i}\in[-W,W]$ and set the hopping integral $t_{h}=1$.

Two electrons. In order to gain a preliminary insight to the spin dynamics, we first study two electrons. The dynamics of a few interacting spinless particles has been studied previously Shepelyansky (1994); Jacquod and Shepelyansky (1997); Schmidtke et al. (2017). Here, we study for illustration $N=2$ electrons with opposite spin projections which propagate on the chain with $L=16$ sites. Assuming that particles are initially at sites $j$ and $l$, $|\psi(0)\rangle=c^{\dagger}_{j\downarrow}c^{\dagger}_{j\uparrow}|0\rangle$, the propagation of $|\psi(t)\rangle$ is obtained via exact diagonalization. Figs. 1a and 1b show, respectively, time–dependence of the local spin $\langle S^{z}_{i}\rangle(t)=\frac{1}{2}\langle\psi(t)|n_{i\uparrow}-n_{i\downarrow}|\psi(t)\rangle$ and density $\langle n_{i}\rangle(t)=\langle\psi(t)|n_{i}|\psi(t)\rangle$ for one configuration of $\varepsilon_{i}$ corresponding to $W=8$. While for such strong disorder, the charge degrees appear to be fully localized, spins undergo oscillations.

Effective spin model.– The coexistence of almost frozen charges and oscillating spins suggests that one can derive an effective spin model. To this end, we use the Anderson states as the basis, i.e., we use the single–particle eigenstates, $\phi_{ia}=\langle i|a\rangle$, of the noninteracting ($U=0$) model. Then,

[TABLE]

Assuming that the charge-dynamics is frozen, the main effect arising from the presence of the Coulomb interaction comes from terms with either $a=a^{\prime}$, $b=b^{\prime}$ or $a=b^{\prime}$, $b=a^{\prime}$, and in both cases $a\neq b$. Then, the Hubbard term in Eq. (2) can be written in a SU(2)–invariant form

[TABLE]

where we use standard density and spin operators: $n_{a}=n_{a\uparrow}+n_{a\downarrow}$, $S^{z}_{a}=\frac{1}{2}(n_{a\uparrow}-n_{a\downarrow})$, $S^{+}_{a}=c^{\dagger}_{a\uparrow}c_{a\downarrow}$, $S^{-}_{a}=c^{\dagger}_{a\downarrow}c_{a\uparrow}$. The effective exchange interaction is ferromagnetic

[TABLE]

In order to test the approximation Eq. (4), we consider $N=2$ electrons with opposite spins, located at sites $j$ and $j+d$. We evaluate the spin–oscillation frequency, $\omega_{2}$, directly from results for the Hubbard model, see Fig. 1a. For the same set of $\varepsilon_{i}$ we then identify Anderson states $a,b$ which maximize $|\phi_{ja}|^{2}$ and $|\phi_{j+d\;b}|^{2}$, respectively. This enables the calculation of $J_{ab}$ from Eq. (4) that should lead to spin oscillations $\langle S^{z}_{j,l}\rangle=\pm\frac{1}{2}\cos(J_{ab}t)$. Figs. 1c and 1d show correlations between $\omega_{2}$ and $J_{ab}$ for various realizations of disorder and various distances $d$ between the electrons. One finds that indeed $\omega_{2}\simeq J_{ab}$ for strong disorder $W\gg 1$ and $d\gg 1$.

For low density of carriers and larger disorder, the maxima of the occupied Anderson states $a$ and $b$ are typically separated by $x_{ab}>\xi$, exceeding the single-particle localization length, $\xi$. Then, one obtains an approximate relation

[TABLE]

The squeezed spin model.– Assuming that charges are frozen to the initial occupations $n_{i}=0,1,2$ it is evident that the effective spin model Eq. (3) acts only on singly occupied sites with $n_{i}=1$. Spin dynamics of the Hubbard model at high temperatures $T\gg W,U$ can be then studied by first randomly positioning $N$ electrons on $L$ sites. This allows us to establish the distribution of distances between the singly occupied sites as well as the distribution of the effective $J_{ab}$, using Eq. (4) or its simplified version, Eq. (5). Due to the exponential decay of $J_{ab}$ we consider only couplings between the neighboring singly occupied sites. The effective Heisenberg model on a squeezed chain then reads

[TABLE]

where the summation is carried out over singly occupied sites $n_{i}=1$, $i\in\{1,...,\tilde{N}\}$ with $\tilde{N}\leq N$. Note that at infinite temperature, the average lattice–spacing in the effective model equals $d=L/\tilde{N}=(\bar{n}-\bar{n}^{2}/2)^{-1}$, where $\bar{n}=N/L$ is the average filling in the original Hubbard model.

In order to establish the probability distribution of $J_{i}$, we first consider a section of length $L\gg 1$, where we randomly choose the continuous positions of $\tilde{N}=L/d$ points and study the regime $L\gg 1$. The probability density for the distances between the neighboring points is $f_{d}(x)=\frac{1}{d}\exp(-x/d)$. While the latter result has formally been obtained for continuous positions of points, it should hold true also for discrete positions of singly occupied sites provided that $d\gg 1$. Using this result one may find the probability density for the random exchange interaction $f_{J}(J)$. To this end, we use Eq. (5) and compare the cumulative distribution functions

[TABLE]

Taking the derivative of Eq. (7) with respect to $y$ and introducing the dimensionless interaction $\tilde{J}=\frac{J}{2U}$ one gets

[TABLE]

It is clear that the interaction $U$ sets the energy scale (and the time scale) of the effective model, whereas the qualitative spin dynamics depends on the ratio between effective localization length, $\lambda$, and inter–particle distance $d$. The important message is that for low doping ($d\gg 1$) and strong disorder ($\lambda\sim 1$) one obtains $\tilde{\lambda}\ll 1$ with the distribution of $\tilde{J}$ being singular at $\tilde{J}=0$. Still, $\lim_{\delta\rightarrow 0^{+}}\int_{0}^{\delta}{\mathrm{d}}\tilde{J}\;f_{\tilde{J}}(\tilde{J})=0$, hence the probability for cutting the Heisenberg chain into disconnected subchains is vanishingly small.

In order to test feasibility and accuracy of Eq. (8) we have numerically generated the distribution of $\tilde{J}=J_{ab}/2U$ also directly from Eq. (4), in the same way as discussed for $N=2$ case. The positions of two electrons $l$ and $j$ have been randomly chosen in such a way that the distance $x=|l-j|$ is distributed according to $f_{d}(x)$ for various $d$. Numerical results for $W=4$ and 8 are shown in Figs. 2a and 2b, respectively. These results have been fitted by Eq.(8), whereby we adjusted a single parameter $\lambda$ for all values of $d$. We have obtained $\lambda\simeq 0.75$ and $\lambda\simeq 0.4$ for $W=4$ and $W=8$, respectively. Although Eq. (8) has been derived for $d\gg 1$, it turns out to remain qualitatively valid also for $d=2$, i.e. for the average distance between singly–occupied sites in the half–filled Hubbard model. We conclude that Eq. (8) accurately describes $f_{\tilde{J}}(\tilde{J})$ at least for small $\tilde{J}$, i.e., in the regime which is essential for the long–time spin dynamics.

Local spin correlations.– We first calculate the time–dependent local spin correlations at infinite temperature,

[TABLE]

where the spin evolution is determined by the effective $H_{H}$. We take the random interaction $J_{i}=\tilde{J}$ as given by Eq. (8), i.e., we express time in units of $1/2U$. $\langle...\rangle_{\rm dis}$ means averaging over various realizations of $J_{i}$ and we use at least $2000$ disorder samples.

Figs.  2c and 2d show $S_{L}(t)$. For longer times and $\tilde{\lambda}<1$ one observes power-law decay $S_{L}(t)\propto(2Ut)^{-\alpha}$ with $\alpha<1/2$, hence the spin dynamics is clearly subdiffusive. In Fig. 3a we demonstrate $\alpha$ obtained from fitting numerical results by $S_{L}(t)\propto t^{-\alpha}$ in the time–window $t\in[10,50]$. The main message coming from these studies is that $\alpha>0$ for arbitrary nonzero $\tilde{\lambda}>0$, i.e., for arbitrary nonzero filling. For $\tilde{\lambda}\ll 1$ we obtain the exponent $\alpha\simeq\tilde{\lambda}$. Still, it should be noted that the distribution Eq. (8) is singular only for $\tilde{\lambda}<1$ which should be the regime of subdiffusion. For $\tilde{\lambda}\ll 1$ the finite size effects are negligible (Fig. 2c) but become significant for larger $\tilde{\lambda}$, Fig. 2d. Nevertheless, for the regime with $\tilde{\lambda}\simeq 1$ (relevant for larger filling $\bar{n}\sim 1$ and/or weaker disorder) our results shown in Fig. 2d are consistent with normal spin diffusion with $\alpha=1/2$, which is also expected in the weakly disordered Hubbard model. Results in Figs. $\ref{fig2}c$, $\ref{fig2}d$ and $\ref{fig3}a$ support the scenario, that the spin excitations spread subdiffusively due to the singular distribution of random exchange interactions, Eq. (8).

Single weak–link scenario.– To explain the relation of the subdiffusive dynamics and the singular distribution of $\tilde{J}_{i}$, we consider a single spin excitation and estimate the time, $t$, in which the excitation spreads over $M$ sites in the effective Heisenberg chain. We assume the weak–link scenario, where the long–time dynamics is govern by rare regions with the smallest $J_{i}$. Similar approach has been used to describe the subdiffusive transport of spinless particles in the vicinity of the MBL transition Agarwal et al. (2015, 2017); Lüschen et al. (2017); Bordia et al. (2017b). Here, we assume that $t\sim 1/(2U\tilde{J}_{m})$, where $\tilde{J}_{m}$ is the weakest exchange out of $\tilde{J}_{i}$ for $i=1,...,M$. The probability that each random $\tilde{J}_{i}$ is larger than $\tilde{J}_{0}$ is

[TABLE]

where $f_{m}(\tilde{J}_{m})$ is the probability density for the smallest interaction. Using Eq. (8) and calculating derivative of Eq. (10) with respect to $\tilde{J}_{0}$ one finds the distribution function $f_{m}(\tilde{J}_{0})=\tilde{\lambda}M\tilde{J}_{0}^{\tilde{\lambda}-1}(1-\tilde{J}^{\tilde{\lambda}}_{0})^{M-1}$. Then, the expectation value of the smallest exchange interaction out of $M$ random $\tilde{J}_{i}$ reads

[TABLE]

In the latter equation we have used formulas for asymptotics at $M\gg 1/\tilde{\lambda}$. So we find the relation between the spread of the spin excitations $\Lambda$ and $t$ as,

[TABLE]

The exponent $\alpha=\tilde{\lambda}$ is the same as previously obtained from numerical studies of the effective Heisenberg model for $\tilde{\lambda}\ll 1$.

Multiple weak–link scenario.– The single–weak link scenario breaks down for $\tilde{\lambda}\sim 1$, where $\alpha\simeq\tilde{\lambda}/2$ instead of $\tilde{\lambda}$, as shown in Fig. 2d. As an alternative explanation for the subdiffusive transport we consider distribution of effective hopping times between neighboring sites in the squeezed spin model. The relevant dimensionless quantity is $\tau=1/\tilde{J}$. Using Eq. (8) one finds the probability density $f_{\tau}(\tau)=\tilde{\lambda}/\tau^{\tilde{\lambda}+1}$. For such broad distribution of hopping times in a classical model of random traps Bouchaud and A. (1990); Machta (1985), one gets subdiffusive transport $\Lambda\propto\left(2Ut\right)^{\alpha}$ where $\alpha=\tilde{\lambda}/(1+\tilde{\lambda})$ for $\tilde{\lambda}<1$. In the latter model, a classical particle may hop between neighboring traps in time $\tau$. The hopping time, $\tau$, is randomly chosen for each site but remains the same for each visit of the same site. This simple model quite accurately reproduces the dynamical exponent $\alpha$ for arbitrary $\tilde{\lambda}$, as shown in Fig. 3a, whereas for $\tilde{\lambda}\ll 1$ it gives the same relation as the single weak–link scenario.

Comparison with the Hubbard model.– Finally, we compare our analytical predictions with numerical results obtained directly for the Hubbard model. The time-dependent local spin correlation function $S_{L}(t)$ at infinite temperature has been obtained using the microcanonical Lanczos method (MCLM) Long et al. (2003); Prelovšek and Bonča (2013) (in analogy to the imbalance correlations presented previously Prelovšek et al. (2016)) for the Hubbard model with $L=18$ and $N=6$, i.e. $d\simeq 3.5$. Results are shown in Figs. 3b,  3c together with analytical prediction, $S_{L}(t)\propto(2Ut)^{-\alpha}$ with $\alpha=\lambda/(\lambda+d)$ and $\lambda$ obtained from fits in Fig. 2b. Despite significant finite–size effects, one observes that the latter estimate correctly describes the subdiffusive spin dynamics in the Hubbard model at low filling $\bar{n}\ll 1$. In particular, the exponent $\alpha$ obtained directly from the Hubbard model weakly depends on $U$.

Moreover, one can consider the validity of the subdiffusion scenario in the full Hubbard model beyond the limit of low filling. It has previously been found that spins reveal a subdiffusive dynamics even for $\bar{n}=1$ Prelovšek et al. (2016). We therefore analyze the MCLM results for $S_{L}(t)\propto t^{-\alpha}$ considering various system sizes $L=14,16,18$ and various numbers of electrons $N$. For the time–window $t\in[1,10]$ we extract the dynamical exponent $\alpha$ and compare with $\alpha=\lambda/(\lambda+d)$, as shown in Fig. 3d. Our approach works even up to $\bar{n}=1$ since the average distance between singly occupied sites $d\geq 2$, while $\lambda<1$ provided that the disorder is sufficiently strong. In particular, for $W=8$ and $\bar{n}\leq 1$ we have estimated that $\tilde{\lambda}\lesssim 0.2$.

Conclusions.– In this paper we presented an explanation for the anomalous spin dynamics in 1D Hubbard model with large potential disorder in the regime of partial MBL, where the charge dynamics appears to be frozen whereas spins exhibit ergodic but subdiffusive transport Prelovšek et al. (2016). We have derived an effective isotropic Heisenberg model with random exchange interactions between neighboring singly–occupied sites $J_{i}$. Our derivation is formally best applicable to the regime of low filling, $\bar{n}\ll 1$, and strong disorder. The main origin of the subdiffusive behavior then appears to be the singular distribution of the effective exchange interaction, $J_{i}$, with the crucial parameter $\tilde{\lambda}=\lambda/d$ representing the ratio of the single-particle Anderson localization length and the average distance between singly occupied sites. Results for the Hubbard model reveal that such scenario seems to remain qualitatively valid beyond the considered limits of low filling, even at $\bar{n}=1$, provided that the disorder is sufficiently strong. It appears that there is no threshold filling $\bar{n}_{c}$, below which also spins would become localized and full MBL would prevail.

There are still questions concerning the dynamics within disordered Hubbard model, being relevant also to cold-atom experiments on MBL Schreiber et al. (2015); Choi et al. (2016); Bordia et al. (2016, 2017b). Our derivation of the effective model, remains on the level of spin dynamics, while charge degrees are assumed to be frozen. It is evident that higher order terms in the Anderson basis, following from Eq. (2), would lead also to the dynamical coupling between charge and spin degrees of freedom. Since the spin dynamics is ergodic, it is not excluded that also charges would eventually delocalize, but then on much larger time scales.