Delocalized charge carriers in strongly disordered t-J model

TL;DR

This paper investigates how electron-magnon interactions affect charge transport in strongly disordered one-dimensional t-J models, revealing potential transient subdiffusive behavior that may evolve into normal diffusion over time.

Contribution

It provides new insights into the transport behavior of strongly disordered t-J models, highlighting the possible transient nature of subdiffusive motion due to electron-magnon interactions.

Findings

Charge carriers remain delocalized under strong disorder without spin excitation localization.

Subdiffusive motion observed up to accessible timescales.

Finite-size scaling suggests a transition from subdiffusive to diffusive transport.

Abstract

We study the influence of the electron-magnon interaction on the particle transport in strongly disordered systems. The analysis is based on results obtained for a single hole in the one-dimensional t-J model. Unless there exists a mechanism that localizes spin excitations, the charge carrier remains delocalized even for a very strong disorder and shows subdiffusive motion up to the longest accessible times. However, upon inspection of the propagation times between neighboring sites as well as a careful finite-size scaling we conjecture that the anomalous subdiffusive transport may be transient and should eventually evolve into a normal diffusive motion.

Equations

$$H$$

$${\sigma }^2(t)=\sum _i{i}^2{\rho }_i(t)-{\left[\sum _{i}i{\rho }_i(t)\right]}^2.$$

$$F_{\alpha }={\int }_0^{\alpha }\mathrm{d}{\alpha }^{\prime }f({\alpha }^{\prime }),$$

$${\int }_{\tau }^{\infty }\mathrm{d}{\tau }^{\prime }f({\tau }^{\prime })=1-F_{\tau },$$

$$H$$

$$O_1$$

$$O_2$$

$$\left\{|{\psi }^{(l)}⟩\right\}={\left(O_1+{\tilde{O}}_2\right)}^l|{\psi }_{(0)}⟩,$$

Full text

Introduction.– The many–body localization (MBL) represents a promising concept of macroscopic devices which do not thermalize 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. (2015a); 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); N. and Huse (2015); Rademaker and Ortuño (2016); Chandran et al. (2015); Ros et al. (2015); Eisert et al. (2015) and may store the quantum information Serbyn et al. (2014a); Nandkishore and Huse (2015). Most of the inherent properties of MBL systems have been investigated using the generic one-dimensional (1D) disordered models of interacting spinless fermionsModak and Mukerjee (2015); Monthus and Garel (2010); Luitz et al. (2015b); Andraschko et al. (2014); Laumann et al. (2014); Huse et al. (2014); Ponte et al. (2015); Lazarides et al. (2015); Vasseur et al. (2015); Serbyn et al. (2014b); Pekker et al. (2014). Emerging characteristic features of MBL systems are: the existence of localized many-body states in the whole energy spectrum that leads to vanishing of d.c. transport at any temperature Berkelbach and Reichman (2010); Barišić and Prelovšek (2010); Agarwal et al. (2015); Gopalakrishnan et al. (2015); Bar Lev et al. (2015); Steinigeweg et al. (2016); Barišić et al. (2016); Kozarzewski et al. (2016), Poisson-like level statistics Oganesyan and Huse (2007), and the logarithmic growth of the entanglement entropy Žnidarič et al. (2008); Bardarson et al. (2012); Kjäll et al. (2014); Serbyn et al. (2015); Luitz et al. (2015a); Serbyn et al. (2013b); Bera et al. (2015). Numerical calculations of dynamical conductivity Agarwal et al. (2015); Steinigeweg et al. (2016); Barišić et al. (2016) and other dynamic properties based on the renormalization-group approach Gopalakrishnan et al. (2015); Vosk et al. (2015); Potter et al. (2015); Luitz et al. (2015a) indicate that in the vicinity of the transition to MBL state the optical conductivity shows a characteristic linear $\omega$-dependence. In the presence of strong disorder but still below the MBL transition, several studies predict a subdiffusive transport Agarwal et al. (2015); Gopalakrishnan et al. (2015); Žnidarič et al. (2016); Lüschen et al. (2016).

The presence of MBL has been rigorously shown so far only for the transverse–field Ising model Imbrie (2016), whereas the indisputable numerical evidence is available mostly for interacting spinless fermions or equivalent spin Hamiltonians. However, in real systems, the particles are coupled to other degrees of freedom and this coupling may be important not only for solids but also for the cold–atom experiments. In particular, the recent experiments Schreiber et al. (2015); Kondov et al. (2015); Bordia et al. (2016) address the problem of MBL in the spin–$1/2$ Hubbard model, where charge carriers are coupled to spin excitation. On the other hand, well established results Basko et al. (2006); Mott (1968a) indicate that phonons destroy the Anderson localization, hence they should destroy the MBL phase as well. Nevertheless, in contrast to phonons, the energy spectrum of many other excitations in the tight–binding models (e.g., the spin excitations) is bounded from above. It remains unclear whether strict MBL survives in the presence of the latter excitations. Solving this problem is important for answering the fundamental question whether MBL exists also in more realistic models including the Hubbard model Mondaini and Rigol (2015); Prelovšek et al. (2016). The preliminary numerical results suggest that charge carriers may indeed be localized despite the presence of delocalized spin excitations Prelovšek et al. (2016). Nonetheless, studies for larger systems and longer times are needed in order to eliminate the transient or finite-size (FS) effects.

We study dynamics of a single charge carrier coupled to spin excitations which propagates in a disordered lattice. Our studies are carried out for the 1D $t$–$J$ model, which should be considered as a limiting case of the Hubbard model for large on-site repulsion. The common understanding of MBL is that it originates from (a single–particle) Anderson insulator Anderson (1958); Mott (1968b); Kramer and MacKinnon (1993); Evers and Mirlin (2008) which persists despite the presence of carrier-carrier interaction Fleishman and Anderson (1980); Basko et al. (2006). The choice of a single particle (hole) in the $t$–$J$ model eliminates the latter interaction, hence, it should act in favour of localization. Note however, that non–trivial many–body physics emerges from interaction between the spin excitations. The dimension of the Hilbert space in the present studies is of the same order as in the commonly studied model of spinless fermions, hence the numerical results are obtained for rather large systems and long times far beyond the limitations of the Hubbard model.

We demonstrate that localization of charge carriers is possible only for localized spin excitations, whereas their dynamics is subdiffusive even for very strong disorder (or diffusive for weak disorder) when spins are delocalized. The latter result resembles the dynamics of interacting spinless fermions for strong disorder but still below the MBL transition Agarwal et al. (2015); Gopalakrishnan et al. (2015); Žnidarič et al. (2016); Lüschen et al. (2016). Here, we demonstrate that the subdiffusive behavior originates from extremely broad distribution of propagation–times for transitions between the neighboring lattice sites. However, this distribution strongly suggests that the subdiffusive behavior is a transient, yet long-lasting phenomenon. The transition to the normal diffusive regime takes place at extremely long times and cannot be observed directly from numerical data.

Model and numerical methods.– We study a single hole (a charge carrier) in the 1D $t$–$J$ model on $L$ sites with periodic boundary conditions

[TABLE]

where $c^{\dagger}_{i\sigma}$ creates an electron with spin $\sigma$ at site $i$, $\vec{S}_{i}$ is the spin operator and $n_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$. The states with doubly occupied sites are excluded ($n_{i\uparrow}n_{i\downarrow}=0$) and, for simplicity, the hopping integral is taken as the energy unit ($t_{h}=1$). The onsite potentials, $\varepsilon_{i}$, are random numbers that are uniformly distributed in the interval $[-W,W]$. In the special case $J_{i}=0$, the spins are frozen (at last in 1D case) and the hole dynamics should be the same as in the 1D Anderson insulators.

Considering the Hamiltonian (LABEL:ham) as a large–$U$ limit of the Hubbard model, one finds that each $J_{i}$ depends on $\varepsilon_{i}$ and $\varepsilon_{i+1}$. However, the disorder in the Hubbard model always enlarges the exchange interaction Maśka et al. (2007), $J_{i}\geq J(\varepsilon_{j}=\rm{const})$, hence such disorder alone should not localize the spin excitations. In order to discuss also a more general case of localized spin excitations, $J_{i}$ and $\epsilon_{i}$ will be set independently of each other.

The transport properties will be discussed mainly from the numerical results for the time propagation of pure states $|\psi_{t}\rangle$. We take $|\psi_{0}\rangle=|\sigma_{1}...\sigma_{j-1}0_{j}\sigma_{j+1}...\sigma_{L}\rangle$ as an initial state, where the position of the hole, $j$, as well as the spin configuration, $\sigma_{i}=\uparrow,\downarrow$, are chosen randomly. The latter choice means that the system is at infinite temperature. The essential information on the charge dynamics will be obtained from the hole density $\rho_{i}(t)=\langle\psi_{t}|1-n_{i\uparrow}-n_{i\downarrow}|\psi_{t}\rangle$. Then, one can define also the mean square deviation of the hole distribution Schmidtke et al. (2016)

[TABLE]

Throughout the paper, the averaging over disorder will be marked by the subscript ”d”. We typically take $10^{3}$ realizations of the disorder.

The toy model — In order to gain a deeper understanding of the anomalous charge dynamics, numerical data for the $t$–$J$ model will be compared with results for a classical particle which randomly walks on the same 1D lattice. As a toy model, we employ the continuous–time random walking in which particle waits for a time $\tau$ on each site $i$ before jumping to the neighboring site $i-1$ or $i+1$. This model is well understood for various distributions of the waiting times $f(\tau)$ Bouchaud and Georges (1990). In particular, if the average waiting time is finite, $\bar{\tau}=\int{\rm d}\tau\;f(\tau)\tau<\infty$, the model shows at long times normal diffusion, $\sigma^{2}(t)\propto t$. However, for a broad distribution of waiting times $f(\tau)\sim 1/\tau^{\mu+1}$ with $0<\mu<1$, $\bar{\tau}$ becomes infinite and one obtains a subdiffusive transport with $\sigma^{2}(t)\propto t^{\mu}$, see Bouchaud and Georges (1990); Montroll and Scher (1973).

Results – First we apply the Lanczos propagation method Park and Light (1986) and study weakly disordered system with homogeneous $J_{i}$, where one expects normal diffusion, i.e., $\sigma^{2}_{d}(t)\propto t$. Such linear behavior is indeed visible in Fig. 1a but only for short times. In order to explain the subsequent break–down of this linear trend we have studied the toy model for exactly the same lattice and $p(\tau)\propto\theta(\tau_{0}-\tau)$. The average waiting time $\bar{\tau}=\tau_{0}/2$ has been tuned to fit the linear regime in the $t$–$J$ model. Figures 1a and 1b show clear similarity between $\sigma^{2}(t)$ in both models. However in the toy model any departure from the normal diffusion must originate from the FS effects. Due to these effects, the numerical results are of physical relevance only for small values of the mean square deviation $\sigma^{2}_{d}<\sigma^{2}_{\rm max}\sim 10$.

In Figs. 1c and 1d we present results for the central problem of this work, i.e., for the charge dynamics in strongly disordered $t$–$J$ model. The power–law dependence $\sigma^{2}_{d}(t)\propto t^{\alpha_{d}}$ is evident over at least two decades of time and the transport is clearly subdiffusive. The exponent $\alpha_{d}\simeq 1/2$ for $W=6$ and further decreases for stronger disorder. Within the studied time window $\sigma^{2}_{d}\ll\sigma^{2}_{\rm max}$. Therefore, we do not expect any essential influence of the FS effects. The latter hypothesis is confirmed by numerical results on larger system ($L=20$) where only slightly larger value of the mean square deviation is obtained ($W=6$), shown also in Fig. 1c. Next, we have repeated the same calculations for various (homogeneous) exchange interactions $J_{i}=J$ and various disorder strengths. For each case we have obtained $\alpha_{d}$ and these exponents are shown in Fig. 1d as a function of $J/W$. Nearly overlapping points on this plot suggest that $\alpha_{d}=\alpha_{d}(J/W)$. Then, the charge localization ($\alpha_{d}=0$) should occur only for $W\rightarrow\infty$ or for $J\rightarrow 0$. Localization is the former case is rather obvious, whereas the latter one is just the Anderson insulator. Otherwise, the transport is subdiffusive or normal diffusive.

The essential question is whether the model in Eq. (LABEL:ham) may show charge localization under some particular conditions. In the following we demonstrate that such localization is indeed possible, provided that spin excitations are as well localized. In order to localize the latter degrees of freedom we put $J_{i}=0$ for every 2nd or every 4th site $i$, otherwise we keep $J_{i}=1$. Results are shown in figure 2a together with the data for the subdiffusive case ($J_{i}=J=0.5$) and the Anderson insulator ($J_{i}=J=0$). Within the time window that is accessible to our numerics, we don’t observe a complete saturation of $\sigma^{2}_{d}(t)$ except for the Anderson insulator. However, since the increase is visibly slower than logarithmic we conclude that that hole is indeed localized. An extremely slow charge dynamics within the localized regime is not very surprising. It has previously been reported also for the MBL phase in a system of interacting spinless fermions Mierzejewski et al. (2016).

From now on, we study the details of the subdiffusive transport in a system with homogeneous exchange interaction and, for simplicity, we set $J_{i}=1$. Fig. 2b shows the exponents $\alpha_{d}$ for various $L$. While the FS effects are not essential they are not negligible either. Therefore, it is important to employ a method which allows to study even larger systems. In a case of single carrier instead of diagonalizing the Hamiltonian in the full Hilbert space we use the limited functional Hilbert space Bonča et al. (2007). Such approach has successfully been applied to the studies on the real–time dynamics of $t$–$J$ and Holstein models Dal Conte et al. (2015); Golež et al. (2014); Mierzejewski et al. (2011); Golež et al. (2012) and it is briefly explained also in the Supplemental material sup . In this approach one accounts for all spin excitations in the closest vicinity of the holde but only for a selected more distant excitations. In contrast to the previous method, $L$ does not represent the geometric size of the lattice but the maximal distance between the hole and the spin excitation. However, in both approaches one is interested in the limit $L\rightarrow\infty$ and the corresponding $1/L$ scaling of $\alpha_{d}$ is shown in Fig. 2b. Both methods obviously give the same extrapolated value of the exponent $\alpha_{d}$. However, diagonalization in the Functional Hilbert space shows much weaker FS effects than the other approach.

Next, we check whether possible isolated cases with a localized hole have been overlooked when discussing results averaged over the disorder. To exclude the latter possibility, we have fitted $\sigma^{2}(t)\propto t^{\alpha}$ independently for each realization of the disorder, thus generating the distribution of the exponents $f(\alpha)$. The calculations have been carried out for times $t\leq 10^{3}$. In Figures 2c and 2d we show the cumulative distribution function,

[TABLE]

which vanishes for small $\alpha$ according to power law $\alpha^{\mu}$. As shown in Fig. 2d, $\mu$ depends on the disorder strength but seems to be free from the FS effects. Therefore, we conclude that $F_{\alpha\rightarrow 0}=0$ also in the thermodynamic limit. In contrast, $F_{\alpha\rightarrow 0}=F_{0}>0$ would indicate localization. For delocalized spin excitations in the $t$–$J$ model, the charge dynamics may be very slow but the hole is never localized at least not in the studied time window $(0,10^{3})$.

It has recently been agued that the SU(2) symmetry precludes conventional MBL Chandran et al. (2014); Potter and Vasseur (2016); Protopopov et al. (2016). In order to explicitly show that the latter mechanism is not responsible for delocalization of charge carriers in the present system, we have also considered the $t$–$J$ Hamiltonian with anisotropic spin–spin exchange interaction. Results in the Supplemental material sup show that breaking the SU(2) symmetry doesn’t lead to the charge localization even for very strong disorder $W=20$.

Finally, we show that the hole dynamics in strongly disordered $t$–$J$ model may be qualitatively understood by studying the classical toy model. The properties of the latter are determined by the distribution of waiting times, hence one should first specify which quantity obtained for the $t$–$J$ model bears the closest resemblance to the classical waiting time. Since the toy model describes sequence of hoppings between the neighboring sites, in the $t$–$J$ model we define $\tau$ as the shortest time for which the mean square deviation (2) equals the lattice constant, $\sigma^{2}(\tau)=1$. Such $\tau$ is well defined for each realization of disorder, and one obtains the distribution of the waiting times $f(\tau)$ in the quantum model. Since we are particularly interested in the large–$\tau$ properties of $f(\tau)$, we study the integrated distribution function

[TABLE]

where $F_{\tau}$is the cumulative distribution. For the algebraically decaying $f(\tau)\sim 1/\tau^{\mu+1}$ one gets $1-F_{\tau}\propto 1/\tau^{\mu}$, where $\mu=1$ is the threshold value for the subdiffusive long–time behavior of the toy model.

Figure 3a shows the integrated distribution of the waiting times obtained in the $t$–$J$ model for the largest accessible systems and various disorder strengths. This distribution closely follows predictions of the toy model. In a system showing normal diffusion ($W=2$) the distribution is very narrow, $1-F_{\tau}$ decays much faster than $1/\tau$ and the average waiting time is quite short $\bar{\tau}\sim 1$. In strongly disordered subdiffusive systems $1-F_{\tau}$ decays slower than $1/\tau$ and $\bar{\tau}$ should be very large, if not infinite. Therefore, our results strongly suggest that the subdiffusive transport originates from very broad distribution of the waiting times. A broad distribution of the propagation times between the neighboring lattices sites has its origin in the disorder strength $W$ which is by far the largest energy scale in the Hamiltonian. However, large disorder is frequently used in the studies of systems showing MBL. Therefore, the present explanation of the subdiffusive transport may apply also to other strongly disordered systems with many–body interactions Agarwal et al. (2015); Gopalakrishnan et al. (2015); Žnidarič et al. (2016) .

The integrated distributions of waiting times shown in Figs. 3a and 3b suggest that in the thermodynamics limit and for sufficiently large $\tau$ the decay of $1-F_{\tau}$ may eventually become faster than $1/\tau$. Then, the average waiting time will be huge but finite and the subdiffusive transport should be a long lasting yet transient phenomenon. The time scale for the onset of the normal diffusion is far beyond the reach of any direct numerical studies of interacting quantum systems. However, such long–time regime can still easily be studied in the toy model. In order to check this scenario, we have fitted numerically obtained results for the waiting times of the $t$–$J$ model, as shown in Fig. 3b, and used this fit in the toy model. The resulting mean square deviation of the particle distribution is shown in Fig. 3c confirming the onset of normal diffusion at $t\gtrsim 10^{4}$ for $W=6$ and $t\gg 10^{5}$ for $W=8$.

The conjecture with respect to the transiency of the subdiffusive transport may also be supported by the analysis of numerical results for the $t$–$J$ model without invoking the toy model. As shown in Fig. 3b , the width of the distributions of the waiting times decreases when $L$ increases. Therefore, one may expect that properly carried out FS scaling may reveal at least a clear tendency for the transition to normal diffusion. In Fig. 3d we show results for $W=8$ and various $L$ together with $\sigma^{2}_{d}(t)$ obtained from linear in $1/L$ extrapolation to $1/L\rightarrow 0$. The slope of extrapolated curve gradually increases with $t$ already in the time–window which is accessible to our numerical procedure.

Conclusions.– We have studied the dynamics of a single hole (charge carrier) in a strongly disordered $t$–$J$ model. Our main result is that localization of the charge carriers should be accompanied by localization of the spin–degrees of freedom, otherwise the charge dynamics is subdiffusive up to the longest times accessible to numerical calculations. This holds true also for $t$–$J$–like Hamiltonians with broken SU(2) symmetry. However, based on the distribution of propagation times between the neighboring sites and after careful finite–size scaling of the mean square deviation we conjecture that the subdiffusive transport is transient and should eventually be replaced by a normal diffusion. According to the latter conjecture, the delocalized magnetic excitations in the thermodynamic limit become an infinite heat-bath which, similarly to electron-phonon coupling Basko et al. (2006); Mott (1968a), restores non–zero albeit very small conductivity. While this conjecture requires further studies, the exceptionally broad distribution of propagation times indicates that utmost care should be taken when formulating the claims on the asymptotic dynamics based on numerical results obtained for times $\sim 10^{3}$ of the inverse hopping integrals.

We expect that our qualitative claims should be valid also for other concentrations of charge carriers since each carrier is coupled to an infinite set of magnetic excitations, provided that the latter excitations remain delocalized. However, an essential open problem is whether/which results reported here for the $t$–$J$ model remain valid also for the Hubbard model. Both models are equivalent provided that the Hubbard repulsion is stronger than all other energy scales including the disorder strength. Therefore, in strongly disordered systems the equivalence of both models is restricted to very strong repulsions when the coupling between charge carriers and spin excitations is too weak to be studied with purely numerical methods.

Acknowledgments. We acknowledge fruitful discussions with Fabian Heidrich-Meisner and Jerzy Łuczka. J.B. acknowledges the support by the program P1-0044 of the Slovenian Research Agency and M.M. acknowledges support from the 2015/19/B/ST2/02856 project of the National Science Centre (Poland).