Beyond fragmented dopant dynamics in quantum spin lattices: Robust localization and non-Gaussian diffusion

TL;DR

The paper addresses how dopants move in a two-leg $t$--$J$ ladder with XXZ spins, revealing robust localization in the Ising limit due to an emergent disordered potential and diffusion when spin exchange is introduced. It employs state-of-the-art matrix-product-state methods (METTS, TDVP, TEBD) to track a single dopant's non-equilibrium dynamics, uncovering self-similar, strongly non-Gaussian diffusion for both charge and spin that cannot be captured by simple hydrodynamics. Diffusion coefficients behave non-monotonically with spin exchange, growing linearly for small $\alpha$ and decreasing as $\alpha^{-1}$ for large $\alpha$, while high-temperature diffusion follows Arrhenius behavior with an activation energy depending weakly on $\alpha$. The study shows that fragmentation is not required for localization and suggests a broader mechanism via disorder-like landscapes, with implications for two-dimensional lattices and quantum simulators exploring strongly correlated dopant dynamics.

Abstract

The motion of dopants in magnetic spin lattices has received tremendous attention for at least four decades due to its connection to high-temperature superconductivity. Despite these efforts, we lack a complete understanding of their behavior, especially out of the equilibrium and at nonzero temperatures. In this paper, we take a significant step towards a much deeper understanding based on state-of-the-art matrix-product-state calculations. In particular, we investigate the non-equilibrium dynamics of a dopant in two-leg $t$--$J$ ladders with antiferromagnetic XXZ spin interactions. In the Ising limit, we find that the dopant is localized for all investigated nonzero temperatures due to an emergent disordered potential, with a localization length controlled by the underlying correlation length of the spin lattice, which increases exponentially with decreasing temperature. The dopant, hereby, only delocalizes asymptotically in the zero temperature limit. This greatly generalizes the localization effect discovered recently in Hilbert space fragmented models. In the presence of spin-exchange processes at rate $α$, the dopant diffuses with a diffusion coefficient, $D_h$, depending non-monotonically on $α$. It initially increases linearly as $D_h \propto α$ for $α\ll 1$ before dropping off as $α^{-1}$ for $α> 1$. Moreover, we show that the underlying spin dynamics at infinite temperature behaves qualitatively the same, albeit with important quantitative differences. We substantiate these findings by showing that the dynamics features self-similar scaling behavior, which strongly deviates from the Gaussian behavior of regular diffusion, especially for weak spin exchange. Finally, we show that the diffusion coefficient $D_h$ follows an Arrhenius relation at high temperatures, whereby it is exponentially suppressed upon cooling.

Figures (14)

• Figure 1: Setup and breaking Hilbert space fragmentation. (a) The system consists of two spin states (red: $\uparrow$, blue: $\downarrow$) and a single dopant -- a hole -- shown in green. The spins interact via Ising interactions $J$ and spin exchange $\alpha J$ and may hop onto the vacant site with amplitudes depending on the direction of the hop ($t_\parallel,t_\perp$). (b) The $t$--$J_z$ limit, i.e. Ising-type interactions ($\alpha = 0$) with only one-dimensional hopping ($t_\perp = 0$), features Hilbert space fragmentation. For a single hole, each block of fixed magnetization (fixed $N_\uparrow,N_\downarrow$) split into Krylov subspaces that are only the length of the ladder, $N_x$, large. (c) For any nonzero rung hopping ($t_\perp > 0$) or any spin exchange ($\alpha > 0$), each block with fixed magnetization collapses to a single Krylov subspace specified by the $U(1)$ spin conservation. (d) High-temperature dynamical phase diagram: while the fragmented phase (HSF) is special to $\alpha = t_\perp = 0$, the dopant remains localized for any nonzero $t_\perp$ and $\alpha = 0$ (blue). For any $\alpha > 0$, the dopant delocalizes diffusively. Moreover, the associated diffusion coefficient is maximal at intermediate spin exchange around $\alpha = 0.6$ as schematically indicated by the orange color gradient.
• Figure 2: Dopant dynamics in the Ising limit. Dopant dynamics at zero temperature (a)-(e), and infinite temperature (f)-(j) in the Ising limit ($\alpha=0$) for indicated values of the rung hopping and $J = 8t_\parallel$. (a) and (f) For $t_\perp = 0$, the spins can only move one site to the left [black long-dashed box]. $t_\perp > 0$ enables Trugman loops shown by the light to dark green arrows around numbered (1-6) $2\times 2$ plaquettes. This moves the entire spin pattern one rung to the left and flips it on its head. [(b) and (c)] Magnetic energy experienced along the specified paths in the zero temperature Néel ground state. (g),(h) Magnetic energy along the specified paths at infinite temperature shown for $50$ spin realizations (gray lines), with three highlighted in color (red, blue and green lines). Thick black lines show the standard deviation, ${\rm std}(E_J)$, averaged over all spin configurations. Thin black lines show the minimal and maximal values of $E_J$. [(d) and (i)] Resulting rms distance dynamics of the dopant featuring long-time ballistic behavior at zero temperature and robust localization at infinite temperature as the rung hopping is turned on. [(e) and (j)] In both cases, the probability of the dopant returning to the origin remains nonzero. Shaded areas show the estimated standard errors on the mean from the $200$ samples of the spin background. In the TEBD calculations, we use a maximum truncation error of $10^{-8}$ and a system sizes of $N_x \times N_y = 61\times 2$ ($T = 0$) and $N_x \times N_y = 25\times 2$ ($T = \infty$).
• Figure 3: Temperature dependence of localization. (a) Dopant rms dynamics for indicated inverse temperatures, $t_\perp = t_\parallel$ and $J = 8t_\parallel$ showing persistent localization at any nonzero temperature. Moreover, a ballistic regime appears at intermediate timescales $\tau t_\parallel \sim 10$ for the lowest nonzero temperatures. This indicates the emergence of quasiparticles, here with lifetimes $\sim 20/t_\parallel$. Shaded areas show estimated standard errors on the mean from $200$ spin samples. For the TEBD, we use a truncation error of $10^{-8}$ and system sizes ranging from $N_x \times N_y = 45\times 2$ ($\beta J = 1$) to $N_x \times N_y = 161\times 2$ ($\beta J = 3.5$). We note that the line for $\beta J = 0$ is almost entirely hidden by the $\beta J = 1, 2$ lines. (b) Corresponding localization length for indicated $t_\perp$ vs the inverse temperature (top axis) as well as the underlying spin-spin correlation length $\xi(\beta J)$ (bottom axis), revealing a linear dependency (red line) for $t_\perp = t_\parallel$ and a saturating localization length (blue line) for $t_\perp=0$.
• Figure 4: Dopant dynamics vs spin exchange. (a) Root-mean-square dynamics of the dopant at zero temperature for indicated values of the spin exchange, $\alpha$, and $J = 5t_\parallel$, showing universal ballistic behavior at short times (short-dashed line) and $\alpha$-dependent ballistic behavior at long times due to the formation of magnetic polaron quasiparticles. (b) Mean-square dynamics at infinite temperature for the same set of parameters. Here, we plot the mean-square distance (rather than the rms in (a)) to make the long-time linear diffusive behavior for any nonzero $\alpha$ apparent. This also means that the initial ballistic behavior is now a parabola. (c),(d) Corresponding probability for the dopant to return to its origin, $n_h({\bf i} = {\bf 0},\tau)$. Short-dashed lines show the initial ballistic behavior, where $d_{\rm rms}(\tau) = \tau\,t_\parallel$. Shaded areas show the estimated standard errors on the mean from 100 spin samples. For the TEBD, we use a truncation error of $10^{-7}$ and system sizes of $N_x\times N_y = 15\times 2, 21\times 2$ at low and high $\alpha$ at $T = 0$, whereas $N_x\times N_y = 11\times 2$ for $T = \infty$.
• Figure 5: Spin vs charge dynamics. (a) Rms spin dynamics at infinite temperature for indicated values of the spin anisotropy, $\alpha$, in the absence of dopants along with diffusive fits (dashed lines). Note that time is now in units of the spin coupling, $J$, and that we include an $\alpha$ dependent time offset $\tau^*_s$ extracted from linear regression of $[d_{\rm rms}(\tau)]^2 \sim 2D_s (\tau - \tau^*_s)$ vs $\tau$. System sizes between $151 \times 2$ and $501 \times 2$ are used, and data end either at the maximal evolution time or when the boundary effects become non-negligible ($S(0,\tau) / S(x_0, \tau) > 10^{-3}$). (b) Comparison of the time offsets in the long-time diffusive behavior [Eq. \ref{['eq.diffusive_behavior']}] for the spins and dopant [the former has been multiplied by $0.1$ to make the scales comparable]. (c) Diffusion coefficients, $D_s,D_h$, for the spins and dopant. For both, we find a linear onset $\sim \alpha$ [solid lines]. At strong spin exchange, the spins follow the expected linear behavior [dashed blue line] from the XX ladder with $D_s \simeq 0.95\alpha J$, whereas the diffusion coefficient for the dopant decreases, seemingly following a $\sim \alpha^{-1}$ drop off [dashed red line]. (d) Ballistic polaronic propagation speed $v_p$ vs $\alpha$ at zero temperature. This seems to follow a quadratic onset at weak spin exchange [solid red line] and a quartic drop-off at strong spin exchange [dashed red line]. We use $J = 5t_\parallel$ and $t_\perp = 0$ as in Fig. \ref{['fig:zero_vs_inf_temp_alpha']}.