Tailoring transport in quantum spin chains via disorder and collisions

TL;DR

This work investigates how disorder and collisional noise jointly shape transport in a disordered XXZ spin chain, exploring single- and multi-excitation dynamics using a stochastic collision model to mimic environmental interactions. By tracking observables such as the inverse participation ratio, inverse ergodicity ratio, and entanglement entropy, the authors identify localization-delocalization transitions and uncover universal features in plateau durations and delocalization times. Key findings include plateaus in localization under time-homogeneous, low-rate noise and a delocalization enhancement with increasing collision rate, with interactions among excitations further modifying the transport landscape. The results provide design principles for tailoring transport via noise and disorder, with potential implications for stroboscopic protocols in quantum devices and for understanding noise-assisted transport in complex biological systems.

Abstract

We systematically investigate the interplay of disorder and time-homogeneous collisional noise in shaping the transport dynamics of an anisotropic XXZ spin chain. Using stochastic collision models to simulate interaction with the environment, we explore the localization-delocalization transitions across regimes with single and multiple excitations. We find that space-homogeneous and low-rate collisions can shape regions where localization sets in the form of subsequent plateaus. The localization process has universal features for the plateaus duration and the delocalization time. Interactions among the excitations favor this process even for tiniest disorder levels. Our findings can be applied to design stroboscopic protocols where sequences of transport and localization can be tailored. We establish relevant connections to noise-engineering of quantum devices in noisy intermediate-scale quantum simulators platforms, and to realistic biological systems where noise and disorder coexist.

Figures (10)

• Figure 1: Sketch of our model. (a) Open quantum spin chain with energy levels affected by disorder in the $[-h, h]$ range. Excitations are depicted as flipped spins (green). Their dynamics is affected by the tunneling rate $J$ (black arrows), interaction strength $\Delta$ (blue ovals) and the addition of a noisy environment. The latter is represented by the auxiliary light-brown qubits colliding with the sites over time. (b)-(c) Histograms displaying the number of collisions undergone over time tJ by the auxiliary qubits (log scale) against spins. This is shown for a noise shape parameter $\nu = 100$ in the time-uniform regime, and collision rates $r_c = 1$ (b) and (c) $r_c = 0.1$. We use the same simulation parameters: number of sites in the spin is $N = 41$ sites, $M=500$ trajectories and final time of simulation $tJ = 30$.
• Figure 2: (De)localization behavior: the case of one excitation. Inverse Participation Ratio (IPR) for a spin chain with $N = 41$ sites. (a) IPR at long time ($tJ = 30$), for large disorder range ($h=10$) and different shape parameters $\nu$ and collision rates $r_c$. (b) IPR vs time for large disorder range ($h=10$) and different collision rates $r_c$. Curves with progressively increasing tickness and color scale refer to increasing values of $r_c=0, 0.1, 0.5, 1, 5, 50, 100$. (c) IPR vs time $tJ$ for low $r_c = 0.1$ and different disorder range $h$. Notice the emergence of plateaus of temporary stabilization of localized behavior. In (b) and (c) the shape parameter is $\nu = 100$ in the time-homogeneous collision regime. The data refer to the case of no anisotropy ($\Delta = 0$ in Eq. \ref{['XXZ_Hamiltonian']}) since they are seen to be not significantly altered by finite $\Delta \neq 0$. The simulations were performed with $M = 500$ trajectories and timestep timestep $dt = 0.02$.
• Figure 3: The case of two excitations. (a) Inverse Ergodicity Ratio (IER) vs time $tJ$ for different collision rates $r_c$ with anisotropy strength $\Delta = 2.5$. Inset: magnetization map over sites $i$ and time $tJ$, showing the spreading of the two excitations over time. The two neighboring excitations are initially separated by two spins with very homogeneous collisions over time ($\nu = 100$) and high disorder range with $h = 10$. The IER has the same trend of the IPR and we focus here on the collision rate regime $0 \leq r_c \leq 10$, proved to be more interesting due to the formation of plateaus. For lower collision rates ($r_c \ll 1$) the system remains localized (IER closer to 1), while increasing $r_c$ leads to a faster delocalization (IER decaying to its limit). (b) IER vs $tJ r_c$ varying the collision rate parameter $0 \leq r_c \leq 1$, at fixed high time-homogeneity ($\nu = 100$), high disorder strength ($h = 10$), and in the presence of anisotropy, considering two excitations initially separated by two spins. Curves with progressively increasing thickness refer to values of $r_c$ increasing from 0.20 to 1.00 by 0.05 each. We see that all curves collapse in the region where plateaus are present since they are naturally shifted with respect to each other. The simulations were performed for a spin chain of $N=20$ sites with two excitations initially separated by two spins with $M = 250$ trajectories and timestep $dt = 0.02$ and final time of simulations $tJ =30$.
• Figure 4: Characterization of the plateaus. (a) 3D plot of plateau duration$D$ in terms of tJ vs disorder range $h \in [0,10]$ and collision rate $r_c \in [0,1]$ (see Appendix \ref{['2excmore']}, Fig. \ref{['fig:2dplot']}, for a 2D version). Different symbols colors and shapes distinguish different levels of collision time-homogeneity (colors) and anisotropy (shapes), as in the legend. (b) Representing all the relevant datasets with only two quantities that collect all the governing physical parameters. Semi-log plot of the dimensionless plateaus area $Z_J D(tJ) r_c$ vs the disorder power per unit collision time $P_h\equiv hr_c$, scaled with the shape parameter $\nu$ (see text for a discussion). The datasets can be identified from the legend: different symbols label the collision rates $r_c = 0.1, 0.2, 0.5, 0.7, 0.9,1$, different colors the disorder strength values $h = 0.5, 1, 5, 10$. Notice that all the datasets collapse in two curves: in essence, no plateaus for low disorder for any other parameter (bluish symbols) and rapidly developing plateaus to a saturating value with increasing disorder power, for slow collision rates and any other parameter (yellow-reddish symbols). As already commented, the anisotropy $\Delta$ does not qualitative impact the results and it is therefore fixed at $\Delta = 2.5$ (that is, other $\Delta$ values would be not distinguishable). Also, adding the data with $r_c > 1$ would result in having an horizontal lineup of data points collapsed at vanishing plateaus area irrespective of all the other parameters, and are thus not reported. The simulations were performed for a spin chain of $N=20$ sites with two excitations initially separated by two spins, with $M = 250$ trajectories, timestep $dt = 0.02$, and final time of simulations $tJ =30$.
• Figure 5: Study of delocalization time $\tau$ (see text, Sec. \ref{['2_exc']}). 3D plot of the complete delocalization time $\tau$ vs $h \in [0,10]$ and $r_c \in [0,10]$. The plane $r_c = h$ separates small from intermediate-to-high $\tau$ regions. Region (1) with small $\tau$ does not exhibit plateaus, while the region with plateaus occurs in two regimes: with intermediate (2) and long (3) $\tau$, corresponding to a more localized/delocalized behavior (see text, Sec. \ref{['2_exc']}). Symbols with different colors represent different levels of collision time-homogeneity as in the legend. The simulations were performed for a spin chain of $N=20$ sites with two excitations initially separated by two spins with $M = 250$ trajectories and timestep $dt = 0.02$ and final time of simulations $tJ =30$. The data with $\tau > 30$ are those for which the system does not succeed to reach complete delocalization in that time.