Boundary-driven Heisenberg chain in the long-range interacting regime: Robustness against far-from-equilibrium effects

We investigate the Heisenberg $XXZ$ chain with long-range interactions in the $Z$ dimension. By applying two magnetic boundary reservoirs, we drive the system out of equilibrium and induce a nonzero steady-state current. The long-range coupled chain shows nearly ballistic transport and linear response for all potential differences of the external reservoirs. In contrast, the common isotropic nearest-neighbor coupling shows negative differential conductivity and a transition from diffusive to subdiffusive transport for a far-from-equilibrium driving. Adding disorder, the change in the transport for nearest-neighbor coupling is therefore highly dependent on the driving. We find for the disordered long-range coupled $XXZ$ chain, any change in the transport behavior is independent of the potential difference and the coupling strengths of the external reservoirs.

Figure 1

Model of the Heisenberg chain coupled to two Lindblad reservoirs at the boundaries: Neighboring spins can flip, which is demonstrated as excitation hopping via ${\left({\sigma }_i^{-}{\sigma }_{i+1}^{+}\right)}^{(†)}$. The interactive Ising part ${\sigma }_i^z{\sigma }_{i+l}^z$ will be either a nearest-neighbor ($l=1$) or a long-range coupling. Two Lindblad reservoirs are applied at the edges of the chain with excitation in- and outscattering rates $\mathrm{\Gamma }$. The chain is driven via a potential difference of the reservoirs $\mu$, which induces a spin current.

Figure 2

Comparison between the nearest-neighbor scenario (a) $\alpha =1000$ and long-range coupling (b) $\alpha =0.5$ for different chain lengths. We show the absolute current $N\langle j\rangle$ versus the driving strength $\mu$. For weak driving, both scenarios are within a linear response regime where the current increases linearly with the driving strength. For nearest-neighbor coupling (a), the current reaches a maximum at a certain driving strength, which is dependent on the system size. After the maximum, the current decreases and shows NDC due to the building of a wide spin blockade. At ${\mu }^{\text{diff}}\approx 0.6$ there is a crossing of the current for different system sizes signifying diffusive transport. In contrast, for long-range coupling (b) the system is still within a linear-response regime for maximal driving.

Figure 3

Relative current $\langle j\rangle$ versus system size $N$ for weak ($\mu =0.02$) and strong ($\mu =1$) external driving. (a) Nearest-neighbor coupling: For weak driving, we obtain $\gamma =0.48$, which is close to the known value $\gamma =0.5$ [33] with superdiffusive transport. For strongest driving the transport changes to either subdiffusive transport ($\gamma =1.87$) or an exponential decay. (b) Long-range coupling: In contrast to the nearest-neighbor scenario, the transport for long-range coupling is independent of the external drive. For both cases the transport is nearly ballistic with $\gamma =0.01$.

Figure 4

Relative current versus the scattering rate $\mathrm{\Gamma }$ for weak (a, b) and strong driving (c, d). For weak driving, the maximum is at $\mathrm{\Gamma }/J=1$ for nearest-neighbor (a) and long-range coupling (b). The current $\langle j\rangle$ decreases with the system size for all $\mathrm{\Gamma }$ values for nearest-neighbor coupling, while it is the same for all investigated system sizes and $\mathrm{\Gamma }$ for long-range coupling. In the far-from-equilibrium situation, the maximum tends to smaller $\mathrm{\Gamma }$ for nearest-neighbor coupling and increasing system size (c). For long-range coupling (d) nothing changes but the value of the current.

Figure 5

Magnetization profile (occupation probability) $\langle {\sigma }_i^z\rangle$ for the different site indexes normalized by the chain length $i/(N-1)$ for weak (a) and maximal driving (b). Red denotes nearest-neighbor and blue long-range coupling. (a) All values of the single-site magnetization are close to $\langle {\sigma }_i^z\rangle \approx 0.5$, there is a linear decrease of the magnetization from the left side up to the right side of the chain. Only the first and the last site do not show this linear behavior due to the in and outscattering. There is a difference in the linear decrease between long-range and nearest-neighbor coupling defining nearly ballistic and superdiffusive transport. (b) Maximal driving: Long-range coupling shows the same magnetization profile as in the weak-driving regime resulting as well in nearly ballistic transport. The magnetization profile for $\alpha =1000$ has changed significantly. The in-and outscattering influence further sites and the building of a wide spin blockade takes place, which is responsible for the current decrease in Fig. 2 and is the reason for the change of the transport.

Figure 6

Illustration of the spin-blockade for nearest-neighbor and the missing NDC for long-range coupling in case of maximal driving. For nearest-neighbor coupling, the spin polarizations accumulate up to the central site. For long-range coupling, the spins interact with its magnetic counterpart at the other side of the chain and thus reducing the respective polarizations.

Figure 7

Benchmark with Ref. [46] of the transition to subdiffusive transport visible via an intersection of curves. The absolute current $N\langle j\rangle$ does not change for all considered system sizes at a disorder strength $h^{\text{diff}}\approx 0.6$. This marks a point with diffusive transport where the system obeys a phenomenological transport law $j=D\mathbf{\nabla }{\sigma }^z$. Before and after $h^{\text{diff}}$ the system shows anomalous transport whereby it changes at $h^{\text{diff}}$ from superdiffusive to subdiffusve transport. Note that here the disorder at the boundary spins is weaker $h_0,h_{N-1}\in \left[-h/2,h/2\right]$ to compare the value of $h^{\text{diff}}$ with Ref. [46].

Figure 8

Comparison between nearest-neighbor and long-range coupling for weak and strong driving. For weak driving, both nearest-neighbor (a) and long-range coupling (b) show diffusive transport at a certain disorder strength. The transition $h^{\text{diff}}$ is at a higher disorder strength for long-range coupling. For maximal driving $\mu =1$, the nearest-neighbor scenario shows transport beyond diffusivity up until $h=0$ (c) while for long-range coupling $h^{\text{diff}}$ does not change significantly (d). Note, that due to the numerical effort, we excluded $N=8$ in (c) because it becomes already clear that the transport is subdiffsive for all investigated disorder strengths.