Dephasing enhanced transport in nonequilibrium strongly correlated quantum systems

A key insight from recent studies is that noise, such as dephasing, can improve the efficiency of quantum transport by suppressing coherent single-particle interference effects. However, it is not yet clear whether dephasing can enhance transport in an interacting many-body system. Here, we address this question by analyzing the transport properties of a boundary driven spinless fermion chain with nearest-neighbor interactions subject to bulk dephasing. The many-body nonequilibrium stationary state is determined using large-scale matrix product simulations of the corresponding quantum master equation. We find dephasing enhanced transport only in the strongly interacting regime, where it is shown to induce incoherent transitions bridging the gap between bound dark states and bands of mobile eigenstates. The generic nature of the transport enhancement is illustrated by a simple toy model, which contains the basic elements required for its emergence. Surprisingly, the effect is significant even in the linear response regime of the full system, and it is predicted to exist for any large and finite chain. The response of the system to dephasing also establishes a signature of an underlying nonequilibrium phase transition between regimes of transport degradation and enhancement. The existence of this transition is shown not to depend on the integrability of the model considered. As a result, dephasing enhanced transport is expected to persist in more realistic nonequilibrium strongly correlated systems.

Figure 1

Spinless fermions hop with amplitude $\tau$ across a chain, subject to a nearest-neighbor density-density interaction of strength $\Delta$, local dephasing at a rate $\gamma$, and boundary driving that injects/ejects fermions at a rate proportional to $\Gamma$ and driving bias $f$. The driving process induces a forward (upper arrows) and a backward (lower arrows) flow of particles, and the bias $f$ determines the imbalance between both.

Figure 2

(a) The current-driving profiles in the weakly interacting regime $\Delta =0.5$ for increasing (as indicated by the arrow) dephasing rates $\gamma$ and $N=16$. (b) Identical plot to (a) but in the strongly interacting regime $\Delta =2$. (Inset) The density $\langle n_j\rangle$ of the system when $\Delta =2$, $f=1$, and $N=16$, corresponding to $\gamma =0,0.05,1.00$.

Figure 3

The current $\langle J\rangle$ at $f=1$, divided by the resulting density difference $\Delta n$ between the ends of the system (excluding the boundary), is plotted against the system size $N-3$ up to $N=100$ sites. The data is for dephasing rates $\gamma =0.5$ ($\times$) and $\gamma =1.0$ ($\blacksquare$). The solid lines show the fitting to $\langle J\rangle /\Delta n=\kappa {(N-3)}^{-\alpha }$ for each $\gamma$. These yield $\kappa =1.288$ and $\alpha =0.863$ for $\gamma =0.5$, and $\kappa =1.228$ and $\alpha =0.958$ for $\gamma =1.0$.

Figure 4

The current as a function of $\gamma$ for driving biases $f=0.1,0.2,0.3$, and $0.4$ (from bottom to top curve) with $\Delta =2$, and $N=40$. The dashed lines correspond to ${\langle J\rangle }_{\Delta =0}$, the noninteracting analytic result.[53, 56]

Figure 5

Optimal dephasing ${\gamma }_{\text{opt}}$ for different sizes of the system, at $f=0.1$, $\Delta =2$, and $\Gamma =1$ (circles). The solid line corresponds to the exponential decay ${\gamma }_{\text{opt}}={\gamma }_{\text{opt}}^{\text{TL}}+a\exp (-N/b)$, with $a=0.252$, $b=85.9$, and ${\gamma }_{\text{opt}}^{\text{TL}}=0.109$. (Inset) Optimal current ${\langle J\rangle }_{\text{opt}}$ as a function of $N$. The solid line corresponds to the power law decay ${\langle J\rangle }_{\text{opt}}=a{N}^{-b}$, with $a=0.0948$ and $b=0.94$. Note that as $N\to \infty$, ${\langle J\rangle }_{\text{opt}}\to 0$.

Figure 6

(a) The energy eigenspectrum of the spinless fermion Hamiltonian governing the chain for $N=12$ and $\Delta =10$. Energies ${\epsilon }_{\nu }$ have been shifted by $\frac{1}{4}\Delta (N-1)$ so that the state $|{\Psi }_D\left(0\right)\rangle =|00\cdots 00\rangle$ has zero energy. The spectrum includes contributions from all particle number sectors $n=0,\cdots ,12$; since the spectra arising from $n=0,\cdots ,5$ are identical to those of $n=7,\cdots ,12$, only the former are shown in addition to $n=6$. The highest lying eigenstate for each sector $|{\Psi }_D\left(n\right)\rangle$ is highlighted and seen to be isolated by a gap of $O\left(\Delta \right)$ from eigenstates composed of break-away configurations (states with the outermost particle of the domain breaking away, or a hole propagating through the domain). The braces give an indication of the location of mobile and flattened bands for $n=6$. (b) For any $f$, the driving alone incoherently connects a quadruplet of configurations $|0\mathbf{x}0\rangle ,|0\mathbf{x}1\rangle ,|1\mathbf{x}0\rangle ,|1\mathbf{x}1\rangle$. The situation for $f=1$ is shown where the driving can be seen to only pump into the so-called dark configuration $|1\mathbf{x}0\rangle$.

Figure 7

The schematic of the toy model and its various processes and properties. These include the nearest-neighbor coherent hopping between the set of states $|1\rangle ,|2\rangle ,\cdots ,|K\rangle$, an energy offset $\Delta$ for state $|K\rangle$, incoherent transitions $|1\rangle \leftrightarrow |s\rangle$ and $|K\rangle \leftrightarrow |s\rangle$ between the boundary states and the auxiliary state $|s\rangle$, and dephasing at a rate $\gamma$. The rates for the incoherent driving transitions $\frac{1}{2}\Gamma (1\pm f)$ are also listed.

Figure 8

(a) The NESS current $\langle J\rangle$, rescaled by ${\Delta }^2$, for $K=20$, as a function of the driving $f$ and dephasing $\gamma$, described by the approximation in Eq. (14). The behavior at $f=1$ with $\gamma$ is emphasized by the additional (red) line. (b) The exact current-driving profiles of the toy-model as a function of $\gamma$ with $\Delta =2$ and $K=20$.

Figure 9

(a) The spectrum ${\epsilon }_{\nu }$ of the Hamiltonian for the toy model for $K=20$ and $\Delta =2$. Eigenstate $\nu =0$ is the highlighted auxiliary state $|s\rangle$ whose energy has been set arbitrarily. The highest-lying eigenstate $\nu =K$ is also highlighted and is the approximate dark state of the model $|{\Psi }_D\rangle$. Beneath this state, separated by a gap $O\left(\right|\Delta \left|\right)$, is a band of eigenstates $|{\Psi }_{\nu }\rangle$ split by the small hopping. (b) The probability distributions $|\langle k|{\Psi }_{\nu }\rangle {|}^2$ of the eigenstates over the configurations $|k\rangle$ are shown. The band of eigenstates is seen to be delocalized over the configurations $|1\rangle ,|2\rangle ,\cdots ,|K-1\rangle$ and expunged from the boundary configuration $|K\rangle$, while the characteristic of a bound state $|{\Psi }_D\rangle$ is predominately peaked at $|K\rangle$ with exponential tail into the bulk.

Figure 10

(a) The optimal dephasing rate ${\gamma }_{\text{opt}}$ as a function of $\Delta$, for $N=40$, $f=0.1$, and $\Gamma =1$. The circles indicate TEBD results, and the solid line corresponds to the fitted function ${\gamma }_{\text{opt}}\propto {(\Delta -{\Delta }_0)}^{\beta }$, with $\beta =0.819$ and ${\Delta }_0=1.07$. The inset plots show the generic behavior of $\langle J\rangle$ with $\gamma$ above and below ${\Delta }_0$. (b) The scaling of ${\Delta }_0$ with $N$ for $f=0.1$ and $\Gamma =1$, along with the fitted power-law shown, where $a=3.906$, $b=1.066$, and $c=0.995$.

Figure 11

The NESS density-density correlations $\left|C\right(r\left)\right|$ across the chain as a function of the fraction distance $r$ about the center for $\Delta =2$ and $f=0.1$. In (a), a moderate dephasing $\gamma =0.05$ is included. The correlations are long-ranged and extend over a larger fraction when increasing $N$. This is evidenced by the kink, coming from $C\left(r\right)$ changing sign, moving to larger $r$. In (b), a strong dephasing $\gamma =1$ is present and the correlations are short ranged, as shown by their diminishing size and extent with increasing $N$.

Figure 12

The half-chain entropy $S$ as a function of $\Delta$ for different $\gamma$'s, $f=0.1$, and $N=40$. The Schmidt coefficients ${\lambda }_{\alpha }$ were normalized so the largest coefficient is ${\lambda }_1=1$.

Figure 13

The current through the system for several staggered potential strengths $B$, for $N=40$, $f=0.1$, and $\Gamma =1$, with (a) $\Delta =0.5$ and (b) $\Delta =2.0$. Inset: energy eigenspectrum for one (blue) and two (red) excitations of a chain of $N=20$, with staggered potential $B=2$ and interaction $\Delta =10$.

Figure 14

(a) A schematic of the system, showing the coherent hopping $\tau$, the nearest-neighbor density-density interaction $\Delta$, dephasing $\gamma$, and the incoherent boundary driving in the strong driving limit $f=1$ where the bias is maximal. In this case, the driving incoherently pumps particles into the system at site 1 and ejects them at site $N$, both at a rate $\Gamma$. (b) For $N=4$ sites and $f=1$, the complete set of driving quadruplets and the transitions induced by coherent hopping between them from the half-filled domain state $|1100\rangle$ is shown. The vertical axis displays the particle number sectors and energy gaps between configurations. We observe that for $N=4$ one hop connects $|1100\rangle$ to $|1010\rangle$, which is a dark configuration of its quadruplet, and a further hop is needed before it is coherently connected to a configuration that is not dark to the $f=1$ driving. This disconnection between the half-filled domain configuration and the driving only occurs for $N\ge 4$, and so this is the smallest size that displays NDC.

Figure 15

For the eigenstate $|{\Psi }_D\left(6\right)\rangle$ for $N=12$ and $\Delta =10$, the exact density deviation ${\delta }_n\left(j\right)$ from the corresponding boundary domain configuration $|B_n\rangle$ is shown $\left(\blacksquare \right)$, along with ${\delta }_n^{\mathrm{ph}}\left(j\right)$ predicted by considering single particle/hole propagation (solid line). The inset shows a schematic of the expected exponential localization of the domain wall.

Figure 16

(a) The probability $p_n$ on a logarithmic scale of the NESS $\rho$ occupying the $n$ particle sector is shown for an exact calculation ($\diamond$) for $N=6$ and $\Delta =10$, and the predicted form (solid line) given in Eq. (A4). (b) The purity $1-\mathrm{tr}\left({\rho }^2\right)$ against $\Delta$ of the NESS $\rho$ on a logarithmic scale for an exact calculation ($\times$) for $N=6$ and the predicted form (solid line) given in Eq. (A5). The inset shows a schematic of the detailed balance approximation applied to compute these predictions.

Figure 17

(a) The NESS current $\langle J\rangle$, rescaled by ${\Delta }^2$, for $K=20$ is shown as a function of the driving $f$ and dephasing $\gamma$ zoomed in around the limit $f=1$, $\gamma =0$ insulating point [see Fig. 8a]. The behavior at $f=1$ with $\gamma$ is emphasized by the additional (red) line. (b) In the strong driving limit $f\sim 1$, the ${\Lambda }_R^{-}$ process $|K\rangle \to |s\rangle$ occurs at a rate $\propto (1-f)\ll 1$ making it slow (as indicated by being faded out), while the ${\Lambda }_R^{+}$ process $|s\rangle \to |K\rangle$ occurs at a rate $\propto f\approx 1$ and so is rapid. The dominant driving process is therefore $|K\rangle \to |s\rangle \to |K\rangle$, which has the effect of destroying any coherence $|K\rangle$ has with other states, such as $|K-1\rangle$.