Dissipation-induced first-order decoherence phase transition in a noninteracting fermionic system

We consider a quantum wire connected to the leads and subjected to dissipation along its length. The dissipation manifests as tunneling into (out of) the chain from (to) a memoryless environment. The evolution of the system is described by the Lindblad equation. Already infinitesimally small dissipation along the chain induces a quantum phase transition (QPT). This is a decoherence QPT: the reduced density matrix of a subsystem in the nonequilibrium steady state (far from the ends of the chain) can be represented as the tensor product of single-site density matrices. The QPT is identified from the jump of the current and the entropy per site as the dissipation becomes nonzero. We also explore the properties of the boundaries of the chain close to the transition point and observe that the boundaries behave as if they undergo a second-order phase transition as a function of the dissipation strength: the particle-particle correlation functions and the response to the electric field exhibit a power-law divergence. Disorder is known to localize one-dimensional systems, but the coupling to the memoryless environment pushes the system back into the delocalized state even in the presence of disorder. Interestingly, we observe a similar transition in the classical dissipative counterflow model: the current has a jump at the ends of the chain introducing an infinitely small dissipation.

Figure 1

The setup of the problem: one-dimensional chain is connected to the source and the drain as in transport experiments. Every site of the chain is coupled to the environment, which models the dissipation from the leakage of the current due to imperfect insulation. The environment consists of two reservoirs at plus/minus infinite chemical potentials coupled at each site of the chain.

Figure 2

(a) The jump of the current, $j$, and the Fano factor, $F$, at infinitesimally small dissipation constant along the chain $d\Gamma =d{\Gamma }^{\left(i\right)}=d{\Gamma }^{\left(o\right)}$ $\left({\Gamma }^{\left(i\right)}={\Gamma }^{\left(o\right)}=1\right)$. (b) Dependence of the current $j$ and the Fano factor $F$ through the ends of the chain on the length $L$ for random dissipation along the chain taken from the range $d{\Gamma }^{\left(i\right)},d{\Gamma }^{\left(o\right)}\in (0,0.04)$ (points with error bars) and for the constant dissipation with the strength $d{\Gamma }^{\left(i\right)}=d{\Gamma }^{\left(o\right)}=0.02$ (points and the dashed lines). Here and everywhere else in the text and the plots the ${\Gamma }_{\mu }$ values are measured in the units of the hopping $t$.

Figure 3

Exponential decay of the current along the chain. (a) Logarithmic scale, different lengths of the system. The currents in the system without randomness in dissipation are represented by the regular sets of points (forming solid lines). Darker, irregularly scattered points represent the current for one realization of the disorder in dissipation along the chain. (b) The current through a dissipative chain after averaging over different disorder realizations. The scale is linear (not logarithmic) to show the standard deviation of the (fluctuating, random) current. Notice that the negative values of the current are physical, because some realization of the (random) couplings $d\Gamma$ can give an overall current flowing in the opposite direction. The couplings at the ends of the chain are ${\Gamma }^{\left(i\right)}={\Gamma }^{\left(0\right)}=1,d\Gamma =0.05$. For the average over disorder $d{\Gamma }_j^{\left(i\right)},d{\Gamma }_j^{\left(o\right)}\in (0,0.1),j\in (2,L-1)$.

Figure 4

Logarithmic plot of the current flowing from the system into the reservoir at the beginning of the chain (denoted by 1) as a function of the hopping rates at the ends of the chain, in the presence of the constant dissipation along the chain, $d{\Gamma }^{\left(i\right)}=d{\Gamma }^{\left(o\right)}=0.02$. Increasing the dissipation makes the current through one end independent of the coupling at the other end of the chain. In this plot we denote ${\Gamma }^{\left(i\right)}={\Gamma }_1,{\Gamma }^{\left(o\right)}={\Gamma }_2$.

Figure 5

(a) The jump of the current at the end of the chain upon switching on the decoherence along the chain in the classical counterflow model. (b) Dependence of the current through the first site of the chain on the length of the chain. Compare to Fig. 3, where analogous behavior is observed for the quantum chain modeled by the Lindblad equation. ${\Gamma }_1^{\left(\mathrm{in}\right)}={\Gamma }_2^{\left(\mathrm{out}\right)}=0.5$

Figure 6

The dependence of the current on the site index for different decoherence rates $d\Gamma$ in logarithmic scale in the classical counterflow model: the exponential decay of the current from the ends toward the middle of the chain is clearly visible, suggesting a similar mechanism of decoherence as in the quantum chain in Fig. 4. ${\Gamma }_1^{\left(\mathrm{in}\right)}={\Gamma }_2^{\left(\mathrm{out}\right)}=0.5,t_{\max }={10}^5.$

Figure 7

Entropy jump at the transition point as a function of the dissipation strength. The dashed line is in agreement with Eq. (20). Inset: dependence of the entropy on the chain length for different dissipation strengths $d\Gamma =0.01,0.02,0.03,0.04,0.05,0.06$ (from top to bottom solid curve respectively), the dash-dotted line corresponds to the entropy in the absence of the coupling to the environment, the point at $d\Gamma =0$ at the main plot.

Figure 8

Main plot: the convergence of the nonlinear response to the electric field for long systems in the bulk of the chain. Solid lines correspond to different coupling strength $d\Gamma =0.005,0.01,0.02,0.03,0.05$ (from top to bottom) and the dashed line is $d\Gamma =0$. Inset: quadratic scaling of $j\left(E\right)-j\left(0\right)$ with the applied electric field (the scale in logarithmic).

Figure 9

Main plot: the convergence of the nonlinear response to the electric field for long systems at the ends of the chain. Solid lines correspond to different coupling strength $d\Gamma =0.005,0.01,0.02,0.03,0.05$ (from top to bottom) and the dashed line is $d\Gamma =0$. Notice that the nonlinear conductivity at the ends points stays nonzero also in the thermodynamic limit. As in Fig. 8, the conductivity is infinite in the absence of dissipation. Inset: scaling of $\sigma$ with disorder strength with power-law fit: $\sigma =\alpha d{\Gamma }^{\beta },\beta =3.161\pm 0.001$.

Figure 10

(a) The lowest eigenvalue ${\lambda }_{\min }$ of the effective Hamiltonian (10) as a function of the system size for the system without (black dashed line) and with dissipation (blue points: averages over the disorder from the range $(0,d\Gamma )$, purple line: constant dissipation with $d\Gamma /2,d\Gamma =0.025)$. (b) The scaling of the lowest eigenvalue with disorder strength, ${\lambda }_{\min }\left(d\Gamma \right)$, and the power-law fit ${\lambda }_{\min }\propto d{\Gamma }^{\beta }$ with $\beta =0.53\pm 0.01$. The couplings to the source and the drain are ${\Gamma }^{\left(i\right)}={\Gamma }^{\left(o\right)}=1$.

Figure 11

Dependence of the correlation length on the coupling to the environment for ${\Gamma }^{\left(i\right)}={\Gamma }^{\left(o\right)}=1$ (dots) and the power-law fit (dashed line). Inset: correlations at one end of the chain as a function of the position for different couplings strengths $d\Gamma =0.005,0.01,0.02$ (from top to bottom: blue, green, red) and exponential fits, which determine the correlation length (dashed lines).

Figure 12

Dependence of the current through a disordered dissipative system on the length of the system for different values of the dissipation along the system, $d\Gamma =0,0.02,0.03,0.05$ (from top to bottom at small $L$: black, blue, red, green; solid lines: $dU=0.3$, dashed lines: $dU=0$; ${\Gamma }_1={\Gamma }_2=1)$. The current through the system is independent of the system length for a sufficiently long system.