Dissipative phase transition of interacting non-reciprocal fermions

TL;DR

We study a one-dimensional interacting fermionic chain subject to non-reciprocal gain and loss by engineering jump operators with relative phases $φ$ and $θ$ in a Lindblad framework. In the non-interacting limit, a line $φ = -θ$ closes the dissipative gap, giving critical, power-law relaxation; introducing interactions opens a many-body dissipative gap and drives a crossover to exponential relaxation, signaling a dissipative phase transition. Despite edge-density localization reminiscent of a skin effect, quantum trajectories reveal volume-law entanglement in the steady state, with reciprocity being dynamically restored above a critical $Δ$. The work highlights how competition between non-reciprocity and interactions yields robust non-equilibrium steady states with distinctive transport and entanglement features.

Abstract

While non-reciprocal couplings are ubiquitous in classical systems, their impact on quantum many-body criticality and entanglement remains largely unexplored. Using exact numerical simulations, we study an interacting fermionic chain subject to non-reciprocal gain and loss. We show that the interplay between dissipation and interactions drives a dissipative phase transition, marked by the opening of a many-body gap and a crossover from power-law to exponential relaxation. The weakly-interacting regime displays non-reciprocal signatures, including nonzero currents and directional charge accumulation reminiscent of the skin effect. Notably, despite this localization, quantum trajectories exhibit volume-law entanglement. Finally, reciprocity is dynamically restored above a critical interaction strength.

Figures (7)

• Figure 1: Sketch of the setup: one-dimensional fermionic chain with nearest-neighbor hoppings, $J$, and density-density interaction, $\Delta$. The gain (loss) bath injects (absorbs) particles to (from) neighboring sites, with relative phase $\theta$ ($\phi$). Together with the hopping term, they realize non-reciprocal gain and loss processes.
• Figure 2: $a)$ - Time evolution of the particle density under PBC (rescaled by the steady-state value) for different values of the dissipative gap $\delta$. $b)$ - Steady-state particle current under PBC as a function of the angles. In $c)$ and $d)$, we depict the particle density dynamics with OBC for different values of the angles, with the initial state a CDW. The inset in $c)$ corresponds to the steady-state configuration. Other parameters: $\Gamma=\kappa=0.1J$.
• Figure 3: Dissipative gap for PBC. $a)$ Gap as a function of the interaction strength ($\phi = -\theta = \pi/2, L = 16$). $b), c)$ Gap as a function of $\phi$ and $\theta$ for $L = 12$ with $\Delta = J$ [$b)$] and $\Delta = 4J$ [$c)$]. Other parameters: $\Gamma = \kappa = 0.1J$.
• Figure 4: Time evolution of $a)$ the particle density (rescaled by the steady-state value) and $b)$ the particle current for different values of the interaction strength $\Delta$ with PBC for $\phi=-\theta=\pi/2$; the system is initialized in the empty state. $I_\alpha(\cdot)$ correspond to the $\alpha$ order Bessel function of the second kind supplementary_mat. Other parameters: $L=16$, $\Gamma=\kappa=0.1J$.
• Figure 5: $a)$ Time evolution of the particle density and $b)$ its value in the steady state for different values of $\Delta$ and with OBC and $\phi =\theta= \pi/2$. The system is initially in a CDW. Other parameters: $\Gamma = \kappa = 0.1J$.