Interaction-Enabled Hartree Fixed Points in Fermionic Resetting Dynamics

Abstract

In resetting dynamics, a system is repeatedly coupled to and decoupled from ancillary degrees of freedom that are reinitialized between interactions. This provides a versatile route to engineer nonequilibrium steady states and constitutes a powerful and analytically transparent framework for studying nonequilibrium dynamics in quadratic fermionic models. The baseline noninteracting resetting scheme yields an affine evolution for the subsystem single-particle density matrix (SPDM), with a clear operational interpretation: a finite environment block E mediates the interaction between the subsystem S and an ideal external thermal reservoir. In this work, we develop a controlled extension of such a framework to weakly interacting systems. We introduce a Hartree mean-field treatment of density-density interactions that preserves closure of the SPDM dynamics while producing genuinely nonlinear behavior. We further construct a completely positive (CP-safe) Gaussian Lindblad embedding that reproduces the resetting dynamics in the noninteracting limit and yields a continuous-time representation of environmental thermalization when interactions are present. Our analytical results are complemented by numerical studies of a ring segmentation geometry and a minimal two-site model, revealing interaction-enabled steady states that cannot be obtained in any purely quadratic setting. Together, these results establish a general and physically consistent route for incorporating weak interactions ino resetting-based approaches to open quantum system.

Figures (7)

• Figure 1: Three-layer architecture underlying both the resetting protocol and its CP-safe GKLS embedding. The subsystem $S$ and environment $E$ form a finite interacting system evolving under $H_0 + H_{\mathrm{Hartree}}[V]$. Only $E$ is directly equilibrated by an external thermal bath at $(\beta,\mu)$ with target SPDM $F_E$. The coupling to the reservoir can be realized stroboscopically as a reset $V_E \to F_E$, or continuously as Gaussian Lindblad dynamics with local jump operators $L_\alpha^\pm$ and damping matrix $\Gamma$ targeting the same $F_E$.
• Figure 2: A schematic of the quadratic fermionic ring comprising $N_E$ sites in the environment (blue) and $N_S$ in the subsystem (red), with $N = N_S + N_E$ total sites. At each reset step the subsystem SPDM block $V_S$ is left intact, the environment block $V_E$ is set to the target $F_E$, and the coherences are reset to zero. This corresponds to the RI protocol of Ref. vieira2020. See text for details.
• Figure 3: Time evolution of the average subsystem occupation $\langle n_S(t)\rangle$ for several interaction strengths $U$ in the resetting protocol. The ring has $N = N_S + N_E = 110$ sites with $N_S = 10$ subsystem sites and $N_E = 100$ environment sites. At each stroboscopic step of duration $\tau$ the environment SPDM is reset to a thermal target $F_E$ while the subsystem block is left unchanged. The mean-field Hamiltonian includes an on-site Hartree shift proportional to the local occupation. As $U$ increases, the steady-state plateau of $\langle n_S(t)\rangle$ shifts, reflecting the Hartree feedback between the subsystem and the environment. The approach to the plateau is smooth and monotone in time.
• Figure 4: Time evolution of the average subsystem occupation $\langle n_S(t)\rangle$ under the CP-safe Hartree SPDM equation \ref{['eq:SPDM-Hartree-CP-main']} for the same ring geometry and parameters as in Fig. \ref{['fig:ring-RI']}. The environment sites experience local Lindblad damping with rate $\gamma$ towards the same thermal target $F_E$ used in the resetting protocol, while the subsystem sites are undamped ($\gamma_\alpha = 0$ on $S$). For the parameter choice used here, the long-time values of $\langle n_S(t)\rangle$ are close to, but not exactly identical with, the plateaus of the resetting dynamics, since the discrete full reset and the continuous GKLS damping realise different bath couplings. Nevertheless, the dependence of the steady-state plateau on $U$ and the direction of the Hartree-induced shift agree with Fig. \ref{['fig:ring-RI']}, showing that the GKLS dynamics provides a CP-safe continuous-time analogue of the Hartree-extended resetting model. The continuous-time evolution exhibits damped oscillations superimposed on the overall relaxation, a generic feature of coherent dynamics in the presence of Markovian dissipation.
• Figure 5: Time evolution of the occupation $n_1(t)=V_{11}(t)$ for the two-site GKLS+Hartree model with interaction strengths $U=0, 0.5, 1.0, 2.0$ (from bottom to top at long times). The parameters are $\varepsilon_1=0$, $\varepsilon_2=0.5$, $J=0.3$, $\gamma_1=\gamma_2=0.5$, and $f_1=0.2$, $f_2=0.8$. For $U=0$ the system relaxes to a noninteracting steady state governed by the quadratic GKLS Liouvillian. As $U$ increases, the Hartree feedback modifies both the transient dynamics and the steady-state value $n_1^{\mathrm{ss}}(U)$, as quantified in Fig. \ref{['fig:two-site-nss']}.