Local Quantum Friction with Pairing: Unitary Dissipation in Large Fermi Systems

Abstract

We present a unitary framework for dissipative quantum dynamics that can be efficiently applied to large-scale Fermi systems. The method introduces local Hermitian operators that emulate frictional forces while strictly preserving the unitarity of time evolution. Unlike approaches based on the Lindblad equation, our formulation scales favorably with system size and can be seamlessly integrated into time-dependent density functional theory frameworks. We demonstrate that energy dissipation arises from the damping of particle currents and pairing-field fluctuations. Furthermore, we develop a variant of the scheme that allows the particle number to vary in time, enabling controlled density scans. The method is generic and versatile, as illustrated by applications to spin-imbalanced unitary Fermi gases and to nuclear matter in the neutron-star crust. The framework can be naturally extended to include stochastic noise, providing a foundation for studying fluctuation-dissipation dynamics and thermalization in strongly interacting Fermi superfluids.

Figures (10)

• Figure 1: Particle-conserving time evolution ($\gamma = 0$) of a state under different dissipation parameter values. Rows (i) and (ii) correspond to initial states with excitation energies of $2\%$ and $25\%$, respectively. Column (a) shows the excitation energy as a function of time. Column (b) displays the difference between the ground-state density and the density of the evolved state at the end of the simulation; colors match the legend in column (a), and the solid thin black line indicates the difference between the ground-state density and the initial density at $t = 0$. Column (c) is analogous to column (b), but shows differences in the absolute value of the pairing field instead of density.
• Figure 2: Evolution of the total energy of an initially uniform 2D state in a time-dependent perturbation \ref{['eq:testBperturb']} with different dissipative parameters that conserve particle number ($\gamma = 0$). The left panel displays the energy dynamics: the solid blue line represents evolution without dissipation, while the dotted red and dashed gray lines correspond to $(\alpha = 0, \beta = 5)$ and $(\alpha = 5, \beta = 0)$, respectively. The black dotted-dashed line depicts the evolution of the amplitude of the external potential. The right panels display snapshots of the density and pairing fields at the times indicated in the energy plot, with the different cooling scenarios separated vertically.
• Figure 3: Particle-changing time evolution with $\gamma = 0.1$, $\alpha = 10$, and $\beta = 5$, for a system initially containing $N(t=0) = 712$ particles toward different target populations $N_0^{\mathrm{end}} \in \{427, 499, 570, 640\}$. We drop $N_{\textrm{req.}}$ to the targeted $N_0^{\mathrm{end}}$ at a constant rate of $N_{\textrm{req.}}'(t) = -2.86\varepsilon_{F}$. Once $N(t) \approx N_0^{\mathrm{end}}$, we resume particle-conserving cooling by setting $\gamma = 0$ and evolve until $t\varepsilon_{F} = 300$. The left panel shows the time evolution of the total energy (solid lines) and particle number (dash–dotted lines). The center and right panels display the final density and pairing-field profiles at $t\varepsilon_{F} = 300$. The ground-state profiles are shown for comparison as dashed lines, with thickness and colors corresponding to the legend below.
• Figure 4: (a) Decrease in the total energy of a spin-imbalanced unitary Fermi gas during dissipative dynamics with parameters $\alpha = 10$ and $\beta = 5$, shown for several spin polarizations $\mathcal{P}$. (b–g) Evolution of the pairing-field magnitude $|\Delta(x, y)|$ under the same conditions. Panels (b), (d), and (f) display the initial pairing-field distributions obtained from static ASLDA calculations for different spin polarizations, while panels (c), (e), and (g) show the corresponding configurations after dissipative evolution. The sequence illustrates the transition from localized ferron states at low spin imbalance, through disordered patterns at intermediate imbalance, to spatially ordered, Larkin–Ovchinnikov-like (LO-like) phases at high imbalance.
• Figure 5: Time-dependent density scan of a Zirconium nucleus ($Z=40$) in a neutron background with increasing density. (a) Total energy as a function of evolution time. (b) Evolution of the neutron background density $\rho_{Bn}$. (c) Total energy per particle as a function of $\rho_{Bn}$ (solid curve), compared with independent static calculations pecak2024WBSkMeff (circles). The dashed line in all frames corresponds to a run where the rate of neutron injection into the system was increased by a factor of five. This demonstrates that a single density scan can accurately reproduce multiple static simulations, provided the background density changes sufficiently slowly.