In open quantum lattice systems, changing the boundary condition would appear to alter both the steady state and the nonzero Liouvillian spectrum. Here we show that boundary conditions can be used to control relaxation without changing the reduced steady state. In a disordered dissipative quantum link chain, the steady state is determined by an accumulated field defined by link-resolved dissipative disorder, and a gauge-generated transformation built from this field gives exact symmetry-resolved steady states with nonuniform, accumulated-field-dependent reduced matter occupations. We then construct a reciprocal cyclic boundary condition that preserves these matter occupations while changing the nonzero Liouvillian spectrum. Consequently, open and cyclic chains relax to the same reduced matter steady-occupation profile with different Liouvillian gaps with the cyclic closure accelerating relaxation. In the strong-dissipation limit, this relaxation difference can be reduced to a spectral comparison of effective exclusion processes with open and cyclic boundaries.
We study a single-particle Lindblad lattice in which nonreciprocal incoherent hopping is combined with a tunable coherent boundary link. The link changes the Hamiltonian boundary condition while the incoherent jump rates are kept fixed, so it does not directly alter the diagonal population generator. Its effect on the Liouvillian gap comes from population-coherence-population feedback: the Hamiltonian couples populations to damped coherences, and eliminating those coherences produces a frequency-dependent self-energy for the slow population branch. For the periodic reference system, where translation symmetry is imposed on both the Hamiltonian and the dissipator, we derive an exact fixed-momentum reduction, a scalar secular equation for the population-connected branch, and the associated hydrodynamic drift and coherence-corrected diffusion. For open finite lattices, exact diagonalization and sparse near-zero eigensolvers show that the coherent boundary link can enhance or suppress the gap in one dimension and produce geometry-dependent responses in two dimensions. We also show that an observed relaxation scale can differ from
1/ΔL when the gap family has weak visibility in the chosen perturbation and observable. The periodic construction gives exact analytical benchmarks, while finite-size simulations in one and two dimensions demonstrate how the same coherence-mediated mechanism controls open-boundary gaps and observable relaxation.
We investigate a generalization of topological order from closed systems to open systems, for which the steady states take the place of ground states. We construct concrete lattice models with steady-state topological order, and characterize them by complementary approaches based on topological degeneracy of steady states, topological entropy, dissipative gauge theory, and relaxation dynamics. We take the topological degeneracy of steady states as the defining property of steady-state topological order, and study its stability using degenerate perturbation theory. We show that while the topological degeneracy is fragile in two dimensions, it is robust against perturbations in three and higher dimensions. We unveil how and why nontrivial steady-state topological order is always endowed with unusual relaxation dynamics. Particularly, the relaxation time is algebraically divergent in the thermodynamic limit, which is indicative of a gapless Liouvillian spectrum. It is shown that steady-state topological order remains definable in the presence of (Liouvillian) gapless modes: Whereas the (Liouvillian) level splitting between topologically degenerate steady states is exponentially small with respect to the system size, the Liouvillian gap between the steady states and the rest of the spectrum decays algebraically as the system size grows. We also investigate the topological phase transition to the trivial phase under additional noise, and find that the lifting of topological degeneracy is accompanied by gapping out the gapless modes. The topological phase transition is also diagnosed from the perspective of dissipative gauge theory, where the topologically ordered phase is identified as the deconfined phase and the trivial phase as the confined phase. Our paper offers a toolbox for investigating open-system topology of steady states.
We unveil a quantum Pontus-Mpemba effect enabled by the Liouvillian skin effect in a dissipative tight-binding chain with asymmetric incoherent hopping and coherent boundary coupling. The skin effect, induced by non-reciprocal dissipation, localizes relaxation modes near the system boundaries and gives rise to non-orthogonal spectral geometry. While such non-normality is often linked to slow relaxation, we show that it can instead accelerate relaxation through a two-step protocol - realizing a quantum Pontus-Mpemba effect. Specifically, we consider a one-dimensional open chain with coherent hopping
J, asymmetric incoherent hoppings
JR=JL, and a controllable end-to-end coupling
ε. For
ε=0, the system exhibits the Liouvillian skin effect, with left and right eigenmodes localized at opposite edges. We compare two relaxation protocols toward the same stationary state: (i) a direct relaxation with
ε=0, and (ii) a two-step (Pontus) protocol where a brief coherent evolution transfers the excitation across the lattice before relaxation. Although both share the same asymptotic decay rate, the two-step protocol relaxes significantly faster due to its reduced overlap with the slow boundary-localized Liouvillian mode. The effect disappears when
JR=JL, i.e., when the skin effect vanishes. Our results reveal a clear connection between boundary-induced non-normality and protocol-dependent relaxation acceleration, suggesting new routes for controlling dissipation and transient dynamics in open quantum systems.
Conformal interfaces provide an important setting for studying universal transmission phenomena in one-dimensional quantum critical systems. While energy and information transmission across such interfaces are characterized by universal quantities in closed systems, the corresponding role of conformal interfaces in dissipative dynamics is less understood. In this work, we study relaxation in locally dissipative quantum critical chains with a conformal interface and show that a universal characterization emerges at the level of individual relaxation modes. We first show that the relaxation coefficient defined in the recent work [1] from the Liouvillian gap can become non-universal for certain boundary conditions because the mode determining the smallest decay rate can change as the interface transmission is varied. To resolve this ambiguity, we introduce a mode-resolved relaxation coefficient
crelax by continuously tracking the same Liouvillian rapidity mode as a function of the interface transmission. Using analytical and numerical calculations for a critical harmonic chain and a critical free-fermion chain, we find that, in the weak-dissipation regime,
crelax follows the same universal dependence on the interface transmission in all cases considered, independent of microscopic details such as the boundary conditions, dissipation strength, and location of the local dissipation. For boundary dissipation, this universal behavior persists even at finite dissipation strength. Our results establish a universal mode-resolved characterization of relaxation across conformal interfaces in open quantum critical systems.
Abstract Relaxation processes in quantum systems coupled to external environments represent one of the most fundamental nonequilibrium phenomena in condensed matter physics. The Lindblad master equation provides a powerful framework for characterizing such open quantum dynamics. In this work, we systematically investigate how different types of quantum jump operators and system geometries influence the Liouvillian gap and the properties of the nonequilibrium steady state (NESS) in finite-size systems. We demonstrate that, due to the intricate structure of the Liouvillian superoperator, multiple NESSs with unphysical characteristics can emerge. The physically meaningful steady state must instead be understood as a superposition of these NESSs that collectively satisfy the required physical constraints. Furthermore, we find that the Liouvillian gap does not necessarily increase monotonically with the system–environment coupling strength. Instead, it can exhibit a nontrivial peak structure, corresponding to a minimum in the relaxation time. The magnitude of this peak is closely related to the symmetry properties of the system. Our results provide a deeper understanding of nonequilibrium behavior in finite quantum systems and offer new insights into the design and control of open quantum dynamics.
It is highly nontrivial to what extent we can deduce the relaxation behavior of a quantum dissipative system from the spectral gap of the Liouvillian that governs the time evolution of the density matrix. We investigate the relaxation processes of a quantum dissipative system that exhibits the Liouvillian skin effect, which means that the eigenmodes of the Liouvillian are localized exponentially close to the boundary of the system, and find that the timescale for the system to reach a steady state depends not only on the Liouvillian gap $\mathrm{\ensuremath{\Delta}}$, but also on the localization length $\ensuremath{\xi}$ of the eigenmodes. In particular, we show that the longest relaxation time $\ensuremath{\tau}$ that is maximized over initial states and local observables is given by $\ensuremath{\tau}\ensuremath{\sim}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{-}1}(1+L/\ensuremath{\xi})$ with
L being the system size. This implies that the longest relaxation time can diverge for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ without gap closing.