Phase-Tunable Liouvillian Topology: Dual Winding Structures and Skin Effects

Abstract

Liouvillian skin effects are usually associated with dissipation whose rates favor unidirectional flow. In this work, we show that directional edge localization can emerge even when population exchange between sites is exactly balanced, with a single tunable phase parameter $α$ deciding their direction and a dual winding structure deciding their nature. Although this number-conserving bath is known for eigenstate selection, we demonstrate here that its phase controls a steady current, which Hamiltonian hopping converts to directional edge localization. Further, we study the spectral winding structure by independently twisting the left (ket) and right (bra) vector indices of the density matrix, revealing two distinct windings whose relationship is dictated by an antiunitary mirror symmetry. Nearest-neighbor jump operators break this symmetry: both windings coincide, giving rise to a common spectral winding $W_0(α)=\pm1$, producing a unipolar skin effect in which the density accumulates at a specific edge. However, next nearest neighbor jump preserve it. The two windings are forced to be opposite, and the ket and bra are pushed to opposite edges, giving a bipolar skin effect of coherences present in eigenmodes, without edge localization of the density in the steady state. Remarkably, in a dimerized chain the phase-controlled density skin effect gives the steady-state zero modes a size-independent weight at the skin edge, accompanied by a kink at the topological transition. In addition, the relaxation time grows linearly with the system size, however at specific phases $α$ where the jump operators are rendered Hermitian, edge-localized eigenmodes with skin character coexist with diffusive relaxation.

Can a number-conserving bath produce a Liouvillian skin effect when its direct population exchange is exactly symmetric? This paper studies a single particle in an SSH chain with phase-dependent bond jumps $O_j=(c_j^\dagger+e^{i\alpha}c_{j+l}^\dagger)(c_j-e^{i\alpha}c_{j+l})$, comparing nearest-neighbor ($l=1$) and next-nearest-neighbor ($l=2$) dissipation. For $l=1$, interference within each jump generates coherences that coherent hopping converts into a circulating current on a ring and directional density accumulation on an open chain. A momentum-transfer-sector determinant calculation gives winding $W_0=+1$ for $0<\alpha<\pi$ and $-1$ for $\pi<\alpha<2\pi$, excluding the Hermitian-jump points; it vanishes at $\alpha=0,\pi$. Independently twisting the ket and bra indices resolves two windings. They coincide at the steady-state reference for $l=1$, while a mirror-chiral antiunitary symmetry enforces opposite windings for $l=2$. Numerical spectra and eigenmodes associate the latter with opposite-edge localization of ket and bra: a skin effect of decaying coherences, not steady-state density. The density skin effect gives the Hamiltonian edge mode at the accumulation boundary size-independent occupation and relaxation time $\tau\propto N$ despite a finite Liouvillian gap. At Hermitian-jump phases, the steady state is maximally mixed and relaxation is diffusive, $\tau\propto N^2$, although some decaying modes remain edge-localized. The results combine symmetry relations, determinant reduction, and finite-size numerics. Stated limitations include no analytic derivation of the phase-dependent winding beyond symmetry constraints, no proof of the winding-to-localization-corner correspondence, and no treatment of finite-density many-particle dynamics.