Shu Long, Chao Yang, Sen Mu, Linhu Li
8 min
Abstract
We establish a symmetry-protected correspondence between band topology of coherent Hamiltonians and Liouvillian spectral winding in Lindblad descriptions of open quantum systems. This allows the Hamiltonian topology to act as a knob for controlling Liouvillian topology and corresponding non-equilibrium dynamics, rather than being passively manipulated by system-environment exchanges. In particular, by exactly solving the Liouvillian spectrum in a class of one-dimensional dissipative lattices, we find that the Hamiltonian band topology constrains the Liouvillian spectral winding and determines the Liouvillian skin effect, provided the Hamiltonian and quantum jump operators respect the same chiral symmetry. We further demonstrate that lattice parity controls the associated bulk-boundary correspondence and the coherence properties of the steady state. Our results unveil a symmetry-enforced topological control of spectral and spatial organization in open quantum systems, providing a unified perspective on topology in Hamiltonian and dissipative dynamics.
Alex: Okay, so both windings flip together at that balance point. And under open boundaries, you see the skin effect switch directions?
Sam: Yes—numerics show both windings jump from negative one to positive one as J zero passes J one. The steady state's average position shifts sharply from left edge to right. Dynamics confirm: start in the middle, density drifts unidirectionally to the predicted edge.
Alex: But the source mentions a catch near the transition—stronger gamma one flipping localization against the winding?
Sam: That's the parity effect. Open chains have an edge defect, unbalancing channels—like missing one rightward leak. Strong gamma one then pumps rightward coherence over the winding's left bias, reversing localization locally. It highlights how boundaries tweak the imprint, yet the core mapping holds in balanced cases.
Alex: So closed-system twists actively dial the open steady states, but edges add this competitive twist. A meaningful control step, with caveats.
Sam: Precisely—this symmetry lets topology program dissipation, pointing to uses like edge lasers where steady light localizes precisely.
Alex: That edge-laser idea sounds practical. How do they confirm this mapping in their calculations—like, what exactly shows the steady state piling up as predicted?
Sam: They look at the full picture across boundary types. Under periodic boundaries—where the chain loops end-to-end like a circle—they plot the Liouvillian spectrum, a map of all decay rates as colored points. Gray dots for open boundaries nest inside colored loops from periodic ones, showing how the winding path encloses the steady-state point at zero. Density matrices confirm: off-diagonal elements cluster near one edge, like a heatmap of probabilities bunched left or right. Start density in the chain's middle; diagonal elements shift unidirectionally to the predicted edge, average position tracing a straight drift.
Alex: Right, but you mentioned that catch with stronger gamma one flipping things near transition. How does that play out? Do they fix it?
Sam: Near balance, open edges create a defect unbalancing pumps. Strong gamma one builds rightward coherence over the left bias, localizing opposite the winding. Yes—they remove one edge site for odd-length chains, restoring parity; position and coherence then match winding exactly.
Alex: And without the symmetry match?
Sam: Chiral-asymmetric jumps break it; steady states follow dissipation strengths alone, windings decouple, localization ignores Hamiltonian twists. Symmetry's essential for the imprint.
Alex: Okay, so symmetry imprints the Hamiltonian's twist onto the Liouvillian's. But how do they nail down that spectral winding number exactly?
Sam: To simplify the chain math, they switch to momentum space using a Fourier transform—a tool that rewrites site-by-site patterns as combinations of waves sweeping across the whole chain, like breaking a song into its bass, drum, and melody tracks. This reveals the Hamiltonian's band topology as momentum varies. In the chiral-symmetric case with jumps only from one sublattice to the other, the steady-state eigenvalue sits at zero but nudges imaginarily as momentum shifts slightly. Its path direction is set by the signed sum of distances times dissipation strengths. They define the Liouvillian spectral winding as the count of loops that path makes around zero—a topological marker linking to skin effect when nonzero. It matches the mapping from Hamiltonian winding.
Alex: So the winding flips with the dissipation imbalance, dialed by the Hamiltonian? Like a phase offset?
Sam: Precisely—the Hamiltonian topology tunes dissipative currents' net direction. This programs steady-state localization actively, as numerics confirm. And it extends to multiple particles, like two hard-core bosons that can't double-occupy sites. Their calculations show the reduced density matrix localizes oppositely in different topological phases, with average position shifting sharply, confirming the skin effect direction flips with the Hamiltonian's control.
Alex: Huh. A clear way to predict open-system piling from closed patterns. Meaningful for those edge lasers.
Sam: Precisely—a meaningful limit, as real setups often have imbalances. Still, the paper suggests potential for topologically programmable quantum simulators, engineering steady-state edge accumulation via coherent patterns. Think edge-state lasers, where this tunes light precisely to boundaries through dissipation.
Alex: A clear advance in predicting open-system behavior from closed ones, with practical paths ahead despite the symmetry demands. Thanks, Sam—that's our look at symmetry-protected control of Liouvillian topological phases via Hamiltonian band topology. Thanks for listening to ResearchPod.