Exact dynamics of quantum dissipative XX models: Wannier-Stark localization in the fragmented operator space

We address dissipative dynamics of the one-dimensional nearest-neighbour XX spin-$1/2$ chain governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. In the absence of dissipation, the model is integrable. We identify a broad class of dissipative terms that generically destroy integrability but leave the operator space of the model fragmented into an extensive number of dynamically disjoint subspaces of varying dimensions. In sufficiently small subspaces, the GKSL equation in the Heisenberg representation can be easily solved, sometimes in a closed analytical form. We provide an example of such an exact solution for a specific choice of dissipative terms. It is found that observables experience the Wannier-Stark localization in the corresponding operator subspace. As a result, the expectation values of the observables are linear combinations of essentially a few discrete decay modes, the long time dynamics being governed by the slowest mode. We examine the complex Liouvillian eigenvalue corresponding to this latter mode as a function of the dissipation strength. We find an exceptional point at a critical dissipation strength that separates oscillating and nonoscillating decay. We also describe a different type of dissipation that leads to a single decay mode in the whole operator subspace. Finally, we point out that our exact solutions of the GKSL equation entail exact solutions of the Schrödinger equation describing the quench dynamics in closed spin ladders dual to the dissipative spin chains.

Figure 1

Dynamics of dissipative XX models. (a) Expectation value of ${\langle {\sigma }_j^x{\sigma }_{j+1}^x\rangle }_t$ after a quench from the initial state (10) for the ${\sigma }^{x,y}$ dissipation (15) (solid) and its approximations by 1, 2, 3, and 4 discrete modes (dashed, from bottom to top). Approximation by five modes is already indistinguishable from the exact result. Left inset: Wannier-Stark localization of an eigenvector of the matrix of Eq. (18). Shown are real (blue circles) and imaginary (magenta squares) parts of the first eigenvector ${\mathcal{U}}_1^n$. Right inset: Liouvillian spectrum within the Onsager subspace. When varying the dissipation strength $\gamma$, eigenvalues closely follow a common trajectory in the complex plane shown by the dashed line (arrows indicate the spectral flow direction when increasing $\gamma$, see the animation in the Supplemental Material [44]). (b) Real and imaginary parts of the eigenvalue ${\lambda }_1$ as a function of the dissipation strength $\gamma$ in the case of ${\sigma }^{x,y}$ dissipation. This eigenvalue determines the leading dissipation mode at large times. The dashed orange line marks the critical value ${\gamma }_c\simeq \pi /2$. For $\gamma <{\gamma }_c$ the spectrum features at least one pair of complex roots leading to oscillating decay modes. For $\gamma >{\gamma }_c$ all Lindbladian eigenvalues are real, leading to pure decay without oscillations.