Determine whether the vacuum–single-excitation coherence sector controls the Liouvillian gap

Establish whether the vacuum–single-excitation coherence sector remains the slowest Liouvillian sector for arbitrary system size, thereby determining when the reduced single-excitation decay rate equals the full Liouvillian gap beyond the small systems tested directly.

Background

The paper studies localized dissipative cooling in excitation-number-conserving quantum systems and derives conditions guaranteeing convergence to a unique zero-excitation state. For a nearest-neighbor Heisenberg spin chain with dissipation applied at one endpoint, the authors reduce the dynamics in the vacuum–single-excitation coherence sector to a non-Hermitian single-excitation effective Hamiltonian.

For system sizes up to the small sizes tested numerically, the reduced single-excitation gap agrees with the full Liouvillian gap to numerical precision. The authors derive an asymptotic reduced-gap scaling proportional to n^{-3}, but do not establish that this sector always provides the slowest decay mode for arbitrary chain length. Resolving this question would show when the reduced decay rate determines the actual relaxation rate of the complete dissipative dynamics.