Hierarchy of Relaxation Timescales in Local Random Liouvillians
To characterize the generic behavior of open quantum many-body systems,
we consider random, purely dissipative Liouvillians with a notion of
locality. We find that the positivity of the map implies a sharp
separation of the relaxation timescales according to the locality of
observables. Specifically, we analyze a spin-1/2 system of size ℓ with
up to n-body Lindblad operators, which are n local in the
complexity-theory sense. Without locality (n=l), the complex Liouvillian
spectrum densely covers a “lemon”-shaped support, in agreement with
recent findings [S. Denisov et al., Phys. Rev. Lett. 123, 140403 (2019),
L. Sa et al., JPA 53, 305303]. However, for local Liouvillians (n find that the spectrum is composed of several dense clusters with random
matrix spacing statistics, each featuring a lemon-shaped support wherein
all eigenvectors correspond to n-body decay modes. This implies a
hierarchy of relaxation timescales of n-body observables, which we
verify to be robust in the thermodynamic limit. Our findings for n
locality generalize immediately to the case of spatial locality,
introducing further splitting of timescales due to the additional
structure. To test our theoretical prediction, we perform experiments
on the IBM quantum computing platform, designing different "waiting
circuits" to inject two body dissipative interactions by two qubit
entangling gates. We find excellent agreement with our theory and
observe the predicted hierarchy of timescales.