Origin of the quantitative discrepancy in the location of the Γ0(μ) minimum

Determine the underlying cause of the quantitative discrepancy in the location of the minimum of the dissipative gap Γ0(μ) for the Lindblad Sachdev-Ye-Kitaev model between (i) the thermodynamic-limit extrapolation of the Liouvillian spectral gap obtained via finite-size scaling of exact diagonalization results, and (ii) the large-N retarded Green’s function analysis; specifically, explain why the Γ0(μ) minimum from the spectral-gap calculation occurs at approximately half the μ value compared to the large-N Green’s function result.

Background

The paper studies the out-of-equilibrium dynamics of the Lindblad Sachdev-Ye-Kitaev (SYK) model, focusing on how exceptional points in the Liouvillian spectrum influence relaxation. A key observable is the dissipative gap Γ0, which sets the slowest decay rate toward the steady state.

The authors compute Γ0(μ) using two approaches: (1) a large-N analysis of the retarded Green’s function, and (2) exact diagonalization of the Liouvillian followed by finite-size scaling to estimate the thermodynamic-limit spectral gap. While both methods qualitatively agree—exhibiting a local maximum followed by a decrease and then a monotonic increase—the quantitative location of the minimum differs.

Specifically, the minimum in Γ0(μ) obtained from the spectral-gap method occurs at a μ value about a factor of two smaller than that inferred from the large-N Green’s function calculation. The authors explicitly state they do not understand the reason for this discrepancy and suggest possible explanations (finite-size limitations or differing contributions from low-lying real eigenvalues), leaving this as an unresolved issue.