Linear response and exact hydrodynamic projections in Lindblad equations with decoupled Bogoliubov hierarchies

Abstract

We consider a class of spinless-fermion Lindblad equations that exhibit decoupled BBGKY hierarchies. In the cases where particle number is conserved, their late time behaviour is characterized by diffusive dynamics, leading to an infinite temperature steady state. Some of these models are Yang-Baxter integrable, others are not. The simple structure of the BBGKY hierarchy makes it possible to map the dynamics of Heisenberg-picture operators on few-body imaginary-time Schrödinger equations with non-Hermitian Hamiltonians. We use this formulation to obtain exact hydrodynamic projections of operators quadratic in fermions, and to determine linear response functions in Lindbladian non-equilibrium dynamics.

Figures (24)

• Figure 1: Complex eigenvalues as a function of total momentum for Model I with $\gamma=3$, $J=1$ and $L=100$. The real parts are approximately $-2\gamma$, indicating that the corresponding modes are strongly damped.
• Figure 2: Real eigenvalues as a function of total momentum for Model I with $\gamma=3$, $J=1$ and $L=100$. The eigenvalues vanish as $(p_n-\pi)^2$ for $p_n\to \pi$, reflecting the existence of a diffusive hydrodynamic mode.
• Figure 3: Real eigenvalues corresponding to bound modes for Model II with $\gamma=3$, $J=1$ and $L=100$. The eigenvalues vanish as $(p_n-\pi)^2$ for $p_n\to \pi$, reflecting the existence of a diffusive hydrodynamic mode.
• Figure 4: Real (red) and imaginary (blue) parts of the eigenvalues (\ref{['EVclass2']}) as a function of total momentum for Model III with $\nu=1$, $\gamma=3$, $J=1$ and $L=100$. The eigenvalues vanish as $(p_n-\pi)^2$ for $p_n\to \pi$, reflecting the existence of a diffusive hydrodynamic mode. The green symbols show the purely real eigenvalues (\ref{['EVclass1']}) for the same parameters.
• Figure 5: Same as Fig. \ref{['fig:boundIIIa']} with $\nu=-1$. Here all eigenvalues (\ref{['EVclass2']}) are real.