Entanglement Hamiltonians in dissipative free fermions and the time-dependent GGE

Abstract

We investigate the dynamics of Entanglement Hamiltonians (EHs) in dissipative free-fermionic systems using a recent operator-based formulation of the quasiparticle picture. Focusing on gain and loss dissipation, we study the post-quench evolution and derive explicit expressions for the EH at the ballistic scale. In the long-time and weak-dissipation regime, the EH is shown to take the form of a time-dependent Generalized Gibbs Ensemble (t-GGE), with a structure that is universal across different initial states of the quench protocol. Within this framework, the emergence of the t-GGE is fully accounted for by the quasiparticle picture, and we argue that this description remains valid whenever the Lindbladian admits an appropriate coarse-grained representation.

Figures (4)

• Figure 1: Matrix elements $k_{j,j+z}$ of the entanglement Hamiltonian, with $z$ ranging between 1 and 4 and $j$ restricted to even values to remove the factor $(-1)^j$. The subsystem size is set to $\ell_A=200$, while the dissipation parameters are $\Gamma=0.01$ and $\overline{n}=0.5$ in order to focus on the constant (in $j$) terms. Symbols represent the exact value computed from the correlation matrix, while the dashed lines are the quasiparticle prediction \ref{['eq:simplified']}. The plots show excellent agreement up to small expected deviations at the boundaries.
• Figure 2: EH in a quench from the Néel state on a lattice of $\ell_A = 200$ sites, with dissipation parameters $\Gamma = 0.01$ and $\overline{n}=0.8$. In order to avoid the oscillating factors, only even values of x are shown. It is clear that for $\overline{n} \neq 0.5$ the solution is no longer constant in $x$, but the extra terms are fully accounted for by the addition of $K_{\rm q}(t)$ and $K_{\rm c}^{\rm d}$.
• Figure 3: Elements of the EH for a quench from a dimer state, with $\Gamma = 0.01$ and $\overline{n}=0.2$, for subsystem size $\ell_A=300$. The dashed analytical curve is obtained simply by considering the quantum part and the diagonal component of the classical part; the agreement shows that the non-diagonal part is strongly suppressed for this initial state, as argued in section \ref{['sec:classicalcontr']}.
• Figure 4: Renyi entropies under dissipative dynamics after a quench from the dimer state, for several values of $\Gamma$ at fixed $\overline{n}=0.3$, for a subsystem of $\ell_A=80$ sites. The symbols represent the exact results obtained through \ref{['eq:renyicorr']}, while the lines are the quasiparticle prediction \ref{['eq:entropyqp']}. The plot shows a perfect agreement even for a relatively small subsystem size. Note the saturation to a $\Gamma$-independent entropy with fixed occupation function $\overline{n}$.