Time-dependent generalized Gibbs ensembles in open quantum systems

Generalized Gibbs ensembles have been used as powerful tools to describe the steady state of integrable many-particle quantum systems after a sudden change of the Hamiltonian. Here, we demonstrate numerically that they can be used for a much broader class of problems. We consider integrable systems in the presence of weak perturbations which break both integrability and drive the system to a state far from equilibrium. Under these conditions, we show that the steady state and the time evolution on long timescales can be accurately described by a (truncated) generalized Gibbs ensemble with time-dependent Lagrange parameters, determined from simple rate equations. We compare the numerically exact time evolutions of density matrices for small systems with a theory based on block-diagonal density matrices (diagonal ensemble) and a time-dependent generalized Gibbs ensemble containing only a small number of approximately conserved quantities, using the one-dimensional Heisenberg model with perturbations described by Lindblad operators as an example.

Figure 1

Exact time evolution of a weakly perturbed Heisenberg model ($N=8,\gamma =0.8,J=1$) for three small values of $\epsilon$. The dashed lines show the result for the time evolution of the block-diagonal density matrix using Eqs. (11) and (12). (a) Decay of the nearest-neighbor spin correlation. (b) For the heat current, rapid oscillations on a timescale of order $1/J=1$ are absent as $J_H$ is a conservation law of $H_0$. For the values of $\epsilon$ shown in the plot, the dashed lines follow the solid lines: the block-diagonal density matrices correctly describe the time evolution with high precision. Inset: at large $\epsilon =1.0$, discrepancies are visible.

Figure 2

Comparison of the results obtained from the truncated GGE (solid line) and the block-diagonal density matrix (dashed line). (a) Antiferromagnetic nearest-neighbor spin correlations. (b) Approximate conservation laws $C_5,-J_H,-H_0$, and $(2/3)C_4$. All quantities change on a timescale of order $1/\epsilon$. Parameters: $\gamma =0.8,J=1,N=14$.

Figure 3

Expectation values of conservation laws in the stationary state ($t\to \infty$) as a function of $\gamma$, which controls the nature of the dissipative terms ($N=14,J=1$). As in Fig. 2, a comparison of the truncated GGE (solid lines) the block-diagonal density matrix (dashed) is shown. For $\gamma =0$, the system heats up to infinite temperature and all conservation laws vanish in the thermodynamic limit. Due to finite-size effects, small finite values are obtained from $H$ and $C_4$.

Figure 4

Steady-state expectation value of the energy $\langle H_0\rangle$ and the heat current $\langle J_H\rangle$ as function of the inverse system size $1/N$ ($N=6,8,10,12,14)$, for the truncated GGE with 2 and 4 conservation laws and the block-diagonal ensemble. For the smallest system size $N=6,8$, we also show the exact results for $\epsilon =0.01$, which practically coincides with the block-diagonal ensemble, and for $\epsilon =1.0$ ($\gamma =1$).

Figure 5

Steady-state expectation value of the energy $\langle H_0\rangle$ and the heat current $\langle J_H\rangle$ as function of the strength $\epsilon$ of the integrability breaking Lindblad terms ($N=8,J=1,\gamma =1$). For $\epsilon \to 0$, the result of the block-diagonal ensemble (dashed line) is recovered.

Figure 6

Left panel: derivative of energy and heat current, $\frac{d\langle H\rangle }{d\epsilon }$ and $\frac{d\langle J_H\rangle }{d\epsilon }$, as function of the strength of perturbation $\epsilon$ calculated for $\gamma =1,J=1$, and $N=8$. Right panel: calculation of the coefficient linear in $\epsilon$ from perturbation theory [51] as function of the broadening $\eta$ for the same parameters.

Figure 7

Effective force field $(F_2,F_3)$ in the vicinity of the steady state (red point) in the plane spanned by the Lagrange parameters ${\lambda }_2=\beta$ and ${\lambda }_3$ using $e^{-\beta H-{\lambda }_3{J}_H}$ as an ansatz for ${\rho }_{\text{GGE}}$ ($N=8$). Left: for $\gamma =0$ the system approaches an infinite temperature state since the Lindblad operator $L^{\left(1\right)}$ is constantly heating the system up. Right: at $\gamma =1$ the system is attracted towards a nonequilibrium state with finite stationary values of both ${\lambda }_2$ and ${\lambda }_3$.

Figure 8

Steady-state expectation values of the energy and the heat current $J_H$ as a function of $\gamma$ parametrizing the type of Markovian coupling. For a system of $N=8$ sites the steady state is shown at $J=1$ for several values of $\epsilon$ and also for the limit $\epsilon \to 0$ where the block-diagonal ensemble becomes exact.