Perturbative approach to weakly driven many-particle systems in the presence of approximate conservation laws

We develop a Liouville perturbation theory for weakly driven and weakly open quantum systems in situations when the unperturbed system has a number of conservations laws. If the perturbation violates the conservation laws, it drives the system to a new steady state which can be approximately but efficiently described by a (generalized) Gibbs ensemble characterized by one Lagrange parameter for each conservation law. The value of those has to be determined from rate equations for conserved quantities. Remarkably, even weak perturbations can lead to large responses of conserved quantities. We present a perturbative expansion of the steady state density matrix; first we give the condition that fixes the zeroth-order expression (Lagrange parameters) and then determine the higher-order corrections via projections of the Liouvillian. The formalism can be applied to a wide range of problems including two-temperature models for electron-phonon systems, Bose condensates of excitons or photons, or weakly perturbed integrable models. We test our formalism by studying interacting fermions coupled to nonthermal reservoirs, approximately described by a Boltzmann equation.

Figure 1

Diagrammatic depiction of the structure of corrections to the zeroth-order density matrix ${\rho }_0$. One draws all possible combinations of open and filled circles starting to the right with a filled circle connected to a small square representing ${\rho }_0$. Then one eliminates all diagrams with a direct connection of two open circles. The order of the diagram is given by the number of filled circles, the sign by the total number of lines. The number of terms to order ${\epsilon }^n$ is given by $2^n$. Note that the corrections to order $n$ in the perpendicular and parallel sector are simply related by the relation $\delta {\rho }_{n,\parallel }=-{\left(\widehat{P}{\widehat{\mathcal{L}}}_1\widehat{P}\right)}^{-1}\left(\widehat{P}{\widehat{\mathcal{L}}}_1\widehat{Q}\right)\delta {\rho }_{n,\perp }$.

Figure 2

Diagrammatic depiction of the corrections $\delta \rho$ to the zeroth-order density matrix ${\rho }_0$ for cases where $\widehat{P}{\widehat{\mathcal{L}}}_1\widehat{P}=0$ (unitary driving). One first draws all possible combinations of open and filled circles starting to the right with a filled circle connected to ${\rho }_0$ (small square). Then one eliminates all diagrams with neighboring open circles, all with the combination open-filled-open, and finally also the combination open-filled-${\rho }_0$ (small square). The order is given by the number of filled minus the number of open circles, the sign by the total number of circles. The number of terms to order ${\epsilon }^n$ is $2^n$. Some diagrams do, however, vanish for monochromatic perturbations; see Appendix pp5-s3.

Figure 3

Expectation values of (a) particle and (b) energy density as a function of perturbation strength $\epsilon$, calculated from the exact steady state density matrix (solid line) or using our perturbation theory up to $k\mathrm{th}$ order in $\epsilon$ on system size $L=4$ and interaction strength $U=0.3$.

Figure 4

Time evolution of the occupation function $f_{e_n}$ shown for $n=1,6,11,\cdots ,41$ ($L=41$) and $\epsilon =0.01$ starting from an initial state with $f_{e_n}=1$ ($f_{e_n}=0$) for states with even (odd) $n$, respectively. (a) On short time scales the system relaxes towards a state with $\beta \approx 0$ and equal occupation of all levels. (b) The time evolution toward the steady state occurs on a time scale set by $1/\epsilon$ and therefore the time axis has been rescaled by a factor $\epsilon$. The points are obtained by solving the time evolution of the Lagrange parameters using Eq. (57) which then determines a Fermi distribution function. The comparison with the exact solution of the Boltzmann equation (lines) shows that this allows for a quantitative description of the slow dynamics for small $\epsilon$.

Figure 5

Force fields calculated using Eq. (56) shown for $L=41$. The force fields determine the time evolution of the Lagrange parameters according to Eq. (57). The red solid line shows the trajectory for the initial conditions used in Fig. 4, where a comparison to the exact solution of the Boltzmann equation is shown. The unique stationary state is indicated by a black circle.

Figure 6

Level occupation $f_{e_n}$ as a function of perturbation strength $\epsilon$. Solid lines are obtained from the exact calculation using the Boltzmann equation, Eq. (48), while dashed lines are obtained from our perturbative approach, including zeroth and first order in $\epsilon$. Only every third $n$ is shown for the system with $L=41$ single-particle states.

Figure 7

(a) Particle and (b) energy density as a function of perturbation strength $\epsilon$. The dashed line shows the result of perturbation theory, including zeroth and first order in $\epsilon$.