Self-consistent projection operator theory for quantum many-body systems

We derive an exact equation of motion for the reduced density matrices of individual subsystems of quantum many-body systems of any lattice dimension and arbitrary system size. Our projection operator based theory yields a highly efficient analytical and numerical approach. Besides its practical use it provides an interpretation and systematic extension of mean-field approaches and an adaption of open quantum systems theory to settings where a dynamically evolving environment has to be taken into account. We show its high accuracy for two significant classes of complex quantum many-body dynamics, unitary evolutions of nonequilibrium states in closed and stationary states in driven-dissipative systems.

Figure 1

Illustration of our approach for a one-dimensional lattice. We consider a quantum many-body system where each subsystem (QS) has some unitary and potentially some nonunitary dynamics (indicated by the rate $\gamma$). The subsystems are coupled via the interaction ${\mathcal{L}}_I$. Within our theory we pick one QS of interest and trace out the remaining constituents.

Figure 2

Application to unitary dynamics of closed quantum many-body systems. (a) $\langle {\sigma }_A^{†}{\sigma }_A\rangle \left(t\right)$; exact value (dashed black) [30], single-site c-MoP (dash-dotted brown), 2-site cluster c-MoP (dotted purple), 4-site cluster c-MoP (dash-dotted blue), and 8-site cluster c-MoP (solid red). (b) $\langle {\sigma }_A^{†}{\sigma }_B\rangle \left(t\right)$; exact value (dashed black) [30], 2-site cluster c-MoP (dotted purple), 4-site cluster c-MoP (dash-dotted blue), and 8-site cluster c-MoP (solid red). The real part of $\langle {\sigma }_A^{†}{\sigma }_B\rangle \left(t\right)$ vanishes for all $t$.

Figure 3

Application to stationary states of driven-dissipative quantum many-body systems. (a) $\mathrm{Tr}\left\{{\sigma }^{†}\sigma {\rho }_{ss}\right\}$, (b) real part, and (c) imaginary part of $\mathrm{Tr}\left\{\sigma {\rho }_{ss}\right\}$ as a function of $ZJ/\gamma$ for $Z=2$, $\Delta =0.6\gamma$, and $\Omega =1.5\gamma$. t-DMRG (dashed green), single-site mean-field (dotted black), two-site cluster mean-field (dash-dotted black), single-site c-MoP (dashed red), and two-site cluster c-MoP (solid red). (d) Trace distances $D({\rho }_{ss}^{\mathrm{DMRG}},{\rho }_{ss}^{\mathrm{MF}})$ (dotted black), $D({\rho }_{ss}^{\mathrm{DMRG}},{\rho }_{ss}^{\mathrm{MF}-\mathrm{cl}})$ (dash-dotted black), $D({\rho }_{ss}^{\mathrm{DMRG}},{\rho }_{ss}^{\mathrm{cM}})$ (dashed red), $D({\rho }_{ss}^{\mathrm{DMRG}},{\rho }_{ss}^{\mathrm{cM}-\mathrm{cl}})$ (solid red), and $D({\rho }_{ss}^{\mathrm{DMRG}},{\rho }_{ss}^{\mathrm{PT}})$ (dashed blue) for the same parameters as (a).

Figure 4

Performance of the method in terms of trace distances from exact solutions for small systems. (a) and (c): $D({\rho }_{ss}^{1\mathrm{D}},{\rho }_{ss}^{\mathrm{MF}})$ and $D({\rho }_{ss}^{1\mathrm{D}},{\rho }_{ss}^{\mathrm{cM}})$, respectively, for $Z=2$, $\Delta =0.5\gamma$, and $N=3$ with periodic boundary conditions as functions of $ZJ/\gamma$ and $\Omega /\gamma$. (b) and (d): $D({\rho }_{ss}^{2\mathrm{D}},{\rho }_{ss}^{\mathrm{MF}})$ and $D({\rho }_{ss}^{2\mathrm{D}},{\rho }_{ss}^{\mathrm{cM}})$, respectively, for $Z=4$, $\Delta =0.5\gamma$, and $N=5$ with periodic boundary conditions as functions of $ZJ/\gamma$ and $\Omega /\gamma$. In the bistable regions of the mean-field approximation we have chosen the branch which is closer to the exact solution.