Emergence of Boltzmann Subspaces in Open Quantum Systems Far from Equilibrium

Single molecule junctions are important examples of complex out-of-equilibrium many-body quantum systems. We identify a nontrivial clustering of steady state populations into distinctive subspaces with Boltzmann-like statistics, which persist far from equilibrium. Such Boltzmann subspaces significantly reduce the information needed to describe the steady state, enabling modeling of high-dimensional systems that are otherwise beyond the reach of current computations. The emergence of Boltzmann subspaces is demonstrated analytically and numerically for fermionic transport systems of increasing complexity.

Figure 1

Three scenarios in a molecular junction: equilibrium (a),(d), near equilibrium (b),(e), and far from equilibrium (c),(f). (a)–(c) Site (green), orbital (purple), and electrode chemical potential (dotted) energies. (d)–(f) Steady-state populations (pluses). Each state (point) is associated with a “Boltzmann line,” $\ln (P^{(st)})=A_{\mathbf{n}}-\beta E$, with $\beta =38$ $1/\mathrm{eV}$. States of different total electron number are colored differently. The model parameters in (a),(d) are $E_0=0.1$, $t=-0.05$, ${\mu }_L={\mu }_R=0.3$, in (b),(e) the changes are ${\mu }_L=0.4$, ${\mu }_R=0.2$, and in (c),(f) the additional change is $E_0=0.16$ (all in eV).

Figure 2

Splitting measure of the Boltzmann subspaces studied in Figs. 1 and 1, where $E_0$ is replaced by a gate voltage, $V_g$. The colors correspond to different electron numbers.

Figure 3

Steady-state currents as functions of ${\mu }_R$ (for ${\mu }_L=0$), in units of the maximal elastic current ($I_0$). Circles: Boltzmann subspaces-based calculations. Colored pluses: full calculations at different couplings to bosonic reservoirs with ${\log }_{10}({\mathrm{\Gamma }}_B/\mathrm{\Gamma })\in \{3,2,1,0,-1,-2,-3\}$ (see legend) and $\hslash {\omega }_B=0.15$ eV. Inset: illustration of the model with the on-site energy values marked in eV.

Figure 4

Steady-state probability distributions ($\ln (P_n^{(st)})$ vs $E_n$) for the model presented in Fig. 3, at high bias (${\mu }_R=0.4$, ${\mu }_L=0\mathrm{eV}$) and strong coupling to bosonic baths [${\log }_{10}({\mathrm{\Gamma }}_B/\mathrm{\Gamma })=3$]. (a), (b), and (c) correspond to increasing Coulomb interaction, $U=0$, 0.1, 0.2 eV, respectively.