Exact Nonequilibrium Steady State of an Open Hubbard Chain

We discuss the current carrying nonequilibrium steady state of an open fermionic Hubbard chain that is strongly driven by Markovian incoherent processes localized at the chain ends. An explicit form of an exact many-body density operator for any value of the coupling parameter is presented. The structure of a matrix product form of the solution is encoded in terms of a novel diagrammatic technique that should allow for generalization to other integrable nonequilibrium models.

Figure 1

A semi-infinite graph $G$ (structure repeating periodically beyond the upper-right corner) showing the allowed transitions for building up the MPO form of NESS for the Hubbard chain. Nodes are in black, edges with multiplicity 1 are in red, and edges with multiplicity 2 are in blue. Each edge $e$ is associated with a physical product-operator $\omega (e)={\sigma }^{b^1}{\tau }^{b^2}$ where $b^{\nu }=0$ ($b^{\nu }=z$) for edges connecting white (black) nodes, where $\nu$ is that Cartesian component which does not change along such $e$ in the diagram. Degenerate edges correspond to operators ${\sigma }^0{\tau }^0$ ($\mu =+1$) and ${\sigma }^z{\tau }^z$ ($\mu =-1$). Insets indicate all possible terms (two in each, orange and brown) for two examples of [$h$, $\omega (e)\otimes \omega (f)$], namely, [$h$, ${\sigma }^{+}{\tau }^{+}\otimes {\sigma }^{+}{\tau }^0$] (a) and [$h$, ${\sigma }^0{\tau }^{-}\otimes {\sigma }^0{\tau }^0$] (b). Full arrows denote valid edge factors, whereas dashed arrows correspond to defect operators.