Infinitely Dimensional Lax Structure for the One-Dimensional Hubbard Model

We report a two-parametric irreducible infinitely dimensional representation of the Lax integrability condition for the Fermi Hubbard chain. In addition to being of fundamental interest, hinting at possible novel quantum symmetry of the model, our construction allows for an explicit representation of an exact steady state many-body density operator for a nonequilibrium boundary-driven Hubbard chain with arbitrary (asymmetric) particle source (sink) rates at the left (right) end of the chain and with arbitrary boundary values of chemical potentials.

Figure 1

Diagrammatic representation of factors of the Lax operator where auxiliary states are labeled by vertices $\mathcal{V}$. Diagrams for ${\mathbf{T}}^s$ are obtained by reflection of those of ${\mathbf{S}}^s$ across the diagonal. Red and blue arrows indicate off diagonal transitions with amplitude $\pm \sqrt{2}$. Red, blue, green, and black open points represent diagonal multiplications by $\omega$, $-\omega$, $\propto \lambda$, 1, 0, respectively, and brown circles represent multiplications by $2\times 2$ matrices $X_k$.