Matrix-valued Boltzmann Equation for the Hubbard Chain

We study, both analytically and numerically, the Boltzmann transport equation for the Hubbard chain with nearest neighbor hopping and spatially homogeneous initial condition. The time-dependent Wigner function is matrix-valued because of spin. The H-theorem holds. The nearest neighbor chain is integrable which, on the kinetic level, is reflected by infinitely many additional conservation laws and linked to the fact that there are also non-thermal stationary states. We characterize all stationary solutions. Numerically, we observe an exponentially fast convergence to stationarity and investigate the convergence rate in dependence on the initial conditions.

💡 Research Summary

The paper investigates the kinetic description of the one‑dimensional Hubbard chain with nearest‑neighbor hopping by deriving a matrix‑valued Boltzmann transport equation for the spin‑½ fermions. Because each particle carries a spin degree of freedom, the Wigner function becomes a 2 × 2 complex matrix W(k,t) depending on momentum k and time t. The authors start from the Hubbard Hamiltonian, apply standard quantum‑mechanical perturbation theory (Fermi’s golden rule) to the two‑particle scattering processes, and obtain a nonlinear collision operator C