Global well-posedness of the spatially homogeneous Hubbard-Boltzmann equation

TL;DR

This paper proves the global existence and uniqueness of solutions for the spatially homogeneous Hubbard-Boltzmann equation, a matrix-valued kinetic equation modeling interacting fermions, under certain conditions on the lattice dispersion relation.

Contribution

It establishes the first rigorous proof of global solutions for the Hubbard-Boltzmann equation with a singular rotation term in the homogeneous setting.

Findings

Global solutions exist for initial data satisfying the Fermi constraint.

Unique physical solutions maintain the Fermi constraint at all times.

Assumptions on the lattice dispersion relation are satisfied by the nearest neighbor Hubbard model in dimensions d >= 3.

Abstract

The Hubbard model is a simplified description for the evolution of interacting spin-1/2 fermions on a d-dimensional lattice. In a kinetic scaling limit, the Hubbard model can be associated with a matrix-valued Boltzmann equation, the Hubbard-Boltzmann equation. Its collision operator is a sum of two qualitatively different terms: The first term is similar to the collision operator of the fermionic Boltzmann-Nordheim equation. The second term leads to a momentum-dependent rotation of the spin basis. The rotation is determined by a principal value integral which depends quadratically on the state of the system and might become singular for non-smooth states. In this paper, we prove that the spatially homogeneous equation nevertheless has global solutions in L^\infty(T^d,C^{2x2}) for any initial data W_0 which satisfies the "Fermi constraint" in the sense that 0 <= W_0 <= 1 almost everywhere. We also prove that there is a unique "physical" solution for which the Fermi constraint holds at all times. For the proof, we need to make a number of assumptions about the lattice dispersion relation which, however, are satisfied by the nearest neighbor Hubbard model, provided that d >= 3. These assumptions suffice to guarantee that, although possibly singular, the local rotation term is generated by a function in L^2(T^d,C^{2x2}).