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

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}).