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arXiv:1207.6926 · Physical Review E

Matrix-valued Boltzmann equation for the Hubbard chain

Martin L. R. Fürst, Christian B. Mendl, Herbert E. Spohn

Published Sep 18, 2012• 31 citations
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Abstract

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 nonthermal 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.

DOI
10.1103/physreve.86.031122

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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 nonthermal 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.
We study the Boltzmann transport equation for the Bose-Hubbard chain in the kinetic regime. The time-dependent Wigner function is matrix-valued with odd dimension due to integer spin. For nearest neighbor hopping only, there are infinitely many additional conservation laws and nonthermal stationary states. Adding longer-range hopping amplitudes entails exclusively thermal equilibrium states. Especially for small next-nearest neighbor hopping amplitudes, we observe prethermalization with two time scales, which can be related to the relative strength of the nearest and next-nearest hopping. We provide a derivation of the Boltzmann equation based on the Hubbard Hamiltonian, including general interactions beyond on-site, and illustrate the results by numerical simulations.
The standard Fermi-Hubbard chain becomes nonintegrable by adding to the nearest neighbor hopping additional longer range hopping amplitudes. We assume that the quartic interaction is weak and investigate numerically the dynamics of the chain on the level of the Boltzmann type kinetic equation. Only the spatially homogeneous case is considered. We observe that the huge degeneracy of stationary states in the case of nearest neighbor hopping is lost and the convergence to the thermal Fermi-Dirac distribution is restored. The convergence to equilibrium is exponentially fast. However for small next-nearest neighbor hopping amplitudes one has a rapid relaxation towards the manifold of quasistationary states and slow relaxation to the final equilibrium state.
We derive and analyze an effective quantum Boltzmann equation in the kinetic regime for the interactions of four distinguishable types of fermionic spin- particles, starting from a general quantum field Hamiltonian. Each particle type is described by a time-dependent, 2 × 2 spin-density ('Wigner') matrix. We show that density and energy conservation laws as well as the H-theorem hold, and enumerate additional conservation laws depending on the interaction. The conserved quantities characterize the thermal (Fermi–Dirac) equilibrium state. We illustrate the approach to equilibrium by numerical simulations in the isotropic three-dimensional setting.
We consider a fermionic Hubbard chain with an additional next-to-nearest neighbor hopping term. We study the thermalization rates of the quasimomentum distribution function within a quantum Boltzmann equation approach. We find that the thermalization rates are proportional to the square of the next-to-nearest neighbor hopping: Even weak next-to-nearest neighbor hopping in addition to nearest neighbor hopping leads to thermalization in a two-particle scattering quantum Boltzmann equation in one dimension. We also investigate the temperature dependence of the thermalization rates, which away from half filling become exponentially small for small temperature of the final thermalized distribution.
We consider the thermalization of the momentum distribution in two one-dimensional models. The first one is the integrable Hubbard model with an additional integrability breaking term in the Hamiltonian, which is a next-to-nearest-neighbor hopping term. This has been previously investigated by Fürst et al. (PRE 86:031122, 2012; PRE 88:012108, 2013). They considered several initial quasiparticle momentum distributions and their relaxation times for different strengths of the next-to-nearest-neighbor hopping amplitude. We find the dependence of the thermalization rates on the model parameters, on the filling and on temperature for any initial quasiparticle momentum distribution in the regime of small integrability breaking using a Boltzmann equation. The second model is an effective model describing a chain of manganite octahedra. There, we investigate a the temperature dependence of the relaxation rates for reasonable choices of the model parameters and the filling.