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arXiv:1306.0934 · Journal of Physics A Mathematical and Theoretical

Derivation of a matrix-valued Boltzmann equation for the Hubbard model

Martin L R Fürst, Jani Lukkarinen, Peng Mei, Herbert E. Spohn

Published Nov 19, 2013• 16 citations
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Abstract

For the spin-1/2 Fermi–Hubbard model, we derive the kinetic equation valid for weak interactions by using a time-dependent perturbation expansion up to second order. In recent theoretical and numerical studies, the kinetic equation has merely been stated without further details. In this contribution, we provide the required background material.

DOI
10.1088/1751-8113/46/48/485002

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For the spin-1/2 Fermi–Hubbard model, we derive the kinetic equation valid for weak interactions by using a time-dependent perturbation expansion up to second order. In recent theoretical and numerical studies, the kinetic equation has merely been stated without further details. In this contribution, we provide the required background material.
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.
In this thesis, the main goal is to study the kinetic dynamics based on Hubbard model in dimensions d>2. We analyse rigorously some basic properties of the dynamics in the following scaling limit. As the coupling constant converges to zero, we rescale the time variable to capture the slow dynamics: we let coupling constant go to zero, time to infinity in such a way that the product of square of the coupling constant and time has a finite limit. The main ingredient of the kinetic limit is that, combining some basic ideas of scattering theory (long time scale) and perturbation theory (small coupling parameter), it automatically selects from the microscopic dynamics some dominating terms. This gives rise to a new description of time evolution (given by quantum Boltzmann equation), whose form can already be postulated from the second order perturbative expansion. The limit equation is expected to capture nontrivial information about the original system. Besides the weak coupling dynamics, we also discuss two complementary results in related Fermion systems.
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 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.
A time-dependent projection-operator formalism is used to derive a kinetic equation for the single-spin density operator in an Ising system of NNN two-state spins in weak interaction with a heat bath. This equation contains a contribution at the self-consistent-field level and also higher-order contributions due to dynamically generated correlations. A detailed evaluation of the self-consistent-field terms is made. Possibilities for generalizing the method to obtain equations for multispin density operators are discussed.
After a sudden disruption, weakly interacting quantum systems first relax to a prethermalized state that can be described by perturbation theory and a generalized Gibbs ensemble. Using these properties of the prethermalized state we perturbatively derive a kinetic equation which becomes a quantum Boltzmann equation in the scaling limit of vanishing interaction. Applying this to interaction quenches in the fermionic Hubbard model we find that the momentum distribution relaxes to the thermal prediction of statistical mechanics. For not too large interaction, this two-stage scenario provides a quantitative understanding of the time evolution leading from the initial pure via a metastable prethermal to the final thermal state.