a Study of Time-Dependent Behavior of Many-Fermion Systems.

Yun-Tak Oh, Yoichi Higashi, Ching-Kit Chan +1 more

The Lang-Firsov Hamiltonian, a well-known solvable model of interacting fermion-boson system with sideband features in the fermion spectral weight, is generalized to have the time-dependent fermion-boson coupling constant. We show how to derive the two-time Green's function for the time-dependent problem in the adiabatic limit, defined as the slow temporal variation of the coupling over the characteristic oscillator period. The idea we use in deriving the Green's function is akin to the use of instantaneous basis states in solving the adiabatic evolution problem in quantum mechanics. With such ``adiabatic Green's function'' at hand we analyze the transient behavior of the spectral weight as the coupling is gradually tuned to zero. Time-dependent generalization of a related model, the spin-boson Hamiltonian, is analyzed in the same way. In both cases the sidebands arising from the fermion-boson coupling can be seen to gradually lose their spectral weights over time. Connections of our solution to the two-dimensional Dirac electrons coupled to quantized photons are discussed.

Birger Stölan

M. Howard Lee, J. M. Hong, N. L. Sharma

Time-dependent behavior of one-dimensional (1D) many-fermion models is obtained by a method of recurrence relations. The Hilbert space of the density-fluctuation operator is two dimensional ($2d$), resulting in a time-independent generalized random force. The relevant Hilbert spaces of 2D and 3D many-fermion models, however, are infinite dimensional and the generalized random forces are consequently time dependent. The structure of these Hilbert spaces provides a picture of time-dependent behavior for 1D models which is fundamentally very different from that for 2D or 3D models.

Robert William Jones

Simone A Hamerla, Götz Silvester Uhrig

We study the time evolution of two fermionic one-dimensional models (spinless fermions with nearest-neighbor repulsion and the Hubbard model) exposed to an interaction quench for short and moderate times. The method used to calculate the time dependence is a semi-numerical approach based on the Heisenberg equation of motion. We compare the results of this approach with the results obtained by bosonization implying power law behavior. Indeed, we find that power laws describe our results well, but our results raise the issue of which exponents occur. For spinless fermions, it seems that the Tomonaga–Luttinger parameters work well, which also describe the equilibrium low-energy physics. But for the Hubbard model this is not the case. Instead, we find that exponents from the bosonization around the initial state work well. Finally, we discuss what can be expected for the long-time behavior.

Sumiyoshi Abe