Equilibration of an isolated Fermi gas in one spatial dimension after an interaction quench is studied. Evaluating Kadanoff-Beym dynamic equations for correlation functions obtained from the two-particle-irreducible effective action in nonperturbative approximation, the gas is seen to evolve to states characterized by thermal as well as nonthermal momentum distributions, depending on the assumed initial conditions. For total energies near the Fermi temperature, stationary power laws emerge for the high-momentum tails while at lower momenta the distributions are of Fermi-Dirac type. The relation found between fluctuations and dissipation exhibits nonthermal final states.
Equilibration of an isolated Fermi gas in one spatial dimension after an interaction quench is studied. Evaluating Kadanoff-Beym dynamic equations for correlation functions obtained from the two-particle-irreducible effective action in nonperturbative approximation, the gas is seen to evolve to states characterized by thermal as well as nonthermal momentum distributions, depending on the assumed initial conditions. For total energies near the Fermi temperature, stationary power laws emerge for the high-momentum tails while at lower momenta the distributions are of Fermi-Dirac type. The relation found between fluctuations and dissipation exhibits nonthermal final states.
In this thesis, far-from-equilibrium dynamics of fermionic quantum gases is discussed utilising functional quantum field theoretical methods. Employing the Schwinger-Keldysh path integral, real-time Schwinger-Dyson/Kadanoff-Baym dynamic equations for the two-point correlation functions are derived from the two-particle irreducible (2PI) effective action. For two specific models, these dynamic equations are investigated further. (a) For an N-fold spin-degenerate ultra-cold Fermi gas, non-perturbative approximation schemes based on either a loop or a 1/N expansion of the 2PI effective action are presented. Adopting these approximations, the long-time evolution of a homogeneous Fermi gas with N=2 after an initial preparation far from thermal equilibrium is thoroughly studied in one spatial dimension. Depending on the total energy, the gas is found to evolve into thermal as well as non-thermal states, the latter becoming manifest in violating the fluctuation-dissipation relation. (b) A similar 1/N expansion is derived for the SU(N) symmetric Kondo lattice model. At leading order, the mean-field dynamic equations of the U=0 Anderson model are recovered. At next-to-leading order (NLO), both spin-flip and direct interactions between localised atoms and conduction band atoms are taken into account non-perturbatively into the dynamic equations. This allows future studies of possibly existing novel phases in coupling regimes where the Kondo screening and RKKY-type interactions are competing.
We study a weak interaction quench in a three-dimensional Fermi gas. We first show that, under some general assumptions on time-dependent perturbation theory, the perturbative expansion of the long-wavelength structure factor S(q) is not compatible with the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q) does develop an analytical component $\ensuremath{\sim}\mathrm{const}+O({q}^{2})$ at $\mathbf{q}\ensuremath{\rightarrow}\mathbf{0}$, as implied by thermalization, but, in contrast, it maintains a nonanalytic part $\ensuremath{\sim}|\mathbf{q}|$ characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity corresponds to persisting power-law decaying density-density correlations, whereas thermalization would predict only an exponential decay. We next consider the case of a dilute gas, where one can obtain nonperturbative results in the interaction strength but at lowest order in the density. We find that in the steady state the momentum distribution jump at the Fermi surface remains finite, though smaller than in equilibrium, up to second order in kFf0, where f0 is the scattering length of two particles in the vacuum. Both results question the emergence of a finite length scale in the quench dynamics as expected by thermalization.
We consider a non-interacting Fermi gas in d dimensions, both in the non-relativistic and relativistic case. The system of size Ld is initially prepared into two halves L and R, each of them thermalized at two different temperatures, TL and TR respectively. At time t=0 the two halves are put in contact and the entire system is left to evolve unitarily. We show that, in the thermodynamic limit, the time evolution of the particle and energy densities is perfectly described by a semiclassical approach which permits to analytically evaluate the correspondent stationary currents. In particular, in the case of non-relativistic fermions, we find a low-temperature behavior for the particle and energy currents which is independent from the dimensionality d of the system, being proportional to the difference TL2−TR2. Only in one spatial dimension (d=1), the results for the non-relativistic case agree with the massless relativistic ones.
Abstract. We consider a non-interacting Fermi gas in d dimensions, both in the non-relativistic and relativistic case. The system of size Ld is initially prepared into two halves L and R, each of them thermalized at two different temperatures, TL and TR respectively. At time t = 0 the two halves are put in contact and the entire system is left to evolve unitarily. We show that, in the thermodynamic limit, the time evolution of the particle and energy densities is perfectly described by a semiclassical approach which permits to analytically evaluate the correspondent stationary currents. In particular, in the case of non-relativistic fermions, we find a low-temperature behavior for the particle and energy currents which is independent from the dimensionality d of the system, being proportional to the difference T 2L − T 2R. Only in one spatial dimension (d = 1), the results for the non-relativistic case agree with the massless relativistic ones. 1.
We outline a kinetic theory of nonthermal fixed points for the example of a dilute Bose gas, partially reviewing results obtained earlier, thereby extending, complementing, generalizing, and straightening them out. We study universal dynamics after a cooling quench, focusing on situations where the time evolution represents a pure rescaling of spatial correlations, with time defining the scale parameter. The nonequilibrium initial condition set by the quench induces a redistribution of particles in momentum space. Depending on conservation laws, this can take the form of a wave-turbulent flux or of a more general self-similar evolution, signaling the critically slowed approach to a nonthermal fixed point. We identify such fixed points using a nonperturbative kinetic theory of collective scattering between highly occupied long-wavelength modes. In contrast, a wave-turbulent flux, possible in the perturbative Boltzmann regime, builds up in a critically accelerated self-similar manner. A key result is the simple analytical universal scaling form of the nonperturbative many-body scattering matrix, for which we lay out the concrete conditions under which it applies. We derive the scaling exponents for the time evolution as well as for the power-law tail of the momentum distribution function, for a general dynamical critical exponent z and an anomalous scaling dimension $\ensuremath{\eta}$. The approach of the nonthermal fixed point is, in particular, found to involve a rescaling of momenta in time t by ${t}^{\phantom{\rule{0.16em}{0ex}}\ensuremath{\beta}}$, with $\ensuremath{\beta}=1/z$, within our kinetic approach independent of $\ensuremath{\eta}$. We confirm our analytical predictions by numerically evaluating the kinetic scattering integral as well as the nonperturbative many-body coupling function. As a side result, we obtain a possible finite-size interpretation of wave-turbulent scaling recently measured by Navon et al.
The dynamics of the Luttinger model after a quantum quench is studied. We compute in detail one- and two-point correlation functions for two types of quenches: from a noninteracting to an interacting Luttinger model and vice versa. In the former case, the noninteracting Fermi gas features in the momentum distribution and other correlation functions are destroyed as time evolves. In the infinite-time limit, equal-time correlations are power laws but the critical exponents are found to differ from their equilibrium values. In all cases, we find that these correlations are well described by a generalized Gibbs ensemble [M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007)], which assigns a momentum-dependent temperature to each eigenmode.