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arXiv:1404.1319 · Journal of Statistical Mechanics Theory and Experiment

Non-equilibrium transport in d-dimensional non-interacting Fermi gases

Mario Collura, Gabriele Martelloni

Published Aug 5, 2014• 32 citations
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Abstract

We consider a non-interacting Fermi gas in $d$ dimensions, both in the non-relativistic and relativistic case. The system of size $L^{d}$ is initially prepared into two halves $\mathcal{L}$ and $\mathcal{R}$, each of them thermalized at two different temperatures, $T_{\mathcal{L}}$ and $T_{\mathcal{R}}$ 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 $T_{\mathcal{L}}^{2}-T_{\mathcal{R}}^{2}$. Only in one spatial dimension ($d=1$), the results for the non-relativistic case agree with the massless relativistic ones.

DOI
10.1088/1742-5468/2014/08/p08006

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We consider a non-interacting Fermi gas in ddd dimensions, both in the non-relativistic and relativistic case. The system of size LdL^{d}Ld is initially prepared into two halves L\mathcal{L}L and R\mathcal{R}R, each of them thermalized at two different temperatures, TLT_{\mathcal{L}}TL​ and TRT_{\mathcal{R}}TR​ respectively. At time t=0t=0t=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 ddd of the system, being proportional to the difference TL2−TR2T_{\mathcal{L}}^{2}-T_{\mathcal{R}}^{2}TL2​−TR2​. Only in one spatial dimension (d=1d=1d=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 study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states. In the non-relativistic case we prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, for short macroscopic times. In the case of relativistic dispersion, we show convergence of the many-body evolution to the relativistic Hartree equation for all macroscopic times. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases.
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.
This thesis considers out-of-equilibrium dynamics of strongly interacting non-relativistic Fermi gases in several two and three dimensional geometries. The tools of second-order hydrodynamics and gauge-gravity duality will be utilized to address this system. Many of the themes of this work are motivated by the observed similarities in transport properties between strongly interacting Fermi gases and other very different strongly interacting quantum fluids such as the quark-gluon plasma, high temperature superconductors, and quantum field theories described by gauge-gravity duality. In particular, these systems all nearly saturate the conjectured lower bound on the ratio of shear viscosity to entropy density η/s≥ℏ/(4πkB)η/s \geq \hbar/(4 πk_B)η/s≥ℏ/(4πkB​) coming from the AdS/CFT correspondence. Among other things, this observation, in conjunction with current experiment and data analysis in atomic, condensed matter, and nuclear physics lends itself to the following questions: How perfect of a fluid is the strongly interacting Fermi gas, and can one find a more stringent constraint on η/sη/sη/s in Fermi gases? Do the similarities in transport properties among strongly interacting quantum systems extend beyond dynamics controlled by the hydrodynamical shear viscosity? In regards to the first question, by utilizing second-order hydrodynamics, it will be demonstrated that higher-order collective modes of a harmonically trapped Fermi gas may serve as a more sensitive probe of the shear viscosity. For the second question, both second-order hydrodynamics and a gravity dual theory are used to make predictions about dynamics occurring on short timescales where hydrodynamics is expected to break down. In particular the appearance of a class of "non-hydrodynamic" collective modes not contained within a Navier-Stokes description of the strongly interacting Fermi gas will be discussed.
We study the time evolution of a one-dimensional interacting fermion system described by the Luttinger model starting from a nonequilibrium state defined by a smooth temperature profile T(x)T(x)T(x). As a specific example we consider the case when T(x)T(x)T(x) is equal to TL{T}_{L}TL​ (TR)({T}_{R})(TR​) far to the left (right). Using a series expansion in $\ensuremath{\epsilon}=2({T}_{R}\ensuremath{-}{T}_{L})/({T}_{L}+{T}_{R})$, we compute the energy density, the heat current density, and the fermion two-point correlation function for all times $t\ensuremath{\ge}0$. For local (delta-function) interactions, the first two are computed to all orders, giving simple exact expressions involving the Schwarzian derivative of the integral of T(x)T(x)T(x). For nonlocal interactions, breaking scale invariance, we compute the nonequilibrium steady state (NESS) to all orders and the evolution to first order in $\ensuremath{\epsilon}$. The heat current in the NESS is universal even when conformal invariance is broken by the interactions, and its dependence on TL,R{T}_{L,R}TL,R​ agrees with numerical results for the XXZXXZXXZ spin chain. Moreover, our analytical formulas predict peaks at short times in the transition region between different temperatures and show dispersion effects that, even if nonuniversal, are qualitatively similar to ones observed in numerical simulations for related models, such as spin chains and interacting lattice fermions.