The dynamic structure function S(k,\ensuremath{\omega}) and the dielectric function \ensuremath{\epsilon}(k,\ensuremath{\omega}) of the two-dimensional electron gas is calculated phenomenologically at
rs=1 and 5 in the memory-function formalism. The density-density response function \ensuremath{\chi}(k,\ensuremath{\omega}) is expressed in terms of that of a noninteracting system and an effective potential. The interaction part of the second-order memory function that determines the effective potential is approximated by a Gaussian in time domain. We use a consistent set of highly accurate ground-state static properties as input, coming from a Green's-function Monte Carlo calculation. The calculated S(k,\ensuremath{\omega}) shows a single-broad-peak structure at intermediate k, which is from the case in three dimensions.
Using the recently developed method of recurrence relations, we have studied the time- and frequency-dependent behavior of the two-dimensional electronic system at long wavelengths. Several families of autocorrelation functions including the relaxation and memory functions are obtained exactly. In addition, other linear-response quantities, e.g., the density-density response function and dynamic structure factor, are obtained. This work contains a detailed application of the method of recurrence relations.
A nonequilibrium occupation distribution relaxes towards the Fermi-Dirac distribution due to electron-electron scattering even in finite Fermi systems. The dynamic evolution of this thermalization process assumed to result from an optical excitation is investigated numerically by solving a Boltzmann equation for the carrier populations using a one-dimensional disordered system. We focus on the short-time-scale behavior. The logarithmically long time scale associated with the glassy behavior of interacting electrons in disordered systems is not treated in our investigation. For weak disorder and short range interaction we recover the expected result that the relaxation rate is enhanced by disorder. For sufficiently strong disorder, however, we find an opposite trend due to the reduction of scattering probabilities originating from the strong localization of the single-particle states. Long-range interaction in this regime produces a similar effect. The relaxation rate is found to scale with the interaction strength, however, the interplay between the implicit and the explicit character of the interaction produces an anomalous exponent.
We show the exact dynamic structure factor S(κ, ω) of the many-electron system by calculating the relaxation function. Our S(κ, ω) is asymptotically exact as κ→∞ and at a particular metallic density r^*_s≈3.5. From the exact solution, we derive two static relations, x^2=3/5y, and x^2=3/16ω^2_ρ[1-g(0)] valid at r^*_s, where x and y are averages of the one-particle kinetic energy and its square, respectively. Our S(κ, ω) is compared with an experimental measurement for Li at κ=2.08κ_F.
The nature of the order as $T\ensuremath{\rightarrow}0$ in the one-dimensional interacting electron gas is investigated. We consider both low-momentum-transfer (
g2 and
g4) and large-momentum-transfer (
g1 and
g3) electron-electron interactions, focusing upon the particular value of
g1 for which Luther and Emery have recently found an exact solution. For this value of ${g}_{1}=\ensuremath{-}\frac{6}{5\ensuremath{\pi}{\ensuremath{\nu}}_{F}}$, we calculate the low-frequency behavior of the zero-temperature response functions explicitly. For the singlet superconducting and charge-density-wave responses, our results are consistent with those of Luther and Emery, implying that these response functions are divergent as $\ensuremath{\omega}\ensuremath{\rightarrow}0$ for certain values of
g2. For the triplet superconducting and spin-density-wave responses, however, our results differ from those of Luther and Emery, as we show explicitly that these responses have a gap for low frequencies, as was predicted by Lee. Furthermore, they do not diverge at the gap edge for any value of
g2. We also consider the effect of interactions between electrons on the same side of the Fermi "surface," and find that the low-momentum-transfer process of that type (
g4) does not change the response behavior qualitatively. For the large-momentum-transfer process between electrons on the same side of the Fermi "surface" (
g3), we may solve the problem exactly for
g2=0 and ${g}_{1}=\ensuremath{-}\frac{6}{5\ensuremath{\pi}{\ensuremath{\nu}}_{F}}$, and find that the ground state of the system exhibits only long-range charge-density-wave order. By a mapping onto the classical two-dimensional Coulomb gas problem, we may extend those results for ${g}_{3}\ensuremath{\ne}0$ to the region
g1<0 and ${g}_{1}\ensuremath{-}2{g}_{2}<|{g}_{3}|$.
Feynman-Vernon influence functional (IF) was originally introduced to describe the effect of a quantum environment on the dynamics of an open quantum system. We apply the IF approach to describe quantum many-body dynamics in isolated spin systems, viewing the system as an environment for its local subsystems. While the IF can be computed exactly only in certain many-body models, it generally satisfies a self-consistency equation, provided the system, or an ensemble of systems, are translationally invariant. We view the IF as a fictitious wavefunction in the temporal domain, and approximate it using matrix-product states (MPS). This approach is efficient provided the temporal entanglement of the IF is sufficiently low. We illustrate the versatility of the IF approach by analyzing several models that exhibit a range of dynamical behaviors, from thermalizing to many-body localized. In particular, we study the non-equilibrium dynamics in the quantum Ising model in both Floquet and Hamiltonian settings. We find that temporal entanglement entropy may be significantly lower compared to the spatial entanglement and analyze the IF in the continuous-time limit. We simulate the thermodynamic-limit evolution of local observables in various regimes, including the relaxation of impurities embedded in an infinite-temperature chain, and the long-lived oscillatory dynamics of the magnetization associated with the confinement of excitations. By incorporating disorder-averaging into the formalism, we analyze discrete time-crystalline response using the IF method. In this case, we find that the temporal entanglement entropy scales logarithmically with evolution time. The IF approach offers a new lens on many-body non-equilibrium phenomena, both in ergodic and non-ergodic regimes, connecting the theory of open quantum systems theory to quantum statistical physics.