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Journal of Physics Condensed Matter

Emergence of Wigner oscillations in a model of real time cooling process: a time-dependent density-functional theory approach

Daniel Vieira

Published Jan 3, 2023• 2 citations
Publisher

Abstract

Abstract Friedel and Wigner oscillations are well known phenomena occurring in quantum systems. Specifically, in a system composed by N confined fermions, the former are characterized by the presence of N / 2 peaks in the density distributions, whereas the last by N peaks. Here, we consider N = 2 fermions harmonically confined in one-dimensional quantum dots. It is known that the transition from the Friedel to the Wigner oscillations is induced by the increment of interaction between the fermions. The increment of temperature, on the other hand, acts on eliminating the oscillations. In this context, by employing a time-dependent density-functional theory formalism, we here obtain the emergence of Wigner oscillations in a model which simulates a real time cooling process.

DOI
10.1088/1361-648x/acafca

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Abstract Friedel and Wigner oscillations are well known phenomena occurring in quantum systems. Specifically, in a system composed by N confined fermions, the former are characterized by the presence of N / 2 peaks in the density distributions, whereas the last by N peaks. Here, we consider N = 2 fermions harmonically confined in one-dimensional quantum dots. It is known that the transition from the Friedel to the Wigner oscillations is induced by the increment of interaction between the fermions. The increment of temperature, on the other hand, acts on eliminating the oscillations. In this context, by employing a time-dependent density-functional theory formalism, we here obtain the emergence of Wigner oscillations in a model which simulates a real time cooling process.
The temperature-induced emergence of Wigner correlations over finite-size effects in a strongly interacting one-dimensional quantum dot is studied in the framework of the spin coherent Luttinger liquid. We demonstrate that, for temperatures comparable with the zero mode spin excitations, Friedel oscillations are suppressed by the thermal fluctuations of higher spin modes. On the other hand, the Wigner oscillations, sensitive to the charge mode only, are stable and become more visible. This behavior has been proved to be robust both in the thermal electron density and in the linear conductance in the presence of a scanning tunnel microscope tip. The latter probe is not directly proportional to the electron density and may confirm the above phenomena with complementary and additional information.
The Wigner function W ( z , k ) for a model of interacting one-dimensional fermions confined to a harmonic trap is evaluated at zero temperature. The model considers two hyperfine states of the same fermionic species and treats the dominant interactions between the two using the bosonization method. Interactions substantially modify the shape of the Wigner function. Irrespective of the sign of the coupling constant, the Friedel oscillations inherent in the Wigner function are enhanced in the k -direction and suppressed in the z -direction.
Density oscillations of confined one-dimensional Fermi gases of contact repulsive interactions in a continuous space are discussed within Bethe-ansatz-based spin-density-functional theory. The results are compared against the exact analytical and the exact diagonalization method. For an unpolarized system, the number of peaks in the density profiles is doubled, signaling the crossover of the 2kF2{k}_{F}2kF​-Friedel to 4kF4{k}_{F}4kF​-Wigner oscillations (with kF{k}_{\mathrm{F}}kF​ being the Fermi wave vector). For both unpolarized and polarized systems, a threshold of the short-range interaction strength can be found where Nf{N}_{f}Nf​-peak Wigner oscillations of a 4kF4{k}_{F}4kF​ wave vector appear in the density profile (Nf{N}_{f}Nf​ is the total particle number).
Motivated by a recent experiment (Pecker et al 2013 Nat. Phys. 9 576), we study the stability, with respect to thermal effects, of Friedel and Wigner density fluctuations for two electrons trapped in a one-dimensional quantum dot. Diagonalizing the system exactly, the finite-temperature average electron density is computed. While the weak and strong interaction regimes display a Friedel oscillation or a Wigner molecule state at zero temperature, which as expected smear and melt as the temperature increases, a peculiar thermal enhancement of Wigner correlations in the intermediate interaction regime is found. We demonstrate that this effect is due to the presence of two different characteristic temperature scales: T(F), dictating the smearing of Friedel oscillations, and T(W), smoothing Wigner oscillations. In the early Wigner molecule regime, for intermediate interactions, T(F) < T(W) leading to the enhancement of the visibility of Wigner oscillations. These results complement those obtained within the Luttinger liquid picture, valid for larger numbers of particles.
Thermal effects on the total charge density are studied for a one-dimensional correlated quantum dot by means of the path integral Monte Carlo method. The competition between Friedel and Wigner oscillations at zero temperature is driven by the ratio between the interaction of electronic strength and the kinetic energy of electrons. At the onset of the formation of a Wigner molecule, we show that thermal enhancement of Wigner oscillations occurs in a range of temperatures, which can be observed in the electron density. We further show that low-temperature Friedel oscillations may change to Wigner oscillations upon an increase in the temperature.
Many-fermion systems bound by hard walls display non-trivial density profiles characterized by so-called Friedel oscillations. In non-interacting systems, the characteristic scale of the oscillations is 2kF2 k^{}_F2kF​, where kFk^{}_FkF​ is the Fermi momentum, while repulsive interactions tend to generate Wigner-crystal oscillations at 4kF4 k^{}_F4kF​. We take a closer look at interacting systems in the presence of strong pairing correlations, such as those realized experimentally with ultracold atoms. We use ab initio lattice Monte Carlo methods to study the density profiles of few- to many-fermion systems in 1D, in the presence of hard-wall boundary conditions and attractive interactions. Additionally, we compute the ground-state energy and the momentum distribution.