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Physical Review B

Nonlinear response formula for an interacting two-dimensional electron gas under a magnetic field and microwave radiation

Tadashi Toyoda, Maho Fujita, Hideki Koizumi, Chao Zhang

Published Jan 24, 2005• 11 citations
Publisher

Abstract

Introducing a canonical transformation of the second quantized field operators for two-dimensional interacting electrons under a uniform static magnetic field and a microwave radiation field, an exact formula that transforms the retarded density response function of the electrons to that of the electrons without the radiation field is derived. The formula provides a new way to treat the effects of the microwave radiation field on the electron density response when the conventional linear response approximation is not applicable.

DOI
10.1103/physrevb.71.033313

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Introducing a canonical transformation of the second quantized field operators for two-dimensional interacting electrons under a uniform static magnetic field and a microwave radiation field, an exact formula that transforms the retarded density response function of the electrons to that of the electrons without the radiation field is derived. The formula provides a new way to treat the effects of the microwave radiation field on the electron density response when the conventional linear response approximation is not applicable.
A linear response expression for the Hall conductance ${\ensuremath{\sigma}}_{\mathrm{xy}}$ of interacting two-dimensional electrons in a strong magnetic field is derived. When the system is dissipationless, ${\ensuremath{\sigma}}_{\mathrm{xy}}$ can be written solely in terms of the density-density response function ${\ensuremath{\chi}}_{\mathrm{nn}}$(q\ensuremath{\rightarrow}) of the electron liquid. Thus, a specific behavior of ${\ensuremath{\chi}}_{\mathrm{nn}}$(q\ensuremath{\rightarrow}) is required on a Hall plateau. This behavior is ${\ensuremath{\chi}}_{\mathrm{nn}}$(q\ensuremath{\rightarrow})\ensuremath{\propto}q2{q}^{2}q2 for small wave vector q, with a proportionality constant given by the Hall quantum number.
Some aspects of the correlations between interacting electrons in a magnetic field are studied by a quantum kinetic approach, taking into account the quantization of the orbital motion of the electrons. The treatment is based on the introduction of an hierarchy of equations of motion for the sss-body density matrices and employing a decoupling scheme, valid in the self-consistent field approximation. In this way the dielectric function is introduced "naturally," and the response of the system to an applied field is calculated. The dielectric function reflects the two main effects of the correlations, i.e., the collective excitations of the system and the dynamical shielding. The dielectric function plays a role also in the "dressing" of the interacting electrons. This is shown by calculating the spectrum of the density fluctuations and the equation which governs the approach to equilibrium of the system. The method is not limited to systems in thermal equilibrium, but also applies to systems in other stationary equilibrium states.
Introduction The calculation of the linear response functions of an electron liquid is a very important, but obviously extremely difficult task. Even after many years of study, and in spite of much progress, a complete solution of the problem is still lacking. In preparation to the study of this difficult problem, we consider in this chapter the main response functions of the non-interacting electron gas, namely, the density–density, spin–spin, and current–current response functions, all of which can be calculated analytically. It turns out that understanding the response of the non-interacting electron gas is an essential prerequisite for understanding the richer and more complex response of the interacting liquid. Indeed, one of the most fruitful ideas in many-body physics is that the response of an interacting system can be pictured as the response of a non-interacting system to an effective self-consistent field, which depends on global properties such as the particle density, the density matrix, the current–density etc. ‥ This is true, in particular, for every dynamical mean field theory , which can be derived from the corresponding static mean field theory (such as the HF theory) with the help of the techniques introduced in Section 4.7. In the first part of this chapter we present a detailed study of the response functions of the homogeneous non-interacting electron gas in three, two, and one dimension. Electron–impurity scattering is included only at the most elementary level, and only to demonstrate its main effect, namely, the emergence of diffusion in the dynamics of density fluctuations. In the next chapter we shall present approximations for the effective self-consistent field.
For both, fundamental physics and novel applications, nonlinear effects are crucial.Powerful laser fields require to treat mobile electrons beyond linear response theory.Phenomena familiar from nonlinear optics are manifest in ultracold Rydberg systems, which have a high potential for advancing the understanding of many-body physics via quantum simulations and promote progress in quantum computation and -information.Various devices and plasmonics systems are based on thin, nearly two dimensional (2D) layers, such as quantum wells, graphene-like structures or transition metal dichalcogenides.We here derive an expression for nonlinear response functions in arbitrary order, and evaluate it explicitly in 2D for the quadratic and the cubic density response of non-interacting fermions.As correlations are of enhanced importance in low dimensions, we next incorporate screened interactions.This results in the generation of 2nd and 3rd harmonics of the collective modes (zero sound and plasmons), damped by multiple particle-hole continua.
Abstract Nonlinear density response theory is revisited focusing on the harmonically perturbed finite temperature uniform electron gas. Within the non-interacting limit, brute force quantum kinetic theory calculations for the quadratic, cubic, quartic and quintic responses reveal a deep connection with the linear response. Careful analysis of the static long wavelength limit led us to conjecture a canonical non-interacting form that expresses arbitrary order nonlinear responses as the weighted sum of the linear responses evaluated at all multiple harmonics. This harmonic expansion is successfully validated against ab initio path integral Monte Carlo simulations.