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Physical review. B, Condensed matter

Translational invariance of the density and current temperature Green’s functions for an electron gas in a uniform magnetic field

Kei Ito, Mariko Fukuda, Tadashi Toyoda

Published Oct 15, 1997• 2 citations
Publisher

Abstract

In the aim of providing a theoretical basis for conventional calculation, the translational invariance of both the density temperature Green's function and the current temperature Green's function is proved for an interacting electron gas in a uniform magnetic field. The proof is based on the previously proposed nonperturbative canonical method [Phys. Rev. B 50, 18 640 (1994)].

DOI
10.1103/physrevb.56.10161

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In the aim of providing a theoretical basis for conventional calculation, the translational invariance of both the density temperature Green's function and the current temperature Green's function is proved for an interacting electron gas in a uniform magnetic field. The proof is based on the previously proposed nonperturbative canonical method [Phys. Rev. B 50, 18 640 (1994)].
Without any approximation, exact relations for the spin-density correlation functions in an interacting electron gas under an arbitrary nonuniform external magnetic field are obtained by making use of the nonperturbative canonical formulation of quantum many-body theory [T. Toyoda, Ann. Phys. (N.Y.) 173, 226 (1987)]. The obtained results contain fff-sum rules, a finite-temperature generalized Ward-Takahashi relation, as well as a finite-temperature version of the Nambu-Goldstone theorem with respect to the spontaneous symmetry breaking of the spin-rotational symmetry.
Thermodynamic properties of the interacting homogeneous electron gas are calculated using a finite-temperature cumulant Green's function approach over a broad range of densities and temperatures up to the warm dense matter regime $T\ensuremath{\sim}{T}_{F}$, where TF{T}_{F}TF​ is the Fermi degeneracy temperature. These properties can be separated into independent particle and exchange-correlation contributions, and our focus here is on the latter. Our approach is based on the Galitskii-Migdal-Koltun and electron number sum rules from the finite temperature many-body Green's function formalism, together with an extension of the cumulant Green's function to finite temperature. Previously this approach yielded exchange-correlation energies and potentials in good agreement with quantum Monte-Carlo calculations. Here the method is extended for various thermodynamic quantities including the chemical potential, total energy, Helmholtz free-energy, electronic equation of state, specific heat, and isothermal compressibility, which optionally include spin dependence. We find that the exchange-correlation contributions are weakly varying at low temperature but exhibit significant temperature dependence in the WDM regime, as well as a crossover from exchange- to correlation-dominated behavior. In contrast to the T=0+T={0}^{+}T=0+ limit, we also find that renormalization effects are largely but not completely suppressed at finite temperature. Comparisons with other approaches at various levels of approximation are also discussed.
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.
We present closed-form, analytical expressions for the thermodynamic properties of an ideal, two-dimensional (2D) charged Fermi or Bose gas in the presence of a uniform magnetic field of arbitrary strength. We consider both the homogeneous quantum gas (in which case our expressions are exact) and the inhomogeneous gas within the local-density approximation. Our results for the Fermi gas are relevant to the current-density-functional theory of low-dimensional electronic systems in magnetic fields. For a 2D charged Bose gas (CBG) in a homogeneous magnetic field, we show that the uniform system undergoes a sharp transition at a critical temperature ${T}_{c}^{\ensuremath{\star}},$ below which there is a macroscopic occupation of the lowest Landau level. An examination of the one-body density matrix, however, reveals the absence of long-range order, thereby indicating that the transition cannot be interpreted to a Bose-Einstein condensate. Nevertheless, for $T<{T}_{c}^{\ensuremath{\star}}$ and weak magnetic fields, the system still exhibits magnetic properties which are practically indistinguishable from those of a condensed, superconducting CBG. We therefore conclude that while a condensate is a sufficient condition for the ideal CBG to exhibit a superconducting state, it may not be a necessary condition.