The spin susceptibility of the electron gas is studied using a functional integral formulation. A Hubbard-Stratonovich transformation is used to decouple the electron-electron interaction. A uniform Fock mean field is chosen, and effects of small fluctuations about this mean field are retained up to second order. The term that gives the usual spin random-phase approximation susceptibility is identified, with the ladder sum contained in the propagator for the field fluctuations. Computations of the static spin susceptibility outline the region of stability as a function of electron-gas coupling parameter and wave vector. These computations parallel those of Hamann and Overhauser [Phys. Rev. 143, 183 (1966)] for the Yukawa interaction. We use the analytic continuation method of Staunton et al. [Phys. Rev. Lett. 82, 3340 (1999)] to extend the calculations to nonzero frequency. The resulting spectral density shows the increase in low-energy spin fluctuations as the spin-density-wave transition is approached, and the development of a pseudogap in the spin fluctuation spectrum near the wave vector of the transition.
We have made a study of the effect of exchange and correlation on spin density waves in the uniform electron gas. In order to take these effects into account, we have used two new magnetic susceptibility formulas which we have recently derived. The first of these is the time-dependent Hartree-Fock (TDHF) formula which contains exchange but no correlation, while the second one is the generalized random phase approximation (GRPA) formula which contains both exchange and correlation. Both of these formulas were evaluated for the uniform electron gas. The TDHF susceptibility was found to have an absolute maximum at a nonzero value of q, while the GRPA susceptibility was found to have an absolute maximum at q = 0. This indicates that when exchange effects are included in the susceptibility, the ground state of the uniform electron gas contains a spin density wave, but when correlation effects are also included, then the spin density wave is no longer present in the ground state of the uniform electron gas.
A simple calculation of the wave-vector- and frequency-dependent paramagnetic spin susceptibility of an interacting electron gas is presented. The equation of motion of the double-time Green's function is solved using a moment-conserving decoupling approximation. For the short-range interaction, our expression for the dynamic spin susceptibility reduces to a previous result of Wolff. In the static and long-wavelength limit, our result for the Stonerenhanced spin susceptibility coincides with that obtained from consideration of the groundstate energy in the Hartree-Fock approximation.
The problem of the many-body enhancement of the static spin susceptibility at long wavelengths and its relation to the quasiparticle effective mass is investigated for a normal electron gas in two-dimensional space as a function of the electronic density. We start from a discussion of the results of the simple Hartree-Fock approximation for various interaction potentials and proceed to develop a complete theory. We find that the effects of the electron-electron interaction are significantly larger than in the familiar three-dimensional case. Our approach is based on a new self-consistent scheme which goes beyond the simple random-phase approximation by explicitly allowing for charge- and spin-fluctuation-induced vertex corrections of the Hubbard type. We show that when the latter are neglected, the many-body enhancement of the spin susceptibility can be cast in a remarkably simple and elegant analytic form.
The problem of the ground state of the electronic system in the Hubbard model for U=∞ is discussed. The author investigates the normal (singlet or nonmagnetic) N state of the electronic system over the entire range of electron densities n⩽1. It is shown that the energy of the N state ɛ 0 (1) (n) in a one-particle approximation, such as (e.g.) the extended Hartree-Fock approximation, is lower than the energy of the saturated ferromagnetic FM state ɛ FM(n) for all n. The dynamic magnetic susceptibility is calculated in the random phase approximation, and it is shown that the N state is stable over the entire range of electron densities: The static susceptibility (ω=0) does not have a band singularity in the zero-wave vector limit q→0. A formally exact representation is obtained for the mass operator of the one-particle Green’s function, and an approximation of this operator is proposed: M k(E)⋍λF(E), where λ=n(1−n)/(1−n/2)z is the kinematic interaction parameter, z is the number of nearest neighbors, and F(E) is the total single-site Green’s function. For an elliptical density of states the integral equation for F(E) is solved exactly, ad it is shown that the spectral intensity rigorously satisfies the sum rule. The calculated energy of the strongly correlated N state ɛ 0(n)<ɛ FM(n) for all n, and in light of this relationship the author discusses the hypothesis that the ground state of the system is the normal (singlet) state in the thermodynamic limit. The electron distribution function at T=0 differs significantly from the Fermi step; it is “smeared” along the entire energy spectrum, and discontinuities do not occur in the region of the chemical potential m.
The two-dimensional (2D) Hubbard model is solved in a new way. The zero-frequency components of the Hubbard interaction are separated out and solved to infinite order to avoid the traditional difficulties associated with perturbation theory in the strong-coupling limit. Specifically the zero-frequency spin interaction is decoupled by the introduction of Stratonovich-Hubbard fields, while the zero-frequency charge interaction is treated by "slave" Hartree-Fock. This approach leads to the formation of large fluctuating spin moments, and as a consequence the one-electron spectral density exhibits a pseudogap. A Metropolis Monte Carlo method is used to perform the configuration averaging over Stratonovich-Hubbard fields. Within single Monte Carlo configurations the non-zero frequency components of the Hubbard interaction are treated by a high-order perturbation method, which incorporates the exchange of dynamical particle-hole fluctuations. Since a single configuration has neither translational symmetry nor spin rotational symmetry, these fluctuations do not have a well-defined crystal momentum or total spin. When the dynamical particle-hole fluctuations are included in the calculation, localized electron pairing results at elevated temperatures. Superconductivity presumably onsets when these localized pairs develop coherence, but this is not studied in the present work. The nature of the pairing interaction and the resulting pair eigenfunctions is studied.
We theoretically investigate the uniform spin susceptibility $\ensuremath{\chi}$ in the superfluid phase of an ultracold Fermi gas in the region of the Bardeen-Cooper-Schrieffer--Bose-Einstein-condensate (BCS-BEC) crossover. In our previous paper [H. Tajima et al., Phys. Rev. A 89, 033617 (2014)], including pairing fluctuations within an extended
T-matrix approximation (ETMA), we showed that strong pairing fluctuations cause the so-called spin-gap phenomenon, where $\ensuremath{\chi}$ is anomalously suppressed even in the normal state near the superfluid phase transition temperature
Tc. In this paper, we extend this work to the superfluid phase below
Tc, to clarify how this many-body phenomenon is affected by the superfluid order. From the comparison of the ETMA $\ensuremath{\chi}$ with the Yosida function describing the spin susceptibility in a weak-coupling BCS superfluid, we identify the region where pairing fluctuations crucially affect this magnetic quantity below
Tc in the phase diagram with respect to the strength of a pairing interaction and the temperature. This spin-gap regime is found to be consistent with the previous pseudogap regime determined from the pseudogapped density of states. We also compare our results with a recent experiment on a
6Li Fermi gas. Since the spin susceptibility is sensitive to the formation of spin-singlet preformed pairs, our results would be useful for the study of pseudogap physics in an ultracold Fermi gas on the viewpoint of the spin degrees of freedom.