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Theoretical and Mathematical Physics

Measuring the coupling constant of polarized fermions via sound wave spectra

Pavel A. Andreev

Published Dec 1, 2022• 2 citations
Publisher

Abstract

The system of hydrodynamic equations is generally limited to the continuity and Euler equations. However, taking higher moments of the distribution function into account improves the description of kinetic properties. We derive the pressure tensor evolution equation (PTEE) for spin-polarized degenerate fermions. We find that the pressure tensor enters the term describing the interaction that generalizes the $$p$$ -wave interaction in the Euler equation. Hence, calculating the interaction for the PTEE allows describing the interaction in the Euler equation more accurately. The proposed model is applied to small-amplitude bulk collective excitations in homogeneous fermions and trapped fermions, yielding a method for experimental measurements of the coupling constant of polarized fermions. It is demonstrated that the anisotropy in the momentum space, manifesting itself in the difference of pressures in the anisotropy direction and the perpendicular directions, leads to a method for determining the coupling constant.

DOI
10.1134/s0040577922120091

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The system of hydrodynamic equations is generally limited to the continuity and Euler equations. However, taking higher moments of the distribution function into account improves the description of kinetic properties. We derive the pressure tensor evolution equation (PTEE) for spin-polarized degenerate fermions. We find that the pressure tensor enters the term describing the interaction that generalizes the ppp -wave interaction in the Euler equation. Hence, calculating the interaction for the PTEE allows describing the interaction in the Euler equation more accurately. The proposed model is applied to small-amplitude bulk collective excitations in homogeneous fermions and trapped fermions, yielding a method for experimental measurements of the coupling constant of polarized fermions. It is demonstrated that the anisotropy in the momentum space, manifesting itself in the difference of pressures in the anisotropy direction and the perpendicular directions, leads to a method for determining the coupling constant.
Usually, hydrodynamic equations are restricted by the continuity and Euler equations. However, the account of the higher moments of the distribution function gives better description of the kinetic properties. Therefore, the pressure tensor evolution equation (PTEE) is derived for spin polarized degenerate fermions. Moreover, it is found that the pressure tensor enters the interaction term generalizing the p-wave interaction in the Euler equation. Hence, the PTEE allows to give a more accurate description of the interaction in Euler equation. Next, the interaction is calculated for the PTEE. The developed model is applied to the small amplitude bulk collective excitations in homogeneous and trapped fermions in order to suggest the methods of experimental measurement of the interaction constant of polarized fermions. It is demonstrated that the anisotropy in the momentum space revealing in the difference of the pressures in the anisotropy direction and the perpendicular directions leads to a method of detection of the interaction constant.
Abstract The possibility of the hydrodynamic description of ultracold fermions via the microscopic derivation of the model is described. Differently truncated hydrodynamic models are derived and compared. All models are based on the microscopic many-particle Schrödinger equation. Minimal coupling models based on the continuity and Euler equations are considered. An extended hydrodynamic model including the independent dynamics of the momentum flux (the pressure evolution) is derived. The influence of the spin polarization is described. The short-range interaction is considered in the isotropic limit. The interaction is considered up to the third order by interaction radius. Therefore, the single-fluid model of spin-1/2 fermions and the two-fluid model of spin-1/2 fermions are under consideration in this paper. Spectra of bulk collective excitations are derived and compared in terms of different models.
Abstract The spectrum of collective excitations of degenerate partially polarized spin-1/2 fermions is considered. The spin-up fermions and the spin-down fermions are considered as different fluids. Corresponding two-fluid hydrodynamics consistent with a non-linear Pauli equation is suggested. An equation of state for the spin current caused by the distribution of particles on different energy levels is suggested for the degenerate regime, where the spin current is caused by the Pauli blocking. Spectrum of three waves is found as a solution of the hydrodynamic equations: two sound waves and one spin wave. Their spectrums are calculated for two regimes: propagation parallel and perpendicular to the direction of the equilibrium spin polarization.
The Heisenberg dynamics of the energy, momentum, and particle densities for fermions with short-range pair interactions is shown to converge to the compressible Euler equations in the hydrodynamic limit. The pressure function is given by the standard formula from quantum statistical mechanics with the two-body potential under consideration. Our derivation is based on a quantum version of the entropy method and a suitable quantum virial theorem. No intermediate description, such as a Boltzmann equation or semi-classical approximation, is used in our proof. We require some technical conditions on the dynamics, which can be considered as interesting open problems in their own right.
The method of many-particle quantum hydrodynamics has been recently developed, particularly this method has been used for an electrically polarized Bose-Einstein condensate. In this paper, we present the development of this method for an electrically polarized three dimensional Fermi gas. We derive corresponding dynamical equations: equation polarization and equation of polarization current evolution as well as the Euler and continuity equations. We study dispersion dependencies of collective excitations in a polarized Fermi gas and consider interference of an equilibrium polarization on dispersion properties.
Phase diagrams and collective excitations are investigated for a trapped imbalanced Fermi gas in two dimensions with sss-wave pairing at finite temperatures. The treatment of collective modes is performed within the hydrodynamic approach using the Euler and continuity equations. The equations of state for different phases are simulated by a polytropic law with parameters obtained using a fit to the microscopic equilibrium distributions of the pressure and density. These are determined within the whole range of the BCS-BEC crossover using a path-integral description and taking into account fluctuations about the saddle point. We focus on the case of imbalanced gases, when the number of ``spin-up'' and ``spin-down'' fermions that form the pair is not equal. The superfluid-to-normal transition in the trapped Fermi gas is governed by the Berezinskii-Kosterlitz-Thouless mechanism. The eigenfrequencies of collective modes behave differently from those in the zero-temperature case. They can be used to study a realistic equation of state of a trapped Fermi gas at finite temperatures.