Three-dimensional Fermi Gas Research Articles

Impurity in a zero-temperature three-dimensional Fermi gas

Impurity in a zero-temperature three-dimensional Fermi gas

Spectral functions of the strongly interacting three-dimensional Fermi gas

Computing dynamical properties of strongly interacting quantum many-body systems poses a major challenge to theoretical approaches. Usually, one has to resort to numerical analytic continuation of results on imaginary frequencies, which is a mathematically ill-defined procedure. Here we present an efficient method to compute the spectral functions of the two-component Fermi gas near the strongly interacting unitary limit directly in real frequencies. To this end, we combine the Keldysh path integral that is defined in real time with the self-consistent T-matrix approximation. The latter is known to predict thermodynamic and transport properties in good agreement with experimental observations in ultracold atoms. We validate our method by comparison with thermodynamic quantities obtained from imaginary-time calculations and by transforming our real-time propagators to imaginary time. By comparison with state-of-the-art numerical analytic continuation of the imaginary-time results, we show that our real-time results give qualitative improvements for dynamical quantities. Moreover, we show that no significant pseudogap regime exists in the self-consistent T-matrix approximation above the critical temperature Tc, an issue that has been under significant debate. We close by pointing out the versatile nature of our method as it can be extended to other systems, like the spin- or mass-imbalanced Fermi gas, other Bose-Fermi models, two-dimensional systems, and systems out of equilibrium. Published by the American Physical Society 2024

Entanglement Entropy of Ground States of the Three-Dimensional Ideal Fermi Gas in a Magnetic Field

We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^3$$\\end{document} subject to a constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order L2ln(L)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2\\ln (L)$$\\end{document}, a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region LΛ⊂R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L\\Lambda \\subset {{\\mathbb {R}}}^3$$\\end{document} as the scaling parameter L tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra ln(L)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ln (L)$$\\end{document} factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom–Sobolev formula in the non-magnetic case. It differs, however, in the sense that the dependence on the boundary, ∂Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Lambda $$\\end{document}, is not solely on its area but on the “surface perpendicular to the direction of the magnetic field”. We utilize a two-term asymptotic expansion by Widom (up to an error term of order one) of certain traces of one-dimensional Wiener–Hopf operators with a discontinuous symbol. This leads to an improved error term of the order L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} of the relevant trace for piecewise C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{C}^{1,\\alpha }$$\\end{document} smooth surfaces ∂Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Lambda $$\\end{document}.

Controllable production of degenerate Fermi gases of Li6 atoms in the crossover from two dimensions to three dimensions

Many-body physics in dimensional crossover systems has attracted much attention, but is yet to explore intensively. Cold atoms provide an excellent experimental platform to address the questions in this area, but technical difficulties still exist. When an optical lattice is applied to tune the dimensionality of trapped cold atoms, it is usually difficult to precisely tune the occupation ratio of the atom in the different lattice bands. In this paper we report the method to tune the occupation ratio of the energy band when transferring a three-dimensional Fermi gas into a one-dimensional optical lattice, where we could tune the occupation ratio in the lowest band from unity to $50%$ quantitatively by jointly varying the trapping potentials of the optical dipole trap and the one-dimensional lattice. This method provides a route to study the dependence of the many-body interaction on the dimensionality by continuously changing the dimensionality of a Fermi gas in the crossover from two dimensions to three dimensions.

Transport in p -wave-interacting Fermi gases

The scattering properties of spin-polarized Fermi gases are dominated by p-wave interactions. Besides their inherent angular dependence, these interactions differ from their s-wave counterparts as they also require the presence of a finite effective range in order to understand the low-energy properties of the system. In this article we examine how the shear viscosity and thermal conductivity of a three-dimensional spin-polarized Fermi gas in the normal phase depend on the effective range and the scattering volume in both the weakly and strongly interacting limits. We show that although the shear viscosity and thermal conductivity both explicitly depend on the effective range near resonance, the Prandtl number which parametrizes the ratio of momentum to thermal diffusivity does not have an explicit interaction dependence both at resonance and for weak interactions in the low-energy limit. In contrast to s-wave systems, p-wave scattering exhibits an additional resonance at weak attraction from a quasi-bound state at positive energies, which leads to a pronounced dip in the shear viscosity at specific temperatures.

Reliability of the Ginzburg–Landau Theory in the BCS-BEC Crossover by Including Gaussian Fluctuations for 3D Attractive Fermions

We calculate the parameters of the Ginzburg–Landau (GL) equation of a three-dimensional attractive Fermi gas around the superfluid critical temperature. We compare different levels of approximation throughout the Bardeen–Cooper–Schrieffer (BCS) to the Bose–Einstein Condensate (BEC) regime. We show that the inclusion of Gaussian fluctuations strongly modifies the values of the Ginzburg–Landau parameters approaching the BEC regime of the crossover. We investigate the reliability of the Ginzburg–Landau theory, with fluctuations, studying the behavior of the coherence length and of the critical rotational frequencies throughout the BCS-BEC crossover. The effect of the Gaussian fluctuations gives qualitative correct trends of the considered physical quantities from the BCS regime up to the unitary limit of the BCS-BEC crossover. Approaching the BEC regime, the Ginzburg–Landau equation with the inclusion of Gaussian fluctuations turns out to be unreliable.

Dicke superradiance of a two-component Fermi gas coupled to a quantized light field

We study the superradiance transition of a two-component three-dimensional Fermi gas interacting with a single-mode light field of Dicke-type coupling. We find that for a noninteracting gas, due to the Fermi blocking, a unique superradiant state with a superradiant outer shell surrounding an inner Fermi sea may appear, and the critical atom-light coupling strength ${g}_{c}$ to trigger the superradiance approaches $\sqrt{{\ensuremath{\omega}}_{c}{E}_{F}/3}$ even for a vanishing transition frequency between a two-spin state (${\ensuremath{\omega}}_{a}\ensuremath{\rightarrow}0$), in contrast to ${g}_{c}\ensuremath{\sim}\sqrt{{\ensuremath{\omega}}_{c}{\ensuremath{\omega}}_{a}}\ensuremath{\rightarrow}0$ for a bosonic or spin system. When the atom-atom attraction is included, we find that the atomic superfluid would compete with the superradiance directly and both orders cannot coexist, giving rise to an interesting ground-state phase diagram with a tricritical point. The resultant phases and phase transitions are characterized by the unique fluctuation spectrum beyond the mean-field level. We further analyze the effects caused by the decay of the light field, which is inevitable for a possible realization in a cavity with a cold atom system. Our results would be beneficial for the understanding of the interplay between Fermi superfluid and superradiance.

From weak to strong: Constrained extrapolation of perturbation series with applications to dilute Fermi systems

We develop a method that uses truncation-order-dependent re-expansions constrained by generic strong-coupling information to extrapolate perturbation series to the nonperturbative regime. The method is first benchmarked against a zero-dimensional model field theory and then applied to the dilute Fermi gas in one and three dimensions. Overall, our method significantly outperforms Pad\'e and Borel extrapolations in these examples. The results for the ground-state energy of the three-dimensional Fermi gas are robust with respect to changes of the form of the re-expansion and compare well with quantum Monte Carlo simulations throughout the BCS regime and beyond.

Third- and fourth-order virial coefficients of harmonically trapped fermions in a semiclassical approximation

Using a leading-order semiclassical approximation, we calculate the third- and fourth-order virial coefficients of nonrelativistic spin-1/2 fermions in a harmonic trapping potential in arbitrary spatial dimensions, and as functions of temperature, trapping frequency and coupling strength. Our simple, analytic results for the interaction-induced changes $\Delta b_3$ and $\Delta b_4$ agree qualitatively, and in some regimes quantitatively, with previous numerical calculations for the unitary limit of three-dimensional Fermi gases.

Polarization in a three-dimensional Fermi gas with Rabi coupling

We investigate the polarization of a two-component three-dimensional fermionic gas made of repulsive alkali-metal atoms. The two pseudo-spin components correspond to two hyperfine states which are Rabi coupled. The presence of Rabi coupling implies that only the total number of atoms is conserved and a quantum phase transition between states dominated by spin-polarization along different axses is possible. By using a variational Hartree–Fock scheme we calculate analytically the ground-state energy of the system and determine analytically and numerically the conditions under which there is this quantum phase transition. This scheme includes the well-known criterion for the Stoner instability. The obtained phase diagram clearly shows that the polarized phase crucially depends on the interplay among the Rabi coupling energy, the interaction energy per particle, and the kinetic energy per particle.

Exchange interactions and itinerant ferromagnetism in ultracold Fermi gases

Itinerant ferromagnetism, i.e. spontaneous polarization of non-localized particles, is expected to occur for strong repulsive interactions in a spin-1/2 Fermi system. However, this state has proven notoriously hard to find experimentally, both in ultracold gases and in solids. This raises questions about the stability of the itinerant ferromagnetic state itself. Here we develop a new approach to describe both the direct and exchange interactions for a general interaction potential in the path-integral formalism and we apply this method to itinerant ferromagnetism in three-dimensional ultracold Fermi gases. We show that the exchange interactions are lost in the Hubbard-Stratonovich transformation and we propose to explicitly include the exchange effects in a new modified interaction potential. In the saddle-point approximation, the effect of interactions can be taken into account using only three parameters. If the interactions become too strong, all saddle points become unstable to density fluctuations. This greatly restricts the area in the phase diagram where uniform itinerant ferromagnetism is expected to occur.

High-momentum tail and universal relations of a Fermi gas near a Raman-dressed Feshbach resonance

In a recent proposal [Jie and Zhang, Phys. Rev. A 95, 060701(R) (2017)], it has been shown that center-of-mass-momentum-dependent two-body interactions can be generated and tuned by Raman coupling the closed-channel bound states in a magnetic Feshbach resonance. Here we investigate the universal relations in a three-dimensional Fermi gas near such a laser modulated $s$-wave Feshbach resonance. Using the operator-product expansion approach, we find that, to fully describe the high-momentum tail of the density distribution up to $q^{-6}$ ($q$ is the relative momentum), four center-of-mass-momentum-dependent parameters are required, which we identify as contacts. These contacts appear in various universal relations connecting microscopic and thermodynamic properties. One contact is related to the variation of energy with respect to the inverse scattering length and determines the leading $q^{-4}$ tail of the high-momentum distribution. Another vector contact appears in the subleading $q^{-5}$ tail, which is related to the velocity of closed-channel molecules. The other two contacts emerge in the $q^{-6}$ tail and are respectively related to the variation of energy with respect to the range parameter and to the kinetic energy of closed-channel molecules. Particularly, we find that the $q^{-5}$ tail and part of the $q^{-6}$ tail of the momentum distribution show anisotropic features. We derive the universal relations and, as a concrete example, estimate the contacts for the zero-temperature superfluid ground state of the system using a mean-field approach.

Triplet pair amplitude in a trapped s-wave superfluid Fermi gas with broken spin rotation symmetry. II. Three-dimensional continuum case

We extend our recent work [Y. Endo et. al., Phys. Rev. 92, 023610 (2015)] for a parity-mixing effect in a model two-dimensional lattice fermions to a realistic three-dimensional ultracold Fermi gas. Including effects of broken local spatial inversion symmetry by a trap potential within the framework of the real-space Bogoliubov-de Gennes theory at $T=0$, we point out that an odd-parity $p$-wave Cooper-pair amplitude is expected to have already been realized in previous experiments on an (even-parity) $s$-wave superfluid Fermi gas with spin imbalance. This indicates that, when one suddenly changes the $s$-wave pairing interaction to an appropriate $p$-wave one by using a Feshbach technique in this case, a non-vanishing $p$-wave superfluid order parameter is immediately obtained, which is given by the product of the $p$-wave interaction and the $p$-wave pair amplitude that has already been induced in the spin-imbalanced $s$-wave superfluid Fermi gas. Thus, by definition, the system is in the $p$-wave superfluid state, at least just after this manipulation. Since the achievement of a $p$-wave superfluid state is one of the most exciting challenges in cold Fermi gas physics, our results may provide an alternative approach to this unconventional pairing state. In addition, since the parity-mixing effect cannot be explained as far as one deals with a trap potential in the local density approximation (LDA), it is considered as a crucial example which requires us to go beyond LDA.

Low-temperature electron-phonon heat transfer in metal films

We consider the deformation potential mechanism of the electron-phonon coupling in metal films and investigate the intensity of the associated heat transfer between the electron and phonon subsystems. The focus is on the temperature region below dimensional crossover $T<T^{\ast}$ where the thermally relevant vibrations are described in terms of a quasi-two-dimensional elastic medium, while electron excitations behave as a three-dimensional Fermi gas. We derive an explicit expression for the power $P\left( T\right) $ of the electron-phonon heat transfer which explains the behavior observed in some experiments including the case of metallic film supported by an insulating membrane with different acoustic properties. It is shown that at low temperatures the main contribution is due to the coupling with Lamb's dilatational and flexural acoustic modes.

Dark soliton solution of the three-dimensional Gross-Pitaevskii equation with an isotropic harmonic potential and nonlinearity in polytropic approximation

We study the three-dimensional Fermi gas in an isotropic harmonic trap during the Bardeen- Cooper-Schrieffer superfluid to Bose-Einstein condensate (BCS-BEC) crossover, which is modeled by using the generalized Gross-Pitaevskii equation (GGPE) in the polytropic approximation. We analytically solved the 3D GGPE with a coupled modulus-phase transformation without introducing any additional integrability constraint, reaching the dark soliton-like solution. We find that the dark soliton identified undergoes an oscillation with a constant period over the whole BCS-BEC crossover region, although the amplitude of the dark soliton varies with polytropic index, demonstrating the peculiar nonlinear properties for the system modeled by using the 3D GGPE.

Induced p-wave superfluidity in imbalanced Fermi gases in a synthetic gauge field

We study the pairing formation and the appearance of induced spin-triplet p-wave superfluidity in dilute three-dimensional imbalanced Fermi gases in the presence of a uniform non-Abelian gauge field. This gauge field generates a synthetic Rashba-type spin–orbit interaction which has remarkable consequences in the induced p-wave pairing gaps. Without the synthetic gauge field, the p-wave pairing occurs in one of the components due to the induced (second-order) interaction via an exchange of density fluctuations in the other component. We show that this p-wave superfluid gap induced by density fluctuations is greatly enhanced due to the Rashba-type spin–orbit coupling.

Repulsive atomic Fermi gas with Rashba spin-orbit coupling: A quantum Monte Carlo study

Thanks to the recent experimental realization and control of artificial gauge fields, spin-orbit (SO) couplings are witnessing an ever increasing interest in the field of cold atoms. However, predicting their effect on spin polarization and energetic properties of interacting systems is a major challenge, due to the complex interplay between spin and position dynamics. In this work we exploit the diffusion Monte Carlo algorithm to compute energetic and polarization properties of a three-dimensional repulsive Fermi gas in the presence of Rashba spin-orbit coupling. We find that SO effects tend to contrast the spin alignment induced by the exchange interaction, slightly shifting the onset of the Stoner instability towards larger values of the scattering length. In addition, polarization and energy properties of the system can be tuned trough a combined control of the repulsive interaction and Rashba coupling.

Loading and detecting a three-dimensional Fermi gas in a one-dimensional optical superlattice

We investigate the procedures of loading and detecting three-dimensional fermionic quantum gases in a one-dimensional optical superlattice potential subjected to a trapping potential. Additionally, we consider the relaxation dynamics after a sudden change of the superlattice potential. We numerically simulate the time-dependent evolution of the continuous system using exact diagonalization of non-interacting fermions. During the loading procedure we analyze the occupation of the instantaneous energy levels and compare the situation in a homogeneous system with the trapped one. Strong differences are found in particular in the evolution of excitations which we trace back to the distinct global density distribution. Starting from an imbalanced state in the superlattice potential, we consider the relaxation dynamics of fermions after a slow change of the superlattice potential and find a bimodule distribution of excitations. To be able to compare with the experimental results we also simulate the measurement sequence of the even and odd local density and find a strong dependence of the outcome on the actual ramp procedure. We suggest how the loading and detecting procedure can be optimized.

Virial coefficients for trapped Bose and Fermi gases beyond the unitary limit: AnS-matrix approach

We study the virial expansion for three-dimensional Bose and Fermi gases at finite temperature using an approximation that only considers two-body processes and is valid for high temperatures and low densities. The first virial coefficients are computed and the second is exact. The results are obtained for the full range of values of the scattering length and the unitary limit is recovered as a particular case. A weak coupling expansion is performed and the free case is also obtained as a proper limit. The influence of an anisotropic harmonic trap is considered using the Local Density Approximation - LDA, analytical results are obtained and the special case of the isotropic trap is discussed in detail.

Absence of thermalization in a Fermi liquid

We study a weak interaction quench in a three-dimensional Fermi gas. We first show that, under some general assumptions on time-dependent perturbation theory, the perturbative expansion of the long-wavelength structure factor $S(\bq)$ is not compatible with the hypothesis that steady-state averages correspond to thermal ones. In particular, $S(\bq)$ does develop an analytical component $\sim const. + O(q^2)$ at $\bq\to\bnot$, as implied by thermalization, but, in contrast, it maintains a non-analytic part $\sim |\bq|$ characteristic of a Fermi-liquid at zero-temperature. In real space, this non-analyticity corresponds to persisting power-law decaying density-density correlations, whereas thermalization would predict only an exponential decay. We next consider the case of a dilute gas, where one can obtain non-perturbative results in the interaction strength but at lowest order in the density. We find that in the steady-state the momentum distribution jump at the Fermi surface remains finite, though smaller than in equilibrium, up to second order in $k_F f_0$, where $f_0$ is the scattering length of two particles in the vacuum. Both results question the emergence of a finite length scale in the quench-dynamics as expected by thermalization.