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arXiv:2105.10608 · Physical Review A

Analytic-normal-mode frequencies for N identical particles: The microscopic dynamics underlying the emergence and stability of excitation gaps from BCS to unitarity

D. K. Watson

Published Sep 27, 2021• 5 citations
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Abstract

The frequencies of the analytic normal modes for $N$ identical particles are studied as a function of system parameters from the weakly interacting BCS regime to the strongly interacting unitary regime. The normal modes were obtained previously from a first-order $L=0$ group theoretic solution of a three-dimensional Hamiltonian with a general two-body interaction for confined identical particles. In a precursor to this study, the collective behavior of these normal modes was investigated as a function of $N$ from few-body systems to many-body systems analyzing the contribution of individual particles to the collective macroscopic motions. A specific case, the Hamiltonian for Fermi gases in the unitary regime, was studied in more detail. This regime is known to support collective behavior in the form of superfluidity and has previously been successfully described using normal modes. Two phenomena that could sustain the emergence and stability of superfluid behavior were revealed, including the behavior of the normal mode frequencies as $N$ increases. In this paper, I focus on a more detailed analysis of these analytic frequencies, extending my investigation to include Hamiltonians with a range of interparticle interaction strengths from the BCS regime to the unitary regime and analyzing the microscopic dynamics that lead to large gaps at unitarity. The results of the present study suggest that in regimes where higher-order effects are small, normal modes can be used to describe the physics of superfluidity from the weakly interacting BCS regime with the emergence of small excitation gaps to unitarity with its large gaps and can offer insight into a possible microscopic understanding of the behavior at unitarity. This approach could thus offer an alternative to the two-body pairing models commonly used to describe superfluidity along this transition.

DOI
10.1103/physreva.104.033320

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The frequencies of the analytic normal modes for NNN identical particles are studied as a function of system parameters from the weakly interacting BCS regime to the strongly interacting unitary regime. The normal modes were obtained previously from a first-order L=0L=0L=0 group theoretic solution of a three-dimensional Hamiltonian with a general two-body interaction for confined identical particles. In a precursor to this study, the collective behavior of these normal modes was investigated as a function of NNN from few-body systems to many-body systems analyzing the contribution of individual particles to the collective macroscopic motions. A specific case, the Hamiltonian for Fermi gases in the unitary regime, was studied in more detail. This regime is known to support collective behavior in the form of superfluidity and has previously been successfully described using normal modes. Two phenomena that could sustain the emergence and stability of superfluid behavior were revealed, including the behavior of the normal mode frequencies as NNN increases. In this paper, I focus on a more detailed analysis of these analytic frequencies, extending my investigation to include Hamiltonians with a range of interparticle interaction strengths from the BCS regime to the unitary regime and analyzing the microscopic dynamics that lead to large gaps at unitarity. The results of the present study suggest that in regimes where higher-order effects are small, normal modes can be used to describe the physics of superfluidity from the weakly interacting BCS regime with the emergence of small excitation gaps to unitarity with its large gaps and can offer insight into a possible microscopic understanding of the behavior at unitarity. This approach could thus offer an alternative to the two-body pairing models commonly used to describe superfluidity along this transition.
The transition from the weakly interacting BCS regime to the strongly interacting unitary regime is explored for ultracold trapped Fermi gases assuming a normal mode description of the gas instead of the conventional Cooper pairing. The Pauli principle is applied ``on paper'' by using specific normal mode assignments. Energies, entropies, critical temperatures, and an excitation frequency are studied and compared to existing results in the literature. These normal modes have been derived analytically for N identical, confined particles from a first-order L=0 group theoretic solution of a three-dimensional Hamiltonian with a general two-body interaction. In previous studies, normal modes were able to describe the unitary regime obtaining ground state energies comparable to benchmark results and thermodynamics quantities in excellent agreement with experiment. In a recent study, the behavior of the normal mode frequencies was investigated for Hamiltonians with a range of interparticle interaction strengths from BCS to unitarity in the first test of this approach beyond the unitary regime, and a microscopic basis of the large excitation gaps and universal behavior at unitarity was proposed. Based on the success of these earlier studies, the current paper continues to explore the ability of normal modes to describe superfluidity along the BCS to unitarity transition. The results confirm earlier conclusions that the physics of superfluidity can be described using normal modes across a wide range of interparticle interaction strengths and offer an alternative to the two-body pairing models commonly used to describe superfluidity along this transition.
Understanding the emergence of collective organizational phenomena is a major goal in many fields of physics from condensed matter to cosmology. Using a recently introduced manybody perturbation formalism for fermions, we propose a mechanism for the emergence of collective behavior, specifically superfluidity, driven by quantum statistics and the enforcement of the Pauli principle through the selection of normal modes. The method, which is called symmetry invariant perturbation theory (SPT), uses group theory and graphical techniques to solve the manybody Schrodinger equation through first order exactly. The solution at first order defines collective coordinates in terms of five N-body normal modes, identified as breathing, center of mass, single particle angular excitation, single particle radial excitation and phonon. A correspondence is established "on paper" that enforces the Pauli principle through the assignment of specific normal mode quantum numbers. Applied in the unitary regime, this normal mode assignment yields occupation only in an extremely low frequency N-body phonon mode at ultralow temperatures. A single particle radial excitation mode at a much higher frequency creates a gap that stabilizes the superfluidity at low temperatures. Coupled with the corresponding values for the frequencies at unitarity obtained by this manybody calculation, we obtain good agreement with experimental thermodynamic results including the lambda transition in the specific heat. Our results suggest that the emergence of collective behavior in macroscopic systems is driven by the Pauli principle and its selection of the correct collective coordinates in the form of N-body normal modes.
Abstract Understanding the universal behavior of strongly-interacting systems of particles has been a major goal in multiple fields of physics from atomic physics to cosmology. Using a recently introduced manybody perturbation formalism for fermions which uses group theoretic and graphical techniques to solve the N -body problem, I study the thermodynamic behavior of the unitary regime for a trapped Fermi gas and investigate the role of the Pauli principle in the emergence of collective behavior, specifically superfluidity at ultralow temperatures. The method, which is called symmetry invariant perturbation theory, has no adjustable parameters and currently offers a solution to the manybody Schrodinger equation through first order in inverse dimensionality exactly. The solution at first order defines collective coordinates in terms of five types of N -body normal modes, identified as symmetric stretch, symmetric bend, single particle angular excitation, single particle radial excitation and phonon. A correspondence is established ‘on paper’ that enforces the Pauli principle through the assignment of specific normal mode quantum numbers. Applied at ultracold temperatures, this normal mode assignment yields occupation only in an extremely low frequency N -body phonon mode. A single particle radial excitation mode at a much higher frequency creates a gap that stabilizes the superfluidity at low temperatures. This radial excitation involves excitation of a single particle out of the synced motion of the phonon mode, rather than the breaking of a fermion pair. Coupled with the corresponding values for the energies at unitarity obtained by this manybody calculation, I obtain good agreement with experimental thermodynamic results. In particular, the lambda transition in the specific heat is clearly seen. Based on these findings, I propose a mechanism for the emergence of superfluidity at ultralow temperatures in this unitary regime that is driven by quantum statistics and the enforcement of the Pauli principle through the selection of normal modes. My results for the calculated thermodynamic quantities suggest that the emergence of some collective behaviors in macroscopic systems is due to the Pauli principle producing manybody pairing and the resulting macroscopic occupation of an N -body phonon mode.
Abstract Identifying possible microscopic mechanisms underlying superfluidity has been the goal of various studies since the introduction of the original BCS theory. Recently a series of papers have proposed microscopic dynamics based on normal modes to describe superfluidity without the use of real-space Cooper pairs. Multiple properties were determined with excellent agreement with experimental data. The group theoretic basis of this general N -body approach has allowed the microscopic behavior underlying these results to be analyzed in detail. This reimagination is now used to reinterpret several interrelated phenomena including Cooper pairs, the Fermi sea, and Pauli blocking. This approach adheres closely to the early tenets of superconductivity/superfluidity which assumed pairing only in momentum space, not in real space. The Pauli principle is used, in its recently revealed role in collective motion, to select the allowed normal modes. The expected properties of superfluidity including the rigidity of the wave function, interactions between the fermions in different pairs, convergence of the momentum and the gap in the excitation spectrum are discussed.
Motivated by recent advances in the creation of few-body atomic Fermi gases with attractive interactions, we study theoretically the few-to-many-particle crossover of pair excitations, which for large particle numbers evolve into a mode that describes amplitude fluctuations of the superfluid order parameter (the “Higgs” mode). Our analysis is based on the hypothesis that salient aspects of the excitation spectrum are captured by interactions between time-reversed pair states in a harmonic oscillator potential. Microscopically, this assumption leads to a Richardson-type pairing model, which is integrable and thus allows a systematic quantitative study of the few-to-many-particle crossover with only minor numerical effort. We first establish a parity effect in the ground-state energy, i.e., a spectral convexity in the energy of open-shell configurations compared to their closed-shell neighbors, which is quantified by a so-called Matveev-Larkin parameter discussed for mesoscopic superconductors, which generalizes the pairing gap to mesoscopic ensembles and which behaves quantitatively differently in a few-body and a many-body regime. The crossover point for this quantity is widely tunable as a function of interaction strength. We then compute the excitation spectrum and demonstrate that the pair excitation energy shows a minimum that deepens with increasing particle number and shifts to smaller interaction strengths, consistent with the finite-size precursor of a quantum phase transition to a superfluid state. We extract a critical finite-size scaling exponent that characterizes the decrease of the gap with increasing particle number. Published by the American Physical Society 2024
We investigate collective modes and free expansions of quasi-one- and quasi-two-dimensional (Q1D and Q2D) ultracold Fermi gases in the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC). We solve a superfluid order parameter equation valid for the BCS-BEC crossover by employing a time-dependent variational method. We take a trial wave function of hybrid Gaussian-parabolic type, which not only reflects the low-dimensional character of the system but also allows an essentially analytical approach for the problem. We present Q1D and Q2D criteria that are valid in various superfluid regimes and show clearly the relation between the maximum condensed particle number and the parameters of the trapping potential as well as the atom-atom interaction. We demonstrate that, due to the small particle number in Q1D and Q2D condensates, the contribution to oscillating frequencies of collective modes by the quantum pressure in the strong-confinement direction is significant and hence the Thomas-Fermi approximation cannot be used. We also show that the free expansion of Q1D and Q2D superfluid Fermi gases in the strong-confinement direction is much faster than that in the weak-confinement direction.