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Physical review. B./Physical review. B

Intrinsic Hamiltonian of composites in many-fermion systems

F. Palumbo

Published Mar 9, 2021• 0 citations
Publisher

Abstract

I determine the intrinsic Hamiltonian of the relative motion of the constituent fermions of bosonic composites in superfluid fermion systems, assuming the effective fermion-fermion potential to be a sum of separable terms. The derivation is based on an expansion that treats composites, quasiparticles, and their interactions on the same footing. The intrinsic Hamiltonian of the composites is expressed in terms of the solution of the gap equation and of the form factors of the potential. It has the Brillouin-Wigner form, namely it is Hermitian and energy-dependent. In such a context, a solid justification is given for discarding negative energies of the composites. As a check, I rederive the dispersion law of the Bogoliubov-Anderson mode and the spectrum of the Bardasis and Schrieffer Hamiltonian. The possible occurrence of gapped modes depending on the form of the fermion-fermion interaction is discussed. The intrinsic wave functions are derived explicitly in the long wave limit, while their full determination requires numerical calculations.

DOI
10.1103/physrevb.103.104506

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I determine the intrinsic Hamiltonian of the relative motion of the constituent fermions of bosonic composites in superfluid fermion systems, assuming the effective fermion-fermion potential to be a sum of separable terms. The derivation is based on an expansion that treats composites, quasiparticles, and their interactions on the same footing. The intrinsic Hamiltonian of the composites is expressed in terms of the solution of the gap equation and of the form factors of the potential. It has the Brillouin-Wigner form, namely it is Hermitian and energy-dependent. In such a context, a solid justification is given for discarding negative energies of the composites. As a check, I rederive the dispersion law of the Bogoliubov-Anderson mode and the spectrum of the Bardasis and Schrieffer Hamiltonian. The possible occurrence of gapped modes depending on the form of the fermion-fermion interaction is discussed. The intrinsic wave functions are derived explicitly in the long wave limit, while their full determination requires numerical calculations.
We present a formalism for the fermionic quasiparticle propagator in a superfluid fermionic system. Starting from a general many-body Hamiltonian confined by the two-body instantaneous interaction, the equation of motion for the fermionic propagator is obtained in the Dyson form. Before making any approximation, the interaction kernel is found to be decomposed into static and dynamical (time-dependent) contributions, where the latter translates to the energy-dependent and the former maps to the energy-independent terms in the energy domain. The three-fermion correlation function, being the heart of the dynamical part of the kernel, is factorized into the two-fermion and one-fermion ones. With the relaxed particle number constraint, the normal propagator is coupled to the anomalous one via both the static and dynamical kernels, which is formalized by introducing the generalized quasiparticle propagator of the Gor'kov type. The dynamical kernel in the factorized form is associated with the quasiparticle-vibration coupling (QVC), with the vibrations unifying both the normal and pairing phonons. The QVC vertices are related to the variations of the Hamiltonian of the Bogoliubov quasiparticles, which can be obtained by the finite amplitude method.
Using the effective action formalism, we investigate collective excitations of a dilute mixture of a Bose gas and a two component Fermi gas when both bosons and fermions have undergone superfluid transitions. We show that there is a repulsion between Bogoliubov and Anderson modes, which has important implications including disappearance of boson superfluidity. We derive an analytic expression for the long-wavelength dispersion relation of the mixture at zero temperature and give a condition for the instability. We also numerically calculate the damping rate due to boson-fermion interaction at finite temperatures and show that the two modes are stable at sufficiently low temperatures.
A consistent finite-temperature microscopic theory for the response of strongly coupled superfluid fermionic systems is formulated. We start from the general many-body Hamiltonian with the vacuum (bare) two-fermion interaction and derive the equation of motion (EOM) for the thermally averaged two-time two-fermion correlation function, which determines the spectrum of the system under study. The superfluidity is introduced via the Bogoliubov transformation of the fermionic field operators, and the entire formalism is carried out in the basis of Bogoliubov's quasiparticles, keeping the complete 4x4 block matrix structure of the two-fermion EOM. Fully correlated static and dynamical interaction kernels of the resulting EOM are discussed. A special focus is then placed on the latter kernel, which is advanced to a factorized form, enabling a minimal truncation of the many-body problem while keeping important effects of emergent collectivity and mapping to the quasiparticle-vibration coupling (qPVC). As in the zero-temperature and non-superfluid cases, the qPVC can be associated with a new order parameter, qPVC vertex. In the thermal superfluid theory, the latter vertex, as well as the components of the dynamical kernel, acquire an extended form including thermally unblocked transition amplitudes.
The problem with the construction of the Gross-Pitaevskii (GP) equation and related wave functions in a medium is associated with the necessity to begin with a description of the configuration space and proceed to a description of the physical space. We show that the balance equations of the number of particles and momentum immediately follow from the multiparticle Schr\"odinger equation. From the obtained set of balance equations, the equation for the wave function in the medium coincides in form with the GP equation, where we can restrict ourselves only by the first-order terms of the interaction radius of the stress tensor of bosons, for dilute gases. As a generalization of the GP equation, we made allowance for the contribution of the third-order terms of the interaction radius to the stress tensor of bosons. For a system of particles that comprises an ultracold mixture of bosons and fermions, a two-kind quantum hydrodynamics is constructed for the third-order terms of the interaction radius. The spectrum of eigenmodes involves additional information on the interparticle interaction as a correction to the Bogoliubov spectrum.
The Higgs and Nambu-Goldstone modes are paradigmatic collective excitations in superconductors and superfluids. These modes are commonly analyzed within the pseudospin formulation of BCS theory, where the dynamics are obtained from the Heisenberg equations of motion for pseudospins. However, this construction becomes inconsistent when directly extended to non-Hermitian systems. In this work, we develop a consistent pseudospin framework for non-Hermitian fermionic superfluids based on the metricized formulation of non-Hermitian quantum mechanics. We apply this formalism to a driven non-Hermitian BCS-type Hamiltonian with complex pairing interaction and analyze its collective excitation spectrum. We find that, in addition to the conventional Higgs (amplitude) mode, the system hosts a novel phase mode that has no counterpart in Hermitian superfluids. Remarkably, this mode is gapped even in the absence of the Anderson-Higgs mechanism, as is the case in neutral superfluids. Furthermore, the resonance spectrum depends explicitly on the initial nongauge phase of the complex order parameter and the dynamical response remains finite at resonance, in contrast to the divergence characteristic of Hermitian systems. These resonances disappear upon the emergence of exceptional points.
The low-energy modes of a superfluid atomic Fermi gas at zero temperature are investigated. The Bose-Einstein-condensate (BEC) side of the superfluid phase is studied in detail. The atoms are assumed to be in only one internal state, so that for a sufficiently diluted gas the pairing of fermions can be considered effective in the l=1 channel only. In agreement with previous works on p-wave superfluidity in Fermi systems, it is found that the px+ipyp_x+ip_ypx​+ipy​ phase represents the lowest energy state in both the Bardeen-Cooper-Schrieffer (BCS) and BEC sides. Our calculations show that at low energy three branches of collective modes can emerge, with different species of dispersion relations: a phonon-like mode, a single-particle-like mode and a gapped mode. A comparison with the Bogoliubov excitations of the corresponding spinor Bose condensate is made. They reproduce the dispersion relations of the excitation modes of the p-wave superfluid Fermi gas to a high accuracy.