Comparative study of quartet superfluid state: Quartet Bardeen-Cooper-Schrieffer theory and generalized Nambu-Gor'kov formalism

TL;DR

This work analyzes quartet superfluidity in a one-dimensional, four-component Fermi gas with a pure four-body interaction, benchmarking quartet-BCS theory against exact four-body results. It shows that a multiple-infinite-product (MIP) ansatz recovers the exact dilute-limit binding energy and yields a self-consistent gap equation that matches the generalized Nambu-Gor'kov formalism, establishing a concrete link between variational and field-theoretical treatments of four-body clustering. In contrast, a single-infinite-product (SIP) ansatz fails to capture the full four-body physics, highlighting the importance of internal momentum structure. Numerically, the quartet superfluid exhibits a crossover from coherent Bogoliubov-like quasiparticles to a continuum-dominated spectrum driven by strong four-body correlations, while a finite quartet order parameter persists across the crossover. The framework provides a path to realistic extensions in higher dimensions and to finite-temperature transitions, with potential implications for nuclear matter and multi-component ultracold gases.

Abstract

We theoretically investigate a quartet superfluid state in fermionic matter by using the quartet Bardeen-Cooper-Schrieffer (BCS) variational theory and the Green's function method. We demonstrate that the quartet BCS theory with the multiple-infinite-product ansatz successfully reproduces an exact four-body result in a one-dimensional four-component Fermi gas at the dilute limit, in contrast to the single-infinite-product ansatz. To see the validity of the quartet BCS state, we derive the self-consistent equation for the quartet superfluid order parameter within the generalized imaginary-time Nambu-Gor'kov formalism, which is found to be consistent with the quartet BCS variational equation. Moreover, by numerically computing the momentum-resolved single-particle spectral function in a one-dimensional system, we discuss how the single-particle spectra evolve with increasing the strength of the four-body cluster formation. We show that a coherent BCS-like quasiparticle branch on the weak-coupling side evolves into a strongly damped, continuum-dominated spectrum in the strong-coupling side, while nonzero quartet superfluid order parameter persists throughout the crossover regime. Our results would be useful for understanding beyond-BCS pairing effects and four-body cluster formations in fermionic systems in an interdisciplinary way.

Figures (4)

• Figure 1: The diagrammatic illustration of the quartet mean-field self-energy in the condensed phase \ref{['eq62']}. The blobs denote the quartet order parameter, and the internal lines represent the effective three-particle propagator Tajima2022Phys.Rev.Research4.L012021.
• Figure 2: (a) The order parameter $\Delta$ as a function of the inverse scattering length $1/a$. (b) The chemical potential $\mu$ as a function of the inverse of scattering length $1/a$. The density is fixed at $4/\pi$.
• Figure 3: Contour plot of the spectral function $A_k(\omega)$ at (a) $1/a=-2.0$, (b) $1/a=0.0$, and (c) $1/a=0.5$, respectively. Here $k_{\rm F}=E_{\rm F}=1$ and the momentum cutoff is $\Lambda=12$. The color bar is in the logarithmic scale, representing the intensity of $A(k,\omega)$.
• Figure 4: (a) Quartet order parameter $\Delta$, and (b) chemical potential $\mu$ as functions of the four-body binding energy $E_{\rm b}$ obtained within the SIP and MIP ansatzes. In the entire regime, the number density is fixed at $4/\pi$ with $k_{\rm F}=E_{\rm F}=1$.