Condensation of Cooper triples

The condensation of Cooper pairs, originating from the Fermi-surface instability due to a weakly attractive interaction between two fermions, opened a new frontier for exploring many-body physics in interdisciplinary contexts. In this work, we discuss the possible condensation of Cooper triples, which are three-body counterparts of Cooper pairs for three-component fermions with a three-body attraction. Although each composite trimer-like state obeys the Fermi-Dirac statistics, its aggregate can form a condensate at zero center-of-mass momentum in the presence of the internal degrees of freedom associated with the relative momenta of constituent particles of momenta close to the Fermi surface. Such condensation can be regarded as bosonization in infinite-component fermions. We propose a variational wave function for the condensate of Cooper triples in analogy with the Bardeen-Cooper-Schrieffer ground state, and we obtain the ground-state energy.

Figure 1

(a) Momentum-space configurations of three fermions of different species on a spherical surface that have zero center-of-mass momentum. The momenta $\mathit{k}$ and ${\mathit{k}}^{\prime }$ of one of the three fermions can take on various directions. (b) The corresponding in-plane configuration of the three fermions of momenta $\mathit{k}, \widehat{R}\mathit{k}$, and ${\widehat{R}}^2\mathit{k}$. The angle between any pair of momenta is $2\pi /3$.

Figure 2

Schematic phase diagram in the plane of the two-body coupling constant $G$ and three-body coupling constant $U_0$ (in arbitrary units). The solid line shows the threshold for the transition between a Cooper pair (CP) and a Cooper triple (CT) state as obtained within the generalized Cooper problem in the weak-coupling limit. $G_{\mathrm{C}}$ is the critical two-body coupling constant for the transition between CP and CT in the absence of $U_0$ [7]. We also plot an eye-guide (dashed line) for the upper bound of CP starting from $(G,U_0)=(G_{\mathrm{C}},0)$. In this work, we consider the weak-coupling region where the Cooper triple formation is dominant, as indicated below the dotted line.