Fermions can exhibit superfluidity/superconductivity by forming Cooper pairs, which undergo Bose-Einstein Condensation (BEC). In the most conventional metallic superconductors, the Cooper pair is formed with zero relative angular momentum (“s-wave pairing”). However, the Cooper pair can be also formed with a non-vanishing relative angular momentum. When the angular momentum of each Cooper pair is aligned, the superfluid as a whole breaks the time-reversal symmetry and is called “chiral superfluid”. One of the superfluid phases of ³He called “superfluid A-phase”, indeed is such a chiral superfluid, in which each Cooper pair has a relative angular momentum 1 (“p+ip-wave pairing”). There is also much evidence that Sr₂RuO₄ is a chiral p+ip-wave superconductor. We can also consider more general chiral superfluids/superconductors, in which each Cooper pair carries the angular momentum ν (ν=1 for p+ip, ν=2 for d+id, etc.).

Fig. 1. The total orbital angular momentum of a d+id-wave (ν=2) chiral superfluid and spectral flow. (a) The energy spectrum in the weak-pairing (BCS) phase, as a function of the angular momentum l. Two gapless edge modes can be seen within the pairing gap. (b) The contribution η_l in each sector with angular momentum l to the quantum number Q, in the weak-pairing (BCS) phase. Non-vanishing contributions are present in the region between the “Fermi angular momenta” where the gapless edge modes cross the zero energy. They make Q non-vanishing, suppressing the total orbital angular momentum from the “ideal” value νN/2. (c) The energy spectrum at a fixed angular momentum l, which lies between the Fermi angular momenta inside the weak-pairing (BCS) phase, as a function of the chemical potential μ. When μ is changed from negative (strong-pairing, BEC phase) to positive (weak-pairing, BCS phase), one of the energy eigenvalues changes sign, as required from the structure of the edge modes. This induces the non-vanishing contribution η_l to Q discussed in (b) above, in the weak-pairing (BCS) phase.

We can then ask a simple question: what is the total orbital angular momentum of the chiral superfluid which consists of N fermions? On one hand, the answer seems obvious: since there would be N/2 Cooper pairs, it must be νN/2. On the other hand, since the pairing energy Δ is usually much smaller than the Fermi energy E_F in Bardeen-Cooper-Schrieffer (BCS) type superfluids, we may also think that only the fermions near the Fermi surface are affected by the pairing. In this picture, we rather expect the total orbital angular momentum is suppressed as (Δ/E_F)^ανN/2, where α>0. There have been many conflicting papers published over the last several decades on this question, but it still remains controversial.

In order to clarify the long-standing puzzle, we studied chiral superfluids in a two-dimensional circular potential (disk), within the standard Bogoliubov-de Gennes framework but without further approximations or assumptions. The Bogoliubov-de Gennes Hamiltonian has anomalous pairing terms and thus does not conserve particle number N nor the total orbital angular momentum L. Nevertheless, as Volovik pointed out, the combination Q = L – νN/2 is still conserved and turns out to be a quite useful quantum number. For each value of ν, there are two phases of chiral superfluid: strong-pairing (BEC) and weak-pairing (BCS) phases, separated by a quantum phase transition. Since L = νN/2 (and thus Q = 0) in the strong pairing limit, and no gap closing occurs within the strong-pairing (BEC) phase, the conservation of the quantum number Q immediately implies that the L = νN/2 holds in the entire strong-pairing (BEC) phase. This is what was naturally expected; the real question is what happens in the weak-pairing (BCS) phase.

We found that, in a p+ip -wave (ν=1) chiral superfluid in the weak-pairing (BCS) phase, L = νN/2 holds exactly for a sufficiently large system. In contrast, in d+id, f+if, etc. (ν≧ 2) chiral superfluids in the weak-pairing (BCS) phase, L is strongly suppressed and is at most of O(Δ/E_F), implying Cooper-pair breaking near the edge. This surprising difference is understood in terms of the spectral flow and the quantum number Q.

References

• [1] Y. Tada, W. Nie, and M. Oshikawa, Phys. Rev. Lett. 114, 195301 (2015).

Authors

• Y. Tada, W. Nie^a, and M. Oshikawa
• ^aTsinghua University