Abstract
In our discussion of interacting nuclear matter we have so far ignored a very important physical effect. We have not included the possibility of superfluidity and/or superconductivity, although we have briefly mentioned the effect of superconductivity on the equation of state of quark matter, see Sect. 2.3. In the following, we shall discuss these effects in more detail. But first let us recapitulate what superconductivity is. Once we have introduced the basic concept we shall see that it may appear in several variants in a compact star. And we will see that it is crucial for the understanding of transport properties of dense matter. And the transport properties of dense matter, in turn, are related to the phenomenology of the star.
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Notes
- 1.
In fact, the fermions are correlated in momentum space, not in real space. Consequently, in the weak-coupling limit, the fermion pairs are not spatially separated bosons. The typical size of a pair is rather larger than the mean distance between fermions. Therefore, one apparently has to be careful to describe the pairs as bosons. However, recent experiments with cold fermionic atoms show that there is no phase transition between the weak-coupling limit (where the pairs are wide spread) and the strong-coupling limit (where the pairs are actual difermions, i.e., bosons). This is the so called BCS-BEC crossover. This observation suggests in particular that it is not too bad to think of the fermion pairs as bosons even in the weak-coupling limit.
- 2.
More precisely, here we compute the fermionic contribution to the specific heat. There may be light Goldstone modes which dominate the specific heat at small temperatures. In this section we ignore such modes for the purpose of illustrating the effect of the fermionic energy gap.
- 3.
- 4.
As already mentioned in the introduction, we neglect the heavy quark flavors although in this section we consider asymptotically large densities. Since we are ultimately interested in extrapolating our results down to compact star densities, we only take u, d, and S quarks into account.
- 5.
Due to the weak interactions this mode acquires a small energy gap in the keV range which we neglect here.
- 6.
Notice that for \(\alpha+2\beta_2<0\) the critical temperature formally becomes imaginary, i.e., the condensate apparently “refuses” to melt. This situation cannot occur for realistic parameters in our case but is an interesting theoretical possibility. See Appendix C in Ref. [10] and references therein for more information.
- 7.
Cooper pairing with mismatched Fermi momenta is an interesting general phenomenon and not only relevant for quark matter, but also in condensed matter physics and atomic physics. See for instance Ref. [16] where mismatched pairing of fermionic atoms is investigated experimentally in an optical trap.
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Schmitt, A. (2010). Superconductivity and Superfluidity in a Compact Star. In: Dense Matter in Compact Stars. Lecture Notes in Physics, vol 811. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12866-0_4
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