# Number Conserving Superconductors

Usual BCS theory used to describe superconductors violates particle number conservation, this is allowed since that theory is written in a grand canonical ensemble (i.e particles can be exchanges between the system and the environment).

My question is: What if the number of particles is forced to be conserved ( by looking at a closed system or working in a canonical ensemble where particles have no where to go ), can superconductivity emerge in such a situation? if yes what is the describing theory ? and if no why no?

The BCS wave function is a superposition of components with definite particle number $N=0,2,4,\ldots$. In the infinite volume limit, the particle number is sharply peaked around $N=nV$, where $V$ is the volume and $n$ is the partcle density. It is intuitively clear that projecting the BCS state on a definite particle number should not make a difference. This has been checked explicitely, see http://arxiv.org/pdf/hep-ph/0110267v1.pdf for a moderately recent paper in the context of pairing in quark matter. Projection on a fixed particle number is obviously important for small systems, most notably in nuclear physics. Neutron matter is superfluid, but actual nuclei have at most 150 neutrons. Projection methods are discussed in standard text books on nuclear physics, for example Ring and Schuck.