What is superconductivity?

A friend pointed out that, while I've written many posts that have to do with superconductivity, I've never really done a concept post about it.  Here's a try, as I attempt to distract myself from so many things happening these days.

The superconducting state is a truly remarkable phase of matter that is hosted in many metals (though ironically not readily in the pure elements (Au, Ag, Cu) that are the best ordinary conductors of electricity - see here for some references).  First, some definitional/phenomenological points:

• The superconducting state is a distinct thermodynamic phase.  In the language of phase transitions developed by Ginzburg and Landau back in the 1950s, the superconducting state has an order parameter that is nonzero, compared to the non-superconducting metal state.   When you cool down a metal and it becomes a superconductor, this really is analogous (in some ways) to when you cool down liquid water and it becomes ice, or (a better comparison) when you cool down very hot solid iron and it becomes a magnet below 770 °C.
• In the superconducting state, at DC, current can flow with zero electrical resistance.  Experimentally, this can be checked by setting up a superconducting current loop and monitoring the current via the magnetic field it produces.  If you find that the current will decay over somewhere between \(10^5\) and \(\infty\) years, that's pretty convincing that the resistance is darn close to zero. 
• This is not just "perfect" conduction.  If you placed a conductor in a magnetic field, turned on perfect conduction, and then tried to change the magnetic field, currents would develop currents that would preserve the amount of magnetic flux through the perfect conductor.  In contrast, a key signature of superconductivity is the Meissner-Oschenfeld Effect:  if superconductivity is turned on in the presence of a (sufficiently small) magnetic field, currents will develop spontaneously at the surface of the material to exclude all magnetic flux from the bulk of the superconductor.  (That is, the magnetic field from the currents will be oppositely directed to the external field and of just the right size and distribution to give \(\mathbf{B}=0\) in the bulk of the material.)  Observation of the bulk Meissner effect is among the strongest evidence for true superconductivity, much more robust than a measurement that seems to indicate zero voltage drop.
• There are two main types of superconductors, uncreatively termed "Type I" and "Type II".  In Type I superconductors, an external \(\mathbf{H} = \mathbf{B}/\mu_{0}\) fails to penetrate the bulk of the material until it reaches a critical field \(H_{c}\), at which point the superconducting state is suppressed completely.  In a Type II superconductor, above some lower critical field \(H_{c,1}\) magnetic flux begins to penetrate the material in the form of vortices, each of which has a non-superconducting ("normal") core.  Above an upper critical field \(H_{c,2}\), superconductivity is suppressed. 
• Interestingly, a lot of this can be "explained" by the London Equations, which were introduced in the 1930s despite a complete lack of a viable microscopic theory of superconductivity.
• Magnetic flux through a conventional superconducting ring (or through a vortex core) is quantized precisely in units of \(h/2e\), where \(h\) is Planck's constant and \(e\) is the electronic charge.  
• (It's worth noting that in magnetic fields and with AC currents, there are still electrical losses in superconductors, due in part to the motion of vortices.)

Physically, what is the superconducting state?  Why does it happen and why does it have the weird properties described above as well as others?  There are literally entire textbooks and semester-long courses on this, so what follows is very brief and non-authoritative.  

• In an ordinary metal at low temperatures, neglecting e-e interactions and other complications, the electrons fill up states (because of the Pauli Principle) starting from the lowest energy up to some highest value, the Fermi energy.  (See here for some mention of this.)   Empty electronic states are available at essentially no energy cost - exciting electrons from filled states to empty states are "gapless".  
• Electrical conduction takes place through the flow of these electronic quasiparticles.   (For more technical readers:  We can think of these quasiparticles like little wavepackets, and as each one propagates around the wavepacket accumulates a certain amount of phase.  The phases of different quasiparticles are arbitrary, but the change in the phase going around some trajectory is well defined.)
• In a superconductor, there is some effective attractive interaction between electrons that we have thus far neglected.  In conventional superconductors, this involves lattice vibrations (as in this wikipedia description), though other attractive interactions are possible.  At sufficiently low temperatures, the ordinary metal state is unstable, and the system will spontaneously form pairs of electrons (or holes).  Those pairs then condense into a single coherent state described by an amplitude \(|\Psi|\) and a phase, \(\phi\), shared by all the pairs.  The conventional theory of this was formulated by Bardeen, Cooper, and Schrieffer in 1957.  A couple of nice lecture note presentations of this are here (courtesy Yuval Oreg) and here (courtesy Dan Arovas), if you want the technical details.  This leads to an energy gap that characterizes how much it costs to create individual quasiparticles.  Conduction in a superconductor takes place through the flow of pairs.  (A clue to this is the appearance of the \(2e\) in the flux quantization.)
• This taking on of a global phase for the pairs of electrons is a spontaneous breaking of gauge symmetry - this is discussed pedagogically for physics students here.  Understanding this led to figuring out the Anderson-Higgs mechanism, btw. 
• The result is a state with a kind of rigidity; precisely how this leads to the phenomenology of superconductivity is not immediately obvious, to me anyway.  If someone has a link to a great description of this, please put it in the comments.  (Interestingly google gemini is not too bad at discussing this.)
• The existence of this global phase is hugely important, because it's the basis for the Josephson effect(s), which in turn has led to the basis of exquisite magnetic field sensing, all the superconducting approaches to quantum information, and the definition of the volt, etc.
• The paired charge carriers are described by a pairing symmetry of their wave functions in real space.  In conventional BCS superconductors, each pair has no orbital angular momentum ("\(s\)-wave"), and the spins are in a singlet state.  In other superconductors, pairs can have \(l = 1\) orbital angular momentum ("\(p\)-wave", with spins in the triplet configuration), \(l = 2\) orbital angular momentum ("\(d\)-wave", with spins in a singlet again), etc.  The pairing state determines whether the energy gap is directionally uniform (\(s\)-wave) or whether there are directions ("nodes") along which the gap goes to zero.  

I have necessarily left out a ton here.  Superconductivity continues to be both technologically critical and scientifically fascinating.  One major challenge in understanding the microscopic mechanisms behind particular superconductors is that the superconducting state itself is in a sense generic - many of its properties (like phase rigidity) are emergent regardless of the underlying microscopic picture, which is amazing.