High density quark matter and color superconductivity

How does matter behave at ultra-high densities? So dense that the the atoms themselves collapse, and nuclei are squeezed together. So dense that a supertanker full of oil would be 1 mm³ in size. Amazingly enough, there are places in the universe where this actually happens: neutron stars. Neutron stars are the remnants of ordinary stars that have exploded spectacularly as supernovae. After the explosion, gravity crushes the remaining matter into a super-dense lump called a neutron star or compact star.

Below the surface of the neutron star, the pressure due to gravity is so extreme that there are no longer any atoms: everything is compressed down to a liquid of neutrons, with a few protons and electrons as well. But for my research I am interested in even higher densities. As you burrow down into the core of the neutron star, the pressure rises relentlessly. We don't know what happens at the center, but if the density there is high enough then the neutrons themselves will be crushed out of existence, liberating the quarks inside. If that happens, the core will consist of a liquid of quarks: quark matter. My research is about the properties of quark matter, which turns out to have remarkable similarities to the state of electrons in a metal, including a type of superconductivity called color superconductivity.

Phase diagram for matter at extreme density/temperature

Physicists often summarize the properties of matter over a range of densities and temperatures by drawing a phase diagram. Even for a commonplace substance like water, the phase diagram is surprisingly complicated when you take into account all the different types of ice.

But I am interested in the phases of matter under far more extreme conditions: trillions of times denser than ice, and trillions of times hotter than room temperature. In this realm we don't have much experimental information to guide us. In principle we should be able to calculate the behavior of matter under such conditions, using the the theory that describes the strong nuclear force, which is the dominant interaction at ultra-high density. Unfortunately, this theory, Quantum ChromoDynamics (QCD) is difficult to work with, and becomes intractable when we introduce the chemical potential µ that is needed to get a finite density of quarks. So we don't really know what the phase diagram looks like, but we can make a reasonable guess:

Conjectured phase diagram of matter at extreme temperature and density:

[phase diagram of QCD]

For a very readable review of the QCD phase diagram, see Simon Hands, "The phase diagram of QCD" (published in Contemp. Phys. 42, 209 (2001)).

Along the horizontal axis the temperature is zero, and the density rises from the onset of nuclear matter through the transition to quark matter. Compact stars are in this region of the phase diagram, although it is not known whether their cores are dense enough to reach the quark matter phase.

Along the vertical axis the temperature rises, taking us through the crossover from the hadronic gas, in which quarks are confined into neutrons and protons, to the quark gluon plasma (QGP), in which quarks and gluons are unconfined. This is the region explored by high-energy heavy-ion colliders such as the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory, and the future Large Hadron Collider (LHC) at CERN.

The yellow-shaded region of the figure is where quark matter, which we expect to be color superconducting, will occur. What do we mean by "color superconducting"? Think of it by analogy with electrons in a metal. At very low temperatures (a few Kelvin), most metals suddenly become superconducting: their resistance drops to zero. This happens because when it is cold enough the electrons can pair up forming Cooper pairs. We think that quarks in quark matter do the same thing.

Superconductivity of electrons and quarks

The reason why we expect quarks to behave like electrons, forming Cooper pairs which produce a superconducting state, is that the mechanism that causes the pairing is very general. All you need is a high density of Fermions (quarks and electrons are both Fermions) with an attractive interaction. Let us see how this works.

Fermions in general

The Fermi sea for free fermions

Fermi sea diagram

Fermions are particles that obey the Pauli exclusion principle, which says that no two fermions can be in the same state. So as you add more and more fermions to a finite-sized box, you have to put them in higher and higher momentum states. For non-interacting fermions at zero temperature you would just end up with a Fermi sea of filled states: all states with energy less than the Fermi energy E_f = µ are filled, and all states above E_f are empty.

(The filled negative energy states form the Dirac sea: removing a particle from the Dirac sea creates an antiparticle. The filled positive energy states form the Fermi sea: removing a particle from the Fermi sea creates a hole.)

But if there is an attractive interaction between the fermions then things are very different. The fermions near the Fermi surface pair up to form lots of Cooper pairs, which settle down in a "condensate". This state, a condensate of Cooper pairs, always forms because you can show that it has lower free energy than the simple Fermi sea depicted above. This was first explained by Bardeen, Cooper, and Schrieffer (BCS).

It is actually quite easy to understand intuitively why a condensate of Cooper pairs forms. The system tries to minimize its "free energy" F = E - µN, where E is the total energy of the system, µ is the chemical potential for quarks, and N is the number of fermions. The Fermi surface is defined by the Fermi energy E_f = µ, at which, if we ignore the attractive interaction, the free energy is minimized, so adding or subtracting a single particle costs zero free energy. For example, adding a particle costs energy E_f because that is the lowest unoccupied state, but it increases fermion number N by 1, so F is unchanged. Now switch on a weak attractive interaction. It still costs no free energy to add a pair of particles (or holes) close to the Fermi surface, but the attractive interaction between them then lowers the free energy of the system. Many such pairs will therefore be created in the modes near the Fermi surface, and these pairs, being bosonic, will form a condensate. The ground state will be a superposition of states with all numbers of pairs, breaking the fermion number symmetry.

Superconductivity of electrons

In the case of electrons, their dominant interaction is electrostatic repulsion, and it is only the presence of a background lattice of positively charged ions in a metal that allows additional attractive phonon-mediated interactions to exist. The resultant Cooper pairing is rather fragile, and easily disrupted by thermal fluctuations, hence metals only become superconducting at very low temperatures. The condensate of Cooper pairs of electrons is charged, and as a result the photon, which couples to electric charge, becomes massive. Superconducting metals therefore contain neither electric nor magnetic fields. A perfect conductor cannot contain electric fields (the charges would rearrange themselves to cancel it), but the special thing about a superconductor is that it expels magnetic fields as well: the Meissner effect.

Color Superconductivity of quarks

For quarks things are very different. The dominant interaction between quarks is the strong interaction, described by QCD, which is very attractive in some channels (after all, QCD binds quarks together to form baryons). This leads us to expect that quarks will form Cooper pairs very readily and that quark matter will generically acquire a condensate of Cooper pairs. Since pairs of quarks cannot be color-neutral, the resulting condensate will break the local color symmetry, making the gluons massive. We call this color superconductivity. Note that the quark pairs play the same role here as the Higgs particle does in the standard model: the color-superconducting phase can be thought of as the Higgs phase of QCD.

Color superconducting phases

Color superconducting quark matter can come in a rich multiplicity of different possible phases, based on different pairing patterns of the quarks. This is possible because quarks come in three different colors, and at the density of a compact star core we expect three different flavors: up, down, and strange. Recent work has concentrated on calculating which type of pairing is favored at which density. This is a complicated problem, in which we must take into account the requirement that bulk matter be neutral with respect to both electric and color charge, as well as equilibration under the weak interaction processes that can turn one quark flavor into another, and finally the strange quark mass. The results so far, starting at the highest densities and working down, are roughly this:

Highest densities: color-flavor-locked (CFL) quark pairing, in which all three flavors participate symmetrically. CFL quark matter has many special properties, including the fact that chiral symmetry is broken by a new mechanism: the quark pairs themselves, instead of the more conventional chiral condensate. There may be kaon condensation.
Very high densities. Gapless CFL, in which holes start to open up in the CFL pairing pattern, leaving some quarks unpaired.
Middle high densities: unknown. Many possibilities have been suggested, including crystalline pairing, two-flavor pairing, single flavor pairing, color-spin locking, etc.
Conventional nuclear matter.

So we know what phase is favored in the limit of infinite density, but the nature of the pairing in quark matter at realistic neutron-star densities is still a vigorously debated question.