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.

The Wikipedia contains articles on color superconductivity and quark matter.
For a short popular article on quark matter, see Frontiers. A more in-depth review, written for Annual Reviews, is available at this http URL.

1. 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.
2. 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 Ef = µ are filled, and all states above Ef 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 Ef = 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 Ef 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.
3. 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.

Other topics of ongoing research include:

* Theoretical questions:
o What are the properties of the suggested crystalline phase?
o Better weak-coupling calculations, include vertex corrections
o Go beyond mean-field approximation, include fluctuations.
* Neutron-star signatures of color superconductivity:
o Equation of state and mass-radius relationship
o Properties of nuclear-quark interface.
o Studies of transport properties
+ conductivity and emissivity (neutrino cooling)
+ shear and bulk viscosity ("r-mode spin-down")

Black Hole

A brief history of black holes

A black hole is an object or region of space where the pull of gravity is so strong that nothing can escape from it, i.e. the escape velocity exceeds the speed of light. The term was coined in 1968 by the physicist John Wheeler. However, the possibility that a lump of matter could be compressed to the point at which its surface gravity would prevent even the escape of light was first suggested in the late 18th century by the English physicist John Michell (c.1724-1793), and then by Pierre Simon, Marquis de Laplace (1749-1827).

Black holes began to take on their modern form in 1916 when the German astronomer Karl Schwarzschild (1873-1916) used Einstein's general theory of relativity to find out what would happen if all the mass of an object were squeezed down to a dimensionless point – a singularity. He discovered that around the infinitely compressed matter would appear a spherical region of space out of which nothing could return to the normal universe. This boundary is known as the event horizon since no event that occurs inside it can ever be observed from the outside. Although Schwarzschild's calculations caused little stir at the time, interest was rekindled in them when, in 1939, J. Robert Oppenheimer, of atomic bomb fame, and Hartland Snyder, a graduate student, described a mechanism by which black holes might actually be created in the real universe. A star that has exhausted all its useful nuclear fuel, they found, can no longer support itself against the inward pull of its own gravity. The stellar remains begin to shrink rapidly. If the collapsing star manages to hold on to a critical amount of mass, no force in the Universe can halt its contraction and, in a fraction of a second, the material of the star is squeezed down into the singularity of a black hole.
Stellar black holes


Artist's impression of Cygnus X-1
In theory, any mass if sufficiently compressed would become a black hole. The Sun would suffer this fate if it were shrunk down to a ball about 2.5 km in diameter. In practice, a stellar black hole is only likely to result from a heavyweight star whose remnant core exceeds the Oppenheimer-Volkoff limit following a supernova explosion.

More than two dozen stellar black holes have been tentatively identified in the Milky Way, all of them part of binary systems in which the other component is a visible star. A handful of stellar black holes have also been discovered in neighboring galaxies. Observations of highly variable X-ray emission from the accretion disk surrounding the dark companion together with a mass determined from observations of the visible star, enable a black hole characterization to be made.

Among the best stellar black hole candidates are Cygnus X-1, V404 Cygni, and several microquasars. The two heaviest known stellar black holes lie in galaxies outside our own. One of these black hole heavyweights, called M33 X-7, is in the Triangulum Galaxy (M33), 3 million light-years from the Milky Way, and has a mass of 15.7 times that of the Sun. Another, whose discovery was announced in October 2007, just a few weeks after that of M33 X-7, is called IC 10 X-1, and lies in the nearby dwarf galaxy, IC 10, 1.8 million light-years away. IC 10 X-1 shattered the record for a stellar black hole with its mass of 24 to 33 times that of the Sun. Given that massive stars lose a significant fraction of their content through violent stellar winds toward the end of their lives, and that interaction between the members of a binary system can further increase the mass loss of the heavier star, it is a challenge to theorists to explain how any star could retain enough matter to form a black hole as heavy as that of IC 10 X-1.

The microquasar V4641 Sagittarii contains the closest known black hole to Earth, with a distance of about 1,500 light-years.

Supermassive, intermediate-mass and mini black holes

Supermassive black holes are known almost certainly to exist at the center of many large galaxies, and to be the ultimate source of energy behind the phenomenon of the active galactic nucleus. At the other end of the scale, it has been hypothesized that countless numbers of mini black holes may populate the universe, having been formed in the early stages of the Big Bang; however, there is yet no observational evidence for them.

In 2002, astronomers found a missing link between stellar-mass black holes and the supermassive variety in the form of middleweight black holes at the center of some large globular clusters. The giant G1 cluster in the Andromeda Galaxy appears to contain a black hole of some 20,000 solar masses. Another globular cluster, 32,000 light-years away within our own Milky Way, apparently harbors a similar object weighing 4,000 solar masses. Interestingly, the ratio of the black hole's mass to the total mass of the host cluster appears constant, at about 0.5%. This proportion matches that of a typical supermassive black hole at a galaxy's center, compared to the total galactic mass. If this result turns out to be true for many more cluster black holes, it will suggest some profound link between the way the two types of black hole form. It is possible that supermassive black holes form when clusters deposit their middleweight black hole cargoes in the galactic centers, and they merge together.


Inside a black hole

According to the general theory of relativity, the material inside a black hole is squashed inside an infinitely dense point, known as a singularity. This is surrounded by the event horizon at which the escape velocity equals the speed of light and that thus marks the outer boundary of a black hole. Nothing from within the event horizon can travel back into the outside universe; on the other hand, matter and energy can pass through this surface-of-no-return from outside and travel deeper into the black hole.

For a non-rotating black hole, the event horizon is a spherical surface, with a radius equal to the Schwarzschild radius, centered on the singularity at the black hole's heart. For a spinning black hole (a much more likely contingency in reality), the event horizon is distorted – in effect, caused to bulge at the equator by the rotation. Within the event horizon, objects and information can only move inward, quickly reaching the singularity. A technical exception is Hawking radiation, a quantum mechanical process that is unimaginably weak for massive black holes but that would tend to cause the mini variety to explode.

Three distinct types of black hole are recognized:

* A Schwarzschild black hole is characterized solely by its mass, lacking both rotation and charge. It possesses both an event horizon and a point singularity.


* A Kerr black hole is formed by rotating matter, possesses a ring singularity, and is of interest in connection with time travel since it permits closed time-like paths (through the ring).


* A Reissner-Nordstrom black hole is formed from non-rotating but electrically-charged matter. When collapsing, such an object forms a Cauchy horizon but whether it also forms closed time-like paths is uncertain.

The equations of general relativity also allow for the possibility of spacetime tunnels, or wormholes, connected to the mouths of black holes. These could act as shortcuts linking remote points of the universe. Unfortunately, they appear to be useless for travel or even for sending messages since any matter or energy attempting to pass through them would immediately cause their gravitational collapse. Yet not all is lost. Wormholes, leading to remote regions in space, might be traversable if some means can be found to hold them open long enough for a signal, or a spacecraft, to pass through.

How black holes work

Escape Velocity

If ball is thrown upwards from the surface of the Earth it reaches a certain height and then falls back. The harder it is thrown, the higher it goes. Laplace calculated the height it would reach for a given initial speed. He found that the height increased faster than the speed, so that the height became very large for a not very great speed. At a speed of 40000 km/h (25000 mph, only 20 times faster than Concorde) the height becomes very great indeed - it tends to infinity, as the mathematician would say. This speed is called the `escape velocity' from the surface of the Earth, and is the speed which must be achieved if a space craft is to reach the Moon or any of the planets. Being a mathematician, Laplace solved the problem for all round bodies, not just the Earth.

He found a very simple formula for the escape velocity. This formula says that small but massive objects have large escape velocities. For example if the Earth could be squeezed and made four times smaller, the escape velocity would need to be twice as large. This surprisingly simple derivation gives exactly the same answer as is obtained from the full theory of relativity.

Light travels at just over 1000 million km/h (670 million mph), and in 1905 Albert Einstein proved in the Special Theory of Relativity that nothing can travel faster than light. The above Laplace formula can be turned around to tell us what radius an object must have if the escape velocity from its surface is to be the speed of light. This particular radius is called the `Schwarzschild radius' in honor of the German astronomer who first derived it from Einstein's theory of gravity (General Theory of Relativity). The formula tells us that the Schwarzschild radius for the Earth is less than a centimeters, compared with its actual radius of 6357 km.


Apparent versus Event Horizon

As a doomed star reaches its critical circumference, an "apparent" event horizon forms suddenly. Why "apparent?" Because it separates light rays that are trapped inside a black hole from those that can move away from it. However, some light rays that are moving away at a given instant of time may find themselves trapped later if more matter or energy falls into the black hole, increasing its gravitational pull. The event horizon is traced out by "critical" light rays that will never escape or fall in. Even before the star meets its final doom, the event horizon forms at the center, balloons out and breaks through the star's surface at the very moment it shrinks through the critical circumference. At this point in time, the apparent and event horizons merge as one: the horizon. For more details, see the caption for the above diagram. The distinction between apparent horizon and event horizon may seem subtle, even obscure. Nevertheless the difference becomes important in computer simulations of how black holes form and evolve. Beyond the event horizon, nothing, not even light, can escape. So the event horizon acts as a kind of "surface" or "skin" beyond which we can venture but cannot see. Imagine what happens as you approach the horizon, and then cross the threshold.
Care to take a one-way trip into a black hole?



The Singularity

At the center of a black hole lies the singularity, where matter is crushed to infinite density, the pull of gravity is infinitely strong, and space-time has infinite curvature. Here it's no longer meaningful to speak of space and time, much less space-time. Jumbled up at the singularity, space and time cease to exist as we know them.
The Limits of Physical Law

Newton and Einstein may have looked at the universe very differently, but they would have agreed on one thing: all physical laws are inherently bound up with a coherent fabric of space and time. At the singularity, though, the laws of physics, including General Relativity, break down. Enter the strange world of quantum gravity. In this bizarre realm in which space and time are broken apart, cause and effect cannot be unraveled. Even today, there is no satisfactory theory for what happens at and beyond the singularity.



It's no surprise that throughout his life Einstein rejected the possibility of singularities. So disturbing were the implications that, by the late 1960s, physicists conjectured that the universe forbade "naked singularities." After all, if a singularity were "naked," it could alter the whole universe unpredictably. All singularities within the universe must therefore be "clothed." But inside what? The event horizon, of course! Cosmic censorship is thus enforced. Not so, however, for that ultimate cosmic singularity that gave rise to the Big Bang