The second and last blog post in this thread is about the analogies between black holes and condensed matter theory, in particular with phases of matter like fluids or superconductors. That is what explains my previous picture…And this post! Are BH geometries exotic classes of “matter”?

At current time (circa 2017), superconductors are not yet completely understood. They are a hard part of quantum many body physics!

Let me begin by something simpler you know from school. Imagine a single particle. Energy is conserved in virtue of symmetries (in particular by time-traslation invariance and the Noether theorem). Mechanical energy is, as you surely remember, the sum of kinetic energy and potential energy. . Imagine now a large number of particles in a closed box. Who wins? Kinetic energy or potential energy? The answer gives you a hint of the state of the matter the particles have! If kinetic energy wins, you get a gas. If potential energy wins, you get a solid, in general regular configurations and ordered configurations are preferred (why? Good question! Let me talk about this in the future). Maximal potential energy configurations are provided by crystals or lattices. I am sure you know the conventional phase diagram for solids, liquids and gases. Here you are two, one simpler, one more complex:

Liquids are more complicated phases of normal matter, where not the kinetic energy or the potential energy wins. Liquids are complex phases “in between” solids and gases. In a liquid, there is a concrete balance between the kinetic energy and the potential energy. It is a complex and complicated balance. Molecules in a liquid become attracted but they are not rigid like a solid!

What about quantum liquids? Quantum mechanics introduces fluctuations via the Heisenberg Uncertainty principle, . Transitions between different phases or states of matter (energy) are related to fluctuations in their particle energies (average temperature and interactions). Quantum phase transitions are phase transitions happening at , at least theorically (we do know we can not reach the absolute zero).

*Example 1.* Mott transitions. Some metals, conductors, become insulators! Hole-particle wires explain conducting metals. The electron-electron repulsion can become strong. If so, a metal could become an insulator, something called Mott insulator. Mott insulators, under certain doping prescriptions, can become superconductors too! Doping is a parameter here. It also happens at low temperatures, close to absolute zero.

*Example 2.* Antiferromagnetism. Electrons have electric charge AND spin. Spin can be aligned (ferromagnetism). The magnetic field correlates with spin. There is also antiferromagnetism. More precisely, there are four phases in ferromagnetism:

- Ferromagnetic phase. Below some critical temperature, spins are completely aligned and parallel to the magnetic field. They form magnetic domains.
- Antiferromagnetic phase. Below some critical temperature, spins are aligned antiparallel in magnetic domains.
- Ferrimagnetic phase. Below some critical temperature, spins are aligned antiparallel and parallel, but the total magnetic field does NOT cancel.
- Paramagnetic phase. Spins are randomonly oriented. It is above any of the critical temperatures mentioned before.

Diamagnetism is another related behaviour. Spins tend to align parallel to external fields. It happens in superconductors or some organic compounds, metals,…

Another quantum phase transition is quark-gluon plasma (QGP). It happened in the early Universe or the LHC (ALICE experiment) and other colliders. Also, you could try to simulate it, but you would need a supercomputer. Expensive? Yes! At least now, circa 2017. The unknown physics of QGP is related to the Hagedorn temperature, the Hagedorn transitions. Some hints of behaviour occurs there as well.

Phases of matter do matter in the History of the Universe. In Quantum Mechanics, there is an order parameter that controls the transition. For instance, antiferromagnetism at absolute zero, increasing the temperature (or changing pressure, doping), can become disordered. Weird phases of matter in between metal-insulator, like those Mott insulators I mentioned above! In between ferromagnetic and antiferromagnetic phases are just like a liquid in between solids and gases. They are phases where quantum fluctuations are important. A natural question arises: do there exist quantum liquids in between ordered and disordered phases? Yes! They exist in materials that many scientists are studying right now! For instance, of course, high temperature superconductivity. Understanding the phases in between conductors and superconductors could hold the key to understand superconductors fully! Even the recently found time crystals are examples of quantum phases! Let me put these words into pictures:

There is a relative importance of motion versus interactions in strange phases of metals. They are quantified by lifetime of excitations. Excitations are ripples in electron-hole pairs. If you have a lifetime , the motion wins whenever . Interactions win whenever . These are the natural units of hole-electron timelifes. It is related to temperature . Just like black holes! Thermal energy is related to kinetic energy. Also in light waves. So, to some extend, you could, in principle, create liquid light or “solid” light (of course, I know about liquid light but not about solid light, yet!). Thus, , and quantum mechanics relate . Long lived particles imply , where T is the temperature. You can also realize a connection between time and temperature. A subtle one! For a quantum liquid, we have . Doping metals, change how they behave. This is well-known in solid-state and condensed matter theory but not in theoretical physics I am going to discuss now!

Black holes were proposed originally by J. Mitchell in 1783, as places where light can not escape! A similar idea was envisioned by Laplace in 1796. The theory of general relativity, in 1915, redefined this concept better. Light also feels gravity. Black holes imply certain irreversibility, a deep concept and notion in physics, similar to the notion of entropy. At last, it seems black hole “are” entropy. We never keep track off all the details and we are doing some coarse-graining. This is the battle macrostates versus microstates. Molecules, atoms and particles are important in the distinction.

The Universe began with a low entropy state. It is crazy, because this is quite unlikely, but we think this is true. This low entropy state evolved towards a state with growing entropy. Now, some questions are got:

- May black holes have an entropy associated to them? Yes! I have written the formula several times in this blog. Hawking is famous by it! Entropy of any BH scales as area:
Moreover, the area theorem states that for black holes, just like entropy in thermodynamics. That entropy is proportional to area and not volume, is the origin of the holographic principle. for black holes as well, where , and Hawking proved that , where is the surface gravity. Therefore, black hole thermodynamics is just like any other thermodynamical theory. Or isn’t it? We are lost the microstates! We don’t know yet the fundamental degrees of freedom of spacetime, i.e., the atoms of space-time. The thermodynamical analogy is very powerful and it is now a solid field completely established. As I mentioned above, and in the previous post, Hawking’s calculation using Quantum Field Theory (QFT) and Quantum Mechanics to black holes proved they have also a temperature. Black hole temperature is a quantum effect! Note the appearance of Planck constant in the Hawking temperature! BH have a huge entropy. Even a solar mass BH has a very big entropy. The formation of black holes releases (hopefully) unknown degrees of freedom. Curiously, a solar mass black hole has about times the sun entropy, and there is also stars in the observed universe! Recently, Verlinde proposed the idea of entropic forces in gravity. Gravity manifests itself as irreversibility from entropy!

- What is a BH made of? What are the BH microstates?
- How do BHs react to perturbations? Charges thrown to a stationary BH do create ripples on it! They could be described by diffusion equations! Heat diffuses nd charges also diffuse through the BH event horizon! BH should be close at thermal equilibrium.
- Are BH a disordered medium? What kind of ordered medium are BHs? Something in between known states? Diffusion equations could provide hints into this. If is the amount of charge (charge density), the diffusion equation reads

The constant says how “quick” or “fast” the “bump relaxation” happens. The more inhomogeneous it is, the faster it relaxes. Are BH like metals? Are BH some type of exotic matter? Are BH topological insulators/superconductors? Are BH a strange phase of space-time? Do BH conduct charges and how? The electrical resitance to the electron motion is an important quantity. We do not expect that astrophysical BH are charged, but accretion processes could provide those charges. What is the BH resistivity? If we connect a source “power” to a BH they are out of equilibrium and they do not evaporate. For instante, imagine a BH connected to some engine or source power. BH could behave like a quantum liquid in some circumstances! Classical BH do have some types of charge (). , , and . How to measure a low T (very massive) BH? We have not even measured the cosmic neutrino background, about , or the cosmic gravitational background, about , so how to measure supermassive BH temperatures? A real problem, indeed…

In summary:

- There are some difficult or hard materials, such as high temperature superconductors, topological insulators/superconductors and many others.
- There are materials that are neither ordered nor disordered.
- Liquids are the classical version of quantum liquids, a harder phase of matter.
- Classical BH have transition phase properties: surface gravity, mass or energy, entropy, charges,…When probed, BH behave strangely, just like a medium with certain time scale , neither long nor short, having not long not short times. It is like the “in between” phases of matter we have introduced here.

Duality is a big idea in current and moder theoretical physics. Traditionally, we, scientists, have been reductionalist. This is the origin of atomism (Democritus, Leucipus). Anaxagoras thought that everything is inside anything. Liquids are precisely where it is NOT clear what the building blocks are. Imagine water molecules in a turbulent river. They are complicated. Duality is the idea that a physical system admits many different descriptions that are fully equivalent. The first example of duality is the electromagnetic duality between electric charges and magnetic charges in vacuum. Another example, quite classical. In Quantum Electrodynamics (QED), you have light (photons) and matter (electrons). . Electromagnetic lines are made up of photons! QED works “nicely” and well because photons don’t interact strongly in normal matter, and . In 1970’s a question arised in Quantum Chromodynamics (QCD): what if the fluxtube of quarks-gluons is more fundamental that the own quarks are gluons? This is, of course, the origin of string theory. Either fluxtubes are fundamental and particles are a derived or emergent concept, or particles are fundamental and fluxtubes are a derived concept. What is then a fundamental or derived concept? It depends on the context! (Just for QM interpreters: contextuality is the key?) Duality is mathematically a simple change of variables from a description using certain degrees of freedom to another variables (the dual variables). Just like phonons in a crystal (solid) are useful, duality holds whenever a field strength is changed to its dual, and the fields transform into other dual fields. In conclusion, duality is, then, the idea that there are different descriptions of the same physics (physical system). It was a hot topic since the 2nd superstring revolution (circa 1995) and the seminal works by Maldacena (circa 1997) about the so-called AdS/CFT correspondence (another duality). The AdS/CFT correspondence is just the claim that the dynamics of certain BH with AdS geometry are dual to certain quantum liquid described by a conformal field theory (CFT), originally N=4 SYM (Super-Yang Mills) theory. Some questions with this condensed matter language would arise:

- Does a quantum liquid described by this AdS/CFT duality exists? Can it be simulated? Can it become superconductor or reach strange phases at low temperatures?
- Does a BH described by AdS/CFT geometry become superconductor or reach strange phases at low temperatures?

The last ultimate question is, of course, is a BH a superconductor or weird type of matter (geometry)? The no-hair theorem, originally by Wheeler, suggests that a BH can not be coated of doped with lots of charges or “stuff”. You can’t just put any outside a BH, if you throw something there, it wants to fall in. There are only (now we have changed a little bit this thought) parameters or charges for a BH. In principle, a BH should radiate off the charges to infinity, but classically it does not happen. This is the information paradox we are eager to solve, since it paves the way to understand quantum gravity! What kind of superconductor can a BH be? Well, that is much more complicated to answer. The first answer would be that it is like a Higgs boson. The Higgs boson has about 125 GeV of mass/energy, and a lifetime . The Higgs field is like a medium through all space and time. . Certain particles in this medium acquire masses. Different particles get a mass. In a superconductor condensate, like a Higgs condensate, mass are electrons, the Higgs boson itself, W bosons, muons,…In a superconductor state, electrons AND photons get an effective mass. The consequences of the photon becoming massive is seen in the force laws. For massless photons, , while for massive photons . If you remember Ohm’s law for electricity, , then , where is the conductivity. This is very similar to mass in a superconducting condensate. If no electric field on matter, and current flows without electric resistance, or conductivity being infinity, we arrive to a superconductor phase. In superstring theory, we also have , so superstring theory has phases as well. The BH in Maldacena’s duality are different in important ways from these examples:

- Box avoids things radiating out to infinity.
- Charged black holes (RN BH, KNdS BH) are different.
- The electric field can create pair particles via Schwinger mechanism. The electron wants to screen the field, gravity wants to antiscreen the field.

The higher the temperature is, the bigger the BH is…But with boxes or boundaries in AdS/CFT. This is a common example of the battle between electromagnetism (and other forces) against gravity. If a BH is big, high temperature implies gravity wins. And it gets bigger. If the BH is small, low temperature limit, the electromagnetic force wins. And it just condensates some bits of charge outside the BH. And it means a superconductive layer outside the BH could arise! And this means hair…If it is soft hair, as Hawking, Strominger and other have been proposing to solve the information paradox is something to be probed in the near future. From a phenomenological viewpoint, the question is what is the total charge seen from a far away observer, and it should be fixed. What sets the superconductor temperature of a BH, if any? What sets the critical temperature in a quantum liquid? Conventional superconductors have strongly correlated electrons (or Cooper pairs). Cooper paires are bosons but they are truly a couple of strongly paired electrons. High temperature charges related to quantum liquids in this BH “in between” phase of spacetime means that BH could be exotic superconductors. Any link to supersymmetry/SUSY? What about entanglement in superconductors or/and quantum liquids?

These questions are why I love physmatics!

P.S.: Epilogue (part II)…

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