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From Black Holes to Superconductors - Part 1

(E) I attended a lecture from Professor Sean Hartnoll from the Stanford Institute of Theoretical Physics, which has many well-known physicists such as Professor Leonard Sussking for his research in String Theory and Professor Andrei Linde for his research in the Inflationary Universe Theory. Professor Hartnoll is working on a new field that attempts to understand some of the properties of Black Holes that should help us better understand the “exotic” phases of low temperature matter such as superconductors.

Superconductors exhibit zero resistivity below a certain critical temperature, can sustain currents over time, and exclude magnetic fields, a phenomenon called the Meissner Effect. High-speed trains use superconductors magnets, cooled with liquid nitrogen, to create both lift and propulsion in order to minimize their friction with their rails. Superconductors are also used in Josephson junctions that are key for the design of the silicon for quantum computers or superconducting quantum circuits.

One of the present challenges for solid-state physicists and electrical engineers is to find new materials that could superconduct at room temperature. The theory of superconductivity has been articulated since 1957 by the BCS theory from Bardeen, Cooper, and Schrieffer. But while high temperature superconductors that can superconduct at 133 Kelvin have been identified for over 30 years, their properties are still not understood.

So could understanding Black Holes lead us to build the next generation of quantum computers?

Following are my notes from the lecture and below the video of the lecture:

Many-Body Physics:

Total Energy = Kinetics Energy + Potential Energy

Gas have high kinetics energy while solids have high potential energy.

Phase Diagram (from Matthieu Marechal/Wikipedia)

Quantum Phase Transition:

Heisenberg’s uncertainty principle: Δx . Δp ≥ ℏ

Quantum phase transition @ T = 0

Example 1: Mott Transition

Change from metal state to insulator state due to dopping

Example 2: Antiferromagnetism

Spins of electrons pointing in opposite directions

Antiferromagnetic Ordering (from Michael Schmid/Wikipedia)

Quantum Liquids:

Quantum liquids exists in high-temperature superconductors.

Liquid exists in a no-man’s land or between solid (order winning) and gas (disorder winning).

Similarly, quantum liquid exists in a no-man’s land or between tendency to be ordered and tendency to be very quantum mechanically delocalized and carrying electrical currents.

Highly simplified: their motion/lifetime ( 𝜏 ) is linked to their interaction/excitation: 𝜏 ~ 1/T

Black Holes (BH):

Concepts from John Michell (1783) and Pierre-Simon LaPlace (1796)

Einstein’s General Relativity (1915): Light also feels gravity - All speeds are lower than C

Law of Irreversibility:

Second law of Thermodynamics: Entropy also increases

Empirical evidence of Black Holes Irreversibility: Nothing get out!

Similarity between the Laws of Thermodynamics and Black Holes Classical Mechanics:

1st law of thermodynamics - more energy more entropy: dE = T dS

2nd law of thermodynamics - entropy increases over time: dS/dt > 0

E = M c2

1st law of Black Holes classical mechanics - the change of mass of the Black Hole is related to its change of area: dM = K dA

2nd law of Black Holes classical mechanics - Hawking’s Area theorem - the horizon area is a non decreasing function of time: dA/dt > 0

M = mass of the BH; A = area of the BH; K = surface gravity of the BH

From comparing the laws of thermodynamics and Black Holes mechanics, we can infer that:

First: S ∝ A : Holographic Principle - the entropy of the Black Hole is proportional to its area

Second: T ∝ K : the temperature of the Black Hole is proportional to its surface gravity

Black Holes Quantum Mechanics:

Now leveraging quantum mechanics, Hawking proposed:

T = ℏ K

As a consequence:

dE = dM = ℏ K dA/ ℏ => S ∝ A/ℏ

Note that the entropy for a Black Hole is 10 to the power of 23 the entropy of our sun!

How Does a Black Hole React to Perturbation?

Diffusion Equation of Fluids: dn/dt = D d2n/dx2

Applied to a Black Holes - using classical mechanics: D ~ c2 ℏ /kT

=> 𝜏 ~ 1/T

So, when you excite a Black Hole, the rate at which the excitement decays is linked to the temperature of the Black Hole.

So could we assume that Black Holes looks like Quantum Liquids?

Note : The picture above is Zapatista, a painting from Jose Clemente Orozco from New York’s MoMA.

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Sunday, March 8, 2015