by Ciro Santilli (@cirosantilli, 32)
Condensed matter physics is one of the best examples of emergence. We start with a bunch of small elements which we understand fully at the required level (atoms, electrons, quantum mechanics) but then there are complex properties that show up when we put a bunch of them together.
Includes fun things like:
As of 2020, this is the other "fundamental branch of physics" besides to particle physics/nuclear physics.
Condensed matter is basically chemistry but without reactions: you study a fixed state of matter, not a reaction in which compositions change with time.
Just like in chemistry, you end up getting some very well defined substance properties due to the incredibly large number of atoms.
Just like chemistry, the ultimate goal is to do de-novo computational chemistry to predict those properties.
And just like chemistry, what we can actually is actually very limited in part due to the exponential nature of quantum mechanics.
Also since chemistry involves reactions, chemistry puts a huge focus on liquids and solutions, which is the simplest state of matter to do reactions in.
Condensed matter however can put a lot more emphasis on solids than chemistry, notably because solids are what we generally want in end products, no one likes stuff leaking right?
But it also studies liquids, e.g. notably superfluidity.
One thing condensed matter is particularly obsessed with is the fascinating phenomena of phase transition.
AMO is a slightly more general area than condensed matter physics, including related phenomena with smaller numbers atoms and optics. The two terms are however sometimes used as synonyms. The term AMO has gained wide usage and acceptability, see e.g.:
If Ciro had had greater foresight, this might have been what he studied at university!
Bibliography:
But then they regained their sanity and put the source code on GitHub: github.com/topocm/topocm_content and is CC BY-SA.
Uses an ungodly combination of Jupyter notebooks and Pelican.
How are the bands measured experimentally?
Why are there gaps? Why aren't bands infinite? What determines the width of gaps?
Bibliography:
It really allows you to do alternating current calculations much as you'd do DC calculations with resistors, quite poweful. It must have been all the rage in the 1950s.
If you adda bit of impurities to certain materials, at low temperatures of a few Kelvin their resistivity actually starts increasing if you go below a certain critical temperature.
Figure 1. Kondo effect graph for gold with added impurities. Source.
The basis of 1970-20XX computers, gotta understand them I guess?
Most notable example: gallium arsenide, see also: gallium arsenide vs silicon.
An important class of semiconductors, e.g. there is a dedicated III-V lab at: École Polytechnique: www.3-5lab.fr/contactus.php
Experiments:
  • "An introduction to superconductivity" by Alfred Leitner originally published in 1965, source: www.alfredleitner.com/
  • Isotope effect on the critical temperature. hyperphysics.phy-astr.gsu.edu/hbase/Solids/coop.html mentions that:
    If electrical conduction in mercury were purely electronic, there should be no dependence upon the nuclear masses. This dependence of the critical temperature for superconductivity upon isotopic mass was the first direct evidence for interaction between the electrons and the lattice. This supported the BCS Theory of lattice coupling of electron pairs.
Lectures:
  • Video 1. 20. Fermi gases, BEC-BCS crossover by Wolfgang Ketterle (2014) Source. Part of the "Atomic and Optical Physics" series, uploaded by MIT OpenCourseWare.
    Actually goes into the equations.
    Notably, youtu.be/O_zjGYvP4Ps?t=3278 describes extremely briefly an experimental setup that more directly observes pair condensation.
  • Video 2. Superconductivity and Quantum Mechanics at the Macro-Scale - 1 of 2 by Steven Kivelson (2016) Source. For the Stanford Institute for Theoretical Physics. Gives a reasonable basis overview, but does not go into the meat of BCS it at the end.
Media:
  • Cool CNRS video showing the condensed wave function, and mentioning that "every pair moves at the same speed". To change the speed of one pair, you need to change the speed of all others. That's why there's not energy loss.
Transition into superconductivity can be seen as a phase transition, which happens to be a second-order phase transition.
Video 1. Superconductor, 4-probe measurement by Frederiksen Scientific A/S (2015) Source. OK experiment, illustrates the educational kit they sell. No temperature control, just dumps liquid nitrogen into conductor and watches it drop. But not too bad either. The kit sale link is broken (obviously, enterprise stuff), but there are no archives unfortunately. But it must be some High-temperature superconductor
TODO!!! Even this is hard to find! A clean and minimal one! Why! All we can find are shittly levitating YBCO samples in liquid nitrogen! Maybe because liquid helium is expensive?
Video 1. First 10T Tape Coil by Mark Benz. Source. Dr. Mark Benz describes the first commercially sold superconducting magnet made by him and colleagues in 1965. The 10 Tesla magnet was made at GE Schenectady and they sold magnets to research facilities world wide before the team formed Intermagnetics General. IGC and Carl Rosner went on to pioneer MRI technology.
We know that superfluidity happens more easily in bosons, and so electrons joins in Cooper pairs to form bosons, making a superfluid of Cooper pairs!
Isn't that awesome!
As of 2020, basically means "liquid nitrogen temperature", which is much cheaper than liquid helium.
Figure 1. Timeline of superconductivity from 1900 to 2015. Source.
Upside: superconducting above 92K, which is above the 77K of liquid nitrogen, and therefore much much cheaper to obtain and maintain than liquid helium.
Downside: it is brittle, so how do you make wires out of it? Still, can already be used in certain circuits, e.g. high temperature SQUID devices.
Discovered in 1988, the first high-temperature superconductor which did not contain a rare-earth element.
Superconductivity is one of the key advances of 21st century technology:
As of 2023 the most important ones economicaly were:
The main application is Magnetic resonance imaging. Both of these are have to be Liquid helium, i.e. they are not "high-temperature superconductor" which is a pain. One big strength they have is that they are metallic, and therefore can made into wires, which is crucial to be able to make electromagnetic coils out of them.
TODO, come on, Internet!
Bibliography.
Video 1. Superconducting Short Circuits across Batteries by Eugene Khutoryansky (2020) Source. Well, internal battery resistance acts as the only resistor, and voltage drops to zero immediately outside of the battery. And you get a huge current.
Main theory to explain Type I superconductors very successfully.
TODO can someone please just give the final predictions of BCS, and how they compare to experiments, first of all? Then derive them.
High level concepts:
  • the wave functions of pairs of electrons (fermions) get together to form bosons. This is a phase transition effect, thus the specific sudden transition temperature.
  • the pairs form a Bose-Einstein condensate
  • once this new state is reached, all pairs are somehow entangled into one big wave function, and you so individual lattice imperfections can't move just one single electron off trajectory and make it lose energy
Discrete quantum effect observed in superconductors with a small insulating layer, a device known as a Josephson junction.
To understand the behaviour effect, it is important to look at the Josephson equations consider the following Josephson effect regimes separately:
A good summary from Wikipedia by physicist Andrew Whitaker:
at a junction of two superconductors, a current will flow even if there is no drop in voltage; that when there is a voltage drop, the current should oscillate at a frequency related to the drop in voltage; and that there is a dependence on any magnetic field
Bibliography:
Some golden notes can be found at True Genius: The Life and Science of John Bardeen page 224 and around. Philip W. Anderson commented:
We were all - Josephson, Pippard and myself, as well as various other people who also habitually sat at the Mond tea and participated in the discussions of the next few weeks - very much puzzled by the meaning of the fact that the current depends on the phase
As part of the course Anderson had introduced the concept of broken symmetry in superconductors. Josephson "was fascinated by the idea of broken symmetry, and wondered whether there could be any way of observing it experimentally."
The inaugural that predicted the Josephson effect.
Published on Physics Letters, then a new journal, before they split into Physics Letters A and Physics Letters B. True Genius: The Life and Science of John Bardeen mentions that this choice was made rather than the more prestigious Physical Review Letters because they were not yet so confident about the results.
Paper by Philip W. Anderson and John M. Rowell that first (?) experimentally observed the Josephson effect.
TODO understand the graphs in detail.
They used tin-oxide-lead tunnel at 1.5 K. TODO oxide of what? Why two different metals? They say that both films are 200 nm thick, so maybe it is: 
   -----+------+------+-----
...  Sn | SnO2 | PbO2 | Pb  ...
   -----+------+------------
          100nm 100nm
A reconstruction of their circuit in Ciro's ASCII art circuit diagram notation TODO: 
DC---R_10---X---G
There are not details of the physical construction of course. Reproducibility lol.
Figure 1. Figure 1 of Probable observation of the Josephson superconducting tunneling effect. TODO what do the dotted lines mean?
Figure 2. Figure 2 of Probable observation of the Josephson superconducting tunneling effect.
This is what happens when you apply a DC voltage across a Josephson junction.
It is called "AC effect" because when we apply a DC voltage, it produces an alternating current on the device.
By looking at the Josephson equations, we see that a positive constant, then just increases linearly without bound.
Therefore, from the first equation:
(1)
we see that the current will just vary sinusoidally between .
This meas that we can use a Josephson junction as a perfect voltage to frequency converter.
Wikipedia mentions that this frequency is , so it is very very high, so we are not able to view individual points of the sine curve separately with our instruments.
Also it is likely not going to be very useful for many practical applications in this mode.
Figure 1. I-V curve of the AC Josephson effect. Source.
Voltage is horizontal, current vertical. The vertical bar in the middle is the effect of interest: the current is going up and down very quickly between , the Josephson current of the device. Because it is too quick for the oscilloscope, we just see a solid vertical bar.
The non vertical curves at right and left are just other effects we are not interested in.
TODO what does it mean that there is no line at all near the central vertical line? What happens at those voltages?
Video 1. Superconducting Transition of Josephson junction by Christina Wicker (2016) Source. Amazing video that presumably shows the screen of a digital oscilloscope doing a voltage sweep as temperature is reduced and superconductivity is reached.
Figure 2. I-V curve of a superconducting tunnel junction. So it appears that there is a zero current between and . Why doesn't it show up on the oscilloscope sweeps, e.g. Video 1. "Superconducting Transition of Josephson junction by Christina Wicker (2016)"?
If you shine microwave radiation on a Josephson junction, it produces a fixed average voltage that depends only on the frequency of the microwave. TODO how is that done more preciesely? How to you produce and inject microwaves into the thing?
It acts therefore as a perfect frequency to voltage converter.
The Wiki page gives the formula: en.wikipedia.org/wiki/Josephson_effect#The_inverse_AC_Josephson_effect You get several sinusoidal harmonics, so the output is not a perfect sine. But the infinite sum of the harmonics has a fixed average voltage value.
And en.wikipedia.org/wiki/Josephson_voltage_standard#Josephson_effect mentions that the effect is independent of the junction material, physical dimension or temperature.
All of the above, compounded with the fact that we are able to generate microwaves with extremely precise frequency with an atomic clock, makes this phenomenon perfect as a Volt standard, the Josephson voltage standard.
TODO understand how/why it works better.
Two equations derived from first principles by Brian Josephson that characterize the device, somewhat like an I-V curve:
(1)
where:
  • : Josephson current
  • : the Josephson phase, a function defined by the second equation plus initial conditions
  • : input voltage of the system
  • : current across the junction, determined by the input voltage
Note how these equations are not a typical I-V curve, as they are not an instantaneous dependency between voltage and current: the history of the voltage matters! Or in other words, the system has an internal state, represented by the Josephson phase at a given point in time.
To understand them better, it is important to look at some important cases separately:
Maximum current that can flow across a Josephson junction, as can be directly seen from the Josephson equations.
Is a fixed characteristic value of the physical construction of the junction.
A function defined by the second of the Josephson equations plus initial conditions.
It represents an internal state of the junction.
A device that exhibits the Josephson effect.
Figure 1. Electron microscope image of a Josephson junction its I-V curve. Source.
TODO is there any relationship between this and the Josephson effect?
This appears to happen to any superconducting loop, because the superconducting wave function has to be continuous.
Video "Superconducting Qubit by NTT SCL (2015)" suggests that anything in between gets cancelled out by a superposition of current in both directions.
The first published experimental observation of the magnetic flux quantum.
The paper that follows it in the journal is also of interest, "Theoretical Considerations Concerning Quantized Magnetic Flux In Superconducting Cylinders" by N. Byers and C. N. Yang, it starts:
In a recent experiment, the magnetic flux through a superconducting ring has been found to be quantized in units of ch/2e. Quantization in twice this unit has been briefly discussed by London' and by Onsager. ' Onsager' has also considered the possibility of quantization in units ch/2e due to pairs of electrons forming quasi-bosons.
So there was some previous confusion about the flux quantum due to the presence of Cooper pairs or not.
Dumping the fitures at: archive.org/details/experimental-evidence-for-quantized-flux-in-superconducting-cylinders One day we can also dump the paper scans when it goes into the public domain in 2056! Public domain scientific paper by year.
Figure 1. Figure 1 of Experimental Evidence for Quantized Flux in Superconducting Cylinders. The legend reads:
(Upper) Trapped flux in cylinder No. 1 as a function of magnetic field in which the cylinder was cooled below the superconducting transition. temperature. The open circles are individual data points. The solid circles represent th, e average value of all data points at a particular value of applied field including all the points plotted and additional data which could not be plotted due to severe overlapping of points. Approximately two hundred data points are represented. The lines are drawn at multiples of hc/2e.
(Lower) Net flux in cylinder No. 1 before turning off the applied field in which it was cooled as a function of the applied field. Open and solid circles have the same significance as above. The lower line is the diamagnetic calibration to which all runs have been normalized. The other lines are translated vertically by successive steps of hc/2e.
Figure 2. Figure 2 of Experimental Evidence for Quantized Flux in Superconducting Cylinders. The legend reads:
(Upper) Trapped flux in cylinder No. 2 as a function of magnetic field in which the cylinder was cooled below the superconducting transition temperature. The circles and triangles indicate points for oppositely directed applied fields. Lines are drawn at multiples of hc/2e.
(Lower) Net flux in cylinder No. 2 before turning off the applied field as a function of the applied field. The circles and triangles are points for oppositely directed applied fields. The lower line is the diamagnetic calibration to which all runs have The other been normalized. lines are translated vertically by successive steps of hc/2e.
The inverse of the magnetic flux quantum.
As mentioned in True Genius: The Life and Science of John Bardeen page 224, the idea of symmetry breaking was a major motivation in Josephson's study of the Josephson effect.
The most practical/precise volt standard.
It motivated the definition of the ampere in the 2019 redefinition of the SI base units
The wiki page en.wikipedia.org/wiki/Josephson_voltage_standard contains amazing schematics of the device, apparently made by the US Government.
Figure 1. Source.
Can be used as a very precise magnetometer.
There are high temperature yttrium barium copper oxide ones that work on liquid nitrogen.
Video 1. Superconducting Quantum Interference Device by Felipe Contipelli (2019) Source. Good intuiotionistic video. Some points deserved a bit more detail.
Video 2. Mishmash of SQUID interviews and talks by Bartek Glowaki. Source.
The videos come from: www.ascg.msm.cam.ac.uk/lectures/. Vintage.
Mentions that the SQUID device is analogous to a double-slit experiment.
One of the segments is by John Clarke.
Video 3. Superconducting Quantum Interference Devices by UNSW Physics (2020) Source.
An experimental lab video for COVID-19 lockdown. Thanks, COVID-19. Presented by a cute and awkward Adam Stewart.
Uses a SQUID device and control system made by STAR Cryoelectronics. We can see Mr. SQUID EB-03 written on the probe and control box, that is their educational product.
As mentioned on the Mr. SQUID specs, it is a high-temperature superconductor, so liquid nitrogen is used.
He then measures the I-V curve on an Agilent Technologies oscilloscope.
Unfortunately, the video doesn't explain very well what is happening behind the scenes, e.g. with a circuit diagram. That is the curse of university laboratory videos: some of them assume that students will have material from other internal sources.
Video 4. The Ubiquitous SQUID by John Clarke (2018) Source.
Two parallel Josephson junctions.
In Ciro's ASCII art circuit diagram notation: 
  |
+-+-+
|   |
X   X
|   |
+-+-+
  |
Specific type of Josephson junction. Probably can be made tiny and in huge numbers through photolithography.
Figure 1. Illustration of a thin-film superconducting tunnel junction (STJ) Source. The superconducting material is light blue, the insulating tunnel barrier is black, and the substrate is green.
Video 1. Quantum Transport, Lecture 14: Josephson effects by Sergey Frolov (2013) Source. youtu.be/-HUVGWTfaSI?t=878 mentions maskless electron beam lithography being used to produce STJs.
Video 1. Alfred Leitner - Liquid Helium II the Superfluid by Alfred Leitner (1963) Source. Original source: www.alfredleitner.com.
Video 2. Ben Miller experiments with superfluid helium by BBC (2011) Source. Just quickly shows the superfluid helium climbing out o the cup, no detailed setup. With professor Robert Taylor from the University of Oxford.
Video 1. Something weird happens when you keep squeezing by Vox (2023) Source. Sodium becomes liquid when you compress it. Weird.
Inward Bound by Abraham Pais (1988) page 282 shows how this can be generalized from the Maxwell-Boltzmann distribution
TODO WTF is this? How is it built? What is special about it?
www.youtube.com/watch?v=PbuiIhr0LVA 7 Different Types of Plastic and Their Uses by Orange Plastics Academy (2018) Does not mention packaging foams.
The wiki comments: en.wikipedia.org/w/index.php?title=Ferromagnetism&oldid=965600553#Explanation
The Bohr–van Leeuwen theorem, discovered in the 1910s, showed that classical physics theories are unable to account for any form of magnetism, including ferromagnetism. Magnetism is now regarded as a purely quantum mechanical effect. Ferromagnetism arises due to two effects from quantum mechanics: spin and the Pauli exclusion principle.
Toy model of matter that exhibits phase transition in dimension 2 and greater. It does not provide numerically exact results by itself, but can serve as a tool to theorize existing and new phase transitions.
Each point in the lattice has two possible states: TODO insert image.
As mentioned at: stanford.edu/~jeffjar/statmech/intro4.html some systems which can be seen as modelled by it include:
  • the spins direction (up or down) of atoms in a magnet, which can undergo phase transitions depending on temperature as that characterized by the Curie temperature and an externally applied magnetic field
    Neighboring spins like to align, which lowers the total system energy.
  • the type of atom at a lattice point in a 2-metal alloy, e.g. Fe-C (e.g. steel). TODO: intuition for the neighbour interaction? What likes to be with what? And aren't different phases in different crystal structures?
Also has some funky relations to renormalization TODO.
TODO what it means to solve an Ising model in general?
stanford.edu/~jeffjar/statmech/lec4.html gives some good notions:
  • is the expectation value of the value. It is therefore a number between -1.0 an and 1.0, -1.0 means everything is always down, 0.0 means half up half down, and 1.0 means all up
  • : correlation between neighboring states. TODO.

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