BCS theory – Close Read

65 years of the BCS theory

Thanks to an arithmetic mistake, I thought 2022 was the 75th anniversary of the invention (or discovery?) of the BCS theory of superconductivity. It’s really the 65th anniversary, but since I’d worked myself up to write about it, I’m going to. 🤷🏽‍♂️ It also helps that the theory is a remarkable fact of nature that make sense of what is weirdly a macroscopic effect of microscopic causes.

There are several ways to classify superconductors – materials that conduct electricity with zero resistance under certain conditions. One of them is as conventional or unconventional. A superconductor is conventional if BCS theory can explain its superconductivity. ‘BCS’ are the initials of the theory’s three originators: John Bardeen, Leon Cooper and John Robert Schrieffer. BCS theory explains (conventional) superconductivity by explaining how the electrons in a material enter a collective superfluidic state.

At room temperature, the valence electrons flow around a material, being occasionally scattered by the grid of atomic nuclei or impurities. We know this scattering as electrical resistance.

An illustration of a lattice of sodium and chlorine atoms in a sodium chloride crystal. Credit: Benjah-bmm27, public domain

The electrons also steer clear of each other because of the repulsion of like charges (Coulomb repulsion).

When the material is cooled below a critical temperature, however, vibrations in the atomic lattice encourage the electrons to become paired. This may defy what we learnt in high school – that like charges repel – but the picture is a little more complicated, and it might make more sense if we adopt the lens of energy instead.

A system will favour a state in which it has lower energy than one in which it has more energy. When two carriers of like charges, like two electrons, approach each other, they repel each other more strongly the closer they get. This repulsion increases the system’s energy (in some form, typically kinetic energy).

In some materials, conditions can arise in which two electrons can pair up – become correlated with each other – across relatively long distances, without coming close to each other, rendering the Coulomb repulsion irrelevant. This correlation happens as a result of the electrons’ effect on their surroundings. As an electron moves through the lattice of positively charged atomic nuclei, it exerts an attractive force on the nuclei, which respond by tending towards the electron. This increases the amount of positive potential near the electron, which attracts another electron nearby to move closer as well. If the two electrons have opposite spins, they become correlated as a Cooper pair, kept that way by the attractive potential imposed by the atomic lattice.

Leon Cooper explained that neither the source of this potential nor its strength matter – as long as it is attractive, and the other conditions hold, the electrons will team up into Cooper pairs. In terms of the system’s energy, the paired state is said to be energetically favourable, meaning that the system as a whole has a lower energy than if the electrons were unpaired below the critical temperature.

Keeping the material cooled to below this critical temperature is important: while the paired state is energetically favourable, the state itself arises only below the critical temperature. Above the critical temperature, the electrons can’t access this state altogether because they have too much kinetic energy. (The temperature of a material is the average kinetic energy of its constituent particles.)

Cooper’s theory of the electron pairs fit into John Bardeen’s theory, which sought to explain changes in the energy states of a material as it goes from being non-superconducting to superconducting. Cooper had also described the formation of electron pairs one at a time, so to speak, and John Robert Schrieffer’s contribution was to work out a mathematical way to explain the formation of millions of Cooper pairs and their behaviour in the material.

The trio consequently published its now-famous paper, ‘Microscopic Theory of Superconductivity’, on April 1, 1957.

(I typo-ed this as 1947 on a calculator, which spit out the number of years since to be 75. 😑 One could have also expected me to remember that this is India’s 75th year of independence and that BCS theory was created a decade after 1947, but the independence hasn’t been registering these days.)

Anyway, electrons by themselves belong to a particle class called fermions. The other known class is that of the bosons. The difference between fermions and bosons is that the former obey Pauli’s exclusion principle while the latter do not. The exclusion principle forbids two fermions in the same system – like a metal – from simultaneously occupying the same quantum state. This means the electrons in a metal have a hierarchy of energies in normal conditions.

However, a Cooper pair, while composed of two electrons, is a boson, and doesn’t obey Pauli’s exclusion principle. The Cooper pairs of the material can all occupy the same state – i.e. the state with the lowest energy, more popularly called the ground state. This condensate of Cooper pairs behaves like a superfluid: practically flowing around the material, over, under and through the atomic lattice. Even when a Cooper pair is scattered off by an atomic nucleus or an impurity in the material, the condensate doesn’t break formation because all the other Cooper pairs continue their flow, and eventually also reintegrate the scattered Cooper pair. This flow is what we understand as electrical superconductivity.

“BCS theory was the first microscopic theory of superconductivity,” per Wikipedia. But since its advent, especially since the late 1970s, researchers have identified several superconducting materials, and behaviours, that neither BCS theory nor its extensions have been able to explain.

When a material transitions into its superconducting state, it exhibits four changes. Observing these changes is how researchers confirm that the material is now superconducting. (In no particular order:) First, the material loses all electric resistance. Second, any magnetic field inside the material’s bulk is pushed to the surface. Third, the electronic specific heat increases as the material is cooled before dropping abruptly at the critical temperature. Fourth, just as the energetically favourable state appears, some other possible states disappear.

Physicists experimentally observed the fourth change only in January this year – based on the transition of a material called Bi-2212 (bismuth strontium calcium copper oxide, a.k.a. BSCCO, a.k.a. bisko). Bi-2212 is, however, an unconventional superconductor. BCS theory can’t explain its superconducting transition, which, among other things, happens at a higher temperature than is associated with conventional materials.

In the January 2022 study, physicists also reported that Bi-2212 transitions to its superconducting state in two steps: Cooper pairs form at 120 K – related to the fourth sign of superconductivity – while the first sign appears at around 77 K. To compare, elemental rhenium, a conventional superconductor, becomes superconducting in a single step at 2.4 K.

A cogent explanation of the nature of high-temperature superconductivity in cuprate superconductors like Bi-2212 is one of the most important open problems in condensed-matter physics today. It is why we still await further updates on the IISc team’s room-temperature superconductivity claim.

At last, physicists report finding the ‘fourth sign’ of superconductivity

Using an advanced investigative technique, researchers at Stanford University have found that cuprate superconductors – which become superconducting at higher temperatures than their better-known conventional counterparts – transition into this exotic state in a different way. The discovery provides new insights into the way cuprate superconductors work and eases the path to discovering a room-temperature superconductor one day.

A superconductor is a material that can transport an electric current with zero resistance. The most well-known and also better understood superconductors are certain metallic alloys. They transition from their ‘normal’ resistive state to the superconducting state when their temperature is brought to a very low value, typically a few degrees above absolute zero.

The theory that explains the microscopic changes that occur as the material transitions is called Bardeen-Cooper-Schrieffer (BCS) theory. As the material crosses its threshold temperature, called the critical temperature, BCS theory predicts four signatures of superconductivity. If these four signatures occur, we can be sure that the material has become superconducting.

First, the material’s resistivity collapses and its electrons begin to flow without any resistance through the bulk – the electronic effect.

Second, the material expels all magnetic fields within its bulk – the magnetic (a.k.a. Meissner) effect.

A magnet levitating above a high-temperature superconductor, thanks to the Meissner effect. Credit: Mai-Linh Doan/Wikimedia Commons, CC BY-SA 3.0

Third, the amount of heat required to excite electrons to an arbitrarily higher energy is called the electronic specific heat. This number is lower for superconducting electrons than for non-superconducting electrons – but it increases as the material is warmed, only to drop abruptly to the non-superconducting value at the critical temperature. This is the effect on the material’s thermodynamic behaviour.

Fourth, while the energies of the electrons in the non-superconducting state have a variety of values, in the superconducting state some energy levels become unattainable. This shows up as a gap in a chart mapping the energy values. This is the spectroscopic effect. (The prefix ‘spectro-‘ refers to anything that can assume a continuous series of values, on a spectrum.)

Conventional superconductors are called so simply because scientists discovered them first and they defined the convention: among other things, they transition from their non-superconducting to superconducting states at very low temperature. Their unconventional counterparts are the high-temperature superconductors, which were discovered in the late 1980s and which transition at temperatures greater than 77 K. And when they do, physicists have thus far observed the corresponding electronic, magnetic and thermodynamic effects – but not the spectroscopic one.

A new study, published on January 26, 2022, has offered to complete this record. And in so doing, the researchers have uncovered new information about how these materials transition into their superconducting states: it is not the way low-temperature superconductors do.

The research team, at Stanford, reportedly did this by studying the thermodynamic effect and connecting it to the material’s spectroscopic effect.

The deeper problem with zeroing in on the spectroscopic effect in high-temperature superconductors is that an electron energy gap shows up before the transition, when the material is not yet a superconductor, and persists into the superconducting phase.

First, recall that at the critical temperature, the electronic specific heat stops increasing and drops suddenly to the non-superconducting value. The specific heat is directly related to the amount of entropy in the system (energy in the system that can’t be harnessed to perform work). The entropy is in turn related to the spectral function – an equation that dictates which energy states the electrons can and can’t occupy. So by studying changes in the specific heat, the researchers can understand the spectroscopic effect.

Second, to study the specific heat, the researchers used a technique called angle-resolved photo-emission spectroscopy (ARPES). These are big words but they have a simple meaning. Photo-emission spectroscopy refers to a technique in which energy-loaded photons are shot into a target material, where they knock out those electrons that they have the energy for. Based on the energies of the electrons knocked out, their position and their momenta, scientists can piece together the properties of the electrons inside the material.

ARPES takes this a step further by also recording the angle at which the electrons are knocked out of the material. This provides an insight into another property of the superconductor. Specifically, another way in which cuprates differ from conventional superconductors is the way in which the electrons pair up. In the latter, the pairs break rotational symmetry, such that the energy required to break up the pair is not equal in all directions.

This affects the way the thermodynamic and spectral effects look in the data. For example, photons fired at certain angles will knock out more electrons from the material than photons incoming at other angles.

The angle-specific measurements of the specific-heat coefficient (y-axis) versus the temperature (x-axis). Credit: https://doi.org/10.1038/s41586-021-04251-2

Taking all this into account, the researchers reported that a cuprate superconductor called Bi-2212 (bismuth strontium calcium copper oxide) transitions to becoming a superconductor in two steps – unlike the single-step transition of low-temperature superconductors.

According to BCS theory, the electrons in a conventional superconductor are encouraged to overcome their mutual repulsion and bind to each other in pairs when two conditions are met: the material’s lattice – the grid of atomic nuclei – has a vibrational energy of a certain frequency and the material’s temperature is lowered. These electron pairs then move around the material like a fluid of zero viscosity, thus giving rise to superconductivity.

The Stanford team found that in Bi-2212, the electrons pair up with each other at around 120 K, but condense into the fluid-like state only at around 77 K. The former gives rise to an energy gap – i.e. the spectroscopic effect – even as the superconducting behaviour itself arises only at the 77-K mark, when the pairs condense.

A small sample of Bi-2212 The side is 1 mm long. Credit: James Slezak, Cornell Laboratory of Atomic and Solid State Physics, CC BY-SA 3.0

There are two distinct feats here: finding the spectroscopic effect and finding the two-step transition. Both – but the first more so – were the product of technological advancements. The researchers obtained their Bi-2212 samples, created with specific chemical compositions so as to help analyse the ARPES data, from their collaborators in Japan, and then studied it with two instruments capable of performing ARPES studies at Stanford: an ultraviolet laser and the Synchrotron Radiation Lightsource.

Makoto Hashimoto, a physicist at Stanford and one of the study’s authors, said in a press statement: “Recent improvements in the overall performance of those instruments were an important factor in obtaining these high-quality results. They allowed us to measure the energy of the ejected electrons with more precision, stability and consistency.”

The second finding, of the two-step transition, is important foremost because it is new knowledge of the way cuprate superconductors ‘work’ and because it tells physicists that they will have to achieve two things – instead of just one, as in the case of conventional, low-temperature superconductors – if they want to recreate the same effects in a different material.

As Zhi-Xun Shen, the researcher who led the study at Stanford, told Physics World, “This knowledge will ultimately help us make better superconductors in the future.”

Featured image: A schematic illustration of an ARPES setup. On the left is the head-on view of the manipulator holding the sample and at the centre is the side-on view. On the right is an electron energy analyser. Credit: Ponor/Wikimedia Commons, CC BY-SA 4.0.

Physicists observe long-expected helium superfluid phase

Physicists have reported that they have finally observed helium 3 existing in a long-predicted type of superfluid, called the ß phase.

This is an important discovery, if it’s borne out, for reasons that partly have to do with its isotope, helium 4. Helium 4 is a fascinating substance because the helium 4 atom is a boson – a type of particle whose quantum properties and behaviour are explained by rules called Bose-Einstein statistics. Helium 3, on the other hand, is a fermion, and fermions are governed by Fermi-Dirac statistics.

Bosons and fermions have one important difference: bosons are allowed to disobey Pauli’s exclusion principle, and by doing so they can assume exotic states of matter rarely found in nature, with many unusual properties.

For example, when helium 4 is cooled below a certain temperature, it becomes a superfluid: a liquid that flows without experiencing any resistance. If you poured a superfluid into a bowl, it will be able to climb the walls of the bowl and spill out without any help. But helium 3 atoms are fermions, so they are bound to obey Pauli’s exclusion principle and can’t become a superfluid.

At least this is what physicists believed for a long time, until the early 1970s, when two independent groups of physicists found – one in theory and the other in experiments – that helium 3 could indeed enter a superfluid phase, but at a temperature 1,000-times lower than the critical temperature of helium 4. The theory group, led by Anthony Leggett at the University of Sussex, had in fact made a significant discovery.

Today, we know that the flow of superfluid helium 4 is analogous to the flow of electrons in a conventional superconductor, which also move around as if they face no resistance from the surrounding atoms. Leggett and co. found that the theory used to explain these superconductors could also be used to explain helium 3 superfluidity. This theory is called Bardeen-Cooper-Schrieffer (BCS) theory, and the materials whose superconductivity it can explain are called BCS superconductors.

Electrons are fermions and cannot ‘super-flow’. But in a BCS superconductor that has been cooled below its critical temperature, some forces in the material cause the electrons to overcome their mutual repulsion (“like charges repel”) and pair up. These electron pairs, while being made of two individual fermions, actually behave like bosons. Similarly, Leggett and co. found that helium 3 atoms could pair up to form a bosonic composite and super-flow.

Over many years, physicists used what they had learnt through these discoveries to expand our understanding of this substance. They found, among other things, that superfluid helium 3 can exist in many phases. The superfluidity would persist in each phase but with different characteristics.

Superfluid helium 3 was first thought to have two phases, called A and B. The temperature-pressure plot below clearly shows the conditions in which each phase emerges.

Credit: E.V. Thuneberg, *Encyclopedia of Condensed Matter Physics*, 2005

When physicists subjected superfluid helium 3 in its A phase to a strong magnetic field, they found another phase that they called A1, whose atom-pairs had different spin characteristics.

In 2015, a group of researchers led by Vladimir Dmitriev, at the P.L. Kapitza Institute for Physical Problems, Moscow, discovered a fourth phase, which they called the polar, or P, phase. Here, they confined helium 3 in a nematic aerogel and exposed the setup to a low magnetic field. Aerogels are ultra-light materials that are extremely porous; nematic means its molecules were arranged in parallel. The aeorogel in the Dmitriev and co. experiment was 98% porous, and whose pores “were much longer than they were wide” (source). That is, the team had found that the shape of the container in which helium 3 was confined also affected the phase of its superfluidity.

In August 2021 (preprint), the same team reported that it had observed a long-expected-to-exist fifth phase called the ß phase.

They reported that they took the setup they used to force superfluid helium 3 into the P phase, but this time exposed it to a high magnetic field. According to their paper, they found that while the superfluid earlier moved into the P phase through a single transition, as the temperature was brought down, this time it did so in two steps. First, it moved into an intermediate phase and then into the P phase. The intermediate is the ß phase.

(If this sounds simple, it wasn’t: the discoveries were each limited by the availability of specially designed instruments capable of picking up on very small-scale changes unavailable to the naked eye. Second, researchers also have had to know in advance what changes they should expect to happen in each phase, and this requires the corresponding theoretical clarity.)

The temperatures at which the phase transition between the two polar phases differ as the magnetic field strength increases. The gap between the two phases is bridged by the ß phase. Source: https://doi.org/10.1103/PhysRevLett.127.265301

I have considerably simplified helium 3’s transition from the ‘normal’ to the superfluid phase in this post. To describe it accurately, physicists use advanced mathematics and associated concepts in high-energy physics. One such concept is symmetry-breaking. When a helium 3 atom pairs up with another to form a bosonic composite, the pair must have a ‘new’ spin and orbital momentum; and their combined wavefunction will also have a ‘new’ phase. All these steps break different symmetries.

There’s a theory called Grand Unification in particle physics, in which physicists expect that at higher and higher energies, the three fundamental forces that affect subatomic particles – the strong-nuclear, the weak-nuclear and the electromagnetic – will combine into a single unified force. Physicists have found in their mathematical calculations that the symmetries that will break in this super-transition resemble those broken by helium 3 during its transition to superfluidity.

Understanding helium 3 can also be rewarding for insights into the insides of neutron stars. Neutron stars are extreme objects – surpassed in their extremeness only by black holes, which exist at the point where known theories of gravitational physics collapse into meaninglessness. A few lakh years after a neutron star is born, it is expected to have cooled sufficiently for its interiors to be composed of superfluids and superconductors.

We may never be able to directly observe these materials in their natural environment. But by studying helium 3’s various phases of superfluidity, we can get a sense of what a neutron star’s innards could be like, and whether their interactions among themselves and the neutrons on the surface could explain these objects’ still-mysterious characteristics.

Featured image: The liquid helium is in the superfluid phase. A thin invisible film creeps up the inside wall of the cup and down on the outside. A drop forms. It will fall off into the liquid helium below. This will repeat until the cup is empty – provided the liquid remains superfluid. Caption and credit: Alfred Leitner, public domain.

A tale of vortices, skyrmions, paths and shapes

There are many types of superconductors. Some of them can be explained by an early theory of superconductivity called Bardeen-Cooper-Schrieffer (BCS) theory.

In these materials, vibrations in the atomic lattice force the electrons in the material to overcome their mutual repulsion and team up in pairs, if the material’s temperature is below a particular threshold (very low). These pairs of electrons, called Cooper pairs, have some properties that individual electrons can’t have. One of them is that all Cooper pairs together form an exotic state of matter called a Bose-Einstein condensate, which can flow through the material with much less resistance than individuals electrons experience. This is the gist of BCS theory.

When the Cooper pairs are involved in the transmission of an electric current through the material, the material is an electrical superconductor.

Some of the properties of the two electrons in each Cooper pair can influence the overall superconductivity itself. One of them is the orbital angular momentum, which is an intrinsic property of all particles. If both electrons have equal orbital angular momentum but are pointing in different directions, the relative orbital angular momentum is 0. Such materials are called s-wave superconductors.

Sometimes, in s-wave superconductors, some of the electric current – or supercurrent – starts flowing in a vortex within the material. If these vortices can be coupled with a magnetic structure called a skyrmion, physicists believe they can give rise to some new behaviour previously not seen in materials, some of them with important applications in quantum computing. Coupling here implies that a change in the properties of the vortex should induce changes in the skyrmion, and vice versa.

However, physicists have had a tough time creating a vortex-skyrmion coupling that they can control. As Gustav Bihlmayer, a staff scientist at the Jülich Research Centre, Germany, wrote for APS Physics, “experimental studies of these systems are still rare. Both parts” of the structures bearing these features “must stay within specific ranges of temperature and magnetic-field strength to realise the desired … phase, and the length scales of skyrmions and vortices must be similar in order to study their coupling.”

In a new paper, a research team from Nanyang Technical University, Singapore, has reported that they have achieved just such a coupling: they created a skyrmion in a chiral magnet and used it to induce the formation of a supercurrent vortex in an s-wave superconductor. In their observations, they found this coupling to be stable and controllable – important attributes to have if the setup is to find practical application.

A chiral magnet is a material whose internal magnetic field “typically” has a spiral or swirling pattern. A supercurrent vortex in an electrical superconductor is analogous to a skyrmion in a chiral magnet; a skyrmion is a “knot of twisting magnetic field lines” (source).

The researchers sandwiched an s-wave superconductor and a chiral magnet together. When the magnetic field of a skyrmion in the chiral magnet interacted with the superconductor at the interface, it induced a spin-polarised supercurrent (i.e. the participating electrons’ spin are aligned along a certain direction). This phenomenon is called the Rashba-Edelstein effect, and it essentially converts electric charge to electron spin and vice versa. To do so, the effect requires the two materials to be in contact and depends among other things on properties of the skyrmion’s magnetic field.

There’s another mechanism of interaction in which the chiral magnet and the superconductor don’t have to be in touch, and which the researchers successfully attempted to recreate. They preferred this mechanism, called stray-field coupling, to demonstrate a skyrmion-vortex system for a variety of practical reasons. For example, the chiral magnet is placed in an external magnetic field during the experiment. Taking the Rashba-Edelstein route means to achieve “stable skyrmions at low temperatures in thin films”, the field needs to be stronger than 1 T. (Earth’s magnetic field measures 25-65 µT.) Such a field could damage the s-wave superconductor.

For the stray-field coupling mechanism, the researchers inserted an insulator between the chiral magnet and the superconductor. Then, when they applied a small magnetic field, Bihlmayer wrote, the field “nucleated” skyrmions in the structure. “Stray magnetic fields from the skyrmions [then] induced vortices in the [superconducting] film, which were observed with scanning tunnelling spectroscopy.”

Experiments like this one reside at the cutting edge of modern condensed-matter physics. A lot of their complexity resides in scientists being able to closely control the conditions in which different quantum effects play out, using similarly advanced tools and techniques to understand what could be going on inside the materials, and to pick the right combination of materials to use.

For example, the heterostructure the physicists used to manifest the stray-field coupling mechanism had the following composition, from top to bottom:

Platinum, 2 nm (layer thickness)
Niobium, 25 nm
Magnesium oxide, 5 nm
Platinum, 2 nm

The next four layers are repeated 10 times in this order:

Platinum, 1 nm
Cobalt, 0.5 nm
Iron, 0.5 nm
Iridium, 1 nm

Back to the overall stack:

Platinum, 10 nm
Tantalum, 2 nm
Silicon dioxide (substrate)

The first three make up the superconductor, the magnesium oxide is the insulator, and the rest (except the substrate) make up the chiral magnet.

It’s possible to erect a stack like this through trial and error, with no deeper understanding dictating the choice of materials. But when the universe of possibilities – of elements, compounds and alloys, their shapes and dimensions, and ambient conditions in which they interact – is so vast, the exercise could take many decades. But here we are, at a time when scientists have explored various properties of materials and their interactions, and are able to engineer novel behaviours into existence, blurring the line between discovery and invention. Even in the absence of applications, such observations are nothing short of fascinating.

Applications aren’t wanting, however.

A quasiparticle is a packet of energy that behaves like a particle in a specific context even though it isn’t actually one. For example, the proton is a quasiparticle because it’s really a clump of smaller particles (quarks and gluons) that together behave in a fixed, predictable way. A phonon is a quasiparticle that represents some vibrational (or sound) energy being transmitted through a material. A magnon is a quasiparticle that represents some magnetic energy being transmitted through a material.

On the other hand, an electron is said to be a particle, not a quasiparticle – as are neutrinos, photons, Higgs bosons, etc.

Now and then physicists abstract packets of energy as particles in order to simplify their calculations.

(Aside: I’m aware of the blurred line between particles and quasiparticles. For a technical but – if you’re prepared to Google a few things – fascinating interview with condensed-matter physicist Vijay Shenoy on this topic, see here.)

We understand how these quasiparticles behave in three-dimensional space – the space we ourselves occupy. Their properties are likely to change if we study them in lower or higher dimensions. (Even if directly studying them in such conditions is hard, we know their behaviour will change because the theory describing their behaviour predicts it.) But there is one quasiparticle that exists in two dimensions, and is quite different in a strange way from the others. They are called anyons.

Say you have two electrons in an atom orbiting the nucleus. If you exchanged their positions with each other, the measurable properties of the atom will stay the same. If you swapped the electrons once more to bring them back to their original positions, the properties will still remain unchanged. However, if you switched the positions of two anyons in a quantum system, something about the system will change. More broadly, if you started with a bunch of anyons in a system and successively exchanged their positions until they had a specific final arrangement, the system’s properties will have changed differently depending on the sequence of exchanges.

This is called path dependency, and anyons are unique in possessing this property. In technical language, anyons are non-Abelian quasiparticles. They’re interesting for many reasons, but one application stands out. Quantum computers are devices that use the quantum mechanical properties of particles, or quasiparticles, to execute logical decisions (the same way ‘classical’ computers use semiconductors). Anyons’ path dependency is useful here. Arranging anyons in one sequence to achieve a final arrangement can be mapped to one piece of information (e.g. 1), and arranging anyons by a different sequence to achieve the same final arrangement can be mapped to different information (e.g. 0). This way, what information can be encoded depends on the availability of different paths to a common final state.

In addition, an important issue with existing quantum computers is that they are too fragile: even a slight interaction with the environment can cause the devices to malfunction. Using anyons for the qubits could overcome this problem because the information stored doesn’t depend on the qubits’ existing states but the paths that they have taken there. So as long as the paths have been executed properly, environmental interactions that may disturb the anyons’ final states won’t matter.

However, creating such anyons isn’t easy.

Now, recall that s-wave superconductors are characterised by the relative orbital angular momentum of electrons in the Cooper pairs being 0 (i.e. equal but in opposite directions). In some other materials, it’s possible that the relative value is 1. These are the p-wave superconductors. And at the centre of a supercurrent vortex in a p-wave superconductor, physicists expect to find non-Abelian anyons.

So the ability to create and manipulate these vortices in superconductors, as well as, more broadly, explore and understand how magnet-superconductor heterostructures work, is bound to be handy.

The Nanyang team’s paper calls the vortices and skyrmions “topological excitations”. An ‘excitation’ here is an accumulation of energy in a system over and above what the system has in its ground state. Ergo, it’s excited. A topological excitation refers to energy manifested in changes to the system’s topology.

On this subject, one of my favourite bits of science is topological phase transitions.

I usually don’t quote from Wikipedia but communicating condensed-matter physics is exacting. According to Wikipedia, “topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending”. For example, no matter how much you squeeze or stretch a donut (without breaking it), it’s going to be a ring with one hole. Going one step further, your coffee mug and a donut are topologically similar: they’re both objects with one hole.

I also don’t like the Nobel Prizes but some of the research that they spotlight is nonetheless awe-inspiring. In 2016, the prize was awarded to Duncan Haldane, John Kosterlitz and David Thouless for “theoretical discoveries of topological phase transitions and topological phases of matter”.

David Thouless in 1995. Credit: Mary Levin/University of Washington

Quoting myself from 2016:

There are four popularly known phases of matter: plasma, gas, liquid and solid. If you cooled plasma, its phase would transit to that of a gas; if you cooled gases, you’d get a liquid; if you cooled liquids, you’d get a solid. If you kept cooling a solid until you were almost at absolute zero, you’d find substances behaving strangely because, suddenly, quantum mechanical effects show up. These phases of matter are broadly called quantum phases. And their phase transitions are different from when plasma becomes a gas, a gas becomes a liquid, and so on.
A Kosterlitz-Thouless transition describes a type of quantum phase transition. A substance in the quantum phase, like all substances, tries to possess as low energy as possible. When it gains some extra energy, it sheds it. And how it sheds it depends on what the laws of physics allow. Kosterlitz and Thouless found that, at times, the surface of a flat quantum phase – like the surface of liquid helium – develops vortices, akin to a flattened tornado. These vortices always formed in pairs, so the surface always had an even number of vortices. And at very low temperatures, the vortices were always tightly coupled: they remained close to each other even when they moved across the surface.
The bigger discovery came next. When Kosterlitz and Thouless raised the temperature of the surface, the vortices moved apart and moved around freely, as if they no longer belonged to each other. In terms of thermodynamics alone, the vortices being alone or together wouldn’t depend on the temperature, so something else was at play. The duo had found a kind of phase transition – because it did involve a change in temperature – that didn’t change the substance itself but only a topological shift in how it behaved. In other words, the substance was able to shed energy by coupling the vortices.

Reality is so wonderfully weird. It’s also curious that some concepts that seemed significant when I was learning science in school (like invention versus discovery) and in college (like particle versus quasiparticle) – concepts that seemed meaningful and necessary to understand what was really going on – don’t really matter in the larger scheme of things.

The awesome limits of superconductors

On June 24, a press release from CERN said that scientists and engineers working on upgrading the Large Hadron Collider (LHC) had “built and operated … the most powerful electrical transmission line … to date”. The transmission line consisted of four cables – two capable of transporting 20 kA of current and two, 7 kA.

The ‘A’ here stands for ‘ampere’, the SI unit of electric current. Twenty kilo-amperes is an extraordinary amount of current, nearly equal to the amount in a single lightning strike.

In the particulate sense: one ampere is the flow of one coulomb per second. One coulomb is equal to around 6.24 quintillion elementary charges, where each elementary charge is the charge of a single proton or electron (with opposite signs). So a cable capable of carrying a current of 20 kA can essentially transport 124.8 sextillion electrons per second.

According to the CERN press release (emphasis added):

The line is composed of cables made of magnesium diboride (MgB2), which is a superconductor and therefore presents no resistance to the flow of the current and can transmit much higher intensities than traditional non-superconducting cables. On this occasion, the line transmitted an intensity 25 times greater than could have been achieved with copper cables of a similar diameter. Magnesium diboride has the added benefit that it can be used at 25 kelvins (-248 °C), a higher temperature than is needed for conventional superconductors. This superconductor is more stable and requires less cryogenic power. The superconducting cables that make up the innovative line are inserted into a flexible cryostat, in which helium gas circulates.

The part in bold could have been more explicit and noted that superconductors, including magnesium diboride, can’t carry an arbitrarily higher amount of current than non-superconducting conductors. There is actually a limit for the same reason why there is a limit to the current-carrying capacity of a normal conductor.

This explanation wouldn’t change the impressiveness of this feat and could even interfere with readers’ impression of the most important details, so I can see why the person who drafted the statement left it out. Instead, I’ll take this matter up here.

An electric current is generated between two points when electrons move from one point to the other. The direction of current is opposite to the direction of the electrons’ movement. A metal that conducts electricity does so because its constituent atoms have one or more valence electrons that can flow throughout the metal. So if a voltage arises between two ends of the metal, the electrons can respond by flowing around, birthing an electric current.

This flow isn’t perfect, however. Sometimes, a valence electron can bump into atomic nuclei, impurities – atoms of other elements in the metallic lattice – or be thrown off course by vibrations in the lattice of atoms, produced by heat. Such disruptions across the metal collectively give rise to the metal’s resistance. And the more resistance there is, the less current the metal can carry.

These disruptions often heat the metal as well. This happens because electrons don’t just flow between the two points across which a voltage is applied. They’re accelerated. So as they’re speeding along and suddenly bump into an impurity, they’re scattered into random directions. Their kinetic energy then no longer contributes to the electric energy of the metal and instead manifests as thermal energy – or heat.

If the electrons bump into nuclei, they could impart some of their kinetic energy to the nuclei, causing the latter to vibrate more, which in turn means they heat up as well.

Copper and silver have high conductance because they have more valence electrons available to conduct electricity and these electrons are scattered to a lesser extent than in other metals. Therefore, these two also don’t heat up as quickly as other metals might, allowing them to transport a higher current for longer. Copper in particular has a higher mean free path: the average distance an electron travels before being scattered.

In superconductors, the picture is quite different because quantum physics assumes a more prominent role. There are different types of superconductors according to the theories used to understand how they conduct electricity with zero resistance and how they behave in different external conditions. The electrical behaviour of magnesium diboride, the material used to transport the 20 kA current, is described by Bardeen-Cooper-Schrieffer (BCS) theory.

According to this theory, when certain materials are cooled below a certain temperature, the residual vibrations of their atomic lattice encourages their valence electrons to overcome their mutual repulsion and become correlated, especially in terms of their movement. That is, the electrons pair up.

While individual electrons belong to a class of particles called fermions, these electron pairs – a.k.a. Cooper pairs – belong to another class called bosons. One difference between these two classes is that bosons don’t obey Pauli’s exclusion principle: that no two fermions in the same quantum system (like an atom) can have the same set of quantum numbers at the same time.

As a result, all the electron pairs in the material are now free to occupy the same quantum state – which they will when the material is supercooled. When they do, the pairs collectively make up an exotic state of matter called a Bose-Einstein condensate: the electron pairs now flow through the material as if they were one cohesive liquid.

In this state, even if one pair gets scattered by an impurity, the current doesn’t experience resistance because the condensate’s overall flow isn’t affected. In fact, given that breaking up one pair will cause all other pairs to break up as well, the energy required to break up one pair is roughly equal to the energy required to break up all pairs. This feature affords the condensate a measure of robustness.

But while current can keep flowing through a BCS superconductor with zero resistance, the superconducting state itself doesn’t have infinite persistence. It can break if it stops being cooled below a specific temperature, called the critical temperature; if the material is too impure, contributing to a sufficient number of collisions to ‘kick’ all electrons pairs out of their condensate reverie; or if the current density crosses a particular threshold.

At the LHC, the magnesium diboride cables will be wrapped around electromagnets. When a large current flows through the cables, the electromagnets will produce a magnetic field. The LHC uses a circular arrangement of such magnetic fields to bend the beam of protons it will accelerate into a circular path. The more powerful the magnetic field, the more accurate the bending. The current operational field strength is 8.36 tesla, about 128,000-times more powerful than Earth’s magnetic field. The cables will be insulated but they will still be exposed to a large magnetic field.

Type I superconductors completely expel an external magnetic field when they transition to their superconducting state. That is, the magnetic field can’t penetrate the material’s surface and enter the bulk. Type II superconductors are slightly more complicated. Below one critical temperature and one critical magnetic field strength, they behave like type I superconductors. Below the same temperature but a slightly stronger magnetic field, they are superconducting and allow the fields to penetrate their bulk to a certain extent. This is called the mixed state.

A hand-drawn phase diagram showing the conditions in which a mixed-state type II superconductor exists. Credit: Frederic Bouquet/Wikimedia Commons, CC BY-SA 3.0

Say a uniform magnetic field is applied over a mixed-state superconductor. The field will plunge into the material’s bulk in the form of vortices. All these vortices will have the same magnetic flux – the number of magnetic field lines per unit area – and will repel each other, settling down in a triangular pattern equidistant from each other.

An annotated image of vortices in a type II superconductor. The scale is specified at the bottom right. Source: A set of slides entitled ‘Superconductors and Vortices at Radio Frequency Magnetic Fields’ by Ernst Helmut Brandt, Max Planck Institute for Metals Research, October 2010.

When an electric current passes through this material, the vortices are slightly displaced, and also begin to experience a force proportional to how closely they’re packed together and their pattern of displacement. As a result, to quote from this technical (yet lucid) paper by Praveen Chaddah:

This force on each vortex … will cause the vortices to move. The vortex motion produces an electric field¹ parallel to [the direction of the existing current], thus causing a resistance, and this is called the flux-flow resistance. The resistance is much smaller than the normal state resistance, but the material no longer [has] infinite conductivity.

1. According to Maxwell’s equations of electromagnetism, a changing magnetic field produces an electric field.

Since the vortices’ displacement depends on the current density: the greater the number of electrons being transported, the more flux-flow resistance there is. So the magnesium diboride cables can’t simply carry more and more current. At some point, setting aside other sources of resistance, the flux-flow resistance itself will damage the cable.

There are ways to minimise this resistance. For example, the material can be doped with impurities that will ‘pin’ the vortices to fixed locations and prevent them from moving around. However, optimising these solutions for a given magnetic field and other conditions involves complex calculations that we don’t need to get into.

The point is that superconductors have their limits too. And knowing these limits could improve our appreciation for the feats of physics and engineering that underlie achievements like cables being able to transport 124.8 sextillion electrons per second with zero resistance. In fact, according to the CERN press release,

The [line] that is currently being tested is the forerunner of the final version that will be installed in the accelerator. It is composed of 19 cables that supply the various magnet circuits and could transmit intensities of up to 120 kA!

While writing this post, I was frequently tempted to quote from Lisa Randall‘s excellent book-length introduction to the LHC, Knocking on Heaven’s Door (2011). Here’s a short excerpt:

One of the most impressive objects I saw when I visited CERN was a prototype of LHC’s gigantic cylindrical dipole magnets. Event with 1,232 such magnets, each of them is an impressive 15 metres long and weighs 30 tonnes. … Each of these magnets cost EUR 700,000, making the ned cost of the LHC magnets alone more than a billion dollars.
The narrow pipes that hold the proton beams extend inside the dipoles, which are strung together end to end so that they wind through the extent of the LHC tunnel’s interior. They produce a magnetic field that can be as strong as 8.3 tesla, about a thousand times the field of the average refrigerator magnet. As the energy of the proton beams increases from 450 GeV to 7 TeV, the magnetic field increases from 0.54 to 8.3 teslas, in order to keep guiding the increasingly energetic protons around.
The field these magnets produce is so enormous that it would displace the magnets themselves if no restraints were in place. This force is alleviated through the geometry of the coils, but the magnets are ultimately kept in place through specially constructed collars made of four-centimetre thick steel.
… Each LHC dipole contains coils of niobium-titanium superconducting cables, each of which contains stranded filaments a mere six microns thick – much smaller than a human hair. The LHC contains 1,200 tonnes of these remarkable filaments. If you unwrapped them, they would be long enough to encircle the orbit of Mars.
When operating, the dipoles need to be extremely cold, since they work only when the temperature is sufficiently low. The superconducting wires are maintained at 1.9 degrees above absolute zero … This temperature is even lower than the 2.7-degree cosmic microwave background radiation in outer space. The LHC tunnel houses the coldest extended region in the universe – at least that we know of. The magnets are known as cryodipoles to take into account their special refrigerated nature.
In addition to the impressive filament technology used for the magnets, the refrigeration (cryogenic) system is also an imposing accomplishment meriting its own superlatives. The system is in fact the world’s largest. Flowing helium maintains the extremely low temperature. A casing of approximately 97 metric tonnes of liquid helium surrounds the magnets to cool the cables. It is not ordinary helium gas, but helium with the necessary pressure to keep it in a superfluid phase. Superfluid helium is not subject to the viscosity of ordinary materials, so it can dissipate any heat produced in the dipole system with great efficiency: 10,000 metric tonnes of liquid nitrogen are first cooled, and this in turn cools the 130 metric tonnes of helium that circulate in the dipoles.

Featured image: A view of the experimental MgB2 transmission line at the LHC. Credit: CERN.

The science in Netflix’s ‘Spectral’

I watched Spectral, the movie that released on Netflix on December 9, 2016, after Universal Studios got cold feet about releasing it on the big screen – the same place where a previous offering, Warcraft, had been gutted. Spectral is sci-fi and has a few great moments but mostly it’s bland and begging for some tabasco. The premise: an elite group of American soldiers deployed in Moldova come upon some belligerent ghost-like creatures in a city they’re fighting in. They’ve no clue how to stop them, so they fly in an engineer to consult from DARPA, the same guy who built the goggles that detected the creatures in the first place. Together, they do things. Now, I’d like to talk about the science in the film and not the plot itself, though the former feeds the latter.

SPOILERS AHEAD

A scene from the film 'Spectral' (2016). Source: Netflix — A scene from the film ‘Spectral’ (2016). Source: Netflix

Towards the middle of the movie, the engineer realises that the ghost-like creatures have the same limitations as – wait for it – a Bose-Einstein condensate (BEC). They can pass through walls but not ceramic or heavy metal (not the music), they rapidly freeze objects in their path, and conventional weapons, typically projectiles of some kind, can’t stop them. Frankly, it’s fabulous that Ian Fried, the film’s writer, thought to use creatures made of BECs as villains.

A BEC is an exotic state of matter in which a group of ultra-cold particles condense into a superfluid (i.e., it flows without viscosity). Once a BEC forms, a subsection of a BEC can’t be removed from it without breaking the whole BEC state down. You’d think this makes the BEC especially fragile – because it’s susceptible to so many ‘liabilities’ – but it’s the exact opposite. In a BEC, the energy required to ‘kick’ a single particle out of its special state is equal to the energy that’s required to ‘kick’ all the particles out, making BECs as a whole that much more durable.

This property is apparently beneficial for the creatures of Spectral, and that’s where the similarity ends because BECs have other properties that are inimical to the portrayal of the creatures. Two immediately came to mind: first, BECs are attainable only at ultra-cold temperatures; and second, the creatures can’t be seen by the naked eye but are revealed by UV light. There’s a third and relevant property but which we’ll come to later: that BECs have to be composed of bosons or bosonic particles.

It’s not clear why Spectral‘s creatures are visible only when exposed to light of a certain kind. Clyne, the DARPA engineer, says in a scene, “If I can turn it inside out, by reversing the polarity of some of the components, I might be able to turn it from a camera [that, he earlier says, is one that “projects the right wavelength of UV light”] into a searchlight. We’ll [then] be able to see them with our own eyes.” However, the documented ability of BECs to slow down light to a great extent (5.7-million times more than lead can, in certain conditions) should make them appear extremely opaque. More specifically, while a BEC can be created that is transparent to a very narrow range of frequencies of electromagnetic radiation, it will stonewall all frequencies outside of this range on the flipside. That the BECs in Spectral are opaque to a single frequency and transparent to all others is weird.

Obviating the need for special filters or torches to be able to see the creatures simplifies Spectral by removing one entire layer of complexity. However, it would remove the need for the DARPA engineer also, who comes up with the hyperspectral camera and, its inside-out version, the “right wavelength of UV” searchlight. Additionally, the complexity serves another purpose. Ahead of the climax, Clyne builds an energy-discharging gun whose plasma-bullets of heat can rip through the BECs (fair enough). This tech is also slightly futuristic. If the sci-fi/futurism of the rest of Spectral leading up to that moment (when he invents the gun) was absent, then the second-half of the movie would’ve become way more sci-fi than the first-half, effectively leaving Spectral split between two genres: sci-fi and wtf. Thus the need for the “right wavelength of UV” condition?

Now, to the third property. Not all particles can be used to make BECs. Its two predictors, Satyendra Nath Bose and Albert Einstein, were working (on paper) with kinds of particles since called bosons. In nature, bosons are force-carriers, acting against matter-maker particles called fermions. A more technical distinction between them is that the behaviour of bosons is explained using Bose-Einstein statistics while the behaviour of fermions is explained using Fermi-Dirac statistics. And only Bose-Einstein statistics predicts the existence of states of matter called condensates, not Femi-Dirac statistics.

(Aside: Clyne, when explaining what BECs are in Spectral, says its predictors are “Nath Bose and Albert Einstein”. Both ‘Nath’ and ‘Bose’ are surnames in India, so “Nath Bose” is both anyone and no one at all. Ugh. Another thing is I’ve never heard anyone refer to S.N. Bose as “Nath Bose”, only ‘Satyendranath Bose’ or, simply, ‘Satyen Bose’. Why do Clyne/Fried stick to “Nath Bose”? Was “Satyendra” too hard to pronounce?)

All particles constitute a certain amount of energy, which under some circumstances can increase or decrease. However, the increments of energy in which this happens are well-defined and fixed (hence the ‘quantum’ of quantum mechanics). So, for an oversimplified example, a particle can be said to occupy energy levels constituting 2, 4 or 6 units but never of 1, 2.5 or 3 units. Now, when a very-low-density collection of bosons is cooled to an ultra-cold temperature (a few hundredths of kelvins or cooler), the bosons increasingly prefer occupying fewer and fewer energy levels. At one point, they will all occupy a single and common level – flouting a fundamental rule that there’s a maximum limit for the number of particles that can be in the same level at once. (In technical parlance, the wavefunctions of all the bosons will merge.)

When this condition is achieved, a BEC will have been formed. And in this condition, even if a new boson is added to the condensate, it will be forced into occupying the same level as every other boson in the condensate. This condition is also out of limits for all fermions – except in very special circumstances, and circumstances whose exceptionalism perhaps makes way for Spectral‘s more fantastic condensate-creatures. We known one such as superconductivity.

In a superconducting material, electrons flow without any resistance whatsoever at very low temperatures. The most widely applied theory of superconductivity interprets this flow as being that of a superfluid, and the ‘sea’ of electrons flowing as such to be a BEC. However, electrons are fermions. To overcome this barrier, Leon Cooper proposed in 1956 that the electrons didn’t form a condensate straight away but that there was an intervening state called a Cooper pair. A Cooper pair is a pair of electrons that had become bound, overcoming their like-charges repulsion because of the vibration of atoms of the superconducting metal surrounding them. The electrons in a Cooper pair also can’t easily quit their embrace because, once they become bound, the total energy they constitute as a pair is lower than the energy that would be destabilising in any other circumstances.

Could Spectral‘s creatures have represented such superconducting states of matter? It’s definitely science fiction because it’s not too far beyond the bounds of what we know about BEC today (at least in terms of a concept). And in being science fiction, Spectral assumes the liberty to make certain leaps of reasoning – one being, for example, how a BEC-creature is able to ram against an M1 Abrams and still not dissipate. Or how a BEC-creature is able to sit on an electric transformer without blowing up. I get that these in fact are the sort of liberties a sci-fi script is indeed allowed to take, so there’s little point harping on them. However, that Clyne figured the creatures ought to be BECs prompted way more disbelief than anything else because BECs are in the here and the now – and they haven’t been known to behave anything like the creatures in Spectral do.

For some, this information might even help decide if a movie is sci-fi or fantasy. To me, it’s sci-fi.

SPOILERS END

On the more imaginative side of things, Spectral also dwells for a bit on how these creatures might have been created in the first place and how they’re conscious. Any answers to these questions, I’m pretty sure, would be closer to fantasy than to sci-fi. For example, I wonder how the computing capabilities of a very large neural network seen at the end of the movie (not a spoiler, trust me) were available to the creatures wirelessly, or where the power source was that the soldiers were actually after. Spectral does try to skip the whys and hows by having Clyne declare, “I guess science doesn’t have the answer to everything” – but you’re just going “No shit, Sherlock.”

His character is, as this Verge review puts it, exemplarily shallow while the movie never suggests before the climax that science might indeed have all the answers. In fact, the movie as such, throughout its 108 minutes, wasn’t that great for me; it doesn’t ever live up to its billing as a “supernatural Black Hawk Down“. You think about BHD and you remember it being so emotional – Spectral has none of that. It was just obviously more fun to think about the implications of its antagonists being modelled after a phenomenon I’ve often read/written about but never thought about that way.

Yoichiro Nambu, the silent revolutionary of particle physics, is dead

The Wire
July 18, 2015

Particle physics is an obscure subject for most people but everyone sat up and took notice when the Large Hadron Collider discovered the particle named after Peter Higgs in 2012. The Higgs boson propelled his name to the front pages of newspapers that until then hadn’t bothered about the differences between bosons and fermions. On the other hand, it also validated a hypothesis he and his peers had made 50 years ago and helped the LHC’s collaborations revitalise their outreach campaigns.

However, much before the times of giant particle colliders – in the late 1950s, in fact – a cascade of theories was being developed by physicists the world over with much less fanfare, and a lot more of the quiet dignity that advanced theoretical physics is comfortable revelling in. It was a silent revolution, and led in part by the mild-mannered Yoichiro Nambu, who passed away on July 5, 2015.

His work and its derivatives gave rise to the large colliders like the LHC at work today, and which might well have laid the foundations of modern particle physics research. Moreover, many of his and his peers’ accomplishments are not easily discussed the way political movements are nor do they aspire to such privileges, but that didn’t make them any less important than the work of Higgs and others.

Yoichiro Nambu also belonged to a generation that marked a resurgence in Japanese physics research – consider his peers: Yoshio Nishina, Masatoshi Koshiba, Hideki Yukawa, Sin-Itiro Tomonaga, Leo Esaki, Makoto Kobayashi and Toshihide Maskawa, to name a few. A part of the reason was a shift in Japan’s dominant political attitudes after the Second World War. Anyway, the first of Nambu’s biggest contributions to particle physics came in 1960, and it was a triumph of intuition.

There was a span of 46 years between the discovery of superconductivity (by Heike Kamerlingh Onnes in 1911) and the birth of a consistent theoretical explanation for it (by John Bardeen, Leon Cooper and John Schrieffer in 1957) because the phenomenon seemed to defy some of the first principles of the physics used to understand charged particles. Nambu was inspired by the BCS theory to attempt a solution for the hierarchy problem – which asks why gravity, among the four fundamental forces, is 10³² times weaker than the strongest strong-nuclear force.

With the help of British physicist Jeffrey Goldstone, Nambu theorised that whenever a natural symmetry breaks, massless particles called Nambu-Goldstone bosons are born under certain conditions. The early universe, around 13.75 billion years ago when it was extremely small, consisted of a uniform pond of unperturbed energy. Then, the pond was almost instantaneously heated to a temperature of 173 billion Suns, when it broke into smaller packets called particles. The symmetry was (thought to be) spontaneously broken and the event was called the Big Bang.

Then, as the universe started to cool, these packets couldn’t reunify into becoming the pond they once made up, evolving instead into distinct particles. There were perturbations among the particles and the resultant forces were mediated by what came to be called Nambu-Goldstone bosons, named for the physicists who first predicted their existence.

Yoichiro in Nambu in 2008. Source: University of Chicago

Nambu was able to use the hypothetical interactions between the Nambu-Goldstone bosons and particles to explain how the electromagnetic force and the weak nuclear force (responsible for radioactivity) could be unified into one electroweak force at higher temperatures, as well as how where the masses of protons and neutrons come from. These were (and are) groundbreaking ideas that helped scientists make sense of the intricate gears that turned then to make the universe what it is today.

Then, in 1964, six physicists (Higgs, Francois Englert, Tom Kibble, Gerald Guralnik, C.R. Hagen, Robert Brout) postulated that these bosons interacted with an omnipresent field of energy – called the Higgs field – to give rise to the strong-nuclear, weak-nuclear (a.k.a. weak) and electromagnetic forces, and the Higgs boson. And when this boson was discovered in 2012, it validated the Six’s work from 1964.

However, Nambu’s ideas – as well as those of the Six – also served to highlight how the gravitational force couldn’t be unified with the other three fundamental forces. In the 1960s, Nambu’s first attempts at laying out a framework of mathematical equations to unify gravity and the other forces gave rise to the beginnings of string theory. But in the overall history of investigations into particle physics, Nambu’s work – rather, his intellect – was a keystone. Without it, the day theorists’ latinate squiggles on paper could’ve become prize-fetching particles in colliders would’ve been farther off, the day we made sense of reality farther off, the day we better understood our place in the universe farther off.

The Osaka City University, where Nambu was a professor, announced his death on July 17, due to an acute myocardial infarction. He is survived by his wife Chieko Hida and son John. Though he was an associate professor at Osaka from 1950 to 1956, he visited the Institute for Advanced Study at Princeton in 1952 to work with Robert Oppenheimer (and meet Albert Einstein). Also, in 1954, he became a research associate at the University of Chicago and finally a professor there in 1958. He received his American citizenship in 1970.

Peter Freund, his colleague in Chicago, described Nambu as a person of incredible serenity in his 2007 book A Passion for Discovery. Through the work and actions of the biggest physicists of the mid-19th century, the book fleshes out the culture of physics research and how it was shaped by communism and fascism. Freund himself emigrated from Romania to the US in the 1960s to escape the dictatorial madness of Ceausescu, a narrative arc that is partially reflected in Nambu’s life. After receiving his bachelor’s degree from the University of Tokyo in 1942, Nambu was drafted into the army and witnessed the infamous firebombing of Tokyo and was in Japan when Hiroshima and Nagasaki were bombed.

The destructive violence of the war that Nambu studied through is mirrored in the creative energies of the high-energy universe whose mysteries Nambu and his peers worked to decrypt. It may have been a heck of a life to live through but the man himself had only a “fatalistic calm”, as Freund wrote, to show for it. Was he humbled by his own discoveries? Perhaps, but what we do know is that he wanted to continue doing what he did until the day he died.

Superconductivity: From Feshbach to Fermi

(This post is continued from this one.)

After a bit of searching on Wikipedia, I found that the fundamental philosophical underpinnings of superconductivity were to be found in a statistical concept called the Feshbach resonance. If I had to teach superconductivity to those who only knew of the phenomenon superfluously, that’s where I’d begin. So.

Imagine a group of students who have gathered in a room to study together for a paper the next day. Usually, there is that one guy among them who will be hell-bent on gossiping more than studying, affecting the performance of the rest of the group. In fact, given sufficient time, the entire group’s interest will gradually shift in the direction of the gossip and away from its syllabus. The way to get the entire group back on track is to introduce a Feshbach resonance: cut the bond between the group’s interest and the entity causing the disruption. If done properly, the group will turn coherent in its interest and to focusing on studying for the paper.

In multi-body systems, such as a conductor harboring electrons, the presence of a Feshbach resonance renders an internal degree of freedom independent of those coordinates “along” which dissociation is most like to occur. And in a superconductor, a Feshbach resonance results in each electron pairing up with another (i.e., electron-vibrations are quelled by eliminating thermal excitation) owing to both being influenced by an attractive potential that arises out of the electron’s interaction with the vibrating lattice.

Feshbach resonance & BCS theory

For particulate considerations, the lattice-vibrations are quantized in the form of hypothetical particles called phonons. As for why the Feshbach resonance must occur the way it does in a superconductor: that is the conclusion, rather implication, of the BCS theory formulated in 1957 by John Bardeen, Leon Neil Cooper, and John Robert Schrieffer.

The BCS theory essentially treats electrons like rebellious, teenage kids (I must be getting old). As negatively charged electrons pass through the crystal lattice, they draw the positively charged nuclei toward themselves, creating an increase in the positive charge density in their vicinity that attracts more electrons in turn. The resulting electrostatic pull is stronger near nuclei and very weak at larger distances. The BCS theory states that two electrons that would otherwise repel each other will pair up in the face of such a unifying electrostatic potential, howsoever weak it is.

This is something like rebellious teens who, in the face of a common enemy, will unite with each other no matter what the differences between them earlier were.

Since electrons are fermions, they bow down to Pauli’s exclusion principle, which states that no two fermions may occupy the same quantum state. As each quantum state is defined by some specific combination of state variables called quantum numbers, at least one quantum number must differ between the two co-paired electrons.

prof_wolfgang_pauli — Prof. Wolfgang Pauli (1900-1958)

In the case of superconductors, this is particle spin: the electrons in the member-pair will have opposite spins. Further, once such unions have been achieved between different pairs of electrons, each pair becomes indistinguishable from the other, even in principle. Imagine: they are all electron-pairs with two opposing spins but with the same values for all other quantum numbers. Each pair, called a Cooper pair, is just the same as the next!

Bose-Einstein condensates

This unification results in the sea of electrons displaying many properties normally associated with Bose-Einstein condensates (BECs). In a BEC, the particles that attain the state of indistinguishability are bosons (particles with integer spin), not fermions (particles with half-integer spin). The phenomenon occurs at temperatures close to absolute zero and in the presence of an external confining potential, such as an electric field.

Since bosons don’t follow Pauli’s exclusion principle, a major fraction of the indistinguishable entities in the condensate may and do occupy the same quantum state. This causes quantum mechanical effects to become apparent on a macroscopic scale.

By extension, the formulation and conclusions of the BCS theory, alongside its success in supporting associated phenomena, imply that superconductivity may be a quantum phenomenon manifesting in a macroscopic scale.

Note: If even one Cooper pair is “broken”, the superconducting state will be lost as the passage of electric current will be disrupted, and the condensate will dissolve into individual electrons, which means the energy required to break one Cooper pair is the same as the energy required to break the composition of the condensate. So thermal vibrations of the crystal lattice, usually weak, become insufficient to interrupt the flow of Cooper pairs, which is the flow of electrons.

The Meissner effect

In this context, the Meissner effect is simply an extrapolation of Lenz’s law but with zero electrical resistance.

Lenz’s law states that the electromotive force (EMF) because of a current in a conductor acts in a direction that always resists a change in the magnetic flux that causes the EMF. In the absence of resistance, the magnetic fields due to electric currents at the surface of a superconductor cancel all magnetic fields inside the bulk of the material, effectively pushing magnetic field lines of an external magnetic potential outward. However, the Meissner effect manifests only when the externally applied field is weaker than a certain critical threshold: if it is stronger, then the superconductor returns to its conducting state.

Now, there are a class of materials called Type II superconductors – as opposed to the Type I class described earlier – that only push some of the magnetic field outward, the rest remaining conserved inside the material in filaments while being surrounded by supercurrents. This state is called the vortex state, and its occurrence means the material can withstand much stronger magnetic fields and continue to remain superconducting while also exhibiting the hybrid Meissner effect.

Temperature & superconductivity

There are also a host of other effects that only superconductors can exhibit, including Cooper-pair tunneling, flux quantization, and the isotope effect, and it was by studying them that a strong relationship was observed between temperature and superconductivity in various forms.

Bardeen_Cooper_Schrieffer — (L to R) John Bardeen, Leon Cooper, and John Schrieffer

In fact, Bardeen, Cooper, and Schrieffer hit upon their eponymous theory after observing a band gap in the electronic spectra of superconductors. The electrons in any conductor can exist at specific energies, each well-defined. Electrons above a certain energy, usually in the valence band, become free to pass through the entire material instead of staying in motion around the nuclei, and are responsible for conduction.

The trio observed that upon cooling the material to closer and closer to absolute zero, there was a curious gap in the energies at which electrons could be found in the material at a particular temperature. This meant that, at that temperature, the electrons were jumping from existing at one energy to existing at some other lower energy. The observation indicated that some form of condensation was occurring. However, a BEC was ruled out because of Pauli’s exclusion principle. At the same time, a BEC-like state had to have been achieved by the electrons.

This temperature is called the transition temperature, and is the temperature below which a conductor transitions into its superconducting state, and Cooper pairs form, leading to the drop in the energy of each electron. Also, the differences in various properties of the material on either side of this threshold are also attributed to this temperature, including an important notion called the Fermi energy: it is the potential energy that any system possesses when all its thermal energy has been removed from it. This is a significant idea because it defines both the kind and amount of energy that a superconductor has to offer for an externally applied electric current.

enrico_fermi — Enrico Fermi, along with Paul Dirac, defined the Fermi-Dirac statistics that governs the behavior all identical particles that obey Pauli’s exclusion principle (i.e., fermions). Fermi level and Fermi energy are concepts named for him; however, as long as we’re discussing eponymy, Fermilab overshadows them all.

In simple terms, the density of various energy states of the electrons at the Fermi energy of a given material dictates the “breadth” of the band gap if the electron-phonon interaction energy were to be held fixed at some value: a direct proportionality. Thus, the value of the energy gap at absolute zero should be a fixed multiple of the value of the energy gap at the superconducting transition temperature (the multiplication factor was found to be 3.5 universally, irrespective of the material).

Similarly, because of the suppression of thermal excitation (because of the low temperature), the heat capacity of the material reduces drastically at low temperatures, and vanishes below the transition temperature. However, just before hitting zero at the threshold, the heat capacity balloons up to beyond its original value, and then pops. It was found that the ballooned value was always 2.5 times the material’s normal heat capacity value… again, universally, irrespective of the material!

The temperature-dependence of superconductors gains further importance with respect to applications and industrial deployment in the context of its possible occurring at higher temperatures. The low temperatures currently necessary eliminate thermal excitations, in the form of vibrations, of nuclei and almost entirely counter the possibility of electrons, or Cooper pairs, colliding into them.The low temperatures also assist in the flow of Cooper pairs as a superfluid apart from allowing for the energy of the superfluid being higher than the phononic energy of the lattice.

However, to achieve all these states in order to turn a conductor into a superconductor at a higher temperature, a more definitive theory of superconductivity is required. One that allows for the conception of superconductivity that requires only certain internal conditions to prevail while the ambient temperature soars. The 1986-discovery of high-temperature superconductors in ceramics by Bednorz and Muller was the turning point. It started to displace the BCS theory which, physicists realized, doesn’t contain the necessary mechanisms for superconductivity to manifest itself in ceramics – insulators at room temperature – at temperatures as high as 125 K.

A firmer description of superconductivity, therefore, still remains elusive. Its construction should not only pave the for one of the few phenomena that hardly appears in nature and natural processes to be fully understood, but also for its substitution against standard conductors that are responsible for lossy transmission and other such undesirable effects. After all, superconductors are the creation of humankind, and only by its hand while they ever be fully worked.

Stages of superconductivity: Researchers’ insight on ‘pseudogap’ an important advance (phys.org)

Related articles