Carbon atoms are made of three types of subatomic particles namely electrons, protons, and neutrons. These atoms have six electrons and six protons while the number of neutrons can be five, six, seven, or even eight. Carbon is the second most abundant element in the human body and it is the 15th most abundant element in Earth’s crust.  Carbon atoms can naturally exist in pure elemental form or in a mixed form with other types of atoms to make a wide variety of organic, inorganic, and organometallic compounds. It is not unreasonable that carbon atoms in different compounds will lead to different properties, however for the sake of demonstrating emergence that results merely from the way we arrange carbon atoms, this essay will focus on systems made solely from carbon atoms. Emergence in these systems is special because it is hierarchical. First, carbon atoms are arranged in some way to get very interesting emergent properties, and then arranging these emergent systems in a special way leads to a higher level of emergent systems with astonishing physical properties.  

In pure elemental form, we can arrange carbon atoms in different ways leading to different allotropes. The most known allotropes are diamond, graphite, fullerene, carbon nanotubes, and graphene. Humans are typically interested in different allotropes because different allotropes have different properties. Thinking about different allotropes of carbon provides very strong evidence for how combing the same thing in different ways can lead to drastically different emergent properties. Although all these allotropes are made of identical atoms graphite is very soft -this is why it is used in making pencils- while diamond is among the hardest materials known - this is why is it used in cutting and drilling applications.  Graphite conducts electricity and is opaque while diamond does not conduct electricity and is transparent. The way we construct the walls of the carbon nanotubes can make them either insulting or conducting.  

One very interesting allotrope is graphene which is a material without a thickness! Graphene is just one layer of graphite in which the carbon atoms are connected together to make a honeycomb lattice structure. Simply put, atoms are arranged to make hexagons. Although graphene is the thinnest possible material, it is nonetheless the strongest material and it is stiffer than diamond. I find this material very interesting basically for two main reasons. First, before its realization in the laboratory, scientists thought that materials without thickness -like graphene- are impossible to exist in the real world. Graphene is a two-dimensional material (literally no thickness for electrons to move in) and the wisdom humans had until the beginning of this century was that two-dimensional materials cannot exist in nature.  This changed when Novoselov and Geim managed to isolate one layer from graphite to make the first-ever two-dimensional material a reality. Before that, the argument was that there are some physical reasons –particularly thermal fluctuations- that will preclude any two-dimensional material from existence. After the isolation of graphene scientists later discovered few dozens more and much more are expected to be found in the lab. This example shows us that a complete knowledge of the basic ingredients of the system – the carbon atoms in our context- not only is not necessarily enough to allow us to predict and understand the properties of emergent systems, it also sometimes gives us the wrong message that some emergent systems are impossible to be realized in real life. Not only it does not help, it actually hurts!

The second reason why graphene is interesting to me is due to the properties it has. Graphene has a special mixture of properties that do not exist together in any other material. In addition to being the thinnest imaginable and strongest material ever measured in the laboratory, it is the most elastically stretchable crystal with up to 20% change in the length of the material. It has the record thermal conductivity and highest electrical current density at room temperature. It is transparent for light and so tight such that nothing can penetrate it, not even Helium atoms. 

Most of these properties exist in graphene because of some special type of electrons that graphene hosts. These electrons are called massless Dirac electrons. They move so fast – 1/300 the speed of light- and they are massless! The key reason why graphene hosts these electrons is that carbon atoms are arranged in a hexagonal pattern. It is not only about the existence of carbon atoms, these special types of electrons emerge because of the way the carbon atoms are connected. Before the realization of graphene, these Dirac electrons were expected to be found only in some very physical circumstances – high-energy systems- and dealing with these circumstances is typically not easy. Nonetheless, graphene provides a great advantage for us. Graphene is a platform to study such electrons at a relatively cheap cost compared to high-energy physics systems. Therefore, we should keep an eye and look for emergent properties in different systems that might have technological and scientific advantages for humanity. 

Graphene already shows emergent properties and now we discuss the second level of emergence where new emergent properties appear on top of the already emergent properties of carbon atoms in graphene. This higher-level emergence appears when one puts two graphene layers on top of each other and twists them relative to each other. Although stacking two graphene sheets on top of each other without any twist angle between them would give properties that can be different from the properties of the original two individual graphene sheets, the whole physics community got a really very big surprise when scientists started to twist them. When the two sheets are twisted with respect to each other, the combined system is called twisted bilayer graphene TBG.  

When the twist angle in TBG is very small – around 1 degree - one can find many emergent properties that exist neither in the individual layers nor when the two layers are combined but without a twist angle. Some of these properties can be explained by knowing the properties of the basic systems while many other properties cannot be explained. These properties include superconductivity, correlated insulation, ferromagnetism among others.   These emergent properties are very surprising. For example, superconductivity has never been seen before in samples that contain only pure carbon in any allotrope of carbon.  Additionally, superconductivity in TBG is phenomenologically similar to superconductivity that appears in a completely different family of materials called cuprate superconductors. Not only superconductivity is not at all expected in a carbon system, it seems somehow similar to the same phenomena that appear in a completely unrelated family of materials. This shows us that emergent phenomena can be universal. More interestingly, once we increase the twist angle between the two layers to reach 30 degrees, we get a new emergent structure and properties that are not related to the properties of TBG at small twist angles. We get a quasi-crystal structure that was once strongly believed not to exist in nature.  It is worth mentioning that this quasi-crystal state in TBG with 30 degrees has properties different than that of the individual layers and different than TBG with a small angle.

One of the things that excites me about this hierarchical emergence is that it defies whatever our intuition might suggest. Naively one would expect that in order to get radically emergent properties, the basics elements of the system should interact with each other very strongly. This is not true in TBG. The interaction between the two layers in TBG is not very strong but leads to electrons that interact with each other very strongly! 

In an attempt to examine the above examples through some common criterion of emergence, I would like to highlight the following points: 
(1)   Unpredictability: The emergent properties in TBG such as superconductivity and ferromagnetism were completely unexpected. The first theoretical study about TBG was in 2007 and between since and 2018 a very large number of theoretical and experimental research papers studied TBG. None of these papers predicted either superconductivity or ferromagnetism. Although there were some theoretical hints that some interesting properties can be discovered in TBG, no theoretical study predicted the emergence of superconductivity in TBG before it was unexpectedly seen in the laboratory by Yuan Cao and Pablo Jarillo-Herrero in 2018. People did not expect superconductivity in pure carbon atoms since carbon-based systems are very common and none of them have shown any sing of superconductivity. Additionally, ferromagnetism was not expected since carbon atoms are very light and therefore they do not have a high value of a property called spin-orbit coupling which is typically reasonable for ferromagnetism. Both superconductivity and ferromagnetism turned out to be exotic versions of both phenomena. This surprise rendered this problem to be probably the hottest problem in condensed matter physics today. The original discovery of superconductivity in TBG has been cited more than 2500 times in just 3 years!  

(2)   Unexplainability: the levels of emergence that appear when we move from carbon atoms (that make up our bodies) to graphene and then to TBG show us a very clear and interesting example of hierarchical emergence in just one example. Although scientists can understand the physics of one emergent step (from carbon to graphene) they currently cannot understand the next step of emergence (from graphene to TBG). Although the brightest minds in condensed matter physics are trying very hard to understand these emergent properties, it seems that the situation is calling for more time and effort. There are probably more than 500 theoretical studi that tried to understand these emergent properties. However, these problems are not solved yet. 

 (3)   Conceptual novelty: although TBG with small angles and TBG with a twist angle of 30 degrees are realized in basically the same system, they have different properties.  Scientists currently not only do not understand both of these sets of properties but also they are not able to connect the two set of properties to each other. This fact of realizing different versions of emergent properties in exactly the same system but without being able to connect between these emergent properties is novel and deserves some pondering. 

(4)   Universality: TBG is interesting not only because we cannot explain it is properties. But also because the emergent properties coming out of this system look somehow similar to some other emergent properties that come out in completely different systems such as high temperatures (high Tc) superconductors.  High Tc superconductors are probably the biggest mystery in condensed matter physics. Solving this mystery can have very immediate effects on our technology. Due to the somehow apparent similarities between the emergent properties in TBG and high Tc superconductors and the relative ease to fabricate and control TBG, some scientists think that understanding TBG  might be easier and can help us understander high Tc superconductivity.   

In many instances in the history of condensed matter physics, the emergence of new properties is associated with the appearance of what we call quasi-particles. Maybe the new properties emerging in TBG can be associated with some new quasi-particles. Regardless of the possible existence of some quasi-particles in TBG, in the next part of this essay, I will discuss the concept of quasi-particles.

Emergence and quasi-particles
Probably a radical idea is that what matters seems to be not the individual components of a system, rather the way they combine together to generate what looks to us like single particles. A combination of a large number of objects will lead to something that acts like exactly one thing. The propagation of sound is one example. What transmits sound is not just the individual atoms, it is rather the collective motion of all atoms together. The collective motion of atoms is called phonons. We say that these phonons are one typical type of quasi-particles. Quasi-particles are simply a large number of particles that behave mathematically and experimentally as if they are just one particle. Quasiparticles are not always irreducible and supervenient. Although quasiparticles are more than just their constituent particles, our understanding of them can in principle be reduced to our understating of their constituent particles. However, this reducibility is not practical and that is why we utilize the concept of quasiparticles.  Although missing a few of the particles that make up the quasi-particle will not affect the overall properties of the quasiparticle, quasiparticles can vanish even without making a change in the supporting lower level.

Quasi-particles are particles in the same sense as elementary particles that appear in the standard model of physics like electrons, protons, neutrons, etc. Quasi-particles have charge, spin, energy, momentum and they can behave as bosons or fermions.  Since we already mentioned superconductivity in this essay, let us have some idea about quasi-particles by discussing the quasi-particles that appear in conventional superconducting materials.  

Our everyday electronic devices function because of some electric current that moves inside “wires” buried inside our devices. The actual current is the movement of electric charge inside the material. Since these wires are typically made of a very large number of core nuclei and electrons, the moving electric charge inside the wires will collide with other electrons and nuclei. These collisions are the reasons why our electronic devices get hot! These collisions are what make our electricity bills more expensive than they could be. The solution to the problem of unwanted collisions of electrons inside wires is to make materials that have moving charges that do not collide with anything else! Some of these materials are called superconductors. There are different types of superconductors. We call superconductors that scientists understand conventional superconductors while the ones that scientists do not understand are called unconventional. 

The reason why conventional superconductors work is because of the emergence of new entities inside these materials. These new entities are called Cooper pairs and each Cooper pair is made of two electrons that coordinate among themselves and move together as if they are attracting each other.   In my opinion, the key feature that makes emergence in this system pronounced is novelty and unpredictability. Nobody could imagine that electors will effectively attract each other! The problem of understanding conventional superconductivity took the bright minds of physics more than 40 years to reach some understanding. They took this long time because they did not imagine that this fascinating physical phenomenon will be due to the emergence of a new entity.  

The list of emerging particles - quasiparticles - is growing and physicists today use them routinely in their quest to understand the behavior of various physical systems. A very recent example of an interesting quasi-particle is called anyons. These anyons were predicted in the early eighties and were firmly confirmed in the middle of 2020. We get anyons in some special circumstances when we put electrons in two-dimensional boxes and expose them to a very high magnetic field. 

According to the standard model of physics, the smallest building block of electric charge that exists in free space is the electron charge. Any other electric charge can just be a multiple of the charge of one electron.  However, when scientists put electrons together to form anyons they get physical objects that have a fraction of the electron charge! The ones discovered last year have a charge of one-third of the charge of the electron.  More interestingly, these quasiparticles are not fermions nor bosons; they are something different. They form their own category, which is called Anyons. This last discovery of the anyons forces me to reconsider whether what we think in physics as fundamental is really fundamental. Maybe they are just the result of emergence. It is probably emergence all the way to the bottom.