We begin with the chariot simile -- a story from early Buddhism about the nature of worldly truth (for a beautiful introduction, see here). The king Milinda goes to the forest to meet the hermit Nagasena. The hermit asks how he came to the forest and Milinda points to his chariot. Nagasena probes further. What precisely is this chariot? Is it the wheels, the axle, the reins or the yoke? Clearly, these are mere parts and no part, by itself, is the chariot. Nor is the chariot something entirely outside of the parts. And yet, we immediately perceive the chariot when we see the parts. We even understand how it operates and the function that it serves. Does this chariot even exist?

This simile extends well beyond chariots to all of the physical world. It underpins every area of physics although it is often not appreciated as such. For example, to describe a planet revolving around a star, one relies on an implicit understanding of a ‘planet’ and a ‘star’. As with the chariot, a planet consists of parts – perhaps rocks, gases and even liquid oceans, none of which is the planet by itself. Likewise, a star consists of hot gases which, by themselves, are not the star. And yet, the words planet and star immediately conjure up mental images of the intended objects. They also convey certain physical properties, perhaps with the star being a luminous sphere. Indeed, every technical term in physics bears a similar weight – of being composed of diverse parts, but functioning as one integral whole in a conventional and functional sense.

What then is a chariot? For that matter, what is a planet, an atom, a wave or a tennis ball? To assign meaning to such terms, we rely on two concepts – scale and complexity. We explore these notions below using the chariot as a simple example.

To understand the concept of scale, let us appeal to our mental image of a chariot. In all likelihood, our image of the chariot can be described as follows. It is a couple of metres in length, breadth and height. It weighs a few hundred kilograms. It moves at speeds of a few tens of km/hr, say 30 km/hr. It is capable of sustained motion over a few hours. These approximate numbers constitute a scale – a window in an abstract space of dimensions. Indeed, the concept of a chariot is inherently tied to a certain scale. For example, it is inconceivable to imagine a chariot that is a few millimetres tall, or one that weighs a thousand tons. Even if a model chariot were built to these measurements, it will simply not function in the conventional chariot-like manner. Likewise, we cannot have a chariot that moves at the speed of sound (~1234 km/hr) as it will simply disintegrate. Nor can we talk about the motion of a chariot over a million year period as any chariot will decay long before then.

Unsurprisingly for a human invention, the chariot is a meaningful idea at human scales. As a counterpoint, consider the world from the point of view of an amoeba. The microbe stretches a few hundred micrometres in length, weighs about a tenth of a milligram and moves at speeds of a few microns per second. Surely, a chariot bears no meaning in its world. Even if we were to construct a scaled down model of a chariot for an amoeba, it would not work in the same way as a human chariot – for instance, forces of surface tension would inhibit its motion. At the other extreme, consider the scale of a planet – 10000 km in length, 10^24 kg in weight and 30 km per second in speed. Once again, there is no meaning to a chariot at this scale. Even if such a humongous chariot were constructed, it would not work in the usual way. The parts would collapse due to their mutual gravitational attraction – crudely, this is the reason why all planets are nearly spherical.

As with the chariot, every ‘object’ is meaningful at a certain scale and not beyond. We cannot have an atom at the scale of a planet. Nor can we have a star at the scale of a human. This is a profound aspect of reality that is not often appreciated. In order to describe any object in the universe, we must necessarily formulate the rules of physics at its scale. As the dominant physical effects can change from one scale to another, physics itself changes with scale. For example, gravity is crucial to describe a planet but is practically absent in the life of an amoeba. Conversely, surface tension is paramount for the amoeba but plays no role for a planet. In other words, the nature of physical reality – the rulebook of physics – changes with the scale.

We now take up the second concept of complexity, once again with the example of a chariot. What is a chariot made of? It is an assembly of a yoke, an axle, four wheels and so on. However, it is not a random assembly of these parts. Rather, a chariot exists only when these parts are put together in a precise fashion. This arrangement ties each part to others in a tight relationship. As the axle turns, so do the wheels; as the yoke is pulled, the body of the chariot moves. To describe this, we say that the chariot exists at a certain level of complexity. This denotes that (a) the chariot is composed of a certain number of parts and (b) these parts interact with one another in precise ways. Without either of these properties, there can be no chariot. Conversely, there can be no chariot at a different level of complexity. Imagine a chariot with twenty axles and fifty wheels. It will certainly not function as we expect a chariot to function, nor will a chariot where the wheel is glued to the yoke.

The notion of complexity extends to every conceivable object. A Helium atom, for example, consists of two protons, two neutrons and two electrons. The protons and the neutrons are tightly bound by the strong nuclear force, while the protons and electrons attract each other via an electric force. If we were to have additional protons, we no longer have the same properties as a Helium atom. Likewise, if the attraction between protons and electrons were to be of a different nature, we would not have a Helium atom as we know it. Let us take a bigger example from the scale of every day experience: a canonball. It may consist of a kilogram or so of iron. The iron atoms are held together by strong binding forces that keep it together as a dense solid. A functioning canonball cannot exist at some other level of complexity. We cannot have a tiny version that weighs a gram, for it would be swept away from its trajectory by the wind. Nor can we have a loosely packed ball of iron filings; it would simply disintegrate when fired.

So far, we have argued that any object exists with reference to a certain scale and a certain level of complexity. Taking a deeper view, we can turn this around into a definition for the notion of an ‘object’. What indeed is an object? It is an ‘effective construct’ that emerges at a certain scale and a certain complexity. The word ‘emerges’ indicates that the object retains its identity and behaves in a regular manner. For instance, the axle, wheels, yoke, etc., stay together so that their composite behaves as one chariot. We can then speak of the chariot as a unit, describing it as moving forward, turning or coming to a halt. For a physicist who seeks to describe the world, this offers a profound simplification. He/she can ignore details about the parts, consider the chariot as one entity and formulate rules for its motion. This analogy carries through to every object in the universe, be it an atom, a lake, a football, a planet, a star or a galaxy. These are all ‘effective constructs’ that emerge at corresponding scales and complexities. To describe any of them, the physicist focuses on the relevant scale and complexity, identifies the dominant phenomena and formulates the rules using the language of mathematics.

This is a powerful idea. Taking it further, we see that world divides itself into layers, each corresponding to a certain scale and complexity. Each layer comes with its own effective constructs or objects. It also has its own rules that describe its native effective constructs. The objects of one layer become the constituents of the next – quarks constitute protons which form atoms that combine into molecules and so on. We have a cascading hierarchy of levels ranging from elementary particles to that of the visible universe. The limits of this hierarchy are posed by our human limitations. At one end, we do not understand (atleast not yet) if quarks are composed of smaller sub-particles. At the other end, we do not know what lies beyond our visible universe. It is quite conceivable that the layers continue beyond. Indeed, this point of view has inspired elaborate branches of physics. String theory is a case in point. It conjectures a hidden base layer of string-like objects which come together at a higher layer to form elementary particles.

The reader can perhaps easily conceive of the objects of one layer coalescing into that of a higher layer. It is more challenging to imagine the rules of one layer giving rise to a different set of rules in a higher layer. However, we know this to be true in many physical contexts. For example, we know that an atom is well described by quantum mechanics. But when several atoms come together to form a piece of chalk, they are described by the simpler rules of classical mechanics. We can safely ignore the quantum nature of atoms (e.g., Schrodinger equation) in a classical description of chalk (e.g., using Newton’s laws of motion). To take another example, let us imagine a physicist who has spent a lifetime on the study of water droplets. He/she will be intimately familiar with the phenomenon of surface tension as it is responsible for the shape of droplets. As the nursery rhyme goes, it is little drops of water that make the ocean. Objects from a lower layer, water drops, come together in large numbers to form an ocean, an object belonging to a higher layer. The droplet-physicist will be quite at sea in this layer (pun intended). The rules of physics in the ocean are markedly different from that in a droplet. Concepts of waves and tides come to play, while surface tension recedes into oblivion.

This notion of emergence is of profound importance to physics, arguably serving as its primary driving force. Crudely speaking, every subfield of physics begins with a set of rules and objects. It then strives to discover new phenomena that emerge when several objects come together. In some cases, the new objects that emerge are not very different from their constituents. For instance, in many ways, a small molecule follows the same rules as the atoms that make it. An atom-physicist can predict some properties of a molecule without too much effort, e.g., both atoms and molecules absorb light at a series of sharp wavelengths. However, in some cases, the new object can take an entirely new form. Its properties can be completely unlike those of the parts. To give a simple example, a metallic magnet ‘emerges’ from a collection of iron atoms. Even if you had studied iron atoms in great detail, you would not be able to predict that a collection of these atoms would conduct electricity. You would be very much surprised that this clump of atoms becomes a magnet that can stick to your fridge. This is an example of an ‘emergent property’, an unpredictably new phenomenon that arises upon changing scale and complexity. In general, such properties are very difficult to explain in a rigorous fashion. Iron serves as a case in point. It is extremely challenging to explain its magnetism starting from knowledge at the level of an atom (here is an old review; here is a relatively recent advanced calculation and here is a lively online debate). While explanations have been offered, they are based on several approximations and cannot necessarily make predictions for other elements.

There has been lively debate among scientists and philosophers regarding emergence. One prominent question is whether emergence is 'strong' or 'weak' (e.g., see here). A second question is whether 'emergence' can be defined in precise terms (e.g., see here). These debates are academic in nature – there is no disagreement on the general fact that emergent properties exist.

We now discuss simple examples of emergence that were discovered recently in my research group. Working with small physical systems, we are able to rigorously demonstrate emergence. In other words, we start from an underlying layer with specific rules and end up with a completely new construct at a higher layer. We discuss results at a broad philosophical level here. Technical details can be found in our paper.

Our discussion is based on magnetic clusters or molecular magnets. These are collections of a small number of ‘spins’ that are coupled to each other. For our purposes, a spin can be viewed as an arrow that can point in any direction. It typically arises from the angular momentum of an electron. Crudely, it can be viewed as the axis about which the electron is spinning. The spins here are the ‘parts’ that constitute a small magnet. These parts are ‘coupled’ to one another by a spin-spin interaction. This interaction tries to oppositely align spins. This is a common feature in many magnets. While we discuss simple theoretical models, these models closely mimic several experimentally studied molecular magnets.

We consider two clusters to illustrate our findings - a dimer and a quadrumer. In the dimer, we have two spins at the ends of a rod. In the quadrumer, we have four spins at the corners of a tetrahedron. These two clusters come form a remarkable sequence of geometric shapes. They are also known as simplex shapes in one and and three dimensions. They share a special property that any pair of corners are separated by the same distance. In other words, every edge in the shape has the same length. Of course, the dimer fits this definition in a comically simple manner as it only has one edge. Nevertheless, the simplex nature of the shape leads to beautiful geometric relations for the energy of the magnet.

We first consider the simplest case, that of the dimer. The energy is lowest when the two spins point in opposite directions. How many such possible configurations do we have? We could fix the first spin to point in an arbitrary direction. The second spin must immediately be chosen to point in the opposite direction. The number of ways to achieve this is simply the number of directions where the the first spin can point. This is clearly infinite. It corresponds to the number of points on the surface of a sphere, if we take the first spin to be an arrow that starts from the centre and ends at some point on the surface. This simple geometric picture leads to a remarkable physical consequence. To see this, we focus on the behaviour of the dimer-magnet at very low energies. This corresponds to a change in ‘scale’ as we have described above. Physically, this corresponds to studying the magnet at very low temperatures. In this situation, the magnet simply moves from one minimum energy configuration to another. That is, it behaves like a particle moving on the surface of a sphere. Indeed, if we look at physical properties such as specific heat, we cannot not tell the dimer-magnet apart from a particle on a sphere. This is an example of an emergent phenomenon. When we change the scale (the energy scale here), the spins come together to form a new object - a bead constrained to move on the surface of a sphere.

This emergent property of the dimer-magnet has long been known, although it has perhaps not been described on these lines. It forms the building block of field theoretic studies of antiferromagnets. This approach was pioneered by FDM Haldane and others. Haldane received the Nobel Prize in part for this contribution. To some skeptics, the morphing of a dimer-magnet into a bead-on-a-sphere may not represent true emergence. The objection essentially is that this could have been predicted (albeit by a very sharp mind), based on the two problems having the same symmetry. There is some truth to this criticism. We now move to the quadrumer where emergence is beyond question - it forms a new object that can not be predicted based on any symmetries.

We consider four spins at the corners of a tetrahedron. Compared to the dimer, this represents an increase in complexity as we have gone from two spins to four. On the same lines as the dimer, we wish to examine this magnet at low energies. We first ask how the energy can be minimized. A simple geometric trick comes into play here. The energy is minimized when the four spin vectors add to zero, according to the standard rules of vector addition. In simpler terms, we visualize the four spins as arrows with the head of one arrow attached to the tail of the next. Starting from the centre, we imagine traversing one arrow after another. If the arrows are such that the tip of the fourth arrow brings us back to the centre, the energy is minimum. To give a simple example, the four spins form the sides of a rhombus so that traversing them brings us back to the starting position. We may even have a ‘buckled’ rhombus where the spins do not all lie in the same plane.

How many such configurations do we have? Clearly, we have an infinite number of choices. In fact, based on some mathematical arguments, this is the number of points on a sheet that lives in a five dimensional space. Although this is not crucial for our discussion here, the nature of this sheet is an interesting question in and of itself. The sheet is not ‘smooth’ as it pinches onto itself at certain points. Leaving that to one side, we come back to our original goal of describing the magnet when the scale is shifted to low energies.

Remarkably, the magnet resembles a particle that moves on the five-dimensional sheet. Thankfully, this in turn reduces to a much simpler picture. The magnet seemingly breaks up into two parts – a new ‘emergent’ spin vector and a rotating sphere! From its physical properties (specific heat, for instance), we cannot tell the magnet apart form a sphere-spin combination. This is a remarkable manifestation of emergent properties. The rotating sphere could perhaps have been predicted beforehand as it has the same symmetries as the magnet. It represents rotating motion of the entire magnet as a whole. However, the new spin is a genuine emergent property. It arises from ‘buckling’ distortions of the rhombus described above. That buckling somehow manifests as a spin is truly emergent - there was no way to predict this at the outset.

We have discussed a simple magnet with four spins. These four spins are locked together by mutual coupling, just as the parts of a chariot are coupled to one another. This represents a rather simple level of complexity. By changing the scale to low energies (say, low temperatures), the magnet takes on a surprising form – resembling a rotating sphere and a separate emergent spin. This offers perhaps the simplest example of emergence. The new spin here is not at all a spin in the original problem of the magnet. It is simply an illusion where certain distortions of the magnet manifest themselves as a new spin. We conclude with a tantalizing possibility – are all spins mere illusions? We know that every electron carries a spin. Could the electron, so far known to be an elementary particle, be made of smaller parts? Could the distortions of these parts manifest as the spin of the electron?

Our paper:

• Quantum spin quadrumer

Physical Review B 97, 054403 (2018); arxiv version here.