# Emergence and Physics

Symmetry plays an important role in physics. Changes in how symmetric a magnet is, for example, change its properties. Some of these changes, such as the appearance of superconductivity or superfluidity, are quite dramatic and give rise to questions about the existence and nature of emergence in physics. The study of soft matter (liquid crystals, polymers and active liquids) raises questions about emergence in contexts that are best described by classical physics rather than quantum physics.

## PART I: Broken Symmetry

Broken symmetry is an important concept in the emergence debate, andexplains the existence of distinct phases of matter, the transitions between these phases and the black hole-like barriers that separate them. The archetypal example is the behavior of a magnet as it is cooled.

### Magnetic Moment

The magnetic moment of a system determines how strongly an applied magnetic field will cause it to rotate around an axis like a spinning top.  The ultimate source of the magnetic moment in magnetic systems is the intrinsic magnetic moment (‘spin’) of electrons, and its direction for one electron in the system is linked to that of the others through electrical repulsion and Pauli exclusion (loosely, two electrons sharing the same magnetic moment in the same system cannot have the same position, momentum or energy).

### Magnetism and symmetry

A magnet is a piece of matter with an arrow, called a “magnetic moment”, placed at regular intervals in a three-dimensional lattice. At high temperature these magnetic moments are disordered, that is, they point in random directions. This state is known as a paramagnet.  Importantly, the physics of the system is the same if we rotate all of these moments through some angle: they pointed in random directions before the rotation and they point in (different) random directions after the rotation, and a device that measures the effect of the magnetic moments (a “magnetometer”) would not be able to tell that we have rotated the moments. Their effects cancelled out before the rotation (since they’re pointing in all possible directions) and will cancel out afterwards. This lack of change (or invariance) in the properties of the system following a rotation is known as a symmetry.

A The paramagnet.
B The same paramagnet with all moments rotated.
Note that we cannot tell the difference between A and B in a typical experiment

### Magnetism and broken symmetry

After the magnet has been cooled below a critical temperature Tc, the magnetic moments spontaneously line up along a unique direction (for iron Tc is 770 °C). This state is known as a ferromagnet(and the change between paramagnet and ferromagnet is known as a phase transition). The properties of the ferromagnetic system measured by the magnetometer will now change if we rotate all of the moments through the same angle. Before the rotation all moments pointed along one direction, now they all point along another. We say that the symmetry has been spontaneously lowered, or “broken”, on cooling through Tc.  Even in the simplest mathematical model of this physics the phases on either side of a symmetry breaking transition are separated by a mathematical singularity in the energy of the system.  This singularity makes it impossible to continuously track the properties of the system through the transition and predict the properties of the broken symmetry state from the more symmetrical state.  Interestingly, there is an analogy between the mathematical singularity in the phase transition and the black hole in cosmology where a black hole singularity prevents observers gaining knowledge of what’s inside the hole. This has led some researchers to call phase transitions black holes in materials!

A the ferromagnet with all moments lined up.
B The ferromagnet with all moments rotated
Note that A and B may be distinguished in an experiment.

### Symmetry breaking and prediction

This mathematical singularity is a fundamental barrier to our knowledge of the broken symmetry state.  Because the properties of the system after symmetry breaking cannot be predicted from the high-temperature state, it is possible to say in some sense that these properties are emergent. This is important: it implies that even if it were possible to exactly solve the Schrödinger equation for every atom in a vessel full of steam (an impossible problem with our current computational capabilities!) the solution would be of limited use since there is nothing to guarantee that it would describe the properties of the ice that would form if we cooled the vessel down in a freezer.

A diagram showing the Free Energy F as the magnetism M is varied for a spin system.
A is the paramagnetic phase; here the energy is lowest when M=0, and so the system is stable in that state.
B is the ferromagnetic state: here there are two possible minimum energies corresponding to a positive and a negative M, and the system will select one of these minima as its ground state.  The existence of multiple possible ground states is characteristic of phases with broken symmetry

The Shrödinger Equation

The Schrödinger equation is the fundamental equation describing the behaviour of matter when the effects of relativity aren’t important but the effects of quantum mechanics are.  It describes the behaviour of a system of particles in terms of how their wavefunctions (see later) change with time.

## PART II: Consequences and generalisations

### Effects of Broken Symmetry

The description of the phase transition in a magnet described above was first formulated by the great 20th century physicist Lev Landau and later popularized by Nobel laureate Philip Anderson. Its popularity stems from the fact that it may also be used to classify an enormous number of physical phenomena (including the distribution of earthquakes, the formation of snowflakes, the length of polymer chains and even the nature of the early Universe) and, as we shall see, reveals a class of emergent phenomena. Anderson notes that the breaking of symmetry in a system by cooling it to below a critical temperature causes the system to develop four new properties:

• Universality: Phase transitions have a number of basic similarities that are independent of the microscopic detail of what’s going on. As a consequence there are a number of basic similarities between the transitions involving liquids and solids, paramagnets and ferromagnets, metals and superconductors and still more exotic examples. This means that measurements of physical quantities involving these very different systems will often display very similar behaviours near a phase transition. For example, systems that are in the same universality class as the ferromagnet will respond to the application of an external field in the same way that a ferromagnet responds to an applied magnetic field, provided that we are close to the phase transition. This is the sense in which broken symmetry unifies an enormous amount of physics.
• Excitations: If we give a very cold solid a little energy something remarkable happens: we create quantum mechanical entities in the materials that act like microscopic particles. (These particles are not the same as the atoms forming the solid; they are something new and different.) We say that the system is in an excited state and there is a spectrum of particle-like excitations. The first truly emergent phenomenon we encounter is found when we give a little energy to the system in its broken symmetry state. We find that upon symmetry breaking new quantum mechanical particle excitations are present that were not there in the unbroken symmetry state. In the case of magnets these new particles are called magnons
• Rigidity: This is similar to the well-known rigidity of solids, which allows you to pick up a ruler by only holding one end. Once a symmetry is broken, attempts to change the arrangement of the underlying components of the system (in magnets, the arrangement of the spins) lead to new forces appearing. This is called generalized rigidity and cannot be predicted from the underlying equations describing the constituent parts of a solid. The new, rigid forces lead to a magnet showing permanent magnetism: a magnet will attempt to keep all of its moments pointing along its chosen direction and forces will appear to maintain this configuration if an attempt is made to change it.
• Defects: In an extended system it is possible to break symmetry in different ways in different parts of the material. For example all of the moments on the left hand side of a material might point up and those on the right might point down. The boundary between these regions is known as a defect. A defect has a basic mathematical similarity with a quantum mechanical particle and these defects may be thought of as additional kinds of excitation that may exist in a solid. Again these are emergent entities that are created through the breaking of symmetry, and can be described using a mathematical language called topology which talks about the overall shapes of objects. In terms of this language, defects are knots of energy that may not be untied in any easy way. Examples of topological defects include the domain wall in a magnet or a liquid crystal and the vortex in the superfluid. There are still more exotic examples including magnetic monopoles and complex particles called skyrmions
A skyrmion configuration in a magnetic system.
Image courtesy of Karin Everschor-Sitte and Matthias Sitte.

### Other kinds of phase transition

The picture of broken symmetry described here relies on things happening as we change temperatures. Such models are, roughly speaking, called classical descriptions (as opposed to quantum mechanical). The classical picture may be extended to a quantum mechanical one (that takes place at zero temperature) where symmetry may be broken by tuning pressure or some other external stimulus, rather than temperature. This leads to the notion of a quantum phase transition which has been the focus of much recent research.

## PART III: Examples

### Superfluidity and superconductivity

Superfluids and superconductors are amongst the most fascinating states of matter yet discovered and are considered by some to be examples of emergence.

• Superfluidity involves the dissipationless flow of particles and is found at very low temperatures in liquid helium. In other words, at very low temperatures, atoms of helium flow without any friction, and so do not lose any energy to the surrounding environment. If you try to pick up such a liquid in a bucket it drains out over the sides immediately!
• Superconductivity is similar in many respects to superfluidity and involves the dissipationless flow of charged particles at very low temperatures (typically a few degrees above absolute zero for many metals). In this case, charge flows in a superconductor without electrical resistance: there is hope that materials showing such superconducting states may solve the energy crisis, since a superconducting wire would be able to transmit electricity very efficiently without generating the waste heat that results from the presence of electrical resistance in ordinary wires.

The symmetry that is broken in the case of the superfluid and superconductor is more subtle than the ones we have met so far, but is arguably more fascinating. In the spirit of the above example of magnetic spin we might ask what exactly is being aligned as we cross the critical temperature. The answer is a quantum mechanical wavefunction.

Superfluid helium.  Note the drop forming at the bottom of the container where the liquid has flowed over the sides due to the lack of friction.

#### Superfluidity and the wavefunction

Let’s consider a superfluid. The quantum wavefunction is a mathematical object that describes all of the known physical content of the superfluid. Think of it as being a bit like a computer into which we can input a position and that outputs two numbers in response:  the first tells us how many particles are present at that position and the second tells us a number called the phase of a wavefunction. (Perhaps surprisingly, this is all that is needed to completely describe a superfluid!) As with the magnetic moment in the example of a magnet, the phase may also be visualized as an arrow, so that the wavefunction tells us that the arrow points in a particular direction at a given point where we’d expect to find a particle.

• The superfluid transition: At temperatures above the superfluid transition we might ask about the phase at a number of positions and we find that it takes random values (just like the paramagnet). If we cool the superfluid through the superfluid transition we find that the arrows all align (just like the ferromagnet). A consequence of this (which is certainly not obvious from the preceding discussion) turns out to be that the particles making up the superfluid lose their identities as well-defined individual objects and we are no longer able to say exactly how many we have at a particular point.
• Emergent particles: The emergent particles of the superfluid phase are called Bogolons and consist of waves of indistinct atoms emerging from the superfluid.
• Effects of emergent rigidity: The gives rise to the most dramatic feature of a superfluid: the flow of particles without resistance. An attempt to change the phase at one end of the sample results in a dissipationless superflow of particles, which acts to cancel out this anomaly in the phase.
• Emergent defect: The emergent defect of a superfluid is the vortex. This is a pattern of phase that winds round in a circle as shown below.

Fig. 2: The vortex defect in a superfluid.

#### Superconductivity and gauge theories

The superconductor is similar to the superfluid except that the particles of a superconductor are electrically charged rather than neutral. This forces us to include a description of electromagnetic fields and electromagnetic particles known as photons, which causes superconductivity to be quite different from superfluidity in a number of ways.

• Gauge theories and emergent particles: Electromagnetism is a gauge theory. This means that there are numerous mathematically equivalent ways to describe the electromagnetic field. This is like the existence of many languages. The underlying description of a tree is identical whether given in English, Japanese or Portuguese, but the sounds are quite different. In gauge theories the emergent particles created from symmetry breaking are very different from the ones created in the previous examples, as those emergent particles disappear! The photons ‘eat’ the normal kind of emergent particles and so aquire a mass (or grow fat), and it is these massive photons that are the emergent particles of the broken symmetry phase in a superconductor. This is a form of the famous Higgs mechanism, only here it appears in a condensed matter system rather than the early universe.
A magnet being levitated above a superconductor as a result of the Meissner effect.
Image by: Mah-Linh Doan

The Higgs Mechanism

At high energies, like those seen in the early universe, the electromagnetic and the weak nuclear force that governs radioactive beta decay are unified into one electroweak force. The Higgs field interacts with the electroweak field in such a way that when the energy of the system becomes small enough it undergoes symmetry breaking, giving rise to an emergent particle that is promptly consumed by the weak portion of the electroweak field, which is a gauge field. This gives the particles that mediate the weak nuclear force the mass they’re observed as having in experiments, and distinguishes them from the photons that mediate the electromagnetic force.

• Effects of generalised rigidity: The emergent rigidity not only gives rise to the flow of electrical current without resistance, which is analagous to the superflow of a superfluid, but also to such striking phenomena as the Meissner effect, where an externally applied magnetic field generates electrical currents inside the superconductor. These currents create a magnetic field that cancels out the applied field, effectively expelling it from the sample. (This allows us to levitate magnets in the laboratory using high-temperature superconductors cooled with liquid nitrogen.) If the field is greater than some critical value there is a phase transition to a non-superconducting state.
• Type II superconductors and emergent defects: In some superconductors an intermediate phase where superconductivity and the magnetic field co-exist in the sample can be observed, where tubes of magnetic flux are allowed to pass through the sample at the centre of vortexes similar to those observed in superfluid states.  These vortexes are the emergent defetrs of the system, and are arranged in a regular pattern, known as an Abrikosov lattice.
Neutron imaging of an Abrikosov lattice in superconductoring
Image courtesy of Don Paul.

### Soft Matter

Soft matter physics is the study of polymers (such as plastics, proteins, and DNA), fluids (solvents, paints, and soaps), membranes (cell membranes) and other chemically active systems. It’s a vast and diverse area, covering everything from the the formation of emulsions (such as liquid paint or milk) to the description of cell membranes and protein folding in biological physics .  In William S. Burrough’s words, we may even view humans as “soft machines”.

A liquid crystal in a nematic phase, where individual molecules are randomly positioned (as in liquids), but exhibit long range order.
Image by: Minutemen

This is a rapidly growing field where there is much evidence for fundamental phenomena emerging from complex, interacting systems. Some of these features are:

• Symmetry breaking: An example of symmetry breaking in soft matter systems is the formation of ‘smetic’ phases in liquid crystals, which are characterised by the formation of layers of molecules that are all aligned in a given direction.
• Classical, not quantum: Soft matter is described by classical physics, rather than quantum mechanics, and provides a different route to finding emergence, compared to the inherently quantum mechanical phenomena of magnetism and superconductivity. Despite this, much of the physics of soft matter may be described in terms of symmetry breaking and order.  Intriguingly, the classical physics of soft matter systems may be related back to quantum mechanics using a mathematical trick known as Wick rotation, whose deep meaning remains an open question.

### Historical

Phillip W. Anderson, More is Different, Science Volume 177, Number 4047 (August 1974), pp 393-396.

### Introductory

Robert B. Laughlin, A Different Universe: Reinventing Physics from the Bottom Down (Basic Books, 2008)

A. Zee, Fearful Symmetry: The Search for Beauty in Modern Physics (Princeton University Press, 2007)

### Intermediate

Stephen Blundell, Magnetism in Condensed Matter (Oxford University Press, 2001)