In a prior article from June, I discussed Noether’s Theorem, which links continuous symmetries to conservation laws. While Noether’s Theorem itself is a captivating concept within physics, the nature of symmetries is equally fundamental and often just as intriguing. A significant portion of theoretical physics revolves around the breaking of these symmetries. But what exactly are symmetries, and what transpires when they are disrupted?

In essence, a symmetry refers to any operation that leaves a system unchanged. These operations can be physical, like rotations or translations, or more abstract, such as time-reversal symmetry or spin space rotation symmetry (which will be elaborated on later). As I elaborated in my previous piece on Noether’s Theorem, symmetries can be classified as either discrete or continuous.

To illustrate, a cube exhibits rotational symmetry at intervals of 90 degrees, but this symmetry does not apply at angles like 45 degrees. Therefore, the cube possesses a discrete rotational symmetry of 90 degrees (and its multiples). In contrast, a sphere has continuous rotational symmetry, as it can be rotated by any angle—no matter how minute—and still retain its appearance.

More broadly, when we analyze a physical system, we define a symmetry as a transformation that leaves the Hamiltonian unchanged. In my earlier article, I defined a system through its Lagrangian; however, the Lagrangian and Hamiltonian are inherently connected via a Legendre transform, which I won’t delve into here. The Hamiltonian is particularly relevant in quantum mechanics, the primary focus of the discussion to follow.

In simple terms, the Hamiltonian describes how a physical system behaves. It incorporates both the position and momentum of a state and dictates the time evolution of that state according to a specific potential. States that remain constant over time under this potential are termed the eigenstates of the Hamiltonian. Just as the eigenstates of the position operator exhibit a well-defined location, the eigenstates of the Hamiltonian possess a well-defined total energy. Consequently, these eigenstates are referred to as energy eigenstates.

A crucial aspect of the Hamiltonian is its Hermitian nature. This can be characterized in various ways, particularly involving matrices and complex conjugation. For our discussion, it suffices to understand that the eigenvalues of the Hamiltonian—representing the energies of the eigenstates—must be real rather than complex. All operators measuring classical quantities, such as position and momentum, are Hermitian, reinforcing that measurements of energy, position, or momentum should yield real values.

While this property might seem trivial, its implications are profound. One significant outcome is that the eigenstates of a Hermitian operator are orthogonal, meaning each eigenstate is independent of the others. This allows us to construct any state as a superposition of eigenstates, a concept I elaborated on regarding position and momentum eigenstates in my previous article.

So, how do symmetries fit into this framework? A symmetry is defined as a transformation that does not alter the Hamiltonian. Given that the eigenstates of a Hamiltonian for a specific potential are uniquely determined, any transformation that leaves the Hamiltonian unchanged must also preserve the eigenstates. If the Hamiltonian governs the time evolution of a state, and that state is a superposition of energy eigenstates unaffected by the transformation, then the time evolution of the state remains unchanged. In this context, a symmetry is a transformation that does not alter the system's behavior!

To clarify this with an example, consider a solitary electron in completely empty space. In this scenario, there are no potentials, and thus no forces acting on the electron. According to Newton’s first law, the electron will continue moving at the velocity it had when introduced to this space.

Now, let's perform a transformation: moving the electron two meters to the right while keeping its velocity constant. What happens to the electron's behavior? Since no potentials are present in this empty space, we expect the electron to continue its motion unaltered by the translation. Therefore, the time evolution of the system—how the electron moves within the space—remains unchanged by the transformation, confirming that translation is indeed a symmetry of the system.

What are some common examples of symmetries? Translational symmetry is quite prevalent, especially in the study of crystals, which are characterized by their periodic structure and thus spatial invariance over specific intervals. Closely related is rotational symmetry, often observed in systems involving rotation. Notably, rotational symmetry plays a significant role in astrophysics, where spherical bodies lead to rotationally symmetric gravitational fields and the elliptical orbits we are familiar with.

In addition to these continuous symmetries, there are also discrete symmetries. One example is time reversal symmetry, which implies that a system behaves identically whether observed in "forward" or "backward" time. The absence of time reversal symmetry is particularly apparent in our experience of time having a "direction," primarily due to this symmetry's lack in many processes, especially thermodynamics.

Another less pronounced symmetry is lattice inversion symmetry, which pertains to crystal lattices and refers to invariance under taking the "mirror image" of the lattice by negating all spatial coordinates.

If a system possesses time reversal symmetry, each energy eigenstate has a corresponding time-reversed partner with the same energy but opposite momentum. Similarly, in systems with inversion symmetry, each energy eigenstate has a partner with opposite momentum whose energy is the same in magnitude but negative.

What occurs when we begin to break these symmetries? First, let's consider the removal of time reversal symmetry. Picture confining an electron to a 2D plane while applying a uniform magnetic field perpendicular to that plane. As a charged particle, the electron experiences a force perpendicular to its velocity, known as the Lorentz force, causing its path to curve within the plane.

If we reverse time in this experiment, the magnetic field remains unchanged, but the initial velocity of the electron is inverted. Consequently, the Lorentz force is now the negative of that experienced in the initial scenario, causing the electron to curve in the opposite direction. This demonstrates that a magnetic field disrupts time reversal symmetry.

This concept can also be applied to electrons in a crystal, a system exhibiting time reversal symmetry. In the simplest case, the time-reversed counterpart of the eigenstates corresponds to states with opposite spins. For each up-spin electron in the crystal, there exists a down-spin electron with the same energy, a phenomenon known as spin degeneracy.

We understand that applying a magnetic field breaks the system's time reversal symmetry, lifting this spin degeneracy. Consequently, under a magnetic field's influence, the energy of up-spin electrons differs from that of down-spin electrons.

Remarkably, this is precisely what we observe! The magnetic field decreases the energy of electrons with spins aligned with the field while increasing the energy of those with spins opposite to the field. The partners now possess different energies, and the degeneracy is lifted. This phenomenon, known as Zeeman splitting, is experimentally evident in the splitting of absorption lines in elemental spectra when subjected to a magnetic field.

What happens if we break inversion symmetry? A noteworthy illustration is found in honeycomb lattices. As mentioned earlier, a honeycomb lattice comprises two distinct types of sites, designated as A and B. When we take the mirror image of a honeycomb lattice, the A sites map onto the B sites. If A and B sites are identical, such as in graphene composed solely of carbon atoms, inversion symmetry is preserved. However, in hexagonal Boron Nitride (hBN), where alternating boron and nitrogen atoms create the lattice, inversion symmetry is broken.

The behavior of electrons on a crystal lattice is typically described using band structure, which illustrates how the energy of the eigenstates of the Hamiltonian varies with the state’s momentum. For simple lattices like graphene and hBN, we generally identify two bands: a lower energy valence band and a higher energy conduction band.

We recognize that time symmetry necessitates each energy eigenstate to have a partner with opposite momentum and the same energy, while inversion symmetry requires that energy eigenstates have partners with opposite momentum and negative energy of equal magnitude. Typically, these conditions can coexist independently across most momenta. However, at certain momenta, they may intersect.

For any crystal lattice, specific momenta are equivalent to their negatives due to the periodic nature of the lattice, referred to as high symmetry points.

If an eigenstate and its time-reversed and inverted counterparts possess the same momentum at high-symmetry points, the energy of the state must equal its own negative. This can only occur if that energy is zero. Consequently, a band crossing must arise at the high symmetry point, as a state in the valence band and its corresponding state in the conduction band share an energy of zero. Such a band crossing is termed symmetry protected, as it directly results from the system's symmetries.

Graphene maintains both time reversal and inversion symmetry, leading to the well-known Dirac points at the K and K’ high symmetry points. However, hBN breaks inversion symmetry, removing the requirement for energy eigenstates to have zero energy, thus indicating that the band structure of hBN should exhibit a bandgap at the K and K’ points. This is again corroborated by experimental observation.

The energy gap arises from the differing energies that electrons experience on Boron and Nitrogen atoms, necessitating additional energy for an electron to transition between sites. In other words, extra energy is required for the electron to conduct electricity. Consequently, in hBN, there exists an energy gap between the valence and conduction bands at the K and K’ high symmetry points.

This contrast in symmetry profoundly alters the electronic characteristics of the two materials. The symmetry-protected Dirac point in graphene results in exceptional conductivity, while hBN behaves as an insulator with a resistivity akin to diamond.

Thus far, we have examined cases where symmetries are disrupted by external influences, such as magnetic fields, or where they are absent from the outset, as in hBN. However, some systems are intrinsically structured to break their own symmetries, a phenomenon known as spontaneous symmetry breaking.

In quantum mechanics, we often start by considering a system's ground state, the behavior of the system at absolute zero temperature, which represents the lowest possible energy state. While direct observation of zero-temperature behavior is impossible, many systems exhibit behaviors in the ground state that persist at observable temperatures or change in predictable ways.

Typically, we expect a system's ground state to reflect the same symmetries as its Hamiltonian. The Hamiltonian governing electrons on a graphene lattice possesses both time reversal and inversion symmetry, and the ground state of electrons residing on carbon atoms exhibits these symmetries as well. Yet, in certain systems, the ground state breaks one or more of the symmetries present in the Hamiltonian. In essence, the natural state of the system does not embody all available symmetries! Since the ground state represents the minimum energy condition, no energy needs to be added to the system to disrupt these symmetries. This process is termed spontaneous symmetry breaking.

A classic instance of spontaneous symmetry breaking is magnetism, which can occur in several ways but generally relies on the alignment of electron spins to confer an overall magnetic moment to a material. In most cases, this alignment is a response to an external magnetic field, but in some materials, electron interactions naturally lead to a permanent alignment of their spins, resulting in a persistent magnetic order. Spins may align parallel, resulting in a ferromagnet like elemental iron, or alternate directions, producing an antiferromagnet.

Typically, such systems exhibit spin space rotational symmetry, meaning there should be no "preferred direction" for electron spins that would lower the system's energy. All magnetic arrangements of a material break this symmetry; however, only ferromagnetic or antiferromagnetic states can achieve this without external influence. Thus, both states exemplify spontaneous symmetry breaking.

Why does this occur? One method of describing systems with interacting spins is the Heisenberg model. This Hamiltonian's energy is contingent solely on the angle between two neighboring spins and a scalar interaction energy. By convention, we assign an overall negative sign to the Hamiltonian. Parallel spins yield a maximal positive contribution, whereas antiparallel spins offer a maximal negative contribution. The spin space rotational symmetry is evident in the Hamiltonian, as the energy does not rely on the orientation of any individual spin but rather on its angle relative to its neighbors.

The ground state is defined as the configuration that minimizes the system's energy. In the Heisenberg model, where we only consider spin interaction energies, the energy is minimized when it is as negative as possible. If the interaction energy is positive, then parallel spins will minimize the overall energy while maintaining the overall negative sign of the Hamiltonian. Consequently, the ground state consists of all electron spins pointing in the same direction.

Conversely, if the interaction energy is negative, the energy is minimized by arranging all spins antiparallel to their neighbors. This configuration allows the negative interaction energy and overall negative sign to cancel, yielding a positive contribution, resulting in alternating up and down spins on neighboring sites. Thus, the distinction between ferromagnets and antiferromagnets lies solely in the sign of the interaction energy.

Coulomb interactions—the potential energy arising from the electric repulsion between like charges, such as negatively charged electrons—contribute positively to the Hamiltonian. In contrast, the electrons' kinetic energy provides a negative contribution. The interaction energy and consequent magnetic ordering often depend on which of these factors prevails, as the electrons' spin properties influence how energy is minimized.

For instance, if the Coulomb repulsion between electrons on the same atom, known as the Hubbard interaction, is particularly strong, the energy is minimized by preventing electrons from occupying the same sites. According to the Pauli exclusion principle, which prohibits two electrons of the same spin from residing on the same site, double occupation can be avoided by ensuring that electrons all maintain the same spin, resulting in a ferromagnetic system. This prediction aligns with the sign of the interaction energy, as the positive Coulomb energy predominates, leading to a positive interaction energy.

Many other fascinating instances of spontaneous symmetry breaking exist, ranging from the Mott insulator transition in condensed matter physics to the spontaneous breaking of gauge symmetries in the early universe, resulting in the Higgs mechanism. But aside from intricate technicalities, what insights can we derive from studying symmetries?

Symmetries serve as invaluable tools in theoretical physics. As demonstrated, we can predict a system's behavior based solely on its symmetries. More importantly, they provide a means to validate our models. If you create a Hamiltonian that fails to uphold a system's symmetry, it's likely that an error has occurred somewhere in your analysis.

Moreover, symmetries possess an inherent beauty. Beyond the implications of Noether’s Theorem, the physics that emerges solely from a system’s symmetries is astonishing. Take graphene, for example: the very property that underpins its technological potential is intrinsically linked to the inversion symmetry of its crystal lattice. The elegance of this connection is, in my opinion, profoundly beautiful.