Exploring the Fascinating World of Broken Symmetries in Physics
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Chapter 1: Understanding Symmetries
In an earlier article, I discussed Noether's Theorem, a remarkable concept linking continuous symmetries to conservation laws. While Noether's insights are profound, symmetries themselves are equally fundamental and often aesthetically pleasing. Moreover, much of theoretical physics stems from the exploration of broken symmetries. So, what exactly are symmetries, and what occurs when they are disrupted?
In basic terms, a symmetry refers to any transformation that leaves a system unchanged. These transformations can be physical, like rotations or translations, or more abstract, such as time-reversal symmetry or spin rotation symmetry (more on these concepts shortly). As elaborated in my previous piece on Noether's theorem, symmetries can be categorized as either discrete or continuous.
For instance, a cube exhibits rotational symmetry at 90-degree intervals, but this symmetry doesn't hold for a 45-degree rotation. Therefore, we classify the cube as having discrete rotational symmetry at 90 degrees (and its multiples). On the other hand, a sphere boasts continuous rotational symmetry, as it can be rotated by any angle, even infinitesimally small ones, without altering its appearance.
In a physical system, we define a symmetry as a transformation that leaves the Hamiltonian unchanged. While my previous article focused on the Lagrangian, the Hamiltonian is often preferred in quantum mechanics, which will be the primary focus here.
To summarize, the Hamiltonian describes how a physical system behaves, detailing its position and momentum, and indicating how that state evolves over time based on a specific potential. States that remain constant over time under this potential are known as eigenstates of the Hamiltonian. Just as position eigenstates have a defined position, Hamiltonian eigenstates have a well-defined total energy, leading to their designation as energy eigenstates.
An essential characteristic of the Hamiltonian is its Hermitian property. This can be articulated in various ways involving matrices and complex conjugation, but fundamentally, it ensures that the eigenvalues of the Hamiltonian—representing the energies of eigenstates—are real numbers rather than complex values. All operators measuring "classical" quantities, like position and momentum, are Hermitian, which aligns with our expectation that we should not measure unreal values for energy, position, or momentum.
While this property might seem trivial, its implications are significant. One notable consequence is that the eigenstates of a Hermitian operator are orthogonal. This independence means that any eigenstate can be expressed as a superposition of other eigenstates, as I have detailed in discussions on position and momentum eigenstates.
So, how do symmetries fit into this framework? We established that a symmetry is a transformation that does not alter the Hamiltonian. Since the eigenstates of a Hamiltonian for a specific potential are uniquely determined, if a transformation leaves the Hamiltonian unchanged, then the eigenstates must also remain unchanged. Consequently, if the Hamiltonian dictates the time evolution of a state, and that state comprises energy eigenstates unaltered by the transformation, then the state’s time evolution remains unaffected. Thus, a symmetry is a transformation that preserves the system's behavior!
Now, let's consider an example. Picture a solitary electron in an empty space devoid of potentials. In this scenario, without any forces acting on the electron, it will continue to move at the same velocity it had when initially placed in that space, adhering to Newton's first law.
Next, let’s perform a transformation by moving the electron two meters to the right while keeping its velocity constant. How does the electron respond? It remains in an unperturbed empty space, so we expect it to continue moving as it did before the translation. Therefore, the system's time evolution—the electron's movement in space—remains unchanged, confirming that translation is indeed a symmetry of the system.
What does this signify for the Hamiltonian? In the absence of potentials, the electron's energy is solely dependent on its velocity (momentum), as it possesses only kinetic energy. Thus, the Hamiltonian can only depend on momentum, not position. Consequently, the translation does not affect the Hamiltonian, fulfilling our definition of symmetry!
The first video titled "Your Daily Equation #25: Noether's Amazing Theorem: Symmetry and Conservation" delves into the intrinsic connection between symmetries and conservation laws, expanding on the significance of Noether's Theorem in physics.
Section 1.1: Types of Symmetries
Translational symmetry is prevalent in various contexts, particularly in crystals characterized by their periodicity and spatial invariance over defined intervals. Closely linked is rotational symmetry, which arises in systems involving rotation. This is especially significant in astrophysics, where spherical bodies give rise to rotationally symmetric gravitational fields and the elliptical orbits we observe.
In addition to continuous symmetries, we encounter several discrete symmetries. One such example is time-reversal symmetry, where a system behaves identically when observed "forwards" or "backwards" in time. The lack of time-reversal symmetry is evident in our experience of time having a "direction," particularly in thermodynamic processes.
Another symmetry is lattice inversion symmetry, which applies to crystal lattices, representing invariance when taking the "mirror image" of the lattice by reversing all spatial coordinates.
If a system possesses time-reversal symmetry, each energy eigenstate has a time-reversed counterpart with the same energy but opposite momentum. Similarly, in systems exhibiting inversion symmetry, each energy eigenstate has a partner with opposite momentum and the same energy magnitude, albeit with a negative sign.
The second video titled "Symmetries & Conservation Laws: A (Physics) Love Story" explores the deep and intricate relationship between symmetries and conservation laws in physics, shedding light on how these concepts intertwine.
Section 1.2: Breaking Symmetries
When symmetries are disrupted, we first examine the removal of time-reversal symmetry. For instance, consider confining an electron to a 2D plane while applying a uniform magnetic field perpendicular to that plane. A charged particle, such as an electron, experiences the Lorentz force when moving in a magnetic field, causing its trajectory to curve.
If we were to reverse time in this scenario, the magnetic field remains unchanged, but the electron's initial velocity would be inverted. Consequently, the Lorentz force acting on the electron would also be reversed, resulting in a path that curves in the opposite direction. Thus, the electron's behavior in the presence of a magnetic field differs when time flows forwards as opposed to backwards, illustrating that a magnetic field breaks time-reversal symmetry!
This concept can also apply to electrons moving within a crystal, a system that typically exhibits time-reversal symmetry. In a straightforward case, the time-reversed counterpart of the eigenstates corresponds to states with opposite spins. For every electron with an "up" spin in the crystal, there exists an electron with a "down" spin at the same energy. This characteristic is known as spin degeneracy.
When a magnetic field is introduced, the time-reversal symmetry of the system is compromised, resulting in the lifting of spin degeneracy. In simpler terms, in a magnetic field, the energy of electrons with spins aligned with the field decreases, while the energy of electrons with spins opposing the field increases. The partners now possess different energies, and the degeneracy is lifted—this phenomenon is referred to as Zeeman splitting, observable in the splitting of absorption lines in elemental spectra under a magnetic field.
Moving on, let’s discuss the breaking of inversion symmetry. A clear illustration of this can be seen in honeycomb lattices. As previously mentioned, such lattices contain two distinct types of sites, referred to as A and B sites. Reflecting the honeycomb lattice yields A sites mapping onto B sites. Inversion symmetry is preserved when A and B sites are identical, as in graphene's carbon-only lattice. However, this symmetry is disrupted in hexagonal Boron Nitride (hBN), where alternating boron and nitrogen atoms create a non-identical arrangement.
We typically describe electron behavior on a crystal lattice using a band structure, which indicates how the energy of the Hamiltonian's eigenstates varies with the state's momentum. For simple lattices like graphene and hBN, we focus on two bands: a lower energy valence band and a higher energy conduction band.
Time symmetry necessitates that energy eigenstates possess partners with opposite momentum and the same energy, while inversion symmetry requires that these eigenstates have partners with opposite momentum and negative energy of the same magnitude. Generally, these two conditions can coexist independently for most momenta. However, there are certain momenta where they collide.
In any crystal lattice, specific momenta correspond to their negatives due to the lattice's periodic nature, and these momenta are referred to as high symmetry points. If an eigenstate and its time-reversed and inverted counterparts share the same momentum, particularly at high-symmetry points, we are left with a requirement that the state's energy must equal its own negative. The only feasible scenario is when that energy is zero, resulting in a band crossing at the high symmetry point, known as symmetry protected, as it arises directly from the system's symmetries.
In graphene, the presence of both time-reversal and inversion symmetries gives rise to the well-known Dirac points at the K and K' high symmetry locations. However, in hBN, the breaking of inversion symmetry negates the necessity for energy eigenstates to have zero energy, leading to an expected energy gap at the K and K' points—a prediction that aligns with experimental observations.
This energy gap arises from the differing energy experiences of electrons on Boron versus Nitrogen atoms, necessitating additional energy for electrons to shift between sites. In essence, hBN exhibits an energy gap between the valence and conduction bands at the K and K' high symmetry points.
This difference in symmetry profoundly impacts the electronic properties of the two materials. The symmetry-protected Dirac point in graphene accounts for its exceptional conductivity, while hBN behaves as an insulator with a resistivity akin to diamond.
Chapter 2: The Role of Spontaneous Symmetry Breaking
Thus far, we have examined scenarios where symmetries are broken by external factors, such as magnetic fields, or where they are absent from the outset, as seen in hBN. However, some systems inherently break their own symmetries, a phenomenon known as spontaneous symmetry breaking.
When studying a quantum system, we often begin with its ground state, representing the system's behavior at zero temperature and its lowest energy configuration. Although we cannot directly observe zero-temperature behavior, many systems exhibit similar behavior at higher temperatures or demonstrate predictable changes.
Generally, we anticipate that a system's ground state will reflect the symmetries of its Hamiltonian. A Hamiltonian describing electrons on a graphene lattice maintains both time-reversal and inversion symmetries, and the corresponding ground state of electrons occupying carbon atoms shares these symmetries. Yet, some systems exhibit ground states that break one or more of the Hamiltonian's symmetries. In simpler terms, the natural state of such systems does not retain all available symmetries! Since the ground state represents the minimum energy configuration, these symmetries can be broken without additional energy input, leading us to term the phenomenon spontaneous symmetry breaking!
A classic example of spontaneous symmetry breaking is magnetism. Magnetism can emerge through various mechanisms, but all depend on the alignment of electron spins, resulting in an overall magnetic moment for the material. Typically, this occurs in response to an external magnetic field, but in certain materials, interactions between electrons can lead to a permanent alignment of their spins, giving rise to inherent magnetic ordering. Spins can align in parallel, resulting in ferromagnets like elemental iron, or alternate directions in antiferromagnets.
These systems typically exhibit spin space rotational symmetry, implying there is no "preferred direction" for electron spins that reduces the system's energy. While all magnetic orderings disrupt this symmetry, only in ferromagnets and antiferromagnets does this occur without external influence. Hence, both represent instances of spontaneous symmetry breaking.
What drives this phenomenon? We can describe interacting spins through the Heisenberg model, where the energy depends solely on the angle between 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, while antiparallel spins generate a maximal negative contribution. The Hamiltonian illustrates spin space rotational symmetry, as energy calculations do not depend on any individual spin's direction but solely on its angular relationship with its neighbors.
Since the ground state minimizes the system's energy, we find that in the Heisenberg model, where we prioritize energies from spin interactions, energy is minimized when it achieves the most negative value. If the interaction energy is positive, parallel spins will minimize energy due to the overall negative sign of the Hamiltonian. Conversely, if the interaction energy is negative, the energy is minimized by arranging spins in an antiparallel fashion, where the negative interaction energy and overall negative sign balance to yield a positive contribution. Thus, a ferromagnet's ground state consists of uniformly aligned spins, while an antiferromagnet features alternating spins.
Coulomb interactions—potential energy arising from the electric repulsion between like charges such as electrons—contribute positively to the Hamiltonian. In contrast, the electrons' kinetic energy results in a negative contribution. The dominance of these factors influences the interaction energy and, consequently, the magnetic ordering.
For example, when the Coulomb repulsion between electrons on the same atom (known as the Hubbard interaction) is particularly strong, energy is minimized by preventing electrons from occupying the same sites. By applying the Pauli exclusion principle, which prohibits two electrons of the same spin from cohabitating a single site, this leads to a scenario where electrons predominantly have the same spin, resulting in a ferromagnetic configuration. We can predict this outcome based on the interaction energy's sign, with the positive Coulomb energy prevailing to create a positive interaction energy.
Numerous fascinating instances of spontaneous symmetry breaking exist, ranging from Mott insulator transitions in condensed matter physics to the spontaneous breaking of gauge symmetries in the early universe, giving rise to the Higgs mechanism. But beyond the technical jargon, what do we gain from studying symmetries?
Symmetries serve as a powerful tool in theoretical physics. As demonstrated, we can derive predictions about a system's behavior solely from its inherent symmetries. Perhaps more crucially, they provide a means of validating our models. If a Hamiltonian fails to preserve a symmetry relevant to the system, it likely indicates an error in our approach.
Additionally, symmetries possess a certain elegance. Beyond Noether's theorem, the phenomena emerging from a system's symmetries are remarkable. Consider graphene: the property underpinning its technological promise is fundamentally intertwined with the inversion symmetry of its crystal lattice. The simplicity of this connection is, in my opinion, truly beautiful.