The type of magnet proposed here does not invoke spin-orbit coupling; in fact, it does not even invoke spin. Instead, the two symmetry-broken states are themselves electronic bands that live on the crystal, and they differ from each other in both momentum space and real space. For this reason, orbital magnetism does not need spin-orbit coupling to support hysteresis, and it can couple to a much wider variety of physical phenomena than spin magnetism can- indeed, anything that affects the electronic band structure or real space wave function is fair game. For this reason we can expect to encounter many of the phenomena we normally associate with spin-orbit coupling in orbital magnets that do not possess it. I would also like to talk briefly about magnetic moments. It has already been said that magnetic moments in orbital magnets come from center-of-mass angular momentum of electrons, which makes them in some ways simpler and less mysterious than magnetic moments derived from electron spin. However, I didn’t tell you how to compute the angular momentum of an electronic band, only that it can be done. It is a somewhat more involved process to do at any level of generality than I’m willing to attempt here- it is described briefly in a later chapter- but suffice to say that it depends on details of band structure and interaction effects, which themselves depend on electron density and, in two dimensional materials, ambient conditions like displacement field. For this reason we can expect the magnitude of the magnetic moment of the valley degree of freedom to be much more sensitive to variables we can control than the magnetic moment of the electron spin, plastic pot manufacturers which is almost always close to 1 µB. In particular, the magnetization of an orbital magnet can be vanishingly small, or it can increase far above the maximum possible magnetization of a spin ferromagnet of 1 µB per electron.

Under a very limited and specific set of conditions we can precisely calculate the contribution of the orbital magnetic moment to the magnetization, and that will be discussed in detail later as well. Finally, I want to talk briefly about coercive fields. The more perceptive readers may have already noticed that we have broken the argument we used to understand magnetic inversion in spin magnets. The valley degree of freedom is a pair of electronic bands, and is thus bound to the two dimensional crystalline lattice- there is no sense in which we can continuously cant it into the plane while performing magnetic inversion. But of course, we have to expect that it is possible to apply a large magnetic field, couple to the magnetic moment of the valley µ, and eventually reach an energyµ · BC = EI at which magnetic inversion occurs. But what can we use for the Ising anisotropy energy EI ? It turns out that this model survives in the sense that we can make up a constant for EI and use it to understand some basic features of the coercive fields of orbital magnets, but where EI comes from in these systems remains somewhat mysterious. It is likely that it represents the difference in energy between the valley polarized ground state and some minimal-energy path through the spin and valley degenerate subspace, involving hybridized or intervalley coherent states in the intermediate regime. But we don’t need to understand this aspect of the model to draw some useful insights from it, as we will see later.Real magnets are composed of constituent magnetic moments that can be modelled as infinitesimal circulating currents, or charges with finite angular momentum. It can be shown that the magnetic fields generated by the sum total of a uniform two dimensional distribution of these circulating currents- i.e., by a region of uniform magnetization- is precisely equivalent to the magnetic field generated by the current travelling around the edge of that two dimensional uniformly magnetized region through the Biot-Savart law. It turns out that this analogy is complete; it is also the case that a two dimensional region of uniform magnetization also experiences the same forces and torques in a magnetic field as an equivalent circulating current.

The converse is also true- circulating currents can be modelled as two dimensional regions of uniform magnetization. The two pictures in fact are precisely equivalent. This is illustrated in Fig. 2.9. It is possible to prove this rigorously, but I will not do so here. One can say that in general, every phenomenon that produces a chiral current can be equivalently understood as a magnetization. All of the physical phenomena are preserved, although they need to be relabeled: Chiral edge currents are uniform magnetizations, and bulk gradients in magnetization are variations in bulk current current density.In the same way that the Berry phase impacts the kinematics of free electrons moving through a two slit interferometer, Berry curvature impacts the kinematics of electrons moving through a crystal. You’ll often hear people describe Berry curvature as a ‘magnetic field in momentum space.’ You already know how electrons with finite velocity in an ambient magnetic field acquire momentum transverse to their current momentum vector. We call this the Lorentz force. Well, electrons with finite momentum in ‘ambient Berry curvature’ acquire momentum transverse to their current momentum vector. The difference is that magnetic fields vary in real space, and we like to look at maps of their real space distribution. Magnetic fields do not ‘vary in momentum space,’ at nonrelativistic velocities they are strictly functions of position, not of momentum. Berry curvature does not vary in real space within a crystal. It does, however, vary in momentum space; it is strictly a function of momentum within a band. And of course Berry curvature impacts the kinematics of electrons in crystals. Condensed matter physicists love to say that particular phenomena are ‘quantum mechanical’ in nature. Of course this is a rather poorly-defined description of a phenomenon; all phenomena in condensed matter depend on quantum mechanics at some level. Sometimes this means that a phenomenon relies on the existence of a discrete spectrum of energy eigenstates.

At other times it means that the phenomenon relies on the existence of the mysterious internal degree of freedom wave functions are known to have: the quantum phase. I hope it is clear that Berry curvature and all its associated phenomena are the latter kind of quantum mechanical effect. Berry curvature comes from the evolution of an electron’s quantum phase through the Brillouin zone of a crystal in momentum space. It impacts the kinematics of electrons for the same reason it impacts interferometry experiments on free electrons; the quantum phase has gauge freedom and is thus usually safely neglected, but relative quantum phase does not, so whenever coherent wave functions are being interfered with each other, scattered off each other, or made to match boundary conditions in a ‘standing wave,’ as in a crystal, we can expect the kinematics of electrons to be affected. We will shortly encounter a variety of surprising and fascinating consequences of the presence of this new property of a crystal. Berry curvature is not present in every crystal- in some crystals there exist symmetries that prevent it from arising- but it is very common, and many materials with which the reader is likely familiar have substantial Berry curvature, including transition metal magnets, black plastic plant pots wholesale many III-V semiconductors, and many elemental heavy metals. It is a property of bands in every number of dimensions, although the consequences of finite Berry curvature vary dramatically for systems with different numbers of dimensions. A plot of the Berry curvature in face-centered cubic iron is presented in the following reference: [84, 90]. We will not be discussing this material in any amount of detail,the only point I’d like you to take away from it is that Berry curvature is really quite common. For reasons that have already been extensively discussed, we will focus on Berry curvature in two dimensional systems.Several chapters of this thesis focus on the properties of a particular class of magnetic insulator that can exist in two dimensional crystals. These materials share many of the same properties with the magnetic insulators described in Chapter 2. They can have finite magnetization at zero field, and this property is often accompanied by magnetic hysteresis. The spectrum of quantum states available in the bulk of the crystal is gapped, and as a result they are bulk electrical and thermal insulators. They have magnetic domain walls that can move around in response to the application of an external magnetic field, or alternatively be pinned to structural disorder. And of course they emit magnetic fields which can be detected by magnetometers.Unlike all trivial insulators and, in particular, trivial magnetic insulators, these magnetic insulators support a continuous spectrum of quantum states within the gap, with the significant caveat that these states are highly localized to the edges of the two dimensional crystalline magnet .

This is the primary consequence of a non-zero Chern number. These quantum states are often referred to as ‘edge states’ or ‘chiral edge states,’ and they have a set of properties that are reasonably easy to demonstrate theoretically. I will describe the origin of these basic properties only qualitatively here; a deep theoretical understanding of their origin is not important for understanding this work, so long as the reader is willing to accept that the presence of these quantum states is a simple consequence of the quantized total Berry curvature of the set of filled bands. Many more details are available in [84]. These materials are known collectively as Chern insulators, magnetic Chern insulators, or Chern magnets. They are, as mentioned, restricted to two dimensional crystals; three dimensional analogues exist but have significantly different properties. The vast majority of this thesis will be spent exploring deeper consequences and subtle but significant implications of the presence of these states. We will start, however, with a discussion of the most basic properties of chiral edge states. Astute readers may have already noticed that all real materials have many electronic bands, and every band has its own Berry curvature Ωn, so the definition provided in equation 3.4 seems to assign a Chern number to each of the bands in a material, not to the material itself. The properties of a particular two dimensional crystal are determined by the total Chern number of the set of filled bands within that crystal, obtained by adding up the Chern numbers of each of its filled bands. The total Chern number determines the number of edge states available at the Fermi level within the gap.In the absence of spin-orbit coupling, every band comes with a twofold degeneracy generated by the spin degree of freedom. Every band can be populated either by a spin up or a spin down electron, and as a result every Bloch state is really a twofold degenerate Bloch state. Adding spinorbit coupling may mix these states but does not break this twofold degeneracy. An important property of the Chern number is that Kramers’ pairs must have opposite-signed Chern numbers equal in magnitude. This is a direct consequence of similar restrictions on Berry curvature within bands. For a magnetic insulator the set of filled bands is a spontaneously broken symmetry, with the system’s conduction and valence bands hysteretically swapping two members of a Kramers’ pair in response to excursions in magnetic field. These two facts together imply that magnetic hysteresis loops of Chern magnets generally produce hysteresis in the total Chern number of the filled bands, precisely following hysteresis in the magnetization of the two dimensional crystal. This hysteresis loop switches the total Chern number of the filled bands between positive and negative integers of equal magnitude. These facts also imply that finite Chern numbers cannot exist in these kinds of systems without magnetism- if both members of a Kramers’ pair are occupied, the system will have a total Chern number of zero.As discussed previously, additional symmetries of the crystalline lattice itself can produce additional degeneracies that can support spontaneous symmetry breaking and magnetism. In most cases similar rules apply to the Chern numbers of these magnets. We will have a lot more to say about the Chern numbers associated with the valley degree of freedom in graphene.