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. Every crystal is defined by a periodic electric potential profile. In two dimensional crystals this is a scalar function of two dimensions over the lattice in real space. Let us switch our focus to momentum space. The periodic electric potential in real space produces a set of functions over momentum space E that define quantum states that electrons within the crystal can occupy. A correctly executed attempt to account for the effects of the Berry phase in crystalline systems produces a new vector-valued function over momentum space Ω that affects the kinematics of electrons in electronic bands. In two dimensional systems Ω is always oriented out-of-plane, but it can be positively or negatively oriented. We call this function the Berry curvature, and it must be accounted for to correctly explain a vast array of electronic phenomena, vertical aeroponic tower garden including electronic transport in metals, electronic transport in insulators, and angular momentum and magnetization in magnetic systems.

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 non-relativistic 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, 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. 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. We call that integer C the Chern number. In so-called ‘trivial’ magnets, this integral vanishes, and the above equation is satisfied by C = 0. This is of course most common in systems with weak or absent Berry curvature. Chromium iodide is a trivial magnetic insulator, and it has C = 0. In the class of magnets we’ll be discussing, the Chern number is a nonzero, signed integer. 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, vertical gardening in greenhouse 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. We have already mentioned the most important consequence of a finite net Chern number: the presence of chiral edge states in the gap of a magnetic insulator. We have not yet discussed the consequences of this state of affairs, and we will do so next. The quantum states available in the bulk of trivial materials, i.e. Bloch states, are delocalized over the entire crystal, and as a result, when Bloch states are present at the Fermi level, electronic transport between any two points in the crystal can occur through the rapid local occupation and depletion of these quantum states. The edge states that appear in Chern magnets support a lower-dimensional analog of this property: they are delocalized quantum states restricted to the edge of a two dimensional crystal, and as a result they support electronic transport along the edge of the crystal through the rapid local occupation and depletion of these semi-localized quantum states. They do not support electronic transportthrough the bulk, and edge states that are not simply connected cannot transmit electrons through the bulk region separating them. As mentioned, the Chern number is a signed integer, and we have not yet discussed the physical meaning of the sign of the Chern number. The edge states in Chern magnets are chiral, meaning that electrons populating a particular edge state can only propagate in one direction around the edge of a two-dimensional crystal. The sign of the Chern number determines the direction or chirality with which propagation of the electronic wave function around the crystal occurs. Electronic bands with opposite Chern numbers produce edge states with opposite chiralities. So in summary, a two dimensional crystal that is a Chern magnet supports electronic transport through chiral edge states that live on its boundaries. These systems remain bulk insulators, and edge states separated by the bulk cannot exchange electrons with each other. The sign of the Chern number is determined by the spin state that is occupied, and thus the chirality of the available edge state is hysteretically switchable, just like the magnetization of the two dimensional magnet.It is important to remember that these quantum states are just as real as Bloch states, and apart from the short list of differences discussed above, they can be analyzed and understood using many of the same tools. For example, in a metallic system, the Fermi level can be raised by exposing a crystal to a large population of free electrons and using an electrostatic gate to draw electrons into the crystal. This process populates previously empty Bloch states with electrons. These Bloch states have a fixed set of allowed momenta associated with their energies, and experiments that probe the momenta of electrons in a crystal will subsequently detect the presence of electrons in newly populated momentum eigenstates.