Preamble

As discussed in Part 1, I present here some notes on my studies. As previously, I use MathJax to render equations — you'll need JavaScript for this. Note that it may take a minute or two to load. I'm also experimenting with rough.js for sketching figures. This article may see minor updates in the coming weeks as I clarify certain points and add references.

Introduction

Part 1 covers the essentials. All I want to do here is detail non-Abelian anyons a little, and introduce the Berry phase, which is a key characteristic of topologically interesting systems. I also want to briefly look at a formalism — the Bogoliubov‒de Gennes formalism — used in this space to describe superconductors.

Non-Abelian anyons and the Kitaev model

I outlined previously the quantum statistics of particle exchange. I'd like to extend that discussion a little here. First, note that in standard 3D systems there are only two possible symmetries — bosons and fermions (I believe I've seen references to more exotic 3D systems which can host anyons). This limitation is simply because adiabatic interchange twice is equivalent to a process where none of the particles move, which is only possible for a sign change of the wavefunction under single interchange (there's an interesting recasting of this discussion in the path integral formulation). In standard 1D systems, particle exchange is impossible — the particles would have to pass through each other. However, in 1D networks braiding can be realised. In 2D, a particle loop that encircles another particle cannot be deformed to a point without cutting the other particle. Hence twice adiabatic interchange can be non-trivial.

Suppose we have two identical particles in 2D. On particle exchange the wavefunction can change by an arbitrary phase

$$\begin{align} \psi \rightarrow e^{i\phi} \psi \end{align}$$

\(\phi\) can take on any value because a second exchange, in the same direction, need not lead back to the initial state. The special cases \(\phi = 0,\pi\) correspond to bosons and fermions.

Consider now \(N\) degenerate states, represented by \(\psi_{1,\dots,N}\). We can describe formally particle exchanges as elements of the braid group. Consider element \(\sigma_1\), which exchanges particles 1 and 2. This element is represented by an \(N \times N\) unitary matrix \(\tau(\sigma_1)\). Exchange is then

$$\begin{align} \psi_\alpha \rightarrow [\tau(\sigma_1)]_{\alpha,\beta} \psi_\beta \end{align}$$

And for exchanging particles 2 and 3

$$\begin{align} \psi_\alpha \rightarrow [\tau(\sigma_2)]_{\alpha,\beta} \psi_\beta \end{align}$$

If \(\tau(\sigma_1)\) and \(\tau(\sigma_2)\) do not commute then the particles obey non-Abelian statistics. Braiding these particles causes non-trivial rotations within the degenerate many-particle Hilbert space. This must be a many-particle system (i.e. \(N>2\)); when there are only two particles the exchange operators must commute and the particles will be Abelian.

Non-Abelian anyons can be combined to produce a different type of anyon in a process called fusion. There is a fairly intuitive language to describe this. The different possible combinations are denoted

$$\begin{align} \psi_\alpha \times \psi_\beta = \sum_\gamma N_{\alpha,\beta}^\gamma \psi_\gamma \end{align}$$

representing particles of species \(\alpha\) and \(\beta\) combine to produce a particle of species \(\gamma\), provided \(N_{\alpha,\beta}^\gamma \neq 0\). Typically \(N_{\alpha,\beta}^\gamma = 0,1\) (there are theories which extend this). Two particles can fuse to produce a particle with statistics \(\phi = 0\) i.e. a trivial particle. This is often called the vacuum and is denoted by 1.

Majorana quasiparticles are Ising anyons. In this model there are three different species: \(1,\sigma,\psi\). 1 represents the trivial particle, \(\sigma\) a Majorana, and \(\psi\) a fermion (these labels can be considered topological charges that an Ising anyon may hold). The fusion rules are

$$\begin{align} &\sigma \times \sigma = 1 + \psi, & &\sigma \times \psi = \sigma, &\psi \times \psi = 1, & &1 \times x = x & &(x \in 1,\sigma,\psi) \end{align}$$

Fusion rules are commutative; given some rules one can then derive all manner of fantastic properties and explore topological quantum information theory. Repeated fusions of the same two anyons do not necessarily result in an anyon of the same type: the resulting anyons may be of several different types each with some probability. Fusion corresponds to measurement.

For completeness, there is a hypothetical anyon called the Fibonacci anyon. The rules are

$$\begin{align} &1 \times \tau = \tau, & &\tau \times \tau = 1 + \tau \end{align}$$

Consider the Kitaev model, detailed in Part 1. Consider Case 2, where Majoranas are coupled to adjacent sites. We have (Dirac) fermion operators

$$\begin{align} d_x = \frac{\gamma_{B,x} + i \gamma_{A,x+1}}{2} \end{align}$$

and Hamiltonian

$$\begin{align} H = 2t\sum_{x=1}^{N-1} d_x^\dagger d_x \end{align}$$

The system is gapped and the cost of adding a fermion is \(2t\). Recall that these fermions are formed from the inner, local modes. There is also a highly non-local fermion, formed from two Majorana modes each at different ends of the chain.

$$\begin{align} c_M = \frac{\gamma_{A,1} + i \gamma_{B,N}}{2} \end{align}$$

This is absent from the Hamiltonian and so the energy is the same whether or not this state is occupied. Hence the ground state is degenerate: if \(|0\rangle\) is the ground state then \(c_M^\dagger\) is also a ground state. We therefore expect non-Abelian statistics.

Statistics of the Majorana modes

Consider the simplest case, with \(N = 1\). There are only two Majorana modes and the braid group has a single generator \(\tau\). \(\tau\) is the operator that exchanges Majorana modes \(A\) and \(B\)

$$\begin{align} \gamma_A \rightarrow \gamma_A^\prime = \tau^\dagger \gamma_A \tau = e^{i\phi} \gamma_B \end{align}$$

The phase \(\phi\) is arbitrary. Choosing \(+1\) then forces the reverse

$$\begin{align} \gamma_B \rightarrow \gamma_B^\prime = \tau^\dagger \gamma_B \tau = e^{i\phi} \gamma_A \end{align}$$

to be \(-1\). This is because when there are only 2 Majorana modes, and the system is isolated, the fermion party must be conserved. The Dirac fermion formed from these two modes is either occupied or unoccupied with

$$\begin{align} i\gamma_A \gamma_B = 1 - 2 c_m^\dagger c_m \end{align}$$

If the right side remains unchanged then \(i \gamma_A \gamma_B\) must also be unchanged. This is only possible with \(\gamma_A \gamma_B = - \gamma_B \gamma_A\). Hence choose the exchange operator to be

$$\begin{align} \tau = \frac{1 + \gamma_A \gamma_B}{\sqrt{2}} \end{align}$$

or, in terms of the fermion number operator \(n = c_M^\dagger c_M\)

$$\begin{align} \tau = e^{i \pi(1 - 2n)/4} \end{align}$$

Clearly \(\tau\) is unitary and works as expected. Further, since \(n\) does not change the statistics is Abelian and it cannot rotate states in the ground state manifold \( (|0\rangle,c_M^\dagger \rangle)\).

Consider \(N = 2\). There are 4 Majorana modes \(\gamma_{A,\dots,D}\) and 2 Dirac fermions:

$$\begin{align} c_1 &= \frac{\gamma_A + i \gamma_B}{2} \\ c_1 &= \frac{\gamma_B + i \gamma_D}{2} \end{align}$$

We have the degenerate ground state

$$\begin{align} | n_1 n_2 \rangle = c_1^\dagger c_2^\dagger | 0,0 \rangle = {|0,0\rangle, |1,0\rangle,|0,1\rangle,|1,1\rangle} \end{align}$$

And operator \(\tau_{A,B}\), which exchanges Majoranas \(A\) and \(B\), leaving \(C\) and \(D\) unchanged (and so on for all other possible exchanges).

If we exchange two Majorana modes from the same fermion there is only a phase change:

$$\begin{align} \tau_{A,B} |n_1,n_2 \rangle &= e^{i \pi ( 1-2n_1 )/4}|n_1,n_2 \rangle \\ \tau_{C,D} |n_1,n_2 \rangle &= e^{i \pi ( 1-2n_2 )/4}|n_1,n_2 \rangle \end{align}$$

i.e. Abelian statistics. Suppose we exchange Majoranas from different fermions. We have

$$\begin{align} \tau_{BC} = \frac{1 + \gamma_B \gamma_C}{\sqrt{2}} = \frac{1}{\sqrt{2}} [ 1 - i(c_1 - c_2^\dagger)(c_2 + c_2^\dagger) ] \end{align}$$

Acting on \(|n_1,n_2\rangle\) does not give a phase change. Instead, we have a rotation

$$\begin{align} \tau_{BC} |n_1,n_2\rangle = \frac{1}{\sqrt{2}}[ |n_1.n_2 \rangle + i(-1)^{n_1} |1 - n_1,1 - n_2 \rangle ] \end{align}$$

i.e. non-Abelian statistics.

Berry Phase

In quantum mechanics a phase difference is acquired by a system during the course of a cyclic adiabatic process, due to the topology of the parameter space of the Hamiltonian. This is the Berry phase. This important in topological phases of matter because it can, as I will detail, explain the integer quantum Hall effect.

Consider a system dependent on various parameters

$$\begin{align} \vec{R} = (R_1,R_2,R_3,\dots) \end{align}$$

These are arbitrary parameters — not necessarily position. Let's consider the evolution of \(\vec{R}(t)\) along a path \(C\). We have the Schrodinger equation

$$\begin{align} \hat{H}(\vec{R}) | n(\vec{R}) \rangle = E_n (\vec{R}) | n(\vec{R}) \rangle \end{align}$$

Now, we normally ignore phase because we measure probability amplitudes and phase is cancelled in \(\psi^*\psi\) i.e.

$$\begin{align} [\hat{H}(\vec{R}),e^{i\phi(\vec{R})}] = 0 \end{align}$$

Let's prepare a system in an initial pure state

$$\begin{align} |n(\vec{R}(t = 0)) \rangle \end{align}$$

We work adiabatically so that the system is always an eigenstate of \(\hat{H}\). Let's consider the phase. In general

$$\begin{align} |\psi(t) \rangle = e^{-\phi(t) }|n(\vec{R}) \rangle \end{align}$$

Hence we analyse the Schrodinger equation

$$\begin{align} \hat{H} (\vec{R}(t)) | \psi(t) \rangle &= i \hbar \frac{d}{dt} | \psi (t) \rangle \\ \Rightarrow E_n (\vec{R}(t)) |n(\vec{R}(t)) \rangle &= \hbar (\frac{d}{dt} \phi(t)) | n(\vec{R}(t)) + i \hbar \frac{d}{dt} | n(\vec{R}(t)) \rangle &\text{(product rule)} \end{align}$$

Next, hit everything from the left with \(\langle (\vec{R}) |\) (assuming normalised states)

$$\begin{align} E_n(\vec{R}(t)) - i \hbar \langle n(\vec{R}(t)) | \frac{d}{dt} | n(\vec{R}(t)) \rangle = \hbar(\frac{d}{dt} \phi(t)) \end{align}$$

And integrate

$$\begin{align} \phi(t) = \frac{1}{\hbar} \int_0^t E_n(\vec{R}(t^\prime)) dt^\prime - i \int_0^t \langle n(\vec{R}(t^\prime)) | \frac{d}{dt^\prime} | n(\vec{R}(t^\prime)) \rangle dt^\prime \end{align}$$

Now, the first term is the dynamical phase (recall \(U = \exp(i \hat{H}(t)))\) and the second phase is \(\gamma_n\), the Berry phase. The Berry phase is often zero, but sometimes not. Let's try and remove the time dependence from \(\gamma_n\). We are interested in working in closed loops so we can rewrite

$$\begin{align} \gamma_n &= i \int_0^{t_{end}} \langle n(\vec{R}(t^\prime)) | \frac{d}{dt^\prime} | n(\vec{R}(t^\prime)) \rangle dt \\ &= i \int_0^{t_{end}} \langle n(\vec{R}(t^\prime)) | \nabla_{\vec{R}} | n(\vec{R}(t^\prime)) \rangle \frac{d\vec{R}}{dt^\prime} dt \end{align}$$

This is a contour in parameter space so we can rewrite

$$\begin{align} \gamma_n = i \int_C \langle n(\vec{R}) | \nabla_{\vec{R}} | n(\vec{R}) \rangle d\vec{R} \end{align}$$

which looks like an integral over expectation values. Let's define a (real) vector potential — the Berry connection

$$\begin{align} A_n(\vec{R}) := i \langle n(\vec{R}) | \nabla_{\vec{R}} | n(\vec{R}) \rangle \end{align}$$

Note that \(A\) must be real because it will be fed back into \(\exp(i \phi)\) and that \(\langle n | \nabla | n \rangle\) is in general imaginary. Note also that we can use Stoke's theorem and deduce the Berry curvature

$$\begin{align} \gamma_n &= \oint_C \langle n(\vec{R}) | \nabla_{\vec{R}} | n(\vec{R}) \rangle d\vec{R} \\ &= \iint_S \nabla \times A_n dS \end{align}$$

Note that \(A_n\) is gauge dependent. Under the gauge transformation \(| n(\vec{R})\rangle \rightarrow \exp(i \xi(\vec{R})) | n(\vec{R}) \rangle \)

$$\begin{align} A_n(\vec{R}) \rightarrow A_n(\vec{R}) - \nabla_{\vec{R}} \xi (\vec{R}) \end{align}$$

It follows that \(\gamma_n\) will be changed by \(- \int_C \nabla_{\vec{R}} \xi(\vec{R}) d\vec{R} = \xi(\vec{R}(0)) - \xi(\vec{R}(t))\) i.e. under gauge transformation \(\gamma_n\) disappears.

At least, that was the conclusion in the early 20th century when the constructors of quantum mechanics analysed this problem. However, we have a loop so

$$\begin{align} \vec{R}(0) = \vec{R}(T) &\Rightarrow | n(\vec{R}(0)) \rangle = | n(\vec{R}(T)) \rangle, \\ \Rightarrow e^{i \xi(\vec{R}(0))} | n(\vec{R}(0)) \rangle &= e^{i \xi(\vec{R}(t))} | n (\vec{R}(0)) \rangle \end{align}$$

Rearrange

$$\begin{align} \xi(\vec{R}(T)) - \xi(\vec{R}(0)) = 2 \pi m, m \in \mathbb{Z} \end{align}$$

What if the object to cancel the Berry phase is not always \(2\pi m\)? If \(\gamma_n \neq 2 \pi\) a gauge transformation cannot cancel the Berry phase! This is a signature; typically, a topological system will have a Berry phase of \(\pm \pi\).

Aside: Two-state systems and the Berry phase

A two level system is by definition

$$\begin{align} | \psi \rangle = \left( \begin{array}{c} c_1 \\ c_2 \end{array} \right) \end{align}$$

The associated Hamiltonian must be a \(2 \times 2 \) matrix and has general form

$$\begin{align} \hat{H} = \left( \begin{array}{cc} a_1 & c - id \\ c + id & a_2 \end{array} \right) \end{align}$$

which can be decomposed

$$\begin{align} \hat{H} = a \sigma_0 + c \sigma_1 + d \sigma_2 + b \sigma_3 = a \sigma_0 + \vec{d} \cdot \vec{\sigma} \end{align}$$

with eigenvalues \(E_\pm = a \pm |\vec{r}|\). It's natural to take a Hamiltonian, diagonalise (possibly in momentum space) and extract the eigenstates. We change basis to the eigenvalues and diagonalise

$$\begin{align} \hat{H} = \left( \begin{array}{cc} E_+ & 0 \\ 0 & E_- \end{array} \right) \end{align}$$

We could now discuss Rabi oscillations and other exciting two-state phenomena.

Suppose we have a two level system of the form

$$\begin{align} \hat{H} = \vec{d}(\vec{R}) \cdot \sigma \end{align}$$

In spherical coördinates

$$\begin{align} \vec{d} (\vec{R}) = | d | (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta) \end{align}$$

with eigenstates

$$\begin{align} | -(R) \rangle &= \left( \begin{array}{c} e^{- i \phi}\sin \theta/2 \\ - \cos \theta/2 \end{array} \right) \end{align}$$

$$\begin{align} | +(R) \rangle &= \left( \begin{array}{c} e^{- i \phi}\cos\theta/2 \\ \sin \theta/2 \end{array} \right) \end{align}$$

We can compute the Berry connections and Berry curvature

$$\begin{align} A_{\theta} &= i \langle -(R) | \partial_\theta | -(R) \rangle = 0, \\ A_{\phi} &= i \langle -(R) | \partial_\phi| -(R) \rangle = \sin^2 \theta/2, \\ F_{\theta \phi} &= \partial_\theta A_\phi - \partial_\phi A_\theta = 1/2 \sin \theta, \end{align}$$

Choose \( = \vec{R}\)

$$\begin{align} \vec{R} = \frac{1}{2} \frac{\vec{R}}{\vec{R}^3} \end{align}$$

This is a singularity, a "magnetic monopole". This is a signature topological defect, and signifies that there is a non-zero Berry phase — and some interesting physics! For example, we can extract the first Chern number. Recognising that \(A\) is the Berry connection, first rewrite

$$\begin{align} F_{xy} (\vec{k}) = \partial_{kx} A_y (\vec{k}) - \partial_{ky} A_x (\vec{k}) \end{align}$$

And integrate

$$\begin{align} C &= \frac{1}{2\pi} \int d \vec{k} F_{xy} (\vec{k}) \end{align}$$

i.e. a sum over all occupied bands equals the first Chern number — an integer topological invariant.

The integer quantum Hall effect

The Hall effect is an electromagnetic phenomena where a secondary voltage is induced in a conductor. Figure 1 sketches the phenomena: a current is applied to a conductor in one direction (\(xx\)), a magnetic field applied normal to the current, and a voltage is induced perpendicular (\(xy\)) to both of these. This is sketched in figure 1. Classically, this effect can be explained simply because electrons feel a force in the presence of a magnetic field — the Lorentz force, \(\vec{F} = q[\vec{E} + (\vec{v} \times \vec{B})] \). The electrons follow a curved path, building up charge along one side of the conductor. This is detected as a voltage. The magnitude and the sign of the Hall voltage are dependent on the material properties of the conductor. The Hall effect is apparent in some quantum systems. There is an integer quantum Hall effect, where the Hall voltage is expressed as the Hall conductivity and is exactly integral \(\sigma_{xy} = (e^2/h) C \). This phenomena can be explained by either the electronic structure of the material, or the topology of the system. There is also a fractional quantum Hall effect, which is not well understood. In this phenomena, the conductivity comes in exact fractions. Models treating this phenomena work with quasiparticles, similar to our Majoranas.

As a brief remark, note that the foundational relation Ohm's law, \(V = IR\) can be expressed in terms of the electric field \(\vec{J} = \rho^{-1} \vec{E} = \sigma \vec{E}\), where \(\rho\) is the resistance and \(\sigma\) the conductivity. Typically the resistance (conductance) is taken as a scalar; however, in the two-dimensional systems of the Hall effect, resistance (conductance) must be considered as tensors. This is simply because the conductance is not uniform across the system.

Figure 2 shows a representative plot of the integer quantum Hall effect. The essential characteristic of this plot is the 'staircase' of resistance. As the magnetic field strength increases, the resistance (conductance) increases near-instantly to the next level. Each level has an integral value, in the appropriate units. This conductance holds true regardless of the geometry of the experiments or the quality (imperfections) of the materials.

Fig. 2. Representative plot of the quantum Hall effect. As in the classical effect, there is an induced voltage. However, here the induced voltage (transformed to resistance) is quantised, with plateaus corresponding to the magnetic field strength. Adapted from ref ².

One explanation is by invoking Landau levels. In a 2D lattice, subject to a magnetic field, electrons occupy discrete energy levels (Landau levels). Each energy level can store some fixed number of electrons. As the magnetic field increases, each level can store more electrons. We can define a filling factor \(f\) to quantify this: when \(f\) is an integer, the Fermi level is between two Landau levels. As higher Landau levels are filled, conduction increases and this filling must be integral. This gives us a hint to the source of the fractional quantum Hall effect: the charge carrier must be some fractionally-charged quasiparticle.

We can also describe this phenomenon topologically. Consider a 2D electron gas cold enough that quantum coherence holds throughout. We can describe the system with a wavefunction and evolve a Hamiltonian. Suppose that the electron gas is confined to a looped ribbon, as in figure 3, with a strong magnetic field normal to its surface. The opposite edges of the ribbon are connect to separate electron reservoirs. We introduce a magnetic flux \(\Phi\) threading the loop. The change in this flux drives the pump: increasing the flux causes current to flow from one reservoir to the other.

Now, the Aharanov-Bohm principle guarantees that the Hamiltonian describing this system is gauge invariant under flux changes by the elementary quantum of magnetic flux, \(\Phi_0 = hc/e\). Hence an adiabatic increase of \(\Phi\) by a single flux quantum is a single cycle of the pump. Hence the charge transported between reservoirs in one pump cycles is the Hall conductance. The Hall conductance can be thought of as a curvature. This is achieved by identifying two angular-dependent parameters which the Hall-effect Hamiltonian depends. One is \(\Phi\), associated with the emf which drives the Hall current. The other is \(\theta\), chosen so that the Hall current takes the form \(I = c \partial_\theta \hat{H}(\Phi,\theta)\). Because of gauge invariance, the Hamiltonian is periodic in both parameters, with period \(\Phi_0\). Now, there is an interesting parameter that can be extracted from the Berry phase. We can define the local adiabatic curvature \(K\) of the ground states in parameter space as the limit of the Berry phase divided by the loop area. This is

$$\begin{align} K = 2 Im \langle \partial_\phi \psi | \partial_\theta \phi \rangle \end{align}$$

for some two-parameter Hamiltonian \(\hat{H}(\Phi,\theta)\). If the ground state is independent of \(\Phi\) and we change \(\Phi\) adiabatically, we can determine the expectation value of the Hall current using the Schrodinger equation

$$\begin{align} \langle \phi | I | \phi \rangle = \hbar c K \dot{\Phi} \end{align}$$

using the adiabatic curvature \(K\). This is a linear relationship between the driving emf, \('{}/c\) and therefore the Hall conductance is \(c K\). Hence there is an elegant geometric interpretation of the Hall conductance as curvature.

We can extend this by noting a remarkable relationship between geometry and topology, quantified with formula by Gauss and Bonnet

$$\begin{align} \frac{1}{2\pi} \int_S K dA = 2(1 - g) \end{align}$$

which is the integral over a surface \(S\) without a boundary (i.e. a torus), with \(K\) the local curvature. The right-hand side is quantised: the integer \(g\) characterises the surface. This has been further generalised by Chern. The striking application here is that the Gauss-Bonnet formula applies just as well to our eigenstates, parameterised by \(\Phi\) and \(\theta\). This gives the first Chern number, a topological quantity because it is invariant under small deformations of the Hamiltonian. Large deformations of the Hamiltonian cause the ground state to cross over other eigenstates. When this happens, the adiabatic curvature diverges and the Chern number is no longer well defined — transitions between Chern number plateaus.

We can approach this a little more formally (the precise technical details are not relevant to my work I feel, so I will only touch on the important points). In a torus, we can derive an expression for the Hall conductivity by adding a flux term to our Hamiltonian, applying first order perturbation theory and working through the algebra. We get

$$\begin{align} \sigma_{xy} = i \hbar \left[ \frac{\partial}{\partial \phi_y} \langle \psi_0 | \frac{\partial \psi_0}{\partial\phi_x} \rangle - \frac{\partial}{\partial \phi_x} \langle \psi_0 | \frac{\partial \psi_0}{\partial \phi_y} \rangle \right] \end{align}$$

We again parameterise our Hamiltonian by \(\phi\) and \(\theta\). We are working in parameter space and considering adiabatic cycles. It is natural then to consider the Berry phase that arises as the parameters are varied. This is described by the Berry connection

$$\begin{align} A(\phi) = - i \langle \psi_0 | \frac{\partial}{\partial \theta} | \psi_0 \rangle \end{align}$$

The field strength (curvature) associated with the Berry connection is

$$\begin{align} F_{xy} = \frac{\partial A_x}{\partial \theta_y} - \frac{\partial A_y}{\partial \theta_x} = i \hbar \left[ \frac{\partial}{\partial \phi_y} \langle \psi_0 | \frac{\partial \psi_0}{\partial\phi_x} \rangle - \frac{\partial}{\partial \phi_x} \langle \psi_0 | \frac{\partial \psi_0}{\partial \phi_y} \rangle \right] \end{align}$$

which is the Hall conductivity. For a torus with fluxes we can write

$$\begin{align} \sigma_xy = - \frac{e^2}{\hbar} F_{xy} \end{align}$$

and when we average over all fluxes

$$\begin{align} \sigma_{xy} = - \frac{e^2}{\hbar} \int_S \frac{d^2 \theta}{(2 \pi)^2} F_{xy} \end{align}$$

The integral over the curvature is the first Chern number

$$\begin{align} C = \frac{1}{2\pi} \int_S d^2 \theta F_{xy} \end{align}$$

which is always integral. Hence we have the integer quantum Hall effect

$$\begin{align} \sigma_{xy} = - \frac{e^2}{h} C \end{align}$$

Chern numbers and energy gaps

A two-state quantum system has Hamiltonian \(\hat{H} = \vec{d} \cdot \vec{\sigma}\). When \(d_z = 0\) we have a Dirac cone and when \(d_z = M k_z\) we have a gapped momentum. This is the massive case. We can derive the Berry connection and curvature to give

$$\begin{align} F_{xy} = \left( \frac{M}{2(m^2 + |k|^2)} \right)^{3/2} \end{align}$$

from which we can derive the Hall conductance

$$\begin{align} \sigma_{xy} &= \frac{e^2}{h} \int \frac{d^2 k}{(2\pi)} F_{xy} \\ &= \frac{e^2}{h} \int_0^{2 \pi} \int_0^\infty \frac{d k}{(2\pi)} \left( \frac{M}{2(m^2 + |k|^2)} \right)^{3/2} \\ &= \frac{e^2}{h} \frac{m}{2} \int_0^\infty dx \left( \frac{m}{2(m^2 + x)} \right)^{3/2} & \text{by substitution} \\ &= \frac{e^2}{h} \frac{m}{2} \int_{m^2}^\infty y^{-3/2} dy \\ &= \frac{e^2}{h} \frac{m}{2} [- \frac{1}{y}]_{m^2}^\infty \\ &= \frac{e^2}{h} \frac{sgn(m)}{2} \end{align}$$

i.e. massive (gapped) Dirac fermions in continuum exhibit a half quantum Hall effect.

In a crystal, the Chern number is associated with the transport of \(k\) around a closed loop — there is a Berry phase associated with the Bloch wavefunction. Note that the Chern number summed over all occupied bands is invariant even if there are degeneracies between occupied bands. Another way of looking at this is that the Chern number is topologically invariant because it does not change when the Hamiltonian changes smoothly.

Chern insulators

A lattice Chern insulator has Hamiltonian

$$\begin{align} \hat{H} = \sin k_x \sigma_x + \sin k_y \sigma_y + (2 + m - \cos k_x - \cos k_y) \sigma_z \end{align}$$

When the gap closes there is a change in Chern number:

• When \(m = 0\), there is gap closing at \(\vec{k} = (0,0)\)
• When \(m = -2\), there is gap closing at \(\vec{k} = (0,\pi),(\pi,0)\)
• When \(m = -4\), there is gap closing at \(\vec{k} = (\pi,\pi)\)

At \(m = \infty\) the system is fully gapped, but it is in a trivial phase because of localisation. We can sketch a phase diagram

Upon further analysis, defects in the energy bands can be extracted, associated with topological phases. Under certain conditions, chiral edge modes appear in topological phases. Note that magnetic fields are associated with an antiunitary operator and break time-reversal symmetry (there's an interesting little discussion to be had about why topologically interesting phases require the breaking of time-reversal symmetry).

Topological Superconductors and the Bogoliubov‒de Gennes Formalism

Topological superconductors are a condensate of Cooper pairs. They are described with a Hamiltonian of the form

$$\begin{align} \hat{H} = \hat{H}_{normal} + \hat{H}_{SC} \end{align}$$

with

$$\begin{align} \hat{H}_{normal} &:= \text{continuum/free gas} \\ &= \sum_{\vec{k},\sigma} \varepsilon(\vec{k}) c_{\vec{k},\sigma}^\dagger c_{\vec{k},\sigma} \\ \hat{H}_{SC} &:= \text{superconductor} \\ &= \sum_{\vec{k}} (\Delta(\vec{k}) c_{\vec{k}\uparrow}^\dagger c_{\vec{k}\downarrow}^\dagger + \Delta(\vec{k})^* c_{-\vec{k}\downarrow} c_{-\vec{k}\uparrow}) \\ &= \sum_{\vec{k}} (\Delta(\vec{k}) c_{\vec{k}\uparrow}^\dagger c_{\vec{k}\downarrow}^\dagger + \text{hermitian conjugate (h.c.)}) \end{align}$$

The continuum Hamiltonian creates and annihilates electrons at all momenta \(\vec{k}\) and spins \(\sigma\). The superconductor Hamiltonian creates and destroys Cooper pairs, weighted by some order parameter \(\Delta{\vec{k}}\). The continuum Hamiltonian can be rewritten by splitting into two parts

$$\begin{align} \hat{H}_{normal} = \frac{1}{2} \sum [\varepsilon(\vec{k}) c^\dagger c - h.c.] \end{align}$$

Then we can use the fermion anticommutation relations and write in the Bogoliubov‒de Gennes form

$$\begin{align} \hat{H}_{BdG} = \frac{1}{2} \sum_\vec{k} (c_{\vec{k} \uparrow}^\dagger,c_{\vec{k} \downarrow}^\dagger,c_{-\vec{k} \uparrow},c_{-\vec{k} \uparrow}) \left( \begin{array}{cccc} \varepsilon(\vec{k}) & 0 & 0 & \Delta \\ 0 & \varepsilon(\vec{k}) & \Delta^* & 0 \\ 0 & -\Delta^* & - \varepsilon(-\vec{k}) & 0 \\ \Delta^* & 0 & 0 & -\varepsilon(\vec{k}) \end{array} \right) (c_{\vec{k} \uparrow}^\dagger,c_{\vec{k} \downarrow}^\dagger,c_{-\vec{k} \uparrow},c_{-\vec{k} \uparrow})^T \end{align}$$

which can be diagonalised or reduced to a \(2 \times 2 \) under certain conditions. This leads to the Bogoliubov quasiparticles, superpositions of electrons and holes. These are single quasiparticle excitations and appear as eigenstates of the Bogoliubov‒de Gennes Hamiltonian. From here we can explore Kitaev's model, as sketched in Part 1. To summarise: find the eigenvalues, and plot, finding gapless phases; then look for zero-energy modes, in real space.

One more, with feeling details

The Bogoliubov‒de Gennes formalism is a mean-field Hamiltonian. It reduces a many-body condensed matter problem into a one-body problem. The correlations between particles in a many-body \(2^N\) Hamiltonian are replaced with an effective one-body potential (e.g. superconducting order parameter). It is related to BCS theory. We proceed:

Start with an interacting Hamiltonian with a local attraction

$$\begin{align} \hat{H} = \hat{H}_0 - g \sum_{i = 1}^N c_{i \uparrow}^\dagger c_{i \downarrow}^\dagger c_{i \downarrow} c_{i \uparrow} \end{align}$$

where \(\hat{H}_0\) is quadratic and we work on an \(N\)-site chain. Then write

$$\begin{align} c_{i \uparrow}^\dagger c_{i \downarrow}^\dagger = \langle c_{i \uparrow}^\dagger c_{i \downarrow}^\dagger \rangle + (c_{i \uparrow}^\dagger c_{i \downarrow}^\dagger - \langle c_{i \uparrow}^\dagger c_{i \downarrow}^\dagger \rangle) \end{align}$$

with the terms in the angle brackets describing fluctuations around the average. We move to a mean-field approximation by discarding second-order fluctuations

$$\begin{align} \hat{H}_{MF} = \hat{H}_0 + \sum_{i =1}^N (\Delta_i c_{i \uparrow}^\dagger c_{i \downarrow}^\dagger + h.c.) \end{align}$$

with \(\Delta_i = -g \langle c_{i \downarrow c_{i \uparrow}}\rangle \). Then introduce the Bogoliubov quasiparticles

$$\begin{align} \gamma_{n \sigma}^\dagger = \sum_{i=1}^N (u_{i,n} c_{i, \sigma}^\dagger + v_{i,n}c_{i,-\sigma}) \end{align}$$

The Bogoliubov quasiparticles then diagonalise the BCS Hamiltonian when \((u_n,v_n)^T\) are eigenstates of the Bogoliubov‒de Gennes Hamiltonian i.e.

$$\begin{align} \hat{H}_{MF} = E_0 + \sum_{n \sigma} E_n \gamma_{n \sigma}^\dagger \gamma_{n \sigma} \end{align}$$

or

$$\begin{align} H_0 u_n + \Delta v_n &= E_n u_n \\ \Delta u_n - H_0 v_n^\dagger &= E_n v_n^* \end{align}$$

with

$$\begin{align} \Delta = g \sum u_n v_n^* \end{align}$$

and the appropriate eigenstates. When you diagonalise a Hamiltonian, you get the eigenstates. We can do this following the usual procedure by hand (or Mathematica), use use a little 'square twice' trick, detailed nicely here.