\section{Introduction}
Symmetries and the resulting conserved quantities underpin all observed physical laws. We are most familiar with the symmetries and phases of matter characterised by local order parameters within Landau-Ginzburg theory \cite{landau1980statistical, cardy1996scaling}, but in the last few decades the exploration of topological phases of matter has lead to many new developments in our understanding of condensed matter physics, culminating in a Nobel prize for Thouless, Haldane and Kosterlitz in 2016. The notion of topology in physics was introduced by Klitzerling and his discovery of the 2D quantum Hall (QH) state \cite{klitzing1980new}, with Thouless et al. explaining the quantization of the Hall conductance in 1982 \cite{PhysRevLett.49.405}. Much progress has been made since, and whereas the QH state explicitly breaks time-reversal (TR) symmetry, new materials obeying TR symmetry have been explored. The first proposals of the 2D topological insulator (TI) – otherwise know as quantum spin Hall states – were remarkably recent (Kane and Mele, 2005 \cite{kane2005z} and Bernevig and Zhang, 2006 \cite{bernevig2006quantum}). The 3D generalisation came soon after in 2007 \cite{fu2007topological} and experiments have shown that these new phases of matter are both realisable and accessible \cite{bernevig2006quantum2, RevModPhys.82.3045, moore2010birth}. All TR invariant TIs in nature (without ground-state degeneracy) fall into two distinct classes, classified by a $Z_2$ topological order parameter. The topologically non-trivial state has a full insulating gap in the bulk and gapless edge or surface states consisting of an odd number of Dirac fermions.
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\\ The main point of study has been the electronic properties of these systems, whilst the interaction of TIs with light has been studied less thoroughly. The presence of surface states will make the material interact very differently with light than if the material were topologically trivial, and experimental verification of interactions with the surface states has been achieved \cite{chen2009experimental}. Very recent work has studied TI nanoparticles (TINPs) – whose size gives both a much higher curvature and higher surface to volume ratio than regular TIs – showing that under the influence of light, the surface state excitations of a TINP are comparable in magnitude to bulk effects, and in fact single photon excitations can have a macroscopic effect on an entire nanoparticle \cite{siroki2016single}. The study of topological insulator nanoparticles (TINPs) with light is so new that there is basically only one paper on this topic. This makes a literature review on the topic fairly restricted, but is highly motivating in that there is plenty of work to be done, making this a very exciting research area.
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\\The outlook of work in this very new area is to better understand the interaction of light with TI nanostructures, especially as already noted, the curvature of the nanostructure can greatly influence the interactions experienced under illumination. As will be revisited throughout the literature review and outlook, the hope is to apply this research to quantum optics, spintronics and nanoplasmonics.
%% From Insulators to Topological Insulators %%
\section{From Insulators to Topological Insulators}
% Topological Order %
\subsection{Topological Order}
Calculating the electronic band structure of materials allows us to categorize them into groups such as metals, semiconductors or insulators. To do so, we calculate the Bloch wave functions of electrons in momentum space, and then by calculating their corresponding energies, we find the band structure \cite{kittel1966introduction}. The chemical potential of the material is the demarcation between occupied and unoccupied states. Regular insulating materials are characterised by the chemical potential residing within an energy gap, nestled between a conduction band and valence band.
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\\The wave functions are composed of basis states defined in a Hilbert space and the Hamiltonian of the system transforms states within that Hilbert space. Topology studies whether objects can be transformed continuously into each other. In condensed matter physics we can ask whether the Hamiltonians of two quantum systems with an energy gap, $H$ and $\tilde{H}$ can be continuously transformed into each other without closing the gap. If that is the case, then we can say that the two systems are `topologically equivalent’.\footnote{If we considered all Hamiltonians without any constraint, every Hamiltonian could be continuously deformed into every other Hamiltonian, and all quantum systems would be topologically equivalent. This changes drastically if we restrict ourselves to systems with an energy gap.}
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\\ We can also introduce more specific criteria, for example that some symmetry inherent in the system be preserved throughout the continuous path which connects two Hamiltonians such as time-reversal symmetry. In order to know whether there is any path which connects $H$ and $\tilde{H}$ without closing the gap, we can count the number of levels below the chemical potential, which we denote $\epsilon = 0$. For gapped Hamiltonians no energy level can move through zero, but otherwise they can move freely. Therefore continuous transformations only exist between Hamiltonians with the same number of energy levels below $\epsilon=0$. Since this number cannot change under continuous transformations inside the set of gapped Hamiltonians, we call it a topological invariant $Q$. Whenever an energy level crosses $\epsilon=0$, this changes the topological invariant, causing a topological phase transition. If two Hamiltonians have a different topological invariant, they must be separated by such a transition and it is impossible to go from one topological phase to another without closing the gap.
% At the Edge
\subsection{At the Edge}
We have so far given no account of the boundary conditions of TIs. For a TI surrounded by a trivially-insulating medium (such as vacuum), we have a boundary on the interface of these two insulators with differing topological order parameters. The change in topological order parameter at the boundary requires the band gap to close, whilst remaining gapped in the bulks of both media. This results in localised boundary states, which traverse the band gap. Physically, the picture we then have of our topological insulator is an insulating bulk, with conducting states localised on the boundary of the material.
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\\Recalling that for the TR-invariant TI, the band gap is protected by demanding that TR-symmetry is preserved. This means that unless a TR-breaking imperfection is applied, the edge states will be robust against perturbations\footnote{Time-reversal breaking perturbations would be for example the addition of a magnetic field-like term}. Other types of TIs will have different symmetries (such as chiral invariance) and thus edge states protected by different symmetries\footnote{Note that the Hamiltonian of the surfaces states is gapless, and symmetry would have to be broken to gap the states}
%% Topological Insulators in 3D %%
\section{Topological Insulators in 3D}
The first 3D TI material that was experimentally identified was Bi${_{1-x}}$Sb$_x$ \cite{hsieh2008topological}, predicted by Fu and Kane \cite{fu2007topological}. It turned out to be poorly suited to detailed topological surface state studied due to a complicated surface state structure – Rashba split surface states are present alongside the topological surface states and predicted surface state structure has not agreed well with experimental results. Instead we focus on the Bi$_2$Se$_3$ family of materials, predicted by Zhang in 2009 \cite{zhang2009topological}, whose model Hamiltonian for this family of materials is used ubiquitously in theoretical studies alongside modifications by Liu et al \cite{liu2010model}.
% Bi2Se3
\subsection{Bi$_2$Se$_3$}
Bi$_2$Se$_3$ consists of five-atom layers stacked along the z direction. Each quintuple layer consists of five atoms per unit cell with two equivalent Se atoms, two equivalent Bi atoms and a third Se atom. The coupling between two atomic layers within a quintuple layer is strong, whilst that between quintuple layers is predominantly due to van der Waals forces and is thus much weaker. Symmetries of the material are comprised of a three-fold rotational symmetry, an additional two-fold rotation symmetry and a reflection symmetry. Experimental confirmation of a single Dirac-cone surface state for Bi$_2$Se$_3$ was reported in 2009 by Xia et al. \cite{lee2012gate}.
% Bulk Behaviour
\subsection{Bulk Behaviour}
As well as the lattice symmetries described above, Bi$_2$Se$_3$ also obeys time-reversal symmetry. The important topological physics occurs near the $\Gamma$ point, so we write down an effective Hamiltonian \cite{liu2010model},
\begin{equation} H(\mathbf{k}) = \epsilon_0 (\mathbf{k})\mathbb{1}_4 +\begin{pmatrix}\mathcal{M}(\mathbf{k}) & A_{1} k_{z} & 0 & A_2 k_{-} \\ A_1 k_z & -\mathcal{M}(\mathbf{k}) & A_{2} k_{-} & 0 \\ 0 & A_{2} k_{+} & \mathcal{M}(\mathbf{k}) & -A_{1} k_{z} \\ A_{2} k_{+} & 0 & -A_{1} k_{z} & -\mathcal{M}(\mathbf{k}) \label{eq:ham}\end{pmatrix}\end{equation}
where $k\pm = k_x \pm i k_y$, $\epsilon_0 (\mathbf{k})=C+D_1 k_z^2+D_2 k_\perp^2$, and $\mathcal{M}(\mathbf{k}) = M-B_1 k_z^2 – B_2 k_\perp^2.$ The parameters in this effective model can be determined by fitting the energy spectrum of the Hamiltonian to that of \textit{ab initio} calculations \cite{zhang2009topological}, with the results for Bi$_2$Se$_3$ presented by Liu et al \cite{liu2010model}.
% At the Surface
\subsection{At the Surface}
The most prominent and perhaps most important property of a TI is the existence of gapless surface states.
3D TR-invariant TIs such Bi$_2$Se$_3$ have 2D surface states. The surface effective model can be obtained by projecting the bulk Hamiltonian (\ref{eq:ham}) onto the surface states, giving a 2D massless Dirac Hamiltonian,
\begin{equation}H_{\mathrm{surf}}(k_x,k_y) = C + A_2 \left( \sigma^x k_y – \sigma^y k_x \right).\label{eq:surface} \end{equation}
The Dirac Hamiltonian gives a linear, relativistic dispersion relation and thus tells us that the surface states are described by massless Dirac fermions. Dirac fermions occur in the two-dimensional electron gas that forms at the surface of topological insulators as a result of the strong spin–orbit interaction existing in the insulating bulk phase. Materials such as Bi$_2$Se$_3$ have a single Dirac cone. Note that true 2D systems will have an even number of Dirac cones, enforced by TR-symmetry (e.g. Graphene, which has 4 Dirac cones). 2D surface systems allow for a single Dirac cone, and as such a system is described as a `holographic metal’. The gapless nature is protected by TR symmetry in Z$_2$ topological insulators. What makes this surface state distinct from ordinary surface states is its helical spin polarization (spin-momentum locking), such that the surface state is spin non-degenerate and the direction of the spin is perpendicular to the momentum vector and is primarily confined in the surface plane. The helical spin polarization of the surface state means that a dissipationless spin current exists on the surface in equilibrium, because there is no net charge flow but the spin angular momentum flows in the direction perpendicular to the spin direction. The spin helicity of the surface state (i.e., whether the up spin is associated with -k or k) determines the spin current direction.
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\\ For Bi$_2$Se$_3$ the surface states can be directly extracted from \textit{ab initio} calculations by constructing maximally localized Wannier functions and calculating the local density of states on an open boundary \cite{zhang2009topological}. The result for the Bi$_2$Se$_3$ family of materials is shown given in \cite{zhang2009topological}, where the single Dirac-cone surface state for the three topologically nontrivial materials can be seen. A mathematical description of how the surface states emerge from the effective model can be found in the literature \cite{linder2009anomalous,zhang2009topological,liu2010model,RevModPhys.83.1057}.
% Electromagnetic properties
\subsection{Electromagnetic Properties}
Similar to the quantized Hall response in QH systems \cite{PhysRevLett.49.405}, the topological structure in TIs should not only lead to robust gapless surface states, but also to unique, quantized electromagnetic response coefficients. The quantized electromagnetic response of 3D TIs turns out to be a topological magnetoelectric effect (TME) \cite{bernevig2013topological}, which occurs when TR symmetry is broken on the surface, but not in the bulk. The TME effect is a generic property of 3D topological insulators. From a topological field theory (TFT) perspective \cite{qi2008topological,ando2013topological}, for Z$_2$ TIs there is an additional E$\cdot$B term in the system Langrangian,
\begin{equation}\mathcal{L} = \frac{1}{8\pi} \left(\epsilon \mathrm{E}^2-\frac{1}{\mu}\mathrm{B}^2 + \frac{\alpha}{4 \pi^2}\theta \mathrm{E}\cdot \mathrm{B} \right),\end{equation}
where $epsilon$ is dielectric constant, $\mu$ is magnetic permeability, $\alpha$ is the fine structure constant and $\theta=0,\pi$ is the Z$_2$ topological invariant. Due to the existence of the $\theta$-term, the constituent equations of the TI become
\begin{align}
\mathrm{D} &= \mathrm{E} + 4 \pi \mathrm{P} – \frac{\alpha \theta}{\pi}\mathrm{B}, \\
\mathrm{H} &= \mathrm{B} + 4 \pi \mathrm{M} – \frac{\alpha \theta}{\pi}\mathrm{E},
\end{align}
where D is electric induction, P is electric polarization and M is magnetization. The most important consequence of this peculiar electromagnetism is that in TIs where $\theta=\pi$, electric field E induces magnetization $4\pi$M$=\alpha$E. Similarly, magnetic field B induces electric polarization $4\pi$P$=\alpha$B. This is the topological magnetoelectric effect to characterize the non-trivial Z$_2$ topology of a TI in its electromagnetic properties. Due to practical difficulties in maintaining the conditions needed for this effect, the TME effect remains to be experimentally discovered\footnote{Another intriguing consequence of the $\theta$-term to be tested in future is the appearance of an image magnetic monopole\cite{qi2009inducing} namely, if the surface of a TI is gapped out by some means and a point charge is placed near the surface, the response of the TI looks as if there is a magnetic monopole in the TI.}. This shows that the interaction of TIs with external fields is still a topic of great interest, and encourages greater study.
% Into the Nanoregime
\subsection{Into the Nano-regime}
Although the work on TIs is voluminous, the properties of topological insulator nanoparticles (TINPs) have scarcely been studied, with the exception of work by Imura et al \cite{imura2012spherical} and Siroki et al \cite{siroki2016single}. Previous work on optical properties of charged nanoparticles was limited to metalic nanoparticles with uniform surface charges, rather than topological insulator nanoparticles. Work on optical properties of TI thin films has been done \cite{wang2013calculation}.
Some experimental work on Bi$_2$Se$_3$ exists, where mechanically triturated n- and p-type Bi$_2$Te$_3$ nanoparticles are employed as nonlinear saturable absorbers to passively mode-lock erbium-doped fiber lasers (EDFLs) for sub-400 fs pulse generations\cite{lin2015using}. This work does not focus on the properties of Bi$_2$Se$_3$ nanoparticles, but rather their application in fiber lasers. Other efforts to produce TI nanstructures are underway, but as with TINPs, the theoretical study of them is limited. Efforts include the production of TIs into nanostructures such as exfoliated flakes from bulk crystals \cite{2010NanoL..10.1209T}, nanoribbons \cite{2010NanoL..10..329K,peng2010aharonov} and nanoplates \cite{2010NanoL..10.2245K}.
%% Topological Insulators and Light %%
\section{Topological Insulator Nanoparticles and Light}
\subsection{Novelty}
As previously stated, the interplay of TINPS and light has only been studied in a single paper by Siroki et al \cite{siroki2016single}. As this paper singlehandedly forms the basis of this topic, we discuss the methodology of the paper here.
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\\This paper gives a theoretical treatment of a Bi$_2$Se$_3$ topological insulator nanoparticle illuminated with THz frequency light. The authors use a model in which the collective response of bulk states is described by the dielectric constant whilst the surface states are described quantum mechanically using time-dependent perturbation theory. Optical cross sections are obtained, highlighting that the scattering and absorption cross sections are significantly modified compared to the case of ordinary insulators. For certain radii and frequencies the absorption cross section increases more than tenfold \cite{bohren1977scattering}. The nanoparticle studied in the paper is spherical, so an important comment to note is that changing the shape of the nanoparticle is expected to drastically change the way in which the TINP interacts with light, as the high surface to volume ratio of a nanoparticle means that the curvature of the surface will have a greater influence on how incoming light interacts with the particle.
\subsection{Methodology}
The paper begins by studying the interaction of a spherical dielectric particle with light. The effective Hamiltonian given in equation \ref{eq:ham} is used, and repeated here for convenience.
\begin{equation} H(\mathbf{k}) = \epsilon_0 (\mathbf{k})\mathbb{1}_4 +\begin{pmatrix}\mathcal{M}(\mathbf{k}) & A_{1} k_{z} & 0 & A_2 k_{-} \\ A_1 k_z & -\mathcal{M}(\mathbf{k}) & A_{2} k_{-} & 0 \\ 0 & A_{2} k_{+} & \mathcal{M}(\mathbf{k}) & -A_{1} k_{z} \\ A_{2} k_{+} & 0 & -A_{1} k_{z} & -\mathcal{M}(\mathbf{k}) \label{eq:ham}\end{pmatrix}\end{equation}
The authors use a simplified form of this Hamiltonian to allow for an analytic solution, in which A$_1=$A$_2$, B$_1=$B$_2$ and $\epsilon_0(\mathbf{k})$ is neglected.
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\\ A derivation of TINP surface states is performed using the envelope function approximation \cite{winkler2003spin,imura2012spherical} and using the resulting surface states and the Hamiltonian given in equation \ref{eq:ham}, the energy spectrum is obtained from the time-independent Schr\”{o}dinger equation. A very interesting result of the finite size of the particle is that the energy spectrum obtained for the surface states are quantised. If the radius of the particle were to increase, the spectrum would tend towards a continuous dispersion relation.
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\\With the envelope functions of the surface states and their energy spectrum obtained, the authors proceed to investigate how they change in the presence of static electric and magnetic fields. Static surface states will not affect the optical properties of a TINP, as the incident field will oscillate at a non-zero frequency. Thus, the model is modified to include time-dependence. The radius of the TINP sets the energy scale for the surface states and allows for the application of perturbation theory. The perturbation due to external light modifies both the wave function and the Hamiltonian. Note that the perturbed wavefunctions will have different surface charge distributions so that the surface of the particle will no longer be neutral. The charged surface will in turn gives rise to a potential in the Hamiltonian which should be taken into account for the solution to be self-consistent consistent. The effect of redistributing surface charge becomes negligible for large particles but for small particles it is significant.
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\\ Solving the system in a self-consistent way, the results show that the polarisability of a TINP is different from that of an ordinary nanoparticle. The topologically protected surface states give rise to an additional term in the polarizability of the TINP, only becoming negligible when the radius of the particle is large (as then the time-dependent surface chage density is low) or when the frequency of the incoming light is very large (as the large detuning makes the transition unlikely). However, for small radii and low detuning, the polarisability is modified which in turn affects the scattering and absorption cross sections and in particular, the absorption cross-section can become an order of magnitude larger.
\subsection{Approximations and limitations}
The most impacting approximation that has been made is in the use of the simplified Hamiltonian from equation \ref{eq:ham}. Bi$_2$Se$_3$ is not an isotropic material, differentiating between the z and x-y directions of the material due to the layered structure. This structure is not encapsulated in the model Hamiltonian used. On a similar note, the dielectric function used to calculate the bulk response is taken to be a scalar rather than a tensor .These two simplifications mean that the delocalised surface states and bulk response have been treated in a simplified fashion. The main features of the TINP are nonetheless captured, making the results of the paper qualitiatively useful. A possibly limitation is found by noting that the energy spacing between the surface states of the TINP is commensurate with k$_B$T, implying that the effects described in the paper will only be observed at low temperatures.
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\\ Due to the analytical method of the paper and self-consistent method to obtain the final results, extending the method to more complicated geometries than a sphere may prove difficult and different methods may need to be employed. An exciting comment on the work is that transitions between the surface states depend on the polarisation of the incident light (and thus the work could be extended to polarizations other than circular). This provides an additional degree of freedom in manipulating these states and may be of interest in the area of quantum information.
% State of the Art
\subsection{Applications and outlook}
The materials described by Landau symmetry-breaking theory have had a substantial impact on technology. For example, ferromagnetic materials that break spin rotation symmetry can be used as the media of digital information storage. A hard drive made of ferromagnetic materials can store gigabytes of information. Liquid crystals that break the rotational symmetry of molecules find wide application in display technology; nowadays one can hardly find a household without a liquid crystal display somewhere in it. Crystals that break translation symmetry lead to well defined electronic bands which in turn allow us to make semiconducting devices such as transistors. Different types of topological orders are even richer than different types of symmetry-breaking orders. This suggests their potential for exciting, novel applications.
\\ In the two-dimensional electron gas at the topological insulator surface, plasmons cannot be excited directly by electromagnetic radiation because their dispersion law is such as to prevent the conservation of momentum in the photon absorption process. How was this overcome in experiment and what was observed?
TINPs with their increased absorption cross sections may be used in photodetectors as well as become a new building block for metamaterials.
In addition, the surface states can act as a screening layer which suppresses absorption inside the particle. These effects may be useful in the areas of plasmonics, cavity electrodynamics and quantum information.
Idea of the project is that TINPs have quantised Dirac cone, and we want to calculate the transition probabilities between the states in the cone. Could also possibly find a configuration of states which allows us to lase or use as a quibit. That would need 4 states to reproduce the usual lasing scheme. In the project I will redo the results of energy states and wave functions for the cylindrical and disk cases, and then do the same for a plane (which will have continuous energy states).
The MSc project will seek to calculate the transition matrix element between edge states in a topological insulator. Such matrix elements will be used in order to calculate the decay rate transition and the polarization properties of the photons emitted. The goal is to understand which are the most/less probable decays. In particular, how the decay rates are affected by the nanoparticle shape? What is the effect of perturbations (defects) on the topological nanoparticle?Can we build a lasing system with the levels of topological nanoparticles?