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In this thesis, a model system of a magnetic topological heterostructure is studied, namely a heterosystem consisting of a single ferromagnetic septuple-layer (SL) of \(MnBi_2Te_4\) on the surface of the three-dimensional topological insulator \(Bi_2Te_3\).
Using MBE and developing a specialized experimental setup, the first part of this thesis deals with the growth of \(Bi_2Te_3\) and thin films of \(MnBi_2Te_4\) on \(BaF_2\)-substrates by the co-evaporation of its binary constituents. The structural analysis is conducted along several suitable probes such as X-ray diffraction (XRD, XRR), AFM and scanning tunnelling electron microscopy (STEM). It is furthermore found that the growth of a single septuple-layer of \(MnBi_2Te_4\) on the surface of \(Bi_2Te_3\) can be facilitated.
By using X-ray absorption and circular magnetic dichroism (XAS, XMCD), the magnetic properties of \(MnBi_2Te_4\) are explored down to the monolayer limit. The layered nature of the vdW crystal and a strong uniaxial magnetocrystalline anisotropy establish stable out-of plane magnetic order at the surface of \(MnBi_2Te_4\), which is stable even down to the 2D limit. Pushing the material system to there, i.e. a single SL \(MnBi_2Te_4\) further allows to study the phase transition of this 2D ferromagnet and extract its critical behaviour with \(T_c \, = \, 14.89~k\) and \(\beta \, = \, 0.484\).
Utilizing bulk crystals of the ferromagnetic \(Fe_3GeTe_2\) as substrate allows to influence, enhance and bias the magnetism in the single SL of \(MnBi_2Te_4\). By growing heterostructures of the type \(MnBi_2Te_4\) -- n layer \(Bi_2Te_3\) -- \(Fe_3GeTe_2\)for n between 0 and 2, it is shown, that a considerable magnetic coupling can be introduced between the \(MnBi_2Te_4\) top-layer and the substrate.
Finally the interplay between topology and magnetism in the ferromagnetic extension is studied directly by angle-resolved photoemission spectroscopy. The heterostructure is found to host a linearly dispersing TSS at the centre of the Brillouin zone. Using low temperature and high-resolution ARPES a large magnetic gap opening of \(\sim\) 35 meV is found at the Dirac point of the TSS. By following its temperature evolution, it is apparent that the scaling behaviour coincides with the magnetic order parameter of the modified surface.
The fascination of microcavity exciton-polaritons (polaritons) rests upon the combination of advanced technological control over both the III-V semiconductor material platform as well as the precise spectroscopic access to polaritonic states, which provide access to the investigation of open questions and complex phenomena due to the inherent nonlinearity and direct spectroscopic observables such as energy-resolved real and Fourier space information, pseudospin and coherence. The focus of this work was to advance the research area of polariton lattice simulators with a particular emphasis on their lasing properties. Following the brief introduction into the fundamental physics of polariton lattices in chapter 2, important aspects of the sample fabrication as well as the Fourier spectroscopy techniques used to investigate various features of these lattices were summarized in chapter 3. Here, the implementation of a spatial light modulator for advanced excitation schemes was presented.
At the foundation of this work is the capability to confine polaritons into micropillars or microtraps resulting in discrete energy levels. By arranging these pillars or traps into various lattice geometries and ensuring coupling between neighbouring sites, polaritonic band structures were engineered. In chapter 4, the formation of a band structure was visualised in detail by investigating ribbons of honeycomb lattices. Here, the transition of the discrete energy levels of a single chain of microtraps to the fully developed band structure of a honeycomb lattice was observed. This study allows to design the size of individual domains in more complicated lattice geometries such that a description using band structures becomes feasible, as it revealed that a width of just six unit cells is sufficient to reproduce all characteristic features of the S band of a honeycomb lattice. In particular in the context of potential technological applications in the realms of lasing, the laser-like, coherent emission from polariton microcavities that can be achieved through the excitation of polariton condensates is intriguing. The condensation process is significantly altered in a lattice potential environment when compared to a planar microcavity. Therefore, an investigation of the polariton condensation process in a lattice with respect to the characteristics of the excitation laser, the exciton-photon detuning as well as the reduced trap distance that represents a key design parameter for polaritonic lattices was performed. Based on the demonstration of polariton condensation into multiple bands, the preferred condensation into a desired band was achieved by selecting the appropriate detuning. Additionally, a decreased condensation threshold in confined systems compared to a planar microcavity was revealed.
In chapter 5, the influence of the peculiar feature of flatbands arising in certain lattice geometries, such as the Lieb and Kagome lattices, on polaritons and polariton condensates was investigated. Deviations from a lattice simulator described by a tight binding model that is solely based on nearest neighbour coupling cause a remaining dispersiveness of the flatbands along certain directions of the Brillouin zone. Therefore, the influence of the reduced trap distance on the dispersiveness of the flatbands was investigated and precise technological control over the flatbands was demonstrated. As next-nearest neighbour coupling is reduced drastically by increasing the distance between the corresponding traps, increasing the reduced trap distance enables to tune the S flatbands of both Lieb and Kagome lattices from dispersive bands to flatbands with a bandwidth on the order of the polariton linewidth. Additionally to technological control over the band structures, the controlled excitation of large condensates, single compact localized state (CLS) condensates as well as the resonant excitation of polaritons in a Lieb flatband were demonstrated. Furthermore, selective condensation into flatbands was realised. This combination of technological and spectroscopic control illustrates the capabilities of polariton lattice simulators and was used to study the coherence of flatband polariton condensates. Here, the ability to tune the dispersiveness from a dispersive band to an almost perfect flatband in combination with the selectivity of the excitation is particularly valuable. By exciting large flatband condensates, the increasing degree of localisation to a CLS with decreasing dispersiveness was demonstrated by measurements of first order spatial coherence. Furthermore, the first order temporal coherence of CLS condensates was increased from τ = 68 ps for a dispersive flatband, a value typically achieved in high-quality microcavity samples, to a remarkable τ = 459 ps in a flatband with a dispersiveness below the polarion linewidth. Corresponding to this drastic increase of the first order coherence time, a decrease of the second order temporal coherence function from g(2)(τ =0) = 1.062 to g(2)(0) = 1.035 was observed. Next to laser-like, coherent emission, polariton condensates can form vortex lattices. In this work, two distinct vortex lattices that can form in polariton condensates in Kagome flatbands were revealed. Furthermore, chiral, superfluid edge transport was realised by breaking the spatial symmetry through a localised excitation spot. This chirality was related to a change in the vortex orientation at the edge of the lattice and thus opens the path towards further investigations of symmetry breaking and chiral superfluid transport in Kagome lattices.
Arguably the most influential concept in solid-state physics of the recent decades is the idea of topological order that has also provided a new degree of freedom to control the propagation of light. Therefore, in chapter 6, the interplay of topologically non-trivial band structures with polaritons, polariton condensates and lasing was emphasised. Firstly, a two-dimensional exciton-polariton topological insulator based on a honeycomb lattice was realised. Here, a topologically non-trivial band gap was opened at the Dirac points through a combination of TE-TM splitting of the photonic mode and Zeeman splitting of the excitonic mode. While the band gap is too small compared to the linewidth to be observed in the linear regime, the excitation of polariton condensates allowed to observe the characteristic, topologically protected, chiral edge modes that are robust against scattering at defects as well as lattice corners. This result represents a valuable step towards the investigation of non-linear and non-Hermitian topological physics, based on the inherent gain and loss of microcavities as well as the ability of polaritons to interact with each other. Apart from fundamental interest, the field of topological photonics is driven by the search of potential technological applications, where one direction is to advance the development of lasers. In this work, the starting point towards studying topological lasing was the Su-Schrieffer-Heeger (SSH) model, since it combines a simple and well-understood geometry with a large topological gap. The coherence properties of the topological edge defect of an SSH chain was studied in detail, revealing a promising degree of second order temporal coherence of g(2)(0) = 1.07 for a microlaser with a diameter of only d = 3.5 µm. In the context of topological lasing, the idea of using a propagating, topologically protected mode to ensure coherent coupling of laser arrays is particularly promising. Here, a topologically non-trivial interface mode between the two distinct domains of the crystalline topological insulator (CTI) was realised. After establishing selective lasing from this mode, the coherence properties were studied and coherence of a full, hexagonal interface comprised of 30 vertical-cavity surface-emitting lasers (VCSELs) was demonstrated. This result thus represents the first demonstration of a topological insulator VCSEL array, combining the compact size and convenient light collection of vertically emitting lasers with an in-plane topological protection.
Finally, in chapter 7, an approach towards engineering the band structures of Lieb and honeycomb lattices by unbalancing the eigenenergies of the sites within each unit cell was presented. For Lieb lattices, this technique opens up a path towards controlling the coupling of a flatband to dispersive bands and could enable a detailed study of the influence of this coupling on the polariton flatband states. In an unbalanced honeycomb lattice, a quantum valley Hall boundary mode between two distinct, unbalanced honeycomb domains with permuted sites in the unit cells was demonstrated. This boundary mode could serve as the foundation for the realisation of a polariton quantum valley Hall effect with a truly topologically protected spin based on vortex charges. Modifying polariton lattices by unbalancing the eigenenergies of the sites that comprise a unit cell was thus identified as an additional, promising path for the future development of polariton lattice simulators.
Diese Veröffentlichung ist eine Einführung in die syntaktischen Strukturen der deutschen Gegenwartssprache und deckt folgende Gebiete ab: Satzdefinition, Wortarten, Topologie deutscher Sätze, valenzabhängige und -unabhängige Satzglieder (Ergänzungen und Angaben), Funktion und Semantik von Dativ- und Genitivkonstruktionen, Hilfs-, Modal- und Modalitätsverben, Funktionsverbgefüge und verbale Wendungen, reflexive Konstruktionen, komplexe Sätze und Satzglieder, Passivkonstruktionen, Temporalität sowie Modalität.
Diese Veröffentlichung ist eine Einführung in die syntaktischen Strukturen der deutschen Gegenwartssprache und deckt folgende Gebiete ab: Satzdefinition, Wortarten, Topologie deutscher Sätze, valenzabhängige und -unabhängige Satzglieder (Ergänzungen und Angaben), Funktion und Semantik von Dativ- und Genitivkonstruktionen, Hilfs-, Modal- und Modalitätsverben, Funktionsverbgefüge und verbale Wendungen, reflexive Konstruktionen, komplexe Sätze und Satzglieder, Passivkonstruktionen, Temporalität sowie Modalität.
Since the prediction of the quantum spin Hall effect in graphene by Kane and Mele, \(Z_2\) topology in hexagonal monolayers is indissociably linked to high-symmetric honeycomb lattices. This thesis breaks with this paradigm by focusing on topological phases in the fundamental two-dimensional hexagonal crystal, the triangular lattice. In contrast to Kane-Mele-type systems, electrons on the triangular lattice profit from a sizable, since local, spin-orbit coupling (SOC) and feature a non-trivial ground state only in the presence of inversion symmetry breaking. This tends to displace the valence charge form the atomic position. Therefore, all non-trivial phases are real-space obstructed. Inspired by the contemporary conception of topological classification of electronic systems, a comprehensive lattice and band symmetry analysis of insulating phases of a \(p\)-shell on the triangular lattice is presented. This reveals not only the mechanism at the origin of band topology, the competition of SOC and symmetry breaking, but sheds also light on the electric polarization arising from a displacement of the valence charge centers from the nuclei, i. e., real-space obstruction. In particular, the competition of SOC versus horizontal and vertical reflection symmetry breaking gives rise to four topologically distinct insulating phases: two kinds of quantum spin Hall insulators (QSHI), an atomic insulator and a real-space obstructed higher-order topological insulator. The theoretical analysis is complemented with state-of-the-art first principles calculations and experiments on trigonal monolayer adsorbate systems. This comprises the recently discovered triangular QSHI indenene, formed by In atoms, and focuses on its topological classification and real-space obstruction. The analysis reveals Kane-Mele-type valence bands which profit from the atomic SOC of the triangular lattice. The realization of a HOTI is proposed by reducing SOC by considering lighter adsorbates. Further the orbital Rashba effect is analyzed in AgTe, a consequence of mirror symmetry breaking, the formation of local angular momentum polarization and SOC. As an outlook beyond topology, the Fermi surface and electronic susceptibility of Group V adsorbates on silicon carbide are investigated.
In summary, this thesis elucidates the interplay of symmetry breaking and SOC on the triangular lattice, which can promote non-trivial insulating phase.
S=N versus S+-N-
(2002)
The main aim of this thesis was to characterise structurally four sulfur-nitrogen compounds in terms of their experimental electron density distribution: Sulfurdiimide S(NtBu)2 (I), sulfurtriimide S(NtBu)3 (II), methyl(diimido)sulfinic acid H(NtBu)2SMe (III) and methylene-bis(triimido)sulfonic acid CH2{S(NtBu)2(HNtBu)}2 (IV). The electron density was determined by multipole refinements on high-resolution X-ray data at low temperatures. The refined densities were analysed by means of Bader’s theory of ‘Atoms in Molecules’ to get information about the bonding types (shared/ closed shell), bond strengths, and the extent of polarisation. The distributions of the static deformation densities, which already showed the most important electronical features as lone-pairs and bonding densities, were calculated for all compounds. The spatial distributions provided a first impression about the bonding properties. The nitrogen lone-pair densities were found to be inclined towards the electropositive sulfur atoms. In II, III and IV the spatial distributions already suggested sp3 hybridisation of the nitrogen atoms. In I gradual differences between the E/Z and Z/Z oriented NtBu groups were visualised. The charge density distribution was analysed along the bond paths, which showed some of the S,N bonds to be considerably bent. In the central part of the thesis detailed topological analyses of the electron density distributions were performed. All BCPs and the related electronical properties as the electron density, the negative Laplacian, the eigenvalues of the Hessian matrix, and several values, which can be deduced from these, were calculated. Due to the low number of comparable published compounds, internal scaling facilitated by III and IV led to system-specific ranking of the S-N and S-C bonds in terms of bond type (shared vs. closed shell), bond order, and bond strength. To quantify bond polarisation a criterion was developed which relates shifts in the BCPs to electron transfer from the electropositive to the electronegative bonding partner. The distributions of the Laplacian were determined for all S-E (E = N, C) bonds because of their fundamental importance for the classification of atomic interactions. Furthermore, the spatial distribution of the negative Laplacian with respect to all important bonds was determined around the central sulfur and nitrogen atoms. The analyses led to detailed information about the S,N interactions. A calculation of the reactive surfaces where the Laplacian equals zero revealed possible reaction pathways of nucleophilic attacks to the central sulfur atoms. All nitrogen atoms in H(NtBu)2SMe (III) as well as in CH2{S(NtBu)2(HNtBu)}2 (IV) are predominantly sp3 hybridised. The S,N bonds should therefore be formulated as S+–N– single bonds, strengthened and shortened by electrostatic reinforcement. In S(NtBu)2 (I) the sp2 hybridisation of the nitrogen atoms was verified. All topological criteria unearthed the inequality of the formally equivalent S=N double bonds. The differences were assigned to the molecular E/Z conformation in the solid state. Interaction between the in-plane lone-pair density of the nitrogen and the sulfur atom located at the same side causes the non-bonding charge concentration at the sulfur atom to be dislocated into the second S–N bond. The existence of a delocalised 3-centres-2-electrons system within the planar SN2 core was assumed to be formed by non-hybridised p-orbitals. An effective delocalisation was found to be possibly disturbed by a weak intermolecular S...S interaction. The interpretation of the S,N interaction in S(NtBu)3 (II) was not straightforward, since the electron density distribution showed both, indicators for multiple bonding as well as for sp3 hybridisation of the nitrogen atoms, which verifies the formulation of a S+–N– bonding mode. The bonding situation in S(NtBu)3 was identified as an intermediate state between that of a delocalised 4-centres-6-electrons system formed by non-hybridised p-orbitals within the planar SN3 unit and that of a S+–N– system.
Adding interactions to topological (non-)trivial free fermion systems can in general have four different effects: (i) In symmetry protected topological band insulators, the correlations may lead to the spontaneous breaking of some protecting symmetries by long-range order that gaps the topological boundary modes. (ii) In free fermion (semi-)metal, the interaction could vice versa also generate long-range order that in turn induces a topological mass term and thus generates non-trivial phases dynamically. (iii) Correlation might reduce the topological classification of free fermion systems by allowing adiabatic deformations between states of formerly distinct phases. (iv) Interaction can generate long-range entangled topological order in states such as quantum spin liquids or fractional quantum Hall states that cannot be represented by non-interacting systems. During the course of this thesis, we use numerically exact quantum Monte Carlo algorithms to study various model systems that (potentially) represent one of the four scenarios, respectively.
First, we investigate a two-dimensional $d_{xy}$-wave, spin-singlet superconductor, which is relevant for high-$T_c$ materials such as the cuprates. This model represents nodal topological superconductors and exhibits chiral flat-band edge states that are protected by time-reversal and translational invariance. We introduce the conventional Hubbard interaction along the edge in order to study their stability with respect to correlations and find ferromagnetic order in case of repulsive interaction as well as charge-density-wave order and/or additional $i$s-wave pairing for attractive couplings. A mean-field analysis that, for the first time, is formulated in terms of the Majorana edge modes suggests that any order has normal and superconducting contributions. For example, the ferromagnetic order appears in linear superposition with triplet pairing. This finding is well confirmed by the numerically exact quantum Monte Carlo investigation.
Second, we consider spinless electrons on a two-dimensional Lieb lattice that are subject to nearest-neighbor Coulomb repulsion. The low energy modes of the free fermion part constitute a spin-$1$ Dirac cone that might be gapped by several mass terms. One option breaks time-reversal symmetry and generates a topological Chern insulator, which mainly motivated this study. We employ two flavors of quantum Monte Carlo methods and find instead the formation of charge-density-wave order that breaks particle-hole symmetry. Additionally, due to sublattices of unequal size in Lieb lattices, this induces a finite chemical potential that drives the system away from half-filling. We argue that this mechanism potentially extends the range of solvable models with finite doping by coupling the Lieb lattice to the target system of interest.
Third, we construct a system with four layers of a topological insulators and interlayer correlation that respects one independent time-reversal and a unitary $\mathbb{Z}_2$ symmetry. Previous studies claim a reduced topological classification from $\mathbb{Z}$ to $\mathbb{Z}_4$, for example by gapping out degenerate zero modes in topological defects once the correlation term is designed properly. Our interaction is chosen according to this analysis such that there should exist an adiabatic deformation between states whose topological invariant differs by $\Delta w=\pm4$ in the free fermion classification. We use a projective quantum Monte Carlo algorithm to determine the ground-state phase diagram and find a symmetry breaking regime, in addition to the non-interacting semi-metal, that separates the free fermion insulators. Frustration reduces the size of the long-range ordered region until it is replaced by a first order phase transition. Within the investigated range of parameters, there is no adiabatic path deforming the formerly distinct free fermion states into each other. We conclude that the prescribed reduction rules, which often use the bulk-boundary correspondence, are necessary but not sufficient and require a more careful investigation.
Fourth, we study conduction electron on a honeycomb lattice that form a Dirac semi-metal Kondo coupled to spin-1/2 degrees of freedom on a Kagome lattice. The local moments are described by a variant of the Balents-Fisher-Girvin model that has been shown to host a ferromagnetic phase and a $\mathbb{Z}_2$ spin liquid at strong frustration. Here, we report the first numerical exact quantum Monte Carlo simulation of the Kondo-coupled system that does not exhibit the negative-sign problem. When the local moments form a ferromagnet, the Kondo coupling induces an anti-ferromagnetic mass term in the conduction-electron system. At large frustration, the Dirac cone remains massless and the spin system forms a $\mathbb{Z}_2$ spin liquid. Owing to the odd number of spins per unit cell, this constitutes a non-Fermi liquid that violates Luttinger's theorem which relates the Fermi volume to the particle density in a Fermi liquid. This phase is a specific realization of the so called 'fractional Fermi liquid` as it has been first introduced in the context of heavy fermion models.
This doctoral thesis investigates magneto-optical properties of mercury telluride layers grown tensile strained on cadmium telluride substrates. Here, layer thicknesses start above the usual quantum well thickness of about 20 nm and have a upper boundary around 100 nm due to lattice relaxation effects. This kind of layer system has been attributed to the material class of three-dimensional topological insulators in numerous publications. This class stands out due to intrinsic boundary states which cross the energetic band gap of the layer's bulk.
In order to investigate the band structure properties in a narrow region around the Fermi edge, including possible boundary states, the method of highly precise time-domain Terahertz polarimetry is used. In the beginning, the state of the art of Teraherz technology at the start of this project is discussed, moving on to a detailed description and characterization of the self-built measurement setup. Typical standard deviation of a polarization rotation or ellipticity measurement are on the order of 10 to 100 millidegrees, according to the transmission strength through investigated samples. A range of polarization spectra, depending on external magnetic fields up to 10 Tesla, can be extracted from the time-domain signal via Fourier transformation.
The identification of the actual band structure is done by modeling possible band structures by means of the envelope function approximation within the framework of the k·p method. First the bands are calculated based on well-established model parameters and from them the possible optical transitions and expected ellipticity spectra, all depending on external magnetic fields and the layer's charge carrier concentration. By comparing expected with measured spectra, the validity of k·p models with varying depths of detail is analyzed throughout this thesis. The rich information encoded in the ellipitcity spectra delivers key information for the attribution of single optical transitions, which are not part of pure absorption spectroscopy. For example, the sign of the ellipticity signals is linked to the mix of Landau levels which contribute to an optical transition, which shows direct evidence for bulk inversion asymmetry effects in the measured spectra.
Throughout the thesis, the results are compared repeatedly with existing publications on the topic. It is shown that the models used there are often insufficient or, in worst case, plainly incorrect. Wherever meaningful and possible without greater detours, the differences to the conclusions that can be drawn from the k·p model are discussed.
The analysis ends with a detailed look on remaining differences between model and measurement. It contains the quality of model parameters as well as different approaches to integrate electrostatic potentials that exist in the structures into the model.
An outlook on possible future developments of the mercury cadmium telluride layer systems, as well as the application of the methods shown here onto further research questions concludes the thesis.
Breaking inversion symmetry in crystalline solids enables the formation of spin-polarized electronic states by spin-orbit coupling without the need for magnetism. A variety of interesting physical phenomena related to this effect have been intensively investigated in recent years, including the Rashba effect, topological insulators and Weyl semimetals. In this work, the interplay of inversion symmetry breaking and spin-orbit coupling and, in particular their general influence on the character of electronic states, i.e., on the spin and orbital degrees of freedom, is investigated experimentally. Two different types of suitable model systems are studied: two-dimensional surface states for which the Rashba effect arises from the inherently broken inversion symmetry at the surface, and a Weyl semimetal, for which inversion symmetry is broken in the three-dimensional crystal structure. Angle-resolved photoelectron spectroscopy provides momentum-resolved access to the spin polarization and the orbital composition of electronic states by means of photoelectron spin detection and dichroism with polarized light. The experimental results shown in this work are also complemented and supported by ab-initio density functional theory calculations and simple model considerations.
Altogether, it is shown that the breaking of inversion symmetry has a decisive influence on the Bloch wave function, namely, the formation of an orbital angular momentum. This mechanism is, in turn, of fundamental importance both for the physics of the surface Rashba effect and the topology of the Weyl semimetal TaAs.
The quantum Hall (QH) effect, which can be induced in a two-dimensional (2D) electron gas by an external magnetic field, paved the way for topological concepts in condensed matter physics. While the QH effect can for that reason not exist without Landau levels, there is a plethora of topological phases of matter that can exist even in the absence of a magnetic field. For instance, the quantum spin Hall (QSH), the quantum anomalous Hall (QAH), and the three-dimensional (3D) topological insulator (TI) phase are insulating phases of matter that owe their nontrivial topology to an inverted band structure. The latter results from a strong spin-orbit interaction or, generally, from strong relativistic corrections. The main objective of this thesis is to explore the fate of these preexisting topological states of matter, when they are subjected to an external magnetic field, and analyze their connection to quantum anomalies. In particular, the realization of the parity anomaly in solid state systems is discussed. Furthermore, band structure engineering, i.e., changing the quantum well thickness, the strain, and the material composition, is employed to manipulate and investigate various topological properties of the prototype TI HgTe.
Like the QH phase, the QAH phase exhibits unidirectionally propagating metallic edge channels. But in contrast to the QH phase, it can exist without Landau levels. As such, the QAH phase is a condensed matter analog of the parity anomaly. We demonstrate that this connection facilitates a distinction between QH and QAH states in the presence of a magnetic field. We debunk therefore the widespread belief that these two topological phases of matter cannot be distinguished, since they are both described by a $\mathbb{Z}$ topological invariant. To be more precise, we demonstrate that the QAH topology remains encoded in a peculiar topological quantity, the spectral asymmetry, which quantifies the differences in the number of states between the conduction and valence band. Deriving the effective action of QAH insulators in magnetic fields, we show that the spectral asymmetry is thereby linked to a unique Chern-Simons term which contains the information about the QAH edge states. As a consequence, we reveal that counterpropagating QH and QAH edge states can emerge when a QAH insulator is subjected to an external magnetic field. These helical-like states exhibit exotic properties which make it possible to disentangle QH and QAH phases. Our findings are of particular importance for paramagnetic TIs in which an external magnetic field is required to induce the QAH phase.
A byproduct of the band inversion is the formation of additional extrema in the valence band dispersion at large momenta (the `camelback'). We develop a numerical implementation of the $8 \times 8$ Kane model to investigate signatures of the camelback in (Hg,Mn)Te quantum wells. Varying the quantum well thickness, as well as the Mn-concentration, we show that the class of topologically nontrivial quantum wells can be subdivided into direct gap and indirect gap TIs. In direct gap TIs, we show that, in the bulk $p$-regime, pinning of the chemical potential to the camelback can cause an onset to QH plateaus at exceptionally low magnetic fields (tens of mT). In contrast, in indirect gap TIs, the camelback prevents the observation of QH plateaus in the bulk $p$-regime up to large magnetic fields (a few tesla). These findings allowed us to attribute recent experimental observations in (Hg,Mn)Te quantum wells to the camelback. Although our discussion focuses on (Hg,Mn)Te, our model should likewise apply to other topological materials which exhibit a camelback feature in their valence band dispersion.
Furthermore, we employ the numerical implementation of the $8\times 8$ Kane model to explore the crossover from a 2D QSH to a 3D TI phase in strained HgTe quantum wells. The latter exhibit 2D topological surface states at their interfaces which, as we demonstrate, are very sensitive to the local symmetry of the crystal lattice and electrostatic gating. We determine the classical cyclotron frequency of surface electrons and compare our findings with experiments on strained HgTe.