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Topological phases of matter represent a new phase which cannot be understood in terms of Landau’s theory of symmetry breaking and are characterized by non-local topological properties emerging from purely local (microscopic) degrees of freedom. It is the non-trivial topology of the bulk band structure that gives rise to topological phases in condensed matter systems. Quantum Hall systems are prominent examples of such topological phases. Different quantum Hall states cannot be distinguished by a local order parameter. Instead, non-local measurements are required, such as the Hall conductance, to differentiate between various quantum Hall states. A signature of a topological phase is the existence of robust properties that do not depend on microscopic details and are insensitive to local perturbations which respect appropriate symmetries. Examples of such properties are the presence of protected gapless edge states at the boundary of the system for topological insulators and the remarkably precise quantization of the Hall conductance for quantum Hall states. The robustness of these properties can be under-stood through the existence of a topological invariant, such as the Chern number for quantum Hall states which is quantized to integer values and can only be changed by closing the bulk gap. Two other examples of topological phases of matter are topological superconductors and Weyl semimetals. The study of transport in various kinds of junctions of these topological materials is highly interesting for their applications in modern electronics and quantum computing. Another intriguing area to study is how to generate new kind of gapless edge modes in topological systems. In this thesis I have studied various aspects of topological phases of matter, such as electronic transport in junctions of topological insulators and topological superconductors, the generation of new kinds of boundary modes in the presence of granularity, and the effects of periodic driving in topological systems. We have studied the following topics. 1. transport across a line junction of two three-dimensional topological insulators, 2. transport across a junction of topological insulators and a superconductor, 3. surface and edge states of a topological insulator starting from a lattice model, 4. effects of granularity in topological insulators, 5. Majorana modes and conductance in systems with junctions of topological superconducting wires and normal metals, and 6. generation of new surface states in a Weyl semimetal in the presence of periodic driving by the application of electromagnetic radiation. A detailed description of each chapter is given below. • In the first chapter we introduce a number of concepts which are used in the rest of the thesis. We will discuss the ideas of topological phases of matter (for example, topological insulators, topological superconductors and Majorana modes, and Weyl semimetals), the renormalization group theory for weak interactions, and Floquet theory for periodically driven systems. • In the second chapter we study transport across a line junction which separates the surfaces of two three-dimensional topological insulators. The velocities of the Dirac electrons on the two surfaces may be unequal and may even have opposite signs. For a time-reversal invariant system, we show that the line junction is characterized by an arbitrary real parameter α; this determines the scattering amplitudes (reflection and transmission) from the junction. The physical origin of α is a potential barrier that may be present at the junction. If the surface velocities have the same sign, edge states exist that propagate along the line junction with a velocity and orientation of the spin which depend on α and the ratio of the velocities. Next, we study what happens if the two surfaces are at an angle φ with respect to each other. We study the scattering and differential conductance across the line junction as functions of φ and α. We also show that there are edge states which propagate along the line junction with a velocity and spin orientation which depend on φ. Finally, if the surface velocities have opposite signs, we find that the electrons must necessarily transmit into the two-dimensional interface separating the two topological insulators. • In the third chapter we discuss transport across a line junction lying between two orthogonal topological insulator surfaces and a superconductor which can have either s-wave (spin-singlet) or p-wave (spin-triplet) pairing symmetry. This junction is more complicated than the line junction discussed in the previous chapter because of the presence of the superconductor. In a topological insulator spin-up and spin-down electrons get coupled while in a superconductor electrons and holes get coupled. Hence we have to use a four-component spinor formalism to describe both spin and particle-hole degrees of freedom. The junction can have three time-reversal invariant barriers on the three sides. We compute the subgap charge conductance across such a junction and study their behaviors as a function of the bias voltage applied across the junction and the three parameters which characterize the barriers. We find that the presence of topological insulators and a superconductor leads to both Dirac and Schrodinger-like features in the charge conductances. We discuss the effects of bound states on the superconducting side on the conductance; in particular, we show that for triplet p-wave superconductors such a junction may be used to determine the spin state of its Cooper pairs. • In the fourth chapter we derive the surface Hamiltonians of a three-dimensional topological insulator starting from a microscopic model. (This description was not discussed in the previous chapters where we directly started from the surface Hamiltonians without deriving them form a bulk Hamiltonian). Here we begin from the bulk Hamiltonian of a three-dimensional topological insulator Bi2Se3. Using this we derive the surface Hamiltonians on various surfaces of the topological insulator, and we find the states which appear on the different surfaces and along the edge between pairs of surfaces. The surface Hamiltonians depend on the orientation of the surfaces and are therefore quite different from the previous chapters. We use both analytical methods based on the surface Hamiltonians (which are derived from the bulk Hamiltonian) and numerical methods based directly on a lattice discretization of the bulk Hamiltonian in order to find surface and edge states. We find that the application of a potential barrier along an edge can give rise to states localized at that edge. These states have an unusual energy-momentum dispersion which can be controlled by applying a potential along the edge; in particular, the velocity of these states can be tuned to zero. The scattering and conductance across the edge are studied as a function of the edge potential. We show that a magnetic field applied in a particular direction can also give rise to zero energy states on certain edges. We point out possible experimental ways of looking for the various edge states. • In the fifth chapter we study a system made of topological insulator (TI) nanocrystals which are coupled to each other. Our theoretical studies are motivated by the following experimental observations. Electrical transport measurements were carried out on thin films of nanocrystals of Bi2Se3 which is a TI. The measurements reveal that the entire system behaves like a single TI with two topological surface states at the two ends of the system. The two surface states are found to be coupled if the film thickness is small and decoupled above a certain film thickness. The surface state penetration depth is found to be unusually large and it decreases with increasing temperature. To explain all these experimental results we propose a theoretical model for this granular system. This consists of multiple grains of Bi2Se3 stacked next to each other in a regular array along the z-direction (the c-axis of Bi2Se3 nanocrystals). We assume translational invariance along the x and y directions. Each grain has top and bottom surfaces on which the electrons are described by Hamiltonians of the Dirac form which can be derived from the bulk Hamiltonian known for this material. We introduce intra-grain tunneling couplings t1 between the opposite surfaces of a single grain and inter-grain couplings t2 between nearby surfaces of two neighboring grains. We show that when t1 < t2 the entire system behaves like a single topological insulator whose outermost surfaces have gapless spectra described by Dirac Hamiltonians. We find a relation between t1, t2 and the surface state penetration depth λ which explains the properties of λ that are seen experimentally. We also present an expression for the surface state Berry phase as a function of the hybridization between the surface states and a Zeeman magnetic field that may be present in the system. At the end we theoretically studied the surface states on one of the side surfaces of the granular system and showed that many pairs of surface states can exist on the side surfaces depending on the length of the unit cell of the granular system. • In the sixth chapter we present our work on junctions of p-wave superconductors (SC) and normal metals (NM) in one dimension. We first study transport in a system where a SC wire is sandwiched between two NM wires. For such a system it is known that there is a Majorana mode at the junction between the SC and each NM lead. If the p-wave pairing changes sign at some point inside the SC, two additional Majorana modes appear near that point. We study the effect of all these modes on the subgap conductance between the leads and the SC. We derive an analytical expression as a function of and the length L of the SC for the energy shifts of the Majorana modes at the junctions due to hybridization between them; the energies oscillate and decay exponentially as L is increased. The energies exactly match the locations of the peaks in the conductance. We find that the subgap conductances do not change noticeably with the sign of . So there is no effect of the extra Majorana modes which appear inside the SC (due to changes in the signs of Δ) on the subgap conductance. Next we study junctions of three p-wave SC wires which are connected to the NM leads. Such a junction is of interest as it is the simplest system where braiding of Majorana modes is possible. Another motivation for studying this system is to see if the subgap transport is affected by changes in the signs of . For sufficiently long SCs, there are zero energy Majorana modes at the junctions between the SCs and the leads. In addition, depending on the signs of the Δ’s in the three SCs, there can also be one or three Majorana modes at the junction of the three SCs. We show that the various subgap conductances have peaks occurring at the energies of all these modes; we therefore get a rich pattern of conductance peaks. Next we study the effects of interactions between electrons (in the NM leads) on the transport. We use a renormalization group approach to study the effect of interactions on the conductance at energies far from the SC gap. Hence the earlier part of this chapter where we studied the transport at an energy E inside the SC gap (so that − < E < Δ) differs from this part where we discuss conductance at an energy E where |E| ≫ . For the latter part we assume the region of three SC wires to be a single region whose only role is to give rise to a scattering matrix for the NM wires; this scattering matrix has both normal and Andreev elements (namely, an electron can be reflected or transmitted as either an electron or a hole). We derive a renormalization group equation for the elements of the scattering matrix by assuming the interaction to be sufficiently weak. The fixed points of the renormalization group flow and their stabilities are studied; we find that the scattering matrix at the stable fixed point is highly symmetric even when the microscopic scattering matrix and the interaction strengths are not symmetric. Using the stability analysis we discuss the dependence of the conductances on the various length scales of the problem. Finally we propose an experimental realization of this system which can produce different signs of the p-wave pairings in the different SCs. • In the seventh chapter we show that the application of circularly polarized electro-magnetic radiation on the surface of a Weyl semimetal can generate states at that surface. The surface states can be characterized by their momenta due to translation invariance. The Floquet eigenvalues of these states come in complex conjugate pairs rather than being equal to ±1. If the amplitude of the radiation is small, we find some unusual bulk-boundary relations: the Floquet eigenvalues of the surface states lie at the extrema of the Floquet eigenvalues of the bulk system when the latter are plotted as a function of the momentum perpendicular to the surface, and the peaks of the Fourier transforms of the surface state wave functions lie at the momenta where the bulk Floquet eigenvalues have extrema. For the case of zero surface momentum, we can analytically derive interesting scaling relations between the decay lengths of the surface states and the amplitude and penetration depth of the radiation. For topological insulators, we again find that circularly polarized radiation can generate states on the surfaces; these states have much larger decay lengths (which can be tuned by the radiation amplitude) than the topological surface states which are present even in the absence of radiation. Finally, we show that radiation can generate surface states even for trivial insulators.

Soft condensed matter (SCM) systems are ubiquitous in nature. SCM systems contain mesoscopic structures in the size range 10 nm to 1 am that are held together by weak entropic forces. These materials are therefore easily perturbed by external fields such as shear, gravity and electric and magnetic fields and are novel systems for studying non-equilibrium phenomena. The elastic constants of these materials are ≈ 109 times smaller than conventional atomic fluids and hence it is possible to measure the viscoelastic response of these materials using commercial instruments such as rheometers. The relaxation time in SCM systems are of the order of milliseconds as compared to atomic systems where relaxation times are of the order of picoseconds. It is easy to study the effect of shear on SCM, as the shear rates attainable by commercial rheometers are of the order of the inverse of their relaxation times. The dynamics of SCM systems and their local rheological properties obtained using the method of probe diffusion can be quantified through dynamic light scattering experiments. The structure of SCM systems can be quantified using diffraction techniques such as small angle x-ray scattering. In this thesis we report experimental studies on the linear and nonlinear rheology and the dynamics of surfactant cetyltrimethylammonium tosylate (CTAT), which forms cylindrical wormlike micelles, studied using bulk rheology and dynamic light scattering (DLS) technique, respectively. We have also studied the phase behaviour of the ternary system formed by cetyltrimethylammonium 3-hydroxy-napthalene 2-carboxylate (CTAHN), sodium bromide (NaBr) and water using small angle x-ray scattering (SAXS). In Chapter 1, we discuss why SCM systems are suitable for studying non-equilibrium phenomena such as the effect of shear on the structure and dynamics of condensed matter. This is followed by a discussion on the chemical structure, phase behaviour and self assembling properties of the amphiphilic molecules in water. We then discuss the intermacromolecular forces such as van der Waals interaction, the screened Coulomb repulsion and hydrophobic and hydration forces. The systems that have been the subject of our experimental studies, viz. CTAT and CTAHN/NaBr/water have also been discussed in detail. This is followed by a theoretical background of linear and nonlinear rheology, dynamic light scattering and small angle x-ray scattering techniques. Next we describe the stress relaxation mechanisms in wormlike micelles. This is followed by a discussion on some standard techniques of nonlinear time series analysis, in particular the evaluation of the delay time L, the embedding dimension m, the correlation dimension ν and the Lyapunov exponent λ. We have also mentioned a few examples of experimental systems where chaos has been observed. We have also discussed in detail the various routes to chaos namely, the period-doubling route, the quasiperiodic route and the intermittency route. The concluding part of this chapter summarises the main results of the thesis. Chapter 2 discusses the experimental apparatus used in our studies. We have discussed the different components of the MCR-300 stress-controlled rheometer (Paar Physica, Germany). The rheo-small angle light scattering experiments and the direct visualisation experiments done using a home-made shear cell are also discussed. Next we describe the various experiments that can be done using a commercial rheometer. The frequency response and flow experiments have been discussed with some examples from our own work on entangled, cylindrical micelles. This is followed by a discussion on the various components of our dynamic light scattering (DLS) setup (Brookhaven Instruments, USA). Particle sizing of submicrometer colloidal spheres using our DLS setup has been discussed with an example of an angle-resolved DLS study of 0.05µm polystyrene colloids. Next we describe the various components of the SAXS setup (Hecus M. Braun, Austria). As an example application of SAXS we have quantified the structure of the lamellar phase formed by the surfactant CTAHN/water. We finally describe the sample preparation methods employed by us for the different experiments. Our nonlinear rheology experiments on viscoelastic gels of surfactant CTAT (cCT AT= 2wt%) in the presence of salt sodium chloride (NaCl) at various concentrations has been discussed in Chapter 3. We observe a plateau in the measured flow curve and this is attributed to a mechanical instability of the shear banding type. The slope of this plateau can be tuned by the addition of salt NaCl. This slope is due to a concentration difference between the shear bands arising from a Helfand-Fredrickson mechanism. This is confirmed by the presence of a “Butterfly” light scattering pattern in SALS experiments performed simultaneously with rheological measurements. We have carried out experiments at six different salt concentrations 10mM < cN aCl<1M, which yield plateau slopes (α) ranging from 0.07 < α < 0.4. We find that a minimum slope of 0.12, corresponding to a salt concentration of 25mM NaCl, is essential to see a “Butterfly” pattern indicating the onset of flow-concentration coupling at this α value. After this we turn our attention to stress/shear rate relaxation experiments. The remainder of this chapter is split in four parts. We show in Part-I that the routes to rheochaos in stress relaxation experiments is via Type-II intermittency. Interestingly in shear rate relaxation, the route is via Type-III intermittency. We also show that flow-concentration coupling is essential to see the route to rheochaos. This section also brings out the crucial role played by orientational ordering of the nematics during rheochaos using SALS measurements performed simultaneously with rheological measurements. In part-II, we study the spatio-temporal dynamics of the shear induced band en route to rheochaos. Our direct visualisation experiments show that the complex dynamics observed in stress/shear rate relaxation measurements during the route to rheochaos is a manifestation of the spatio-temporal dynamics of the high shear band. In part-III, we describe the results of our stress/shear rate relaxation measurements at a fixed shear rate/stress with temperature as the control parameter and thereby control the micellar length. We see the Type-II intermittency route to rheochaos in stress relaxation measurements and the Type-III intermittency route to rheochaos in shear rate relaxation measurements. We conclude this section by showing the results of linear rheology measurements carried out at different temperatures. We estimate the mean micellar length ¯L, reptation time τrepand the breaking time τbreak. We show that L¯ increases by ≈ 58%, as the sample goes through the route to rheochaos. In Part-I of this chapter we had only qualitatively discussed the correlations between the measured time series of stress and the VH scattered intensity during the Type-II intermittency route to rheochaos. In part-IV we have attempted to quantify the correlations between the two time series using the technique of linear and nonlinear Granger causality. We have also studied the phase space dynamics of the two time series using the technique of Cross Recurrence Plots. We show that there exists a causal feedback mechanism between the stress and the VH intensity with the latter having a stronger causal effect. We have also shown that the bivariate time series share similar phase space dynamics using the method of Cross Recurrence Plots. In chapter 4, we have studied the dynamics of wormlike micellar gels of surfactant CTAT using the DLS technique. We report an interesting result in the dynamics of these systems: concentration fluctuations in semidilute wormlike-micelle solutions of the cationic surfactant Cetyltrimethylammonium Tosylate (CTAT) at wavenumber q have a mean decay rate α qz, with z -̃1.8, for a wide range of surfactant concentrations just above the overlap value c∗. The process we are seeing is thus superdiffusive, like a L´evy flight, relaxing on a length scale L in a time of order less than L2 . The rheological behaviour of this system is highly non-Maxwellian and indicates that the micelle-recombination kinetics is diffusion-controlled (DC) (micelles recombine with their original partners). With added salt (100mM NaCl) the rheometric behaviour turns Maxwellian, indicating a crossover to a mean-field (MF) regime (micelles can recombine with any other micellar end). The concentration fluctuations, correspondingly, show normal diffusive behaviour. The stress relaxation time, moreover is about twenty times slower without salt than with 100mM NaCl. Towards the end of this chapter, we propose an explanation of these observations based on the idea that stress due to long-lived orientational order enhances concentration fluctuations in DC regime. In the previous chapter we had studied the dynamics of wormlike micellar gels of pure CTAT 2wt% and found superdiffusive relaxation of concentration fluctuations due to a nonlinear coupling of long-lived stress and orientational fluctuations to the con- centration. In chapter 5 we present results from dynamic light scattering experiments to quantify the diffusive motion of polystyrene (PS) colloids in the same system. This chapter is split in two parts. In Part-I, we discuss dynamics of PS particles of radius 115 nm and 60 nm in CTAT 2wt%. The radius of the colloidal spheres is comparable to the mesh size ξ = 80 nm of the wormlike micellar network and hence we are probing the network dynamics. We find that ∆r2(t) is wavevector independent at small and large lag times. However at intermediate times, we find an anomalous wavevector dependence which we believe arises from the rapid restructuring of the gel network. This anomalous wavevector dependence of ∆r2(t) disappears as the temperature is increased. In Part-II we discuss the dynamics of PS particles of radius 25 nm and 10 nm, smaller than ξ, in CTAT 1wt% & 2wt%. We once again find an anomalous wavevector dependence of ∆r2(t) at intermediate times for the 2wt% sample. Surprisingly, at large times the particle motion is not diffusive, rather ∆r2(t) saturates. We do not have a clear understanding of this as yet. Also for the 10 nm particle, the motion at small lag times is superdiffusive. The motion of these particles is probably influenced by the superdiffusion of concentration fluctuations observed in pure CTAT 2wt% system (chapter 4). In chapter 6, we report the observation of an intermediate mesh phase with rhom- bohedral symmetry, corresponding to the space group R¯3m, in the ternary system consisting of CTAHN/NaBr/water. It occurs at lower temperatures between a random mesh phase (LDα ) and a lamellar phase (Lα) on increasing the surfactant concentration φs. The micellar aggregates, both in the intermediate and random mesh phases, are found to be made up of a two-dimensional network of rod-like segments, with three rods meeting at each node. SAXS studies also show the presence of small angle peaks corresponding to ad−spacing of 25 nm. Freeze fracture electron microscopy results shows that this peak may correspond to the presence of nodule like structures with no long-range correlations. The thesis concludes with a summary of main results and a brief discussion of the scope for future work in Chapter 7.

We present a novel approach to the investigation of transport coefficients in condensed matter systems, which is based on a pertinent time-convolutionless (TCL) projection operator technique. In this context we analyze in advance the convergence of the corresponding perturbation expansion and the influence of the occurring inhomogeneity. The TCL method is used to establish a formalism for a consistent derivation of a Boltzmann equation from the underlying quantum dynamics, which is meant to apply to non-ideal quantum gases. We obtain a linear(ized) collision term that results as a finite non-singular rate matrix and is thus adequate for further considerations, e.g., the calculation of transport coefficients. In the work at hand we apply the provided scheme to numerically compute the diffusion coefficient of an atomic wire and especially analyze its dependence on certain model properties, in particular on the width of the wire.