etd@IISc Collection:
http://etd.iisc.ernet.in/2005/37
2018-06-21T15:32:14ZAspects of Higher Spin Theories Conformal Field Theories and Holography
http://etd.iisc.ernet.in/2005/3643
Title: Aspects of Higher Spin Theories Conformal Field Theories and Holography
Authors: Raju, Avinash
Abstract: This dissertation consist of three parts. The first part of the thesis is devoted to the study of gravity and higher spin gauge theories in 2+1 dimensions. We construct cosmological so-lutions of higher spin gravity in 2+1 dimensional de Sitter space. We show that a consistent thermodynamics can be obtained for their horizons by demanding appropriate holonomy conditions. This is equivalent to demanding the integrability of the Euclidean boundary CFT partition function, and reduces to Gibbons-Hawking thermodynamics in the spin-2 case. By using a prescription of Maldacena, we relate the thermodynamics of these solutions to those of higher spin black holes in AdS3. For the case of negative cosmological constant we show that interpreting the inverse AdS3 radius 1=l as a Grassmann variable results in a formal map from gravity in AdS3 to gravity in flat space. The underlying reason for this is the fact that ISO(2,1) is the Inonu-Wigner contraction of SO(2,2). We show how this works for the Chern-Simons actions, demonstrate how the general (Banados) solution in AdS3 maps to the general flat space solution, and how the Killing vectors, charges and the Virasoro algebra in the Brown-Henneaux case map to the corresponding quantities in the BMS3 case. Our results straightforwardly generalize to the higher spin case: the flat space higher spin theories emerge automatically in this approach from their AdS counterparts. We also demonstrate the power of our approach by doing singularity resolution in the BMS gauge as an application. Finally, we construct a candidate for the most general chiral higher spin theory with AdS3 boundary conditions. In the Chern-Simons language, the left-moving solution has Drinfeld-Sokolov reduced form, but on the right-moving solution all charges and chemical potentials are turned on. Altogether (for the spin-3 case) these are 19 functions. Despite this, we show that the resulting metric has the form of the “most general” AdS3 boundary conditions discussed by Grumiller and Riegler. The asymptotic symmetry algebra is a product of a W3 algebra on the left and an affine sl(3)k current algebra on the right, as desired. The metric and higher spin fields depend on all the 19 functions.
The second part is devoted to the problem of Neumann boundary condition in Einstein’s gravity. The Gibbons-Hawking-York (GHY) boundary term makes the Dirichlet problem for gravity well defined, but no such general term seems to be known for Neumann boundary conditions. In our work, we view Neumann boundary condition not as fixing the normal derivative of the metric (“velocity”) at the boundary, but as fixing the functional derivative of the action with respect to the boundary metric (“momentum”). This leads directly to a new boundary term for gravity: the trace of the extrinsic curvature with a specific dimension-dependent coefficient. In three dimensions this boundary term reduces to a “one-half” GHY term noted in the literature previously, and we observe that our action translates precisely to the Chern-Simons action with no extra boundary terms. In four dimensions the boundary term vanishes, giving a natural Neumann interpretation to the standard Einstein-Hilbert action without boundary terms. We also argue that a natural boundary condition for gravity in asymptotically AdS spaces is to hold the renormalized boundary stress tensor density fixed, instead of the boundary metric. This leads to a well-defined variational problem, as well as new counter-terms and a finite on-shell action. We elaborate this in various (even and odd) dimensions in the language of holographic renormalization. Even though the form of the new renormalized action is distinct from the standard one, once the cut-off is taken to infinity, their values on classical solutions coincide when the trace anomaly vanishes. For AdS4, we compute the ADM form of this renormalized action and show in detail how the correct thermodynamics of Kerr-AdS black holes emerge. We comment on the possibility of a consistent quantization with our boundary conditions when the boundary is dynamical, and make a connection to the results of Compere and Marolf. The difference between our approach and microcanonical-like ensembles in standard AdS/CFT is emphasized.
In the third part of the dissertation, we use the recently developed CFT techniques of Rychkov and Tan to compute anomalous dimensions in the O(N) Gross-Neveu model in d = 2 + dimensions. To do this, we extend the “cow-pie contraction” algorithm of Basu and Krishnan to theories with fermions. Our results match perfectly with Feynman diagram computations.2018-05-29T18:30:00ZConformal Bootstrap : Old and New
http://etd.iisc.ernet.in/2005/3616
Title: Conformal Bootstrap : Old and New
Authors: Kaviraj, Apratim
Abstract: Conformal Field Theories (CFT) are Quantum Field Theories characterized by enhanced (conformal) symmetries. They are interesting to Theoretical Physicists because they occur at critical points in phase transitions of various systems and also in the world sheet formulation of String Theory. CFTs allow Operator Product Expansion (OPE) in their correlators. The idea of Conformal Bootstrap is to solely use the conformal symmetries and crossing symmetry in the OPE to solve a conformal led theory and not explicitly use a lagrangian. Solving a CFT is equivalent to obtaining the anomalous dimensions and OPE coe client’s of the operators. The work presented in this thesis shows how ideas of bootstrap can be used to get analytic results for dimensions and OPE coe client’s of various operators in CFTs.
In the conventional bootstrap program, the OPE in the direct (s-) channel is compared with the OPE of a crossed (t-) channel. This requirement of crossing symmetry is called the bootstrap equation. The flow of logic is somewhat reversed in the \new" idea that is formulated in this thesis. The trick is to expand a CFT correlator in terms of Witten diagrams, in all channels. This is a manifestly crossing symmetric description, and is in contrast to the usual expansion in terms of conformal blocks, which is in only one channel. For convenience we work with the Mellin transforms of Witten diagrams. For consistency of the Witten diagrams expansion with the conformal block expansion in a certain channel, we require the satisfaction of some equations, which we call the bootstrap equations in Mellin space. This scheme was rest chalked out by Polyakov in 1973, where he proposed the use of \unitary amplitudes" to expand a correlator. The unitary amplitudes had similar symmetry and analytic properties as the Witten diagrams. Even though he did not take his idea forward, replacing unitary amplitudes with Witten diagrams seems to work very well for obtaining analytic results.
The working of bootstrap equations in Mellin space is demonstrated for the 4 Wilson-Fisher fixed point in d = 4 , O(N) theory at Wilson-Fisher point (in d = 4 ), as well as with large N (in general d), and large spin operators in strongly coupled and weakly coupled theories. For the case of global symmetry we have also analysed the somewhat unexplored case of cubic anisotropy. The results are obtained as perturbative series in , 1=N, or 1=` as applicable, and they are consistent with known results in literature. We also obtain various new results, for instance the OPE coe client’s of general higher spin operators. These results are otherwise very di cult to end from Feynman diagrams, but in this approach they come out very simply, essentially by solving some algebraic equations. We also show the use of the conventional bootstrap strategy, for analytically obtaining anomalous dimensions of large spin operators having higher twists, in a O(N) theory, by working in the light cone limit.
One can question the validity of the proposal of using Witten diagrams to expand a correlator. One such issue is convergence of the sum over Witten diagrams. Convergence can be shown to hold for the operator spectrum we have worked with. Also there are operators that might upset convergence under some conditions. Resolutions of such cases, and ways to improve convergence have also been discussed.
The conventional bootstrap method has been very successful in giving numerical results in nonpertur-bative CFTs, like the 3 dimensional Ising model. Numerical analysis can also be made possible with the new bootstrap in Mellin space approach. Having a convergent basis of expansion improves the prospect of numeric. The goal is to formulate a bootstrap scheme that, under a single framework, can make most of all the CFT properties. It should be systematic, so that one can obtain anomalous dimensions and OPE coe client’s of all operators up to any desired order, and works for all strongly/weakly coupled and perturbative/nonpertur-bative CFTS, both analytically and numerically. Finally, the use of Witten diagrams also indicates the possibility of Ising CFT or weakly coupled CFTs having connections with AdS/CFT, and hence String Theory. It does seem we have a right direction towards achieving our goal.2018-05-24T18:30:00ZProbing the Beyond Standard Model Physics in Top Quark and Dark Matter Sectors
http://etd.iisc.ernet.in/2005/3578
Title: Probing the Beyond Standard Model Physics in Top Quark and Dark Matter Sectors
Authors: Mendiratta, Gaurav
Abstract: The Standard Model (SM) of particle physics provides the theoretical framework to describe the fundamental interactions among elementary constituents of matter. SM is supported by experiments to a high degree of accuracy, up to parts per-mil for the electroweak (EW) sector and parts-per-trillion for QED alone, but it still remains incomplete. Many observed phenomena lack explanation in the framework of the SM and its particles. They indicate the possibility of existence of particles and interactions beyond the SM (BSM). These phenomena include dark matter (DM), dark energy and baryonic asymmetry of the universe. In addition, a quantum description of gravity is still lacking.
The top quark has the largest mass among the SM particles. Due to it’s heavy mass, top quark is the only colored particle which does not hadronize and hence its properties are directly accessible by studying it’s decay particles. The order one Yukawa coupling of the top quark also imbibes it with an important role in the behavior of the SM couplings at higher energy scales where possible BSM physics may contribute. As a result, precision measurements of top quark properties may provide a glimpse into BSM physics and hence making these measurements is one of the core aims of the Large Hadron Collider.
In stark contrast with top quark physics is the elusive, dark matter (DM) of the universe. There exists a lot of observational evidence for it but, as of yet, with no clue with regards to its particle properties and interactions. Compelling evidence for the existence of DM comes from measurements based on cosmic microwave background radiation, astrophysical observations of distribution of visible matter in galaxy clusters, galactic cluster collisions (e.g. bullet cluster), gravitational lensing, galactic rotation curves, structure formation simulations, to name a few. It is interesting to investigate the possibility that there may be a connection between top quark and DM.
In this thesis, we extend the SM with simplified models to study BSM physics at colliders and also to explain the DM puzzle. Here, we use the Top quark as a laboratory for constructing generic probes of BSM and also of the dark sector physics. In Chapter 1, we introduce some relevant background and salient aspects of the SM framework on which the following BSM theories are built. In Chapter 2 we explore an s channel and a t-channel simplified model in the context of top quark pair production using asymmetries constructed with kinematic variables of the top decay products. In Chapter 3, we then propose a simplified model which includes a colored scalar as the mediator between DM and SM particles, termed gluphillic scalar dark matter (GSDM). Monojet process is one of the primary channels to probe DM at hadron colliders. In Chapter 3, the discussion of monojet process at the Large Hadron Collider (LHC) is limited to the effective field theory (EFT) framework. In Chapter 4 we discuss collider processes in GSDM model with complete loop calculations for the diagrams involving the mediating colored scalar. We also compare the loop calculation with the EFT results to find the range of applicability of the EFT.
The top quark study in Chapter 2 was initially inspired from an interesting observation made in 2008 which suggested a deviation from the SM in the forward-backward asymmetry (FBA) of a pair produced top quark. The value of FBA measured at the time was 18% ±12%. This value deviated by more than 1σ with respect to the SM leading order (LO) value of 5%. The deviation was observed by both the detectors at Tevatron, D0 and CDF, and it’s significance increased with additional data in 2012. Recent analyses of the data by D0 is now in better agreement with the latest effective-NNNLO calculations. However, the FBA measurements by CDF are still in tension with those by D0 and the value predicted by theoretical calculations. Inspired by this puzzle, which may be on its way to getting solved, we have been able to construct effective probes of BSM physics for the on-going and future searches of BSM in the top quark sector. In our analyses, we studied correlations among observables which can distinguish between different sources of BSM contributions in the top quark pair production. As a template, we use an s-channel and a t-channel mediator, both of which leave very different signatures in the kinematic asymmetry correlations. The simplified models considered by us also included parity breaking interactions which lead to polarized top quarks, providing another probe into the underlying production process. We find that all the kinematic distributions of the decay lepton get influenced by the polarization of the top quark.
We show that these correlations can distinguish well between the template models of axigluon and diquark. In general, all of these observables also provide a probe into the polarization of the top quark and therefore any chiral couplings with the mediator. However, the lepton polar angle asymmetry measured in the lab frame is special in that it can not only probe the longitudinal polarization as other observables but is also sensitive to the transverse polarization of the top quark. We also show the effectiveness of the proposed top quark kinematic observables, to distinguish between s and t-channel BSM physics models, in future searches for BSM particles at the run-II LHC.
In a large verity of dark matter (DM) models the simplest candidate is the model of a singlet scalar particle. The scalar may couple to the standard model in a number of ways via any of the SM particles. Such models with BSM Yukawa interactions or gauge sector extensions are strongly constrained from both the direct detection and collider precision measurements. The remaining models either predict a very heavy dark matter, completely out of reach of collider searches or introduce an unnaturally weak coupling with the SM particles giving no justifications for the small numbers. An interesting corner of the space of possible DM models which has been under-explored so far includes interactions of DM particles with gluons. Although DM particles cannot themselves be charged or colored, a colored scalar mediator can allow this interaction. One such model arises when we consider the scalar DM in presence of a colored scalar particle, for example the one from t-channel model above. Such colored scalars are generically present in a number of BSM theories including SUSY and GUT. How-ever, without the need for any additional gauge symmetries, the two scalars would interact with each other via the marginal operators.
In Chapter 3 we study a SM singlet scalar DM candidate which couples to SM via a colored scalar particle. In the GSDM model, DM and mediator interact via the quartic, marginal operator. DM annihilation cross-section of the order of weak interactions (∼ 0.1pb) is predicted to explain the observed dark matter relic density if arising from thermal production of a WIMP DM candidate of mass ∼ 100 GeV. On investigating the GSDM model, we find that it allows a large annihilation cross-section and is still compatible with direct detection bounds. This is so because the annihilation cross-section to a pair of colored scalars proceeds via a tree-level interaction, whereas the interaction with SM particles proceeds via loop diagrams involving the colored scalars.
Our work shows that this model is compatible with the observed relic density of DM when the mediating particle is lighter than DM for a large range of the couplings. For masses of the DM and the mediator less then ∼ 50 GeV, the DM can also be lighter than the mediator where the annihilation then proceeds via loop interactions. This region of parameter space is strongly constrained from the collider physics bounds on a colored scalar particle. These bounds become much weaker in the case where the colored scalar does not couple to quarks and hence cannot decay. The bounds coming from long-lived colored scalars become relevant in those cases and also constrain the light mass window.
A colored scalar interacting with quarks must do so without violating the strong flavor constraints. We consider the scalar in the framework of a class of models termed minimally flavor violating (MFV) and also assume that it couples only to the right handed up-sector quarks. Such a particle would couple to the top quark and would be observable at the LHC pair production of the top quark. We find constraints on a color triplet particle in such a case and show the coupling and mass regions allowed. Constraints from the decays to light quarks are interpreted from dijet process searches and limit the mass of a color-triplet scalar above 350 GeV. The primary process for direct search of stable particles produced at a collider is a single jet in association with missing transverse energy (MET). We find that in an effective field theory (EFT) framework, very weak bounds are obtained on the mediating scale.
In Chapter 4, we perform complete loop calculations for processes involving colored scalar particles and DM at LHC in order to explore the GSDM model at LHC and FCC (Future Circular Collider). The EFT is valid only for mediator masses much heavier than the momentum transfer or the MET cuts. We show the region of applicability of the EFT by comparing it with respect to the loop induced calculation. We analyze the monojet + missing transverse energy (MET) process to find the expected bounds from LHC 13 TeV run-II. We calculate the reach of the LHC in the high luminosity run in the future and also the reach of the FCC to explore the GSDM model. We perform all our calculations for a number of representations of the colored mediator from a triplet to dimension 15. As expected, collider constraints are only significant when the dark matter is light enough (mDM ∼ 10 GeV) to be copiously produced at the LHC. We find that in the high luminosity run, LHC can probe the scalar triplet particle up-to 50 GeV mass in the monojet process though a dimension 15 particle can be probed up to 150 GeV. With an order of magnitude higher beam energy, FCC can rule out much larger parameter space or provide observational evidence for TeV scale mediating particles. In conclusion, this thesis adds to the growing body of literature which points towards BSM discoveries around the corner at high luminosity LHC in the top physics and in dark sector physics. We have also proposed avenues for precision BSM studies at the next generation colliders.2018-05-21T18:30:00ZStudies of Topological Phases of Matter : Presence of Boundary Modes and their Role in Electrical Transport
http://etd.iisc.ernet.in/2005/3571
Title: Studies of Topological Phases of Matter : Presence of Boundary Modes and their Role in Electrical Transport
Authors: Deb, Oindrila
Abstract: 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.2018-05-20T18:30:00Z