270 research outputs found
Field theoretic studies of quantum spin systems in one dimension
No full textWe describe some field theoretic methods for studying quantum spin systems in one dimension. These include the nonlinear
s-model approach which is particularly useful for large values of the spin, the idea of Luttinger liquids and bosonization which are more useful for
small values of spin such as spin-1/2, and the technique of low-energy effective Hamiltonians which can be useful if the system under consideration
is perturbatively close to an exactly solvable model. We apply these techniques to similar spin models, such as spin chains with dimerization and
frustration, and spin ladders in the presence of a magnetic field. This comparative study illustrates the relative strengths of the different methods.Field theoretic studies of quantum spin systems in one dimension
Diptiman Sen
Centre for High Energy Physics, Indian Institute of Science, Bangalore-560 012, India
E-mail : [email protected] for High Energy Physics, Indian Institute of Science, Bangalore-560 012, Indi
Transport In Quasi-One-Dimensional Quantum Systems
No full textThis thesis reports our work on transport related problems in mesoscopic physics using analytical as well as numerical techniques. Some of the problems we studied are: effect of interactions and static impurities on the conductance of a ballistic quantum wire[1], aspects of quantum charge pumping [2, 3, 4], DC and AC conductivity of a (dissipative) quantum Hall (edge) line junctions[5, 6], and junctions of three or more Luttinger liquid (LL)quantum wires[7].
This thesis begins with an introductory chapter which gives a brief glimpse of the underlying physical systems and the ideas and techniques used in our studies. In most of the problems we will look at the physical effects caused by e-e interactions and static scattering processes.
In the second chapter we study the effects of a static impurity and interactions on the conductance of a 1D-quantum wire numerically. We use the non-equilibrium Green’s function (NEGF) formalism along with a self-consistent Hartree-Fock approximation to numerically study the effects of a single impurity and interactions between the electrons (with and without spin) on the conductance of a quantum wire [1]. We study the variation of the conductance with the wire length, temperature and the strength of the impurity and electron-electron interactions. We find our numerical results to be in agreement with the results obtained from the weak interaction RG analysis. We also discover that bound states produce large density deviations at short distances and have an appreciable effect on the conductance which is not captured by the renormalization group analysis.
In the third chapter we use the equations of motion (EOM) for the density matrix and Floquet scattering theory to study different aspects of charge pumping of non-interacting electrons in a one-dimensional system. We study the effects of the pumping frequency, amplitude, band filling and finite bias on the charge pumped per cycle, and the spectra of the charge and energy currents in the leads[2]. The EOM method works for all values of parameters, and gives the complete time-dependences of the current and charge at any site of the system. In particular we study a system with oscillating impurities at several sites and our results agree with Floquet and adiabatic theory where these are applicable, and provides support for a mechanism proposed elsewhere for charge pumping by a traveling potential wave in such systems. For non-adiabatic and strong pumping, the charge and energy currents are found to have a marked asymmetry between the two leads, and pumping can work even against a substantial bias. We also study one-parameter charge pumping in a system where an oscillating potential is applied at one site while a static potential is applied in a different region [3]. Using Floquet scattering theory, we calculate the current up to second order in the oscillation amplitude and exactly in the oscillation frequency. For low frequency, the charge pumped per cycle is proportional to the frequency and therefore vanishes in the adiabatic limit. If the static potential has a bound state, we find that such a state has a significant effect on the pumped charge if the oscillating potential can excite the bound state into the continuum states or vice versa.
In the fourth chapter we study the current produced in a Tomonaga-Luttinger liquid (TLL) by an applied bias and by weak, point-like impurity potentials which are oscillating in time[4]. We use bosonization to perturbatively calculate the current up to second order in the impurity potentials. In the regime of small bias and low pumping frequency, both the DC and AC components of the current have power law dependences on the bias and pumping frequencies with an exponent 2K−1 for spinless electrons, where Kis the interaction parameter. For K<1/2, the current grows large for special values of the bias. For non-interacting electrons with K= 1, our results agree with those obtained using Floquet scattering theory for Dirac fermions. We also discuss the cases of extended impurities and of spin-1/2 electrons.
In chapter five, we present a microscopic model for a line junction formed by counter or co-propagating single mode quantum Halledges corresponding to different filling factors and calculate the DC [5] and AC[6] conductivity of the system in the diffusive transport regime. The ends of the line junction can be described by two possible current splitting matrices which are dictated by the conditions of both lack of dissipation and the existence of chiral commutation relations between the outgoing bosonic fields. Tunneling between the two edges of the line junction then leads to a microscopic understanding of a phenomenological description of line junctions introduced by Wen. The effect of density-density interactions between the two edges is considered exactly, and renormalization group (RG) ideas are used to study how the tunneling parameter changes with the length scale. The RG analysis leads to a power law variation of the conductance of the line junction with the temperature (or other energy scales) and the line junction may exhibit metallic or insulating phase depending on the strength of the interactions. Our results can be tested in bent quantum Hall systems fabricated recently.
In chapter six, we study a junction of several Luttinger Liquid (LL) wires. We use bosonization with delayed evaluation of boundary conditions for our study. We first study the fixed points of the system and discuss RG flow of various fixed points under switching of different ‘tunneling’ operators at the junction. Then We study the DC conductivity, AC conductivity and noise due to tunneling operators at the junction (perturbative).We also study the tunneling density of states of a junction of three Tomonaga-Luttinger liquid quantum wires[7]. and find an anomalous enhancement in the TDOS for certain fixed points even with repulsive e-e interactions
Living on the Edge A Study of Boundary Modes In Two-dimensional Topological Systems
No full textIn the last few decades an enormous amount of research has been carried out on some novel
phases of matter called topological phases which are beyond the paradigm of Landau’s theory
of symmetry breaking. One of the earliest breakthroughs in this field was the discovery of
the quantum Hall effect. A topological system has some properties which are immune to slight
perturbations which obey the symmetries of the unperturbed system. Topological systems can be
characterised by means of a topological invariant, such as the Chern number in two-dimensional
systems. Topological phases can be found in a variety of systems and have been studied both
theoretically and experimentally over the last several years.
Topological insulators (TIs) are materials which have gapped states in the bulk and gapless states
on the boundaries which are protected by some symmetries. Materials such as bismuth selenide
and bismuth telluride exhibit such properties and are examples of topological insulators in three
dimensions. The surfaces of these materials host conducting states which are robust against
impurities. An interesting property of these surface states is “spin-momentum locking”. This
is responsible for preventing backscattering of these surface modes from scalar (non-magnetic)
impurities.
In two dimensions, topologically protected one-dimensional edge states are found to exist in
graphene nanoribbons with a spin-orbit coupling (SOC). This was one of the earliest theoretically
proposed examples of the quantum spin Hall effect. Though the intrinsic SOC in graphene
is weak, placing it in proximity to a TI is known to induce a stronger SOC giving rise to some
very interesting phenomena, some of which are discussed in this thesis. Topological phases
can also be seen in some models involving interacting spins such as the kagome lattice spin
model which is presented in this thesis. In this case, it is the magnons or spin waves which are
topological in nature
To summarise, this thesis deals with topological phases and edge modes in three different systems
1. Surface states of three-dimensional topological insulators,
2. Graphene in the presence of Kane-Mele and Rashba spin-orbit couplings,
3. Spin waves (magnons) on a kagome lattice.
In all these cases localised states are found to reside on the boundaries of the system or along
potential barriers
Quantum Spin Chains And Luttinger Liquids With Junctions : Analytical And Numerical Studies
No full textWe present in this thesis a series of studies on the physical properties of some one dimensional systems. In particular we study the low energy properties of various spin chains and a junction of Luttinger wires. For spin chains we specifically look at the role of perturbations like frustrating interactions and dimerisation in a nearest neighbour chain and the formation of magnetisation plateaus in two kinds of models; one purely theoretical and the other motivated by experiments. In our second subject of interest we study using a renormalisation group analysis the effect of spin dependent scattering at a junction of Luttinger wires. We look at the physical effects caused by the interplay of electronic interactions in the wires and the scattering processes at the junction. The thesis begins with an introductory chapter which gives a brief glimpse of the ideas and techniques used in the specific problems that we have worked on. Our work on these problems is then described in detail in chapters 25. We now present a brief summary of each of those chapters.
In the second chapter we look at the ground state phase diagram of the mixed-spin sawtooth chain, i.e a system where the spins along the baseline are allowed to be different from the spins on the vertices. The spins S1 along the baseline interact with a coupling strength J1(> 0). The coupling of the spins on the vertex (S2) to the baseline spins has a strength J2. We study the phase diagram as a function of J2/J1 [1]. The model exhibits a rich variety of phases which we study using spinwave theory, exact diagonalisation and a semi-numerical perturbation theory leading to an effective Hamiltonian. The spinwave theory predicts a transition from a spiral state to a ferrimagnetic state at J2S2/2J1S1 = 1 as J2/J1 is increased. The spectrum has two branches one of which is gapless and dispersionless (at the linear order) in the spiral phase. This arises because of the infinite degeneracy of classical ground states in that phase. Numerically, we study the system using exact diagonalisation of up to 12 unit cells and S1 = 1 and S2 =1/2. We look at the variation of ground state energy, gap to the lowest excitations, and the relevant spin correlation functions in the model. This unearths a richer phase diagram than the spinwave calculation. Apart from revealing a possibility of the presence of more than one kind of spiral phases, numerical results tell us about a very interesting phase for small J2. The spin correlation function (for the spin1/2s) in this region have a property that the nextnearest-neighbour correlations are much larger than the nearest neighbour correlations. We call this phase the NNNAFM (nextnearest neighbour antiferromagnet) phase and provide an understanding of this phase by deriving an effective Hamiltonian between the spin1/2s. We also show the existence of macroscopic magnetisation jumps in the model when one looks at the system close to saturation fields.
The third chapter is concerned with the formation of magnetisation plateaus in two different spin models. We show how in one model the plateaus arise because of the competition between two coupling constants, and in the other because of purely geometrical effects. In the first problem we propose [2] a class of spin Hamiltonians which include as special cases several known systems. The class of models is defined on a bipartite lattice in arbitrary dimensions and for any spin. The simplest manifestation of such models in one dimension corresponds to a ladder system with diagonal couplings (which are of the same strength as the leg couplings). The physical properties of the model are determined by the combined effects of the competition between the ”rung” coupling (J’ )and the ”leg/diagonal” coupling (J ) and the magnetic field. We show that our model can be solved exactly in a substantial region of the parameter space (J’ > 2J ) and we demonstrate the existence of magnetisation plateaus in the solvable regime. Also, by making reasonable assumptions about the spectrum in the region where we cannot solve the model exactly, we prove the existence of first order phase transitions on a plateau where the sublattice magnetisations change abruptly. We numerically investigate the ladder system mentioned above (for spin1) to confirm all our analytical predictions and present a phase diagram in the J’/J - B plane, quite a few of whose features we expect to be generically valid for all higher spins.
In the second problem concerning plateaus (also discussed in chapter 3) we study the properties of a compound synthesised experimentally [3]. The essential feature of the structure of this compound which gives rise to its physical properties is the presence of two kinds of spin1/2 objects alternating with each other on a helix. One kind has an axis of anisotropy at an inclination to the helical axis (which essentially makes it an Ising spin) whereas the other is an isotropic spin1/2 object. These two spin1/2 objects interact with each other but not with their own kind. Experimentally, it was observed that in a magnetic field this material exhibits magnetisation plateaus one of which is at 1/3rd of the saturation magnetisation value. These plateaus appear when the field is along the direction of the helical axis but disappear when the field is perpendicular to that axis.
The model being used for the material prior to our work could not explain the existence of these plateaus. In our work we propose a simple modification in the model Hamiltonian which is able to qualitatively explain the presence of the plateaus. We show that the existence of the plateaus can be explained using a periodic variation of the angles of inclination of the easy axes of the anisotropic spins. The experimental temperature and the fields are much lower than the magnetic coupling strength. Because of this quite a lot of the properties of the system can be studied analytically using transfer matrix methods for an effective theory involving only the anisotropic spins. Apart from the plateaus we study using this modified model other physical quantities like the specific heat, susceptibility and the entropy. We demonstrate the existence of finite entropy per spin at low temperatures for some values of the magnetic field.
In chapter 4 we investigate the longstanding problem of locating the gapless points of a dimerised spin chain as the strength of dimerisation is varied. It is known that generalising Haldane’s field theoretic analysis to dimerised spin chains correctly predicts the number of the gapless points but not the exact locations (which have determined numerically for a few low values of spins). We investigate the problem of locating those points using a dimerised spin chain Hamiltonian with a ”twisted” boundary condition [4]. For a periodic chain, this ”twist” consists simply of a local rotation about the zaxis which renders the xx and yy terms on one bond negative. Such a boundary condition has been used earlier for numerical work whereby one can find the gapless points by studying the crossing points of ground states of finite chains (with the above twist) in different parity sectors (parity sectors are defined by the reflection symmetry about the twisted bond). We study the twisted Hamiltonian using two analytical methods. The modified boundary condition reduces the degeneracy of classical ground states of the chain and we get only two N´eel states as classical ground states. We use this property to identify the gapless points as points where the tunneling amplitude between these two ground states goes to zero. While one of our calculations just reproduces the results of previous field theoretic treatments, our second analytical treatment gives a direct expression for the gapless points as roots of a polynomial equation in the dimerisation parameter. This approach is found to be more accurate. We compare the two methods with the numerical method mentioned above and present results for various spin values.
In the final chapter we present a study of the physics of a junction of Luttinger wires (quantum wires) with both scalar and spin scattering at the junction ([5],[6]). Earlier studies have investigated special cases of this system. The systems studied were two wire junctions with either a fully transmitting scattering matrix or one corresponding to disconnected wires. We extend the study to a junction of N wires with an arbitrary scattering matrix and a spin impurity at the junction. We study the RG flows of the Kondo coupling of the impurity spin to the electrons treating the electronic interactions and the Kondo coupling perturbatively. We analyse the various fixed points for the specific case of three wires. We find a general tendency to flow towards strong coupling when all the matrix elements of the Kondo coupling are positive at small length scales. We analyse one of the strong coupling fixed points, namely that of the maximally transmitting scattering matrix, using a 1/J perturbation theory and we find at large length scales a fixed point of disconnected wires with a vanishing Kondo coupling. In this way we obtain a picture of the RG at both short and long length scales. Also, we analyse all the fixed points using lattice models to gain an understanding of the RG flows in terms of specific couplings on the lattice. Finally, we use to bosonisation to study one particular case of scattering (the disconnected wires) in the presence of strong interactions and find that sufficiently strong interactions can stabilise a multichannel fixed point which is unstable in the weak interaction limit
Topological Phases, Majorana Modes, Dynamics and Transport in One-Dimensional Systems
No full textThis thesis presents work done on topological phases, Majorana modes, dynamics and
transport in various system like the Kitaev models in one and two dimensions and systems
with junctions of a p-wave superconductor and normal metal leads in one dimension. The
systems we have studied are a one-dimensional spin-1/2 model placed in a transverse
magnetic field [1,2], a lattice model of spinless electrons with p-wave superconductivity
(Kitaev chain) [3,4], the Kitaev model of spin- 1/2’s placed on the sites of a honeycomb
lattice [5], and continuum and lattice models of one-dimensional systems with junctions
of a p-wave superconductor and normal metal leads [6]
Studies of Topological Phases of Matter : Presence of Boundary Modes and their Role in Electrical Transport
No full textTopological 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
Electronic Transport in Low-Dimensional Systems Quantum Dots, Quantum Wires And Topological Insulators
No full textThis thesis presents the work done on electronic transport in various interacting and non-interacting systems in one and two dimensions. The systems under study are: an interacting quantum dot [1], a non-interacting quantum wire and a ring in which time-dependent potentials are applied [2], an interacting quantum wire and networks of multiple quantum wires with resistive regions [3, 4], one-dimensional edge stages of a two-dimensional topological insulator [5], and a hybrid system of two-dimensional surface states of a three-dimensional topological insulator and a superconductor [6].
In the first chapter, we introduce a number of concepts which are used in the rest of the thesis, such as scattering theory, Landauer conductance formula, quantum wires, bosonization, topological insulators and superconductor.
In the second chapter, we study transport through a quantum dot with interacting electrons which is connected to two reservoirs. The quantum dot is modeled by two sites within a tight-binding model with spinless electrons. Using the Lippman-Schwinger method, we write down an exact two-particle wave function for the dot-reservoir system with the interaction localized in the region of the dot. We discuss the phenomena of two-particle resonance and rectification.
In the third chapter, we study pumping in two kinds of one-dimensional systems:
(i) an infinite line connected to reservoirs at the two ends, and (ii) an isolated ring. The infinite line is modeled by the Dirac equation with two time-independent point-like backscatterers that create a resonant barrier. We demonstrate that even if the reservoirs are at the same chemical potential, a net current can be driven through the channel by the application of one or more time-dependent point-like potentials. When the left-right symmetry is broken, a net current can be pumped from one reservoir to the other by applying a time-varying potential at only one site. For a finite ring, we model the system by a tight-binding model. The ring is isolated in the sense that it is not connected to any reservoir or environment. The system is driven by one or more time-varying on-site potentials. We develop an exact method to calculate the current averaged over an infinite amount of time by converting it to the calculation of the current carried by certain states averaged over just one time period. Using this method, we demonstrate that an oscillating potential at only one site cannot pump charge, and oscillating potentials at two or more sites are necessary to pump charge. Further we study the dependence of the pumped current on the phases and the amplitudes of the oscillating potentials at two sites.
In the fourth chapter, we study the effect of resistances present in an extended region in a one-dimensional quantum wire described by a Tomonaga-Luttinger liquid model. We combine the concept of a Rayleigh dissipation function with the technique of bosonization to model the dissipative region. In the DC limit, we find that the resistance of the dissipative patch adds in series to the contact resistance. Using a current splitting matrix M to describe junctions, we study in detail the conductances of: a three-wire junction with resistances and a parallel combination of resistances. The conductance and power dissipated in these networks depend in general on the resistances and the current splitting matrices that make up the network. We also show that the idea of a Rayleigh dissipation function can be extended to couple two wires; this gives rise to a finite transconductance analogous to the Coulomb drag.
In the fifth chapter, we study the effect of a Zeeman field coupled to the edge states of a two-dimensional topological insulator. These edge states form two one-dimensional channels with spin-momentum locking which are protected by time-reversal symmetry. We study what happens when time-reversal symmetry is broken by a magnetic field which is Zeeman-coupled to the edge states. We show that a magnetic field over a finite region leads to Fabry-P´erot type resonances and the conductance can be controlled by changing the direction of the magnetic field. We also study the effect of a static impurity in the patch that can backscatter electrons in the presence of a magnetic field.
In the sixth chapter, we use the Blonder-Tinkham-Klapwijk formalism to study trans-port across a line junction lying between two orthogonal topological insulator surfaces and a superconductor (which can have either s-wave or p-wave pairing). The charge and spin conductances across such a junction and their behaviors as a function of the bias voltage applied across the junction and various junction parameters are studied. Our study reveals that in addition to the zero conductance bias peak, there is a non-zero spin conductance for some particular spin states of the triplet Cooper pairs. We also find an unusual satellite peak (in addition to the usual zero bias peak) in the spin conductance for a p-wave symmetry of the superconductor order parameter
Critical Quantum Spin Chains as Conformal Field Theories
No full textThe author presents a series of one-dimensional quantum Hamiltonians which, at their critical points, realise the minimal unitary series of conformal field theories with central charge less than 1. The models consist of ferromagnetically coupled SU(2) spins in a transverse magnetic field. He shows how the infinite spin (free boson) limit can be obtained by using the Holstein-Primakoff transformation. The analysis can be generalised to SU(3) quantum chains which realise a different series of conformal field theories at criticality
Transport and criticality in topological systems and spin models
No full textThis thesis presents work done on transport in topological insulators and graphene-based systems, and quantum criticality in one- and two-dimensional spin
models. In particular we study the following: transport on surfaces of
three-dimensional topological insulators in the presence of time-independent and
time-dependent barriers, Majorana modes in a one-dimensional topological insulator in proximity with a -wave superconductor, the phase diagram of the
Hubbard model on a triangular lattice periodically driven by an in-plane electric
field, quantum criticality of a Ising model with three-spin interactions and
a transverse field, the origin of spin-orbit coupling in a graphene-WSe heterostructure, and a prediction of edge states in trilayer graphene.
In the first chapter, we give a brief introduction to the concepts relevant to the
rest of the thesis such as topological insulators, superconductivity, Floquet theory for studying periodically driven Hamiltonians, graphene and the spin-orbit coupling terms, quantum phase transitions, and the transverse field Ising model.
In the second chapter, we consider a thin-film topological insulator (TI) in which the top and the bottom surfaces are separated by a small distance. The hybridisation
between the states on the top and bottom surfaces of this system is characterized by a coupling strength . We study the various features of transport when a potential or
magnetic barrier is applied on one of the surfaces. We find that the conductance of this system oscillates with the barrier strength with the period of oscillations
varying with the coupling strength . This gives us an indirect way of estimating the extent of hybridisation in such thin films by looking at the
conductance. The period of these oscillations changes from to as increases from zero to a value close to the energy of the incident electrons. Next we
study the effects of a magnetic barrier, and we find that the conductance reaches a
non-zero and -dependent value as the barrier strength
is increased. This is in sharp contrast to the behavior of the
conductance of a single TI surface where it approaches zero with increasing magnetic
barrier strength. We also find oscillations in the case of a magnetic barrier for
large barrier widths. The period of these oscillations depends on .
In the third chapter, we consider a similar magnetic barrier whose strength is periodically driven in time. We explore the behaviour of the conductance as a function of the driving parameters. Such a barrier can be realised by shining linearly
polarised light over a region of width on the surface of a TI.
We find that the conductance of
this system exhibits a number of interesting features like prominent peaks and dips as the parameters of the system are varied. This also paves the way to have an optical
(electromagnetic) control over the electrical current in such junctions where we can go from a high-conductance regime to a low-conductance regime or vice versa by tuning the
amplitude and frequency of the light. We also see that this system can act as a frequency detector or an optically controlled switch as a function of the incident
energy of the electron.
In the fourth chapter, we consider a model of a TI which is now constricted to a narrow and long strip running along the direction. We study what happens to the Majorana
modes when such a system is placed in proximity to an -wave superconductor. This model hosts a spin-dependent chirality and only has a right-moving spin-up and a left-moving
spin-down branch. We find that this leads to a number of unusual features, such as only one zero energy Majorana mode at each end of a finite system, a single Andreev bound
state at a Josephson junction instead of two states, and multiple Shapiro steps for particular frequencies of an AC driving.
In the fifth chapter, we study a Hubbard model on a triangular lattice at half-filling in the limit of large interaction. At half-filling, this is known to
describe a Heisenberg spin Hamiltonian with equal nearest-neighbour couplings. We study the effects of driving this system periodically with an in-plane electric field. Taking
the driving to be the perturbation, we find, using Floquet perturbation theory, that the effective Hamiltonian up to third order has two-spin Heisenberg couplings with different
magnitudes in the three different directions of the triangular lattice. We also get a three-spin interaction chiral term in the third order with its sign being opposite on
up- and down-pointing triangles.
We study the ground state phase diagram as a function of the three couplings
using exact diagonalization. We find that driving leads to new phases in the system apart from the spiral phase. We have three collinear ordered phases, one
coplanar ordered phase, and three disordered (spin-liquid) phases. These phases are distinguished by looking at the peaks of the static spin structure function in
the Brillouin zone, the ground state fidelity susceptibility, the minimum value of the correlation function in real space, and the crossings of the
energies of the ground state and first excited state.
In the sixth chapter, we consider a one-dimensional Ising model with a three-spin interaction with a transverse field of magnitude . We find that this model has
duality and a second-order phase transition at the self-dual point . We find from finite-size scaling that the correlation length exponent is
close to in this model. Having a dynamical critical exponent and a central charge , we find that the model displays weak universality and lies somewhere in
the middle of the Ashkin-Teller line of models, with the two extreme
limits of the line being the transverse field Ising and four-state Potts models.
Unlike the transverse Ising model,
our model is non-integrable, with the level spacing statistics being governed
by the Wigner-Dyson Gaussian orthogonal ensemble. We also find
that this model has a subset of zero energy states which are
rather special as they are independent of the value of
and have very low entanglement entropy compared to the states in the neighbourhood of the energy eigenvalues. These states are quantum many-body scars and they
violate the eigenstate thermalisation hypothesis (ETH).
Chapters and describe works done in collaboration with some experimental groups. In Chapter , we study the system of graphene-WSe heterostructure
where we have a strong proximity-induced spin-orbit coupling. The quantum Shubnikov-de Haas (SdH) oscillations observed experimentally show a beating implying
the presence of two closely spaced frequencies. The energy dispersion thus extracted is then studied theoretically using an effective Hamiltonian with all possible spin-orbit
couplings present. The Fermi velocity of the sample is about times that of pristine graphene. The data fitting and perturbation calculations show that the
spin-splitting
energy of nearly meV comes dominantly from the valley-Zeeman and Rashba
spin-orbit couplings in the system. In chapter , we study a system of trilayer graphene under the influence of a perpendicular electric field. The non-local
and local resistance measurements done in this system show a scaling relation given by with for a range of values of the
displacement field. The
value of is seen to be close to 1 up to temperatures around which the bulk gap closes in the system. This strongly suggests that the transport is dominated in this
sample by edge modes. We study a theoretical model for trilayer graphene with displacement fields consistent with the experiments, and show that in this regime the
valley Chern number is non-zero with a large value of for a given valley and a given spin. We also show that the system host zig-zag edge
modes for the displacement fields of interest, although they are not protected from backscattering. A simple resistor circuit model that mimics the inter-valley
scattering through dissipation then explains the linear relation between the non-local and local resistances.
At the end, we summarise our results and discuss possible future
studies in these areas of research
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