etd@IISc Collection:http://hdl.handle.net/2005/372016-09-21T15:32:28Z2016-09-21T15:32:28ZTransport In Quasi-One-Dimensional Quantum SystemsAgarwal, Amit Kumarhttp://hdl.handle.net/2005/11072011-04-01T20:30:30Z2011-03-31T18:30:00ZTitle: Transport In Quasi-One-Dimensional Quantum Systems
Authors: Agarwal, Amit Kumar
Abstract: This 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.2011-03-31T18:30:00ZTopics In Noncommutative Gauge Theories And Deformed Relativistic TheoriesChandra, Nitinhttp://hdl.handle.net/2005/24682015-08-12T10:11:55Z2015-08-11T18:30:00ZTitle: Topics In Noncommutative Gauge Theories And Deformed Relativistic Theories
Authors: Chandra, Nitin
Abstract: There is a growing consensus among physicists that the classical notion of spacetime has to be drastically revised in order to nd a consistent formulation of quantum mechanics and gravity. One such nontrivial attempt comprises of replacing functions of continuous spacetime coordinates with functions over noncommutative algebra. Dynamics on such noncommutative spacetimes (noncommutative theories) are of great interest for a variety of reasons among the physicists. Additionally arguments combining quantum uncertain-ties with classical gravity provide an alternative motivation for their study, and it is hoped that these theories can provide a self-consistent deformation of ordinary quantum field theories at small distances, yielding non-locality, or create a framework for finite truncation of quantum field theories while preserving symmetries.
In this thesis we study the gauge theories on noncommutative Moyal space. We nd new static solitons and instantons in terms of the so-called generalized Bose operators (GBO). GBOs are constructed to describe reducible representation of the oscillator algebra. They create/annihilate k-quanta, k being a positive integer. We start with giving an alternative description to the already found static magnetic flux tube solutions of the noncommutative gauge theories in terms of GBOs. The Nielsen-Olesen vortex solutions found in terms of these operators also reduce to the ones known in the literature. On the other hand, we nd a class of new instanton solutions which are unitarily inequivalent to the ones found from ADHM construction on noncommutative space. The charge of the instanton has a description in terms of the index representing the reducibility of the Fock space representation, i.e., k. After studying the static soliton solutions in noncommutative Minkowski space and the instanton solutions in noncommutative Euclidean space we go on to study the implications of the time-space noncommutativity in Minkowski space. To understand it properly we study the time-dependent transitions of a forced harmonic oscillator in noncommutative 1+1 dimensional spacetime. We also provide an interpretation of our results in the context of non-linear quantum optics. We then shift to the so-called DSR theories which are related to a different kind of noncommutative ( -Minkowski) space. DSR (Doubly/Deformed Special Relativity) aims to search for an alternate relativistic theory which keeps a length/energy scale (the Planck scale) and a velocity scale (the speed of light scale) invariant. We study thermodynamics of an ideal gas in such a scenario.
In first chapter we introduce the subjects of the noncommutative quantum theories and the DSR. Chapter 2 starts with describing the GBOs. They correspond to reducible representations of the harmonic oscillator algebra. We demonstrate their relevance in the construction of topologically non-trivial solutions in noncommutative gauge theories, focusing our attention to flux tubes, vortices, and instantons. Our method provides a simple new relation between the topological charge and the number of times the basic irreducible representation occurs in the reducible representation underlying the GBO. When used in conjunction with the noncommutative ADHM construction, we nd that these new instantons are in general not unitarily equivalent to the ones currently known in literature.
Chapter 3 studies the time dependent transitions of quantum forced harmonic oscillator (QFHO) in noncommutative R1;1 perturbatively to linear order in the noncommutativity . We show that the Poisson distribution gets modified, and that the vacuum state evolves into a \squeezed" state rather than a coherent state. The time evolutions of un-certainties in position and momentum in vacuum are also studied and imply interesting consequences for modelling nonlinear phenomena in quantum optics.
In chapter 4 we study thermodynamics of an ideal gas in Doubly Special Relativity. We obtain a series solution for the partition function and derive thermodynamic quantities. We observe that DSR thermodynamics is non-perturbative in the SR and massless limits. A stiffer equation of state is found. We conclude our results in the last chapter.2015-08-11T18:30:00ZStudy Of CP-Violation And Determination Of Higgs Boson Properties At Future CollidersSingh, Ritesh Khttp://hdl.handle.net/2005/13662011-08-18T07:16:20Z2011-08-17T18:30:00ZTitle: Study Of CP-Violation And Determination Of Higgs Boson Properties At Future Colliders
Authors: Singh, Ritesh K2011-08-17T18:30:00ZQuantum Spin Chains And Luttinger Liquids With Junctions : Analytical And Numerical StudiesRavi Chandra, Vhttp://hdl.handle.net/2005/6522010-04-28T23:35:18Z2010-03-09T11:44:35ZTitle: Quantum Spin Chains And Luttinger Liquids With Junctions : Analytical And Numerical Studies
Authors: Ravi Chandra, V
Abstract: We 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.2010-03-09T11:44:35Z