No description available.

We investigate a special case of the quantum integrable Heisenberg spin chain known as Gaudin model. The Gaudin model is an important example of quantum integrable systems. We study the Gaudin model for the Lie algebra s[z(<C). The key problem is to find the spectrum and the corresponding eigenvectors of the commuting Hamiltonians. The standard method to solve this type of classical problem was introduced by H. Bethe and is known as the Bethe-Ansatz. Bethe's technique has proven to be very powerful in various areas of modem many-body theory and statistical mechanics. [19], [14], [4] Following Sklyanin's ideas in [19], we derive the Bethe-Ansatz equations for sl2(<C). Solving the Bethe-Ansatz equations is equivalent to finding polynomial solutions of the Lame differential equation, which has a meaning in electrostatics. We derive this equation for sl2(<C), and investigate its special cases. We discuss classical and more recent results on the Gaudin spin chain for sl2(<C) and provide numerical evidence for new observations in the real case of the Lame equation. Using roots of classical polynomials known as Jacobi polynomials, which are solutions to a special case of the Lame equation, we numerically approximate solutions to the Lame equation in more complicated settings. We discuss the Gaudin model associated to the Lie algebra sl3(C). Using the Bethe-Ansatz equations for sl3(C), we provide solutions in special cases. / Thesis / Master of Science (MSc)

Dimerized antiferromagnetic spin-1/2 chains and ladders demonstrate quantum critical phase transition, the existence or absence of which is dependent on the dimerization and the dimerization pattern of the chain and the ladder, respectively. The gapped phases can not be distinguished by the conventional Landau long-range order parameters. However, they possess non-local topological string order parameters which can be used to classify different phases. We utilize the self-consistent free fermionic approximation and some standard results for exactly solved models to analytically calculate the string order parameters of dimerized spin chains. As a complement parameter the gapped phases possess the topological number, called the winding number and they are characterized by different integer values of the winding number. In order to calculate the string order parameters and winding numbers in dimerized spin chains and two-leg ladders we use analytical methods such as the Jordan-Wigner transformation, mean-field approximation, duality transformations, and some standard results available for the exactly 1D solve models. It is shown that the winding number provides the complementary framework to the string order parameter to characterize the topological gapped phases.

Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.

Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.

Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

Quantum spin liquids (QSLs) may roughly be defined as states possessing sufficiently high quantum fluctuations that they impede long range magnetic order. Various electron interactions are currently being studied in order to physically realize such states in condensed matter systems since they can host fractionalized excitations. The purpose of our study is to examine two interactions established as important in the literature while not having been paired together. We consider a bond-dependant $J$-$\Gamma$ ladder, comprised of an alternating symmetric exchange of spin components, mediated by $\Gamma$, along with a Heisenberg interaction controlled by $J$. By parameterizing these couplings by an angle $\phi$, we produce a phase diagram of the system using the Infinite Density Matrix Renormalization Group (iDMRG) numerical technique. In order to classify the phases, we search for discontinuities in the entanglement spectrum for bonds along one of the legs and the rungs of the ladder while also looking at divergences in the susceptibility of the energy. These criteria reveal a possible 10 phases hosted by the system, with 7 of them showing some form of magnetic ordering seen directly from the spin correlations and by applying magnetic fields in appropriate directions. Moreover, known points in the phase diagram can be adiabatically connected to other points within the same phase by tuning $J$ or $\Gamma$. The remaining three phases however show no obvious long-range magnetic order while also having large contributions to the entanglement spectrum. Such phases, showing interesting initial signs, are discussed further in our study. / Thesis / Master of Science (MSc)

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.

Superconformal field theories (SCFTs) are an important class of quantum field theories. These SCFTs have been a significant component in exploring and comprehending the fundamental framework of quantum field theories. In the vast realm of quantum field theories, integrability plays a crucial role, providing powerful analytic tools that allow us to solve certain physical quantities exactly. In this thesis, we focus on the representation theory of the algebraic structure in six-dimensional (6D) SCFTs and investigate the intricate interplay between 6D SCFTs and integrability. To begin, we delve into the fundamental concepts of representation theory, establishing a solid foundation for our subsequent analysis. The discussion then will move on to all possible generators in the SCFTs, explaining how they are realized in terms of bosonic and fermionic oscillators. Finally, we investigate spin chains and their application in 6D SCFTs. We demonstrate that symmetry arguments derived from representation theory are not sufficient to establish the integrability of the spin chains in 6D SCFTs. This conclusion does not imply the absence of integrable systems within 6D SCFTs; rather, it suggest there are other potential methods available e.g., correlation functions, to explore the appearance of integrable systems in 6D SCFTs. / Superkonforma fältteorier (SCFTs) är en viktig klass av kvantfältteorier. Dessa SCFTs utgör en viktig komponent för att utforska och förstå det fundamentala ramverket för kvantfältteorin. Inom det stora riket av kvantfältteori spelar integrabilitet en avgörande roll, vilket tillhandahåller kraftfulla analytiska verktyg som gör att vi kan lösa vissa fysiska storheter exakt. I denna avhandling fokuserar vi på representationsteorin av den algebraiska strukturen i sexdimensionella (6D) SCFTs och undersöker det intrikat samspelet mellan 6D SCFTs och integrabilitet. Till att börja med kommer vi att fördjupa oss i de grundläggande begreppen inom representationsteori och skapa en gedigen grund för vår efterföljande analys.  Diskussionen kommer sedan att gå vidare till alla möjliga generatorer i SCFTs, och förklarar hur de realiseras i termer av bosoniska och fermioniska oscillatorer. Slutligen kommer spinnkedjor och dess tillämpningar i 6D SCFTs att undersökas. Vi kommer visa att symmetriargument som härleds från representationsteori inte är  tillräckliga för att fastställa integrerbarhet av spinnkedjor i 6D SCFTs. Denna slutsats innebär inte att integrerbara system inte existerar inom 6D SCFTs, utan föreslår att det finns andra potentiella metoder, till exempel korrelationsfunktioner, för att utforska existensen av integrerbara system i 6D SCFTs.

In this thesis, we investigate the entanglement and magnetic properties of frustrated spin systems and correlated electronic properties of conjugated carbon systems. In chapter 1, we present different approaches to solve the time-independent, nonrelativistic Schr¨odinger equation for a many-body system. We start with the full non-relativistic Hamiltonian of a multi nuclear system to describe the Born - Oppenheimer approximation which allows the study of electronic Hamiltonian which treats nuclear positions parametrically. We then also describe ab initio techniques such as the Hartree-Fock Method and density functional theories. We then introduce model Hamiltonians for strongly correlated systems such as the Hubbard, Pariser-Parr-Pople and Heisenberg models, and show how they result from the noninteracting one-band tight-binding model. In chapter 2, we discuss various numerical techniques like the exact diagonalization methods and density matrix renormalization group (DMRG) method. We also discuss quantum entanglement and the success of DMRG which can be attributed to the area law of entanglement entropy. In chapter 3, we study here different regions in phase diagrams of the spin-1/2, spin-1 and spin-3/2 one-dimensional antiferromagnetic Heisenberg systems with nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions and dimerization (d ). Frustration arises for specific relative signs of the interactions J1 and J2. In particular, we analyze the behavior of the bipartite entanglement entropy and fidelity at the gapless to gapped phase transitions and across the lines separating different phases in the J2−d plane. All the calculations in this work are based on exact diagonalizations of finite systems. In chapter 4, we study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (JF 1 ) and antiferromagnetic (JA 1 ) nearest-neighbor interactions and a ferromagnetic next-nearest-neighbor interaction (JF 2 ). In this model frustration is present due to non-zero JF 2 . The model with site spin s behaves like a Haldane spin chain with site spin 2s in the limit of vanishing JF 2 and large JF 1 /JA 1 . We show that the exact ground state of the model can be found along a line in the parameter space. For fixed JF 1 , the phase diagram in the space of JA 1 −JF 2 is determined using numerical techniques complemented by analytical calculations. A number of quantities, including the structure factor, energy gap, entanglement entropy and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and potentially important feature of this model is that it can exhibit a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian. In chapter 5, we study correlated electronic properties of zigzag and armchair fused naphthalenes and polyperylene systems in the presence of long-range electronelectron interactions. We find that the ground state of zigzag fused naphthalene system is a higher spin state, while the ground state of armchair fused naphthalene is a singlet. The spin gap of polyperylene is unusually small and the ground state is a singlet. Our calculations of optical gap and two-photon gap suggest that polyperylene should exhibit fluorescence. From the charge gap calculation, we predict that in zigzag fused naphthalene and polyperylene systems, excitons are weakly binding. Peierls type of distortion is negligible in zigzag fused naphthalene and polyperylene systems, however, in armchair fused naphthalene system, interior bonds have tendency to distort in low-lying excited states. In chapter 6, we study the ground state spin of the Heisenberg spin-1/2 nearestneighboring antiferromagnetic exchange models of systems with fused odd member rings. In particular, we compute the ground state spin of fused three and five membered rings as well as fused five membered rings. In the thermodynamic limit, the ground state of the fused three and five membered system is a higher spin state, while fused five membered system shows a singlet ground state, for all system sizes.