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

In this dissertation, a low energy theory approach is applied to the studies of Dark Matter direct detection experiments and two-dimensional Quantum Chromodynamics (QCD) spectra. We build a general framework of non-relativistic effective field theory of Dark Matter direct detection using non-relativistic operators. Any Dark Matter particle theory can be translated into the coefficients of an effective operator and any effective operator can be related to a most general description of the nuclear response. Response functions are evaluated for common Dark Matter targets. Based on the effective field theory we perform an analysis of the experimental constraints on the full parameter space of elastically scattering Dark Matter. We also formulate an analytic approach to solving two-dimensional gauge theories. We find that in theories with confinement, in a conformal operator basis, the decoupling of high scaling-dimension operators from the low-energy spectrum occurs exponentially fast in their scaling-dimension. Consequently the low-energy spectrum of a strongly coupled system like QCD can be calculated using a truncated conformal basis, to an accuracy parametrized exponentially by the cutoff dimension. We apply the conformal basis approach in two models, a two-dimensional QCD with an adjoint fermion at large N, and a two-dimensional QCD with a fundamental fermion at finite N. It is shown that the low energy spectrum converges efficiently in both cases.

Quantum Field Theory (QFT) is the language that describes a wide spectrum of physics. However, it is notoriously hard at strong coupling regime. We approach this problem in an old Quantum Mechanical method - keep a finite number of states and diagonalize the Hamiltonian as a finite-size matrix. To study a QFT, we take the Hamiltonian to be the Conformal Field Theory as the Ultraviolet fixed point of the theory's Renormalization Group Flow, deformed by a relevant operator. We use a recent framework known as the Lightcone Conformal Truncation (LCT), where we use conformal basis and lightcone quantization. As an application of the method, we study the two dimensional Supersymmetric (SUSY) Gross-Neveu-Yukawa Model. The model is expected to have a critical point in the universality class of tri-critical Ising model, a massive phase and a massless SUSY-breaking phase. We use the LCT to compute the spectrum and the spectral density of the theory at all couplings and map the entire phase diagram.

No description available.

No description available.

No description available.

This thesis begins with the fundamentals of conformal field theory in three dimensions. The general properties of the conformal bootstrap are then reviewed. The three-dimensional Ising model is presented from the perspective of the renormalization group, after which the conformal field theory aspect at the critical point is discussed. Finally, the bootstrap programme is applied to the three-dimensional Ising model using numerical techniques, and the results analysed.

Abstract The collective field theory technique provides a method of tackling problems with two N × N matrices in the large N limit. The collective field background from one matrix is first found, then the second matrix is introduced into this background as an impurity. Within the context of the AdS/CFT correspondence, this technique can be used to describe gauge theory states in the BMN limit. This dissertation starts by developing the collective field theory technique, firstly in general variables, then for one matrix, and subsequently for two matrices. It goes on to introduce a Yang-Mills interaction term, where two variable identifications are considered. The first is the more traditional angular momentum eigenstate model. The second is a model that directly uses two of the Higgs scalars. This model has been mentioned in the literature, but has not been considered in great depth. The exact two impurity spectrum is found, and the multi-impurity spectrum is found to first order. The resulting energy values match a spectrum that has been found for giant magnons.