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Perturbative and non-perturbative studies in low dimensional quantum field theory

Anna Lishman

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

A relevant perturbation of a conformal field theory (CFT) on the half-plane, by both a bulk and boundary operator, often leads to a massive theory with a particle description in terms of the bulk S-matrix and boundary reflection factor R. The link between the particle basis and the CFT in the bulk is usually made with the thermodynamic Bethe ansatz effective central charge C(_eff). This allows a conjectured S-matrix to be identified with a specific perturbed CFT. Less is known about the links between the reflection factors and conformal boundary conditions, but it has been proposed that an exact, off-critical version of Affleck and Ludwig's g-function could be used, analogously to C(_eff), to identify the physically realised reflection factors and to match them with the corresponding boundary conditions. In the first part of this thesis, this exact g-function is tested for the purely elastic scattering theories related to the ADET Lie algebras. Minimal reflection factors are given, and a method to incorporate a boundary parameter is proposed. This enables the prediction of several new flows between conformal boundary conditions to be made. The second part of this thesis concerns the three-parameter family of PT-symmetric Hamiltonians H(M,o,1) = p(^2) – (ix) (^2M) – α(ix) The positions where the eigenvalues merge and become complex correspond to quadratic and cubic exceptional points. The quasi-exact solvability of the models for M = 3 is exploited to exploreaway from M = 3 is investigated using both numerical and perturbative approaches.

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A relevant perturbation of a conformal field theory (CFT) on the half-plane, by both a bulk and boundary operator, often leads to a massive theory with a particle description in terms of the bulk S-matrix and boundary reflection factor R. The link between the particle basis and the CFT in the bulk is usually made with the thermodynamic Bethe ansatz effective central charge C(_eff). This allows a conjectured S-matrix to be identified with a specific perturbed CFT. Less is known about the links between the reflection factors and conformal boundary conditions, but it has been proposed that an exact, off-critical version of Affleck and Ludwig's g-function could be used, analogously to C(_eff), to identify the physically realised reflection factors and to match them with the corresponding boundary conditions. In the first part of this thesis, this exact g-function is tested for the purely elastic scattering theories related to the ADET Lie algebras. Minimal reflection factors are given, and a method to incorporate a boundary parameter is proposed. This enables the prediction of several new flows between conformal boundary conditions to be made. The second part of this thesis concerns the three-parameter family of PT-symmetric Hamiltonians H(M,o,1) = p(^2) – (ix) (^2M) – α(ix) The positions where the eigenvalues merge and become complex correspond to quadratic and cubic exceptional points. The quasi-exact solvability of the models for M = 3 is exploited to exploreaway from M = 3 is investigated using both numerical and perturbative approaches.
This thesis is concerned with perturbed conformal field theory, the thermodynamic Bethe ansatz technique and applications to statistical mechanics. In particular, the phase space of two dimensional Z(_N)-symmetric statistical models is examined using these techniques. The aim of the first two chapters is to review some general material concerning statistical mechanics, perturbed conformal field theory, integrable two-dimensional quantum field theory and the thermodynamic Bethe ansatz (TBA) technique. In the third chapter Z(_N)-symmetric statistical theories are discussed and the known features of the phase space of such models are surveyed. The field content of the conformal models in this space (called parafermionic models) is investigated in order to determine which perturbations can be used to investigate the phase space. In the fourth and fifth chapters TBA equations are proposed to describe massless and massive renormalisation flows from the Z(_N)-symmetric conformal theories under self-dual Z(_N)-symmetric perturbations. According to the sign of the perturbation parameter the infrared limits are shown to be either conformal c = 1 or massive theories. The ground state energies of these models can be discovered in all perturbative regimes via the TBA method and the results agree with perturbation theory in ultraviolet and infrared limits. Results from detailed studies of the N = 5, 6..10 models are presented throughout. It is also deduced that the parafermionic models lie exactly at the bifurcation point of the first-order transition region into the Kosterlitz-Thouless region of the Z(_N)-symmetric phase space. The sixth and seventh chapters deal solely with massive perturbations. In chapter six, results from the TBA equations are used to deduce the mass spectrum and the vacuum structure of the underlying scattering theory. In chapter seven, proposals for the massive S-matrices are made. For N odd the mass spectra proposed by the TBA method and that predicted by the S-matrix approach (using the minimality principle) differ. It is suggested therefore, that the N odd S-matrices contain zeroes in the physical strip, violating the minimality principle.
This thesis is concerned with two-dimensional models that are integrable in the presence of a boundary and whose spectrum in the bulk is constituted of massless particles. Although there is already a vast literature on the subject (e.g. Kondo and Callan-Rubakov models), the common minimal denominator in all these situations is the fact that the bulk theory is conformal invariant and it is the boundary that is responsible for the broken scale invariance. Here, our purpose is to consider the alternative situation, where the boundary respects the conformal invariance of the theory and the renormalization group trajectory is controlled by a bulk perturbation. The model in question is the principal chiral model at level k = 1. We propose the set of permissible boundary conditions suggested by the symmetries of the problem and compute the corresponding minimal reflection matrices. For one of the boundary conditions we compute the boundary ground state energy and the boundary entropy using the technique of boundary thermodynamic Bethe ansatz. In the infrared limit our results are shown to be in complete agreement with the predictions of the boundary conformal field theory approach. Finally, we consider the classical supersymmetric Liouville theory on the half-line and compute the boundary conditions compatible with the superconformal invariance. We construct an infinite set of commuting integrals of motion using Lax-pair techniques and discuss some aspects of the quantum theory as well as its relation to the super Korteweg-de Vries equation.
This thesis is concerned with various aspects of perturbed conformal field theory and the methods used to calculate finite-size effects of integrable quantum field theories. Nonlinear integral equations are the main tools to find the exact ground-state energy of a quantum field theory. The thermodyamic Bethe ansatz (TBA) equations are a set of examples and are known for a large number of models. However, it is also an interesting question to find exact equations describing the excited states of integrable models. The first part of this thesis uses analytical continuation in a continuous parameter to find TBA like equations describing the spin-zero excited states of the sine-Gordon model at coupling β(^2) = 16π/3. Comparisons are then made with a further type of nonlinear integral equation which also predicts the excited state energies. Relations between the two types of equation are studied using a set of functional relations recently introduced in integrable quantum field theory. A relevant perturbation of a conformal field theory results in either a massive quantum field theory such as the sine-Gordon model, or a different massless conformal field theory. The second part of this thesis investigates flows between conformal field theories using a nonlinear integral equation. New families of flows are found which exhibit a rather unexpected behaviour. The final part of this thesis begins with a review of a connection between integrable quantum field theory and properties of certain ordinary differential equations of second- and third-order. The connection is based on functional relations which appear on both sides of the correspondence; for the second-order case these are exactly the functional relations mentioned above. The results are extended to include a correspondence between n(^th) order differential equations and Bethe ansatz system of SU(n) type. A set of nonlinear integral equations are derived to check the results.
This thesis is concerned with two-dimensional models that are integrable in the presence of a boundary and whose spectrum in the bulk is constituted of massless particles. Although there is already a vast literature on the subject (e.g. Kondo and Callan-Rubakov models), the common minimal denominator in all these situations is the fact that the bulk theory is conformal invariant and it is the boundary that is responsible for the broken scale invariance. Here, our purpose is to consider the alternative situation, where the boundary respects the conformal invariance of the theory and the renormalization group trajectory is controlled by a bulk perturbation. The model in question is the principal chiral model at level k = 1. We propose the set of permissible boundary conditions suggested by the symmetries of the problem and compute the corresponding minimal reflection matrices. For one of the boundary conditions we compute the boundary ground state energy and the boundary entropy using the technique of boundary thermodynamic Bethe ansatz. In the infrared limit our results are shown to be in complete agreement with the predictions of the boundary conformal field theory approach. Finally, we consider the classical supersymmetric Liouville theory on the half-line and compute the boundary conditions compatible with the superconformal invariance. We construct an infinite set of commuting integrals of motion using Lax-pair techniques and discuss some aspects of the quantum theory as well as its relation to the super Korteweg-de Vries equation.
This thesis contains three main parts, which are largely independent. In the first part we deal with the boundary bootstrap in supersymmetric factorized scattering theory. We give a description of supersymmetry in the case when the space is a half-line and present rules for the determination of the representations in which higher level boundary bound states transform, and for the determination of the supersymmetric one-particle reflection matrix factors for the higher level boundary bound states. These rules apply under the condition that the bulk particles transform in the kink or in the boson-fermion representation. Examples for the application of these rules to specific models are also given. In the second part we investigate the problem whether the TCSA spectrum can be approximated by the spectrum of the original Hamiltonian operator in which the coefficients of the terms are suitably changed. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. Another truncation method that preserves the solvability of the model is also considered. The results of perturbative and numerical calculations show that the above approximation is possible and that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method. In the third part we investigate the phase structure of the two- and three-frequency sine-Gordon models using the TCSA. In the case of the three-frequency model the tricritical point, several points of the critical line and a few points of the line of first order transition are found.
We have gathered quite a large amount of technology to construct conformal boundary conditions and study their properties. So far, most of the tools relied on detailed knowledge of the underlying algebraic structure of conformal field theories (CFTs), such as representations of symmetry algebras and modular transformations. In almost all cases, this route will lead to a restricted class of boundary conditions only, favouring those that preserve a lot of symmetry. In “real life applications” of boundary CFT, arising in string theory, in condensed matter physics, or in the study of integrable models, we are often faced with situations where we start from a simple given boundary condition and then “deform” by “turning on perturbations”. In this chapter, we will discuss methods aimed at making this procedure well defined and at determining the perturbed boundary theory. Most of the time, we will restrict ourselves to pure boundary perturbations, which leave the bulk CFT untouched. We will sometimes be led to new boundary conditions that would have been difficult to find directly. The total space of boundary CFTs takes the form of a fibration over the moduli space of bulk theories, with the fibre over each point made up from all boundary conditions (branes) that exist for that bulk theory. Marginal boundary deformations, if they exist, will generate, starting from a single boundary condition, whole continuous families within this boundary moduli space.