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 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.
RSOS models and their associated RZMS scattering QFT models, this is certainly not the case, and here the corresponding TBA equations have been derived only in a few special cases.This shortcoming provides the primary motivation for the work contained in this thesis.After presenting a brief review of the various techniques for deriving the TBA equations associated with the ground state energy for the 0 lj3 perturbations of the minimal CFT models, attempts are made at extending these techniques to the derivation of the TBA equations associated with the ground state energy of the 0 i;2 and perturbations of the minimal CFT models.More recently, the functional relation approach for constructing the ground state TBA equations was extended to the derivation of TBA-like integral equations which correspond to excited state energy levels.In principle, this approach can be used to determine the complete energy spectrum of the perturbed CFT, but in practice this is a very difficult undertaking and only the lowest few excited states are rela tively easily to obtain.In the final chapter of this thesis, this approach is used to determine the TBA-like equations corresponding to the lowest three energy levels of a particular simple perturbed CFT, notably the non-unitary minimal CFT perturbed by its operator <^2,i-L ist o f P u b lic a tio n sThe bulk of the original research content of this thesis is contained in Chapters 4, 5 and 6, with some numerical results tabulated in Appendices B and C. Various parts of this research are, or are soon to be, published as follows.The ground state TBA results of Chapter 5, appear in the paper:• R. M. Ellem, and V. V. Bazhanov: Thermodynamic Bethe Ansatz for the subleading magnetic perturbation of the tricritical Ising Model, Nucl.Phys.B512 (1998), 563-580.The ground state and excited state TBA results of Sections 4.1, 5.1, 5.4 and Chapter 6, and the transfer matrix fusion relations in Section 4.2 and Chapter 6, appear in the paper:• R. M. Ellem, and V. V. Bazhanov: Excited State TBA for the < /> 2,i Perturbed A43;5 Model.To be submitted.The general 5-matrix amplitudes and bootstrap properties of Section 4.3, will ap pear in the paper:• R. M. Ellem: Scattering Structure of the (f)\i2 and (f)2,i Perturbed Minimal Conformal Field Theory Models. Currently in preparation.An extension of the integral equation techniques introduced in Chapter 6, to the fusion equation for the regime 1+ dilute A 2 model described in Section 4.2.1, will appear in the paper:• R. M. Ellem: Excited State TBA for the 02jl Perturbed M .%,4Model.Currently in preparation.
The study of Finite Size Effects in Quantum Field Theory allows the extraction of precious perturbative and non-perturbative information. The use of scaling functions can connect the particle content (scattering theory formulation) of a QFT to its ultraviolet Conformal Field Theory content. If the model is integrable, a method of investigation through a nonlinear integral equation equivalent to Bethe Ansatz and deducible from a light-cone lattice regularization is available. It allows to reconstruct the S-matrix and to understand the locality properties in terms of Bethe root configurations, thanks to the link to ultraviolet CFT guaranteed by the exact determination of scaling function. This method is illustrated in practice for Sine-Gordon / massive Thirring models, clarifying their locality structure and the issues of equivalence between the two models. By restriction of the Sine-Gordon model it is also possible to control the scaling functions of minimal models perturbed by Phi_1,3
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.
Abstract We study the thermodynamic Bethe ansatz (TBA) equations for pure T T ¯ perturbations of some simple integrable quantum field theories with a single bosonic or fermionic particle, in particular the massive sinh-Gordon model and its ultraviolet (UV) limit which is a deformation of the conformally invariant free massless boson. Whereas the TBA equations for T T ¯ deformations of massive theories are in principle known, the TBA equations we propose for the deformations of conformal field theories are relatively new and require a special factorization in rapidity variables of the CDD factor for the scattering of the massless particles. The latter TBA equations can be solved exactly and reproduce the known results for the ground state energy on a cylinder of circumference R which were previously obtained using different methods based for instance on the Burgers differential equation. Special attention is paid to the c -theorem in this context which is discussed in some detail. For positive infra-red central charge c IR , for flows consistent with the c -theorem the ground state energy develops a (previously known) square-root singularity toward the UV, which strongly suggests the theories are UV incomplete in these physically important cases. We suggest that the singularity indicates a tachyonic vacuum instability. Other cases with c IR < 0 do not have this singularity are interpreted as being UV complete with c UV = 0. We extend our results to a continuously variable c IR by introducing a chemical potential and suggest this as a possible toy model for the T T ¯ perturbed Liouville theory.
In this thesis, we review recent progresses on Nonlinear Integral Equation approach to finite size effects in two dimensional integrable quantum field theories, with emphasis to Sine-Gordon/Massive Thirring model and restrictions to minimal models perturbed by Φ1,3. Exact calculations of the dependence of energy levels on the size are presented for vacuum and many excited states.