Abstract We construct an effective Quantum Field Theory for the wrapping effects in 1+1 dimensional models of factorised scattering. The recently developed graph-theoretical approach to TBA gives the perturbative desctiption of this QFT. For the sake of simplicity we limit ourselves to scattering matrices for a single neutral particle and no bound state poles, such as the sinh-Gordon one. On the other hand, in view of applications to AdS/CFT, we do not assume that the scattering matrix is of difference type. The effective QFT involves both bosonic and fermionic fields and possesses a symmetry which makes it one-loop exact. The corresponding path integral localises to a critical point determined by the TBA equation.
Abstract We construct an effective Quantum Field Theory for the wrapping effects in 1+1 dimensional models of factorised scattering. The recently developed graph-theoretical approach to TBA gives the perturbative desctiption of this QFT. For the sake of simplicity we limit ourselves to scattering matrices for a single neutral particle and no bound state poles, such as the sinh-Gordon one. On the other hand, in view of applications to AdS/CFT, we do not assume that the scattering matrix is of difference type. The effective QFT involves both bosonic and fermionic fields and possesses a symmetry which makes it one-loop exact. The corresponding path integral localises to a critical point determined by the TBA equation.
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 series of lectures given at KIAS in Nov 2018, I briefly introduce the concepts of integrability, at classical and quantum level. Then I elaborate on the factorised scattering theory emerging in 1+1 integrable QFT. The Thermdynamic Bethe Ansatz (TBA) is a well known method to get exact information from the scattering theory, in order to identify the underlying QFT. Here I illustrate how to get TBA in the case of the prototypical Sine-Gordon model, particularly interesting to show the techniques when the S-matrix is not diagonal.
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.
Abstract The thermodynamics of a quantum field theory in an infinite volume can be determined by its S-matrix. This idea, originally proposed by R. Dashen, S.K. Ma, and H.J. Berstein, has been widely used to study the thermal properties of the integrable field theories in (1 + 1) dimensions. The reason consists of the particularly simple properties of these scattering matrices, as discussed in the previous two chapters, and the possibility to generalize to the relativistic case the thermodynamics Bethe ansatz (TBA) techniques successfully applied to non-relativistic problems by C.N. Yang and C.P. Yang. In the TBA approach, the derivation of the thermodynamics of a purely elastic scattering theory reduces to finding the solution of a set of nonlinear integral equations that rule the energies of the particle excitations and their statistical distribution.
We present the TBA equations for the exact spectrum of multi-magnon local operators in the D-dimensional bi-scalar fishnet CFT. The mixing matrix of such operators is given in terms of fishnet planar graphs of multiwheel and multispiral type. These graphs probe the two key building blocks of the TBA approach, the magnon dispersion relation and scattering matrix, which we obtain by diagonalizing suitable graph-building operators. We also obtain the dual version of the TBA equations, which relates, in the continuum limit, D-dimensional graphs to two-dimensional sigma models in AdS_{D+1}.
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.