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Reflecting boundaries and massless factorized scattering in two dimensions

João Nuno Prata

Published Jan 1, 1998• 0 citations
OpenAlex

Abstract

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.

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Topics in 2D integrable field theories with boundary interactions

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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 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.
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.
Recent years witnessed an extensive development of the theory of the critical point in two-dimensional statistical systems, which allowed to prove {\it existence} and {\it conformal invariance} of the {\it scaling limit} for two-dimensional Ising model and dimers in planar graphs. Unfortunately, we are still far from a full understanding of the subject: so far, exact solutions at the lattice level, in particular determinant structure and exact discrete holomorphicity, play a cucial role in the rigorous control of the scaling limit. The few results about not-integrable (interacting) systems at criticality are still unable to deal with {\it finite domains} and {\it boundary corrections}, which are of course crucial for getting informations about conformal covariance. In this thesis, we address the question of adapting constructive Renormalization Group methods to non-integrable critical systems in d=1+1d= 1+1d=1+1 dimensions. We study a system of interacting spinless fermions on a one-dimensional semi-infinite lattice, which can be considered as a prototype of the Luttinger universality class with Dirichlet Boundary Conditions. We develop a convergent renormalized expression for the thermodynamic observables in the presence of a quadratic {\it boundary defect} counterterm, polynomially localized at the boundary. In particular, we get explicit bounds on the boundary corrections to the specific ground state energy.
In this paper we study the boundary effects for off-critical integrable field theories which have close analogues with integrable lattice models. Our models are the coset conformal field theories perturbed by an integrable boundary and bulk operators. The boundary interactions are encoded into the boundary reflection matrix. Using the thermodynamic Bethe ansatz (TBA) method, we verify the flows of the conformal BCs by computing the boundary entropies. These flows of the BCs have direct interpretations for the fusion restricted solid-on-solid (RSOS) lattice models. For super conformal field theories (CFTs) ( k = 2) we show that these flows are possible only for the Neveu-Schwarz sector and are consistent with the lattice results. The models we consider cover a wide class of integrable models. In particular, we show how the impurity spin is screened by electrons for the k -channel Kondo model by taking the limit. We also study the problem using an independent method based on the boundary roaming TBA. Our numerical results are consistent with the boundary CFTs and RSOS TBA analysis.
We study different aspects of integrable boundary quantum field theories, focusing mostly on the ``boundary sine-Gordon model'' and its applications to condensed matter physics. The first part of the review deals with formal problems. We analyze the classical limit and perform semi-classical quantization. We show that the non-relativistic limit corresponds to the Calogero-Moser model with a boundary potential. We construct a lattice regularization of the problem via the XXZ chain. We classify boundary bound states. We generalize the Destri de Vega method to compute the ground state energy of the theory on a finite interval. The second part deals with some applications to condensed matter physics. We show how to compute analytically time and space dependent correlations in one-dimensional quantum integrable systems with an impurity. Our approach is based on a description of these systems in terms of massless scattering of quasiparticles. Correlators follow then from matrix elements of local operators between multiparticle states -- the massless form-factors. Although, in general an infinite sum of these form-factors has to be considered, we find that for the current, spin and energy operators only a few (two or three) are necessary to obtain an accuracy of more than 1\%. Our results hold for arbitrary impurity strength, in contrary to the perturbative expansions in the coupling constants. As an example, we compute the frequency dependent condunctance, at zero temperature, in a Luttinger liquid with an impurity, and also discuss the susceptibility in the Kondo model and the time-dependent properties of the two-state problem with dissipation.