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arXiv:hep-th/9805101 · Journal of Physics A Mathematical and General

Boundary flows in general coset theories

Changrim Ahn, Chaiho P. Rim

Published Jan 1, 1999• 59 citations
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Abstract

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.

DOI
10.1088/0305-4470/32/13/004

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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.
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We show how a lattice approach can be used to derive Thermodynamic Bethe Ansatz (TBA) equations describing all excitations for boundary flows. The method is illustrated for a prototypical flow of the tricritical Ising model by considering the continuum scaling limit of the A4 lattice model with integrable boundaries. Fixing the bulk weights to their critical values, the integrable boundary weights admit two boundary fields xi\\xixi and eta\\etaeta which play the role of the perturbing boundary fields phi1,3\\phi_{1,3}phi1,3​ and phi1,2\\phi_{1,2}phi1,2​ inducing the renormalization group flow between boundary fixed points. The excitations are completely classified in terms of (m,n) systems and quantum numbers but the string content changes by certain mechanisms along the flow. For our prototypical example, we identify these mechanisms and the induced map between the relevant finitized Virasoro characters. We also solve the boundary TBA equations numerically to determine the flows for the leading excitations.
Boundary phases and thermodynamics of the Kondo spin-sss chain: from overscreened Kondo to boundary-bound states
We study a spin-12\frac1221​ impurity coupled to the boundary of a strongly correlated spin-sss Takhtajan--Babujian chain, an integrable model whose low-energy physics is described by a perturbed SU(2)2sSU(2)_{2s}SU(2)2s​ Wess--Zumino--Witten conformal field theory. While boundary conformal field theory determines the low-energy universality class of the weak-coupling regime, exact Bethe Ansatz methods reveal a sequence of boundary quantum phase transitions in which impurity-bound states emerge and reorganize the Hilbert space into multiple excitation towers built on distinct boundary configurations. This tower restructuring provides the organizing principle for a rich boundary phase diagram extending beyond the conventional Kondo regime. Weak antiferromagnetic coupling realizes the overscreened 2s2s2s-channel Kondo universality class, whereas stronger couplings generate localized boundary modes and qualitatively new screening mechanisms. To describe the resulting thermodynamics, we develop a generalized thermodynamic Bethe Ansatz framework that captures the multi-tower structure across all regimes. The impurity entropy reproduces the boundary conformal field theory prediction in the overscreened Kondo regime but develops pronounced nonmonotonic temperature dependence once boundary-bound states appear, in quantitative agreement with large-scale finite-temperature matrix-product-operator simulations. Complementary dynamical calculations reveal sharp threshold features in the impurity spectral function that directly track the underlying tower structure. Together, boundary conformal field theory, exact Bethe Ansatz, generalized thermodynamic Bethe Ansatz, and tensor-network simulations provide a unified description of impurity screening, boundary-bound-state formation, and excitation-tower reconstruction in a correlated spin-sss chain.
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.