Boundary states and finite size effects in sine-Gordon model with Neumann boundary condition

The sine-Gordon model with Neumann boundary condition is investigated. Using the bootstrap principle the spectrum of boundary bound states is established. Somewhat surprisingly it is found that Coleman–Thun diagrams and bound state creation may coexist. A framework to describe finite size effects in boundary integrable theories is developed and used together with the truncated conformal space approach to confirm the bound states and reflection factors derived by bootstrap.     

Quantum field theories and critical phenomena on defects

We derive a nonlinear integral equation (NLIE) for some bulk excited states of the sine-Gordon model on a finite interval with general integrable boundary interactions, including boundary terms proportional to the first time derivative of the field. We use this NLIE to compute numerically the dimensions of these states as a function of scale, and check the UV and IR limits analytically. We also find further support for the ground-state NLIE by comparison with boundary conformal perturbation theory (BCPT), boundary truncated conformal space approach (BTCSA) and the boundary analogue of the Lüscher formula.     

Boundary renormalisation group flows of the supersymmetric Lee–Yang model and its extensions

In this paper we examine the supersymmetric Lee–Yang model in the presence of boundaries. We determine the reflection factors for the Neveu–Schwarz type boundary conditions from the reduction of the supersymmetric sine-Gordon model and check them by using boundary truncated conformal space approach in the massless case. We explore the boundary renormalisation groups flows using boundary TBA and TCSA.     

One-point functions in perturbed boundary conformal field theories

We consider the one-point functions of bulk and boundary fields in the scaling Lee–Yang model for various combinations of bulk and boundary perturbations. The one-point functions of the bulk fields are analysed using the truncated conformal space approach and the form-factor expansion. Good agreement is found between the results of the two methods, though we find that the expression for the general boundary state given by Ghoshal and Zamolodchikov has to be corrected slightly. For the boundary fields we use thermodynamic Bethe ansatz equations to find exact expressions for the strip and semi-infinite cylinder geometries. We also find a novel off-critical identity between the cylinder partition functions of models with differing boundary conditions, and use this to investigate the regions of boundary-induced instability exhibited by the model on a finite strip.     

Boundary form factors in finite volume

We describe the volume dependence of matrix elements of local boundary fields to all orders in inverse powers of the volume. Using the scaling boundary Lee–Yang model as testing ground, we compare the matrix elements extracted from boundary truncated conformal space approach to exact form factors obtained using the bootstrap method. We obtain solid confirmation for the boundary form factor bootstrap, which is different from all previously available tests in that it is a non-perturbative and direct comparison of exact form factors to multi-particle matrix elements of local operators, computed from the Hamiltonian formulation of the quantum field theory.     

Spectrum of local boundary operators from boundary form factor bootstrap

Using the recently introduced boundary form factor bootstrap equations, we map the complete space of their solutions for the boundary version of the scaling Lee–Yang model and sinh-Gordon theory. We show that the complete space of solutions, graded by the ultraviolet behaviour of the form factors can be brought into correspondence with the spectrum of local boundary operators expected from boundary conformal field theory, which is a major evidence for the correctness of the boundary form factor bootstrap framework.     

Finite size effects in the spin-1 XXZ and supersymmetric sine-Gordon models with Dirichlet boundary conditions

Starting from the Bethe Ansatz solution of the open integrable spin-1 XXZ quantum spin chain with diagonal boundary terms, we derive a set of nonlinear integral equations (NLIEs), which we propose to describe the boundary supersymmetric sine-Gordon model BSSG⁺ with Dirichlet boundary conditions on a finite interval. We compute the corresponding boundary S matrix, and find that it coincides with the one proposed by Bajnok, Palla and Takács for the Dirichlet BSSG⁺ model. We derive a relation between the (UV) parameters in the boundary conditions and the (IR) parameters in the boundary S matrix. By computing the boundary vacuum energy, we determine a previously unknown parameter in the scattering theory. We solve the NLIEs numerically for intermediate values of the interval length, and find agreement with our analytical result for the effective central charge in the UV limit and with boundary conformal perturbation theory.     

On the renormalisation group for the boundary truncated conformal space approach

In this paper we continue the study of the truncated conformal space approach to perturbed boundary conformal field theories. This approach to perturbation theory suffers from a renormalisation of the coupling constant and a multiplicative renormalisation of the Hamiltonian. We show how these two effects can be predicted by both physical and mathematical arguments and prove that they are correct to leading order for all states in the TCSA system. We check these results using the TCSA applied to the tri-critical Ising model and the Yang–Lee model. We also study the TCSA of an irrelevant (non-renormalisable) perturbation and find that, while the convergence of the coupling constant and energy scales are problematic, the renormalised and rescaled spectrum remain a very good fit to the exact result, and we find a numerical relationship between the IR and UV couplings describing a particular flow. Finally we study the large coupling behaviour of TCSA and show that it accurately encompasses several different fixed points.     

Breather boundary form factors in sine-Gordon theory

A previously conjectured set of exact form factors of boundary exponential operators in the sinh-Gordon model is tested against numerical results from boundary truncated conformal space approach in boundary sine-Gordon theory, related by analytic continuation to sinh-Gordon model. We find that the numerical data strongly support the validity of the form factors themselves; however, we also report a discrepancy in the case of diagonal matrix elements, which remains unresolved for the time being.     

Form factors in finite volume I: Form factor bootstrap and truncated conformal space

We describe the volume dependence of matrix elements of local fields to all orders in inverse powers of the volume (i.e., only neglecting contributions that decay exponentially with volume). Using the scaling Lee–Yang model and the Ising model in a magnetic field as testing ground, we compare them to matrix elements extracted in finite volume using truncated conformal space approach to exact form factors obtained using the bootstrap method. We obtain solid confirmation for the form factor bootstrap, which is different from all previously available tests in that it is a non-perturbative and direct comparison of exact form factors to multi-particle matrix elements of local operators, computed from the Hamiltonian formulation of the quantum field theory. We also demonstrate that combining form factor bootstrap and truncated conformal space is an effective method for evaluating finite volume form factors in integrable field theories over the whole range in volume.     

Excited states by analytic continuation of TBA equations

We suggest an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the thermodynamic Bethe ansatz equations for its ground state. The idea relies on analytic continuation through complex values of the coupling constant, and an analysis of the monodromies that the equations and their solutions undergo. For the scaling Lee-Yang model, we find equations in this way for the one- and two-particle states in the spin-zero sector, and suggest various generalisations. Numerical results show excellent agreement with the truncated conformal space approach, and we also treat some of the ultraviolet and infrared asymptotics analytically.     

NLIE of Dirichlet sine-Gordon model for boundary bound states

We investigate boundary bound states of sine-Gordon model on the finite-size strip with Dirichlet boundary conditions. For the purpose we derive the nonlinear integral equation (NLIE) for the boundary excited states from the Bethe ansatz equation of the inhomogeneous XXZ spin 1/2 chain with boundary imaginary roots discovered by Saleur and Skorik. Taking a large volume (IR) limit we calculate boundary energies, boundary reflection factors and boundary Lüscher corrections and compare with the excited boundary states of the Dirichlet sine-Gordon model first considered by Dorey and Mattsson. We also consider the short distance limit and relate the IR scattering data with that of the UV conformal field theory.     

Minimal model boundary flows and c=1 CFT

We consider perturbations of unitary minimal models by boundary fields. Initially we consider the models in the limit as c→1 and find that the relevant boundary fields all have simple interpretations in this limit. This interpretation allows us to conjecture the IR limits of flows in the unitary minimal models generated by the fields φ_rr of ‘low’ weight. We check this conjecture using the truncated conformal space approach. In the process we find evidence for a new series of integrable boundary flows.     

Universal boundary reflection amplitudes

For all affine Toda field theories we propose a new type of generic boundary bootstrap equations, which can be viewed as a very specific combination of elementary boundary bootstrap equations. These equations allow to construct general solutions for the boundary reflection amplitudes, which are valid for theories related to all simple Lie algebras, that is simply laced and non-simply laced. We provide a detailed study of these solutions for concrete Lie algebras in various representations. The boundary bootstrap equations relating different types of exited boundary states are not automatically solved by our expressions.     

Reflection factors and exact g-functions for purely elastic scattering theories

We discuss reflection factors for purely elastic scattering theories and relate them to perturbations of specific conformal boundary conditions, using recent results on exact off-critical g-functions. For the non-unitary cases, we support our conjectures using a relationship with quantum group reductions of the sine-Gordon model. Our results imply the existence of a variety of new flows between conformal boundary conditions, some of them driven by boundary-changing operators.     

Conformally coupled scalars, instantons, and vacuum instability in 4D anti-de Sitter space

We show that a scalar field conformally coupled to AdS gravity in four dimensions with a quartic self-interaction can be embedded into M theory. The holographic effective potential is exactly calculated, allowing us to study nonperturbatively the stability of AdS4 in the presence of the conformally coupled scalar. It is shown that there exists a one-parameter family of conformal scalar boundary conditions for which the boundary theory has an unstable vacuum. In this case, the bulk theory has instanton solutions that mediate the decay of the AdS4 space. These results match nicely with the vacuum structure and the existence of instantons in an effective three-dimensional boundary model.     

Integrable boundary conditions for the B_2 model

New integrable B_2 model with off-diagonal boundary reflections is proposed. The general solutions of the reflection matrix for the B_2 model are obtained by using the fusion technique. Wefind that the reflection matrix has 7 free boundary parameters, which are used to describe the degree of freedom of boundary couplings, without breaking the integrability of the system. The new quantization conditions will induce the novel structure of the energy spectrum and the boundary states. The corresponding boundary effects can be studied based on the results in this paper. Meanwhile, the reflection matrix of high rank models associated with Bnalgebra can also be obtained by using the method suggested in this paper.