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arXiv:1502.03976

Lee-Yang Model in Presence of Defects

Omar El Deeb

Published Feb 13, 2015• 4 citations
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Abstract

I choose the Lee-Yang model and go through different approaches to analyze it using the form factor approach and the bootstrap program, the lattice description and the lattice TBA equations for a full understanding of the model. The bootstrap program aims to explicitly solve 1+1 dimensional integrable quantum field theories. In the first step, called the S-matrix bootstrap, the scattering matrix, is determined from its properties. In developing a defect form factor program the first step is the T-matrix bootstrap. Interacting integrable defect theories are purely transmitting and topological. We analyze both operators localized in the bulk and also on the defect. By finding their solutions, the spectral representation of any correlator can be determined. Next we consider the model on the lattice. The lattice approach allows to obtain both massive and massless excited TBA equations by studying the continuum scaling limit of the associated integrable lattice models. The most important input from the lattice approach is an insight into the analytic structure of the excited state solutions of the TBA equations. Previously this structure had to be guessed. In this thesis we turn our attention to the simplest example of a non-unitary minimal theory, namely, the Lee-Yang minimal model M(2,5). We study the Lee-Yang model on the lattice. We analyze the periodic, boundary and the seam cases in both massive and massless regimes. We derive their ground state TBA equations and analyze the flows between different boundary and defect conditions. Finally, in the large volume limit, we calculate the Lüscher correction terms of the TBA energy of a one-particle state. We display those ideas in the end of the thesis as this is an important direction of work in string theory, particularly in the AdS/CFT correspondence.

DOI
10.48550/arxiv.1502.03976

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I choose the Lee-Yang model and go through different approaches to analyze it using the form factor approach and the bootstrap program, the lattice description and the lattice TBA equations for a full understanding of the model. The bootstrap program aims to explicitly solve 1+1 dimensional integrable quantum field theories. In the first step, called the S-matrix bootstrap, the scattering matrix, is determined from its properties. In developing a defect form factor program the first step is the T-matrix bootstrap. Interacting integrable defect theories are purely transmitting and topological. We analyze both operators localized in the bulk and also on the defect. By finding their solutions, the spectral representation of any correlator can be determined. Next we consider the model on the lattice. The lattice approach allows to obtain both massive and massless excited TBA equations by studying the continuum scaling limit of the associated integrable lattice models. The most important input from the lattice approach is an insight into the analytic structure of the excited state solutions of the TBA equations. Previously this structure had to be guessed. In this thesis we turn our attention to the simplest example of a non-unitary minimal theory, namely, the Lee-Yang minimal model M(2,5). We study the Lee-Yang model on the lattice. We analyze the periodic, boundary and the seam cases in both massive and massless regimes. We derive their ground state TBA equations and analyze the flows between different boundary and defect conditions. Finally, in the large volume limit, we calculate the Lüscher correction terms of the TBA energy of a one-particle state. We display those ideas in the end of the thesis as this is an important direction of work in string theory, particularly in the AdS/CFT correspondence.
We consider the non-unitary Lee-Yang minimal model M(2,  5) \mathrm{\mathcal{M}}\left(2,\;5\right) M(2,5) in three different finite geometries: (i) on the interval with integrable boundary conditions labelled by the Kac labels (r, s) = (1, 1), (1, 2), (ii) on the circle with periodic boundary conditions and (iii) on the periodic circle including an integrable purely transmitting defect. We apply φ1,3 integrable perturbations on the boundary and on the defect and describe the flow of the spectrum. Adding a Φ1,3 integrable perturbation to move off-criticality in the bulk, we determine the finite size spectrum of the massive scattering theory in the three geometries via Thermodynamic Bethe Ansatz (TBA) equations. We derive these integral equations for all excitations by solving, in the continuum scaling limit, the TBA functional equations satisfied by the transfer matrices of the associated A4 RSOS lattice model of Forrester and Baxter in Regime III. The excitations are classified in terms of (m, n) systems. The excited state TBA equations agree with the previously conjectured equations in the boundary and periodic cases. In the defect case, new TBA equations confirm previously conjectured transmission factors.
We analyze the defect scaling Lee–Yang model from the perturbed defect conformal field theory (DCFT) point of view. First the defect Lee–Yang model is solved by calculating its structure constants from the sewing relations. Integrable defect perturbations are identified in conformal defect perturbation theory. Then pure defect flows connecting integrable conformal defects are described. We develop a defect truncated conformal space approach (DTCSA) to analyze the one parameter family of integrable massive perturbations in finite volume numerically. Fusing the integrable defect to an integrable boundary the relation between the IR and UV parameters can be derived from the boundary relations. We checked these results by comparing the spectrum for large volumes to the scattering theory.
The purpose of the "bootstrap program" is to construct integrable quantum field theories in 1+1 dimensions in terms of their Wightman functions explicitly. As an input the integrability and general assumptions of local quantum field theories are used. The object is to be achieved in tree steps: 1) The S-matrix is obtained using a qualitative knowledge of the particle spectrum and the Yang-Baxter equations. 2) Matrix elements of local operators are calculated by means of the "form factor program" using the S-matrix as an input. 3) The Wightman functions are calculated by taking sums over intermediate states. The first step has been performed for a large number of models and also the second one for several models. The third step is unsolved up to now. Here the program is illustrated in terms of the sine-Gordon model alias the massive Thirring model. Exploiting the "off-shell" Bethe Ansatz we propose general formulae for form factors. For example the n-particle matrix element for all higher currents are given and in particular all eigenvalues of the higher conserved charges are calculated. Furthermore quantum operator equations are obtained in terms of their matrix elements, in particular the quantum sine-Gordon field equation. Exact expressions for the finite wave function and mass renormalization constants are calculated.
The form factors of the descendant operators in the massive Lee–Yang model are determined up to level 7. This is first done by exploiting the conserved quantities of the integrable theory to generate the solutions for the descendants starting from the lowest non-trivial solutions in each operator family. We then show that the operator space generated in this way, which is isomorphic to the conformal one, coincides, level by level, with that implied by the S -matrix through the form factor bootstrap. The solutions that we determine satisfy asymptotic conditions carrying the information about the level that we conjecture to hold for all the operators of the model.
We developed the theory of finite volume form factors in the presence of integrable defects. These finite volume form factors are expressed in terms of the infinite volume form factors and the finite volume density of states and incorporate all polynomial corrections in the inverse of the volume. We tested our results, in the defect Lee–Yang model, against numerical data obtained by truncated conformal space approach (TCSA), which we improved by renormalization group methods adopted to the defect case. To perform these checks we determined the infinite volume defect form factors in the Lee–Yang model exactly, including their vacuum expectation values. We used these data to calculate the two point functions, which we compared, at short distance, to defect CFT. We also derived explicit expressions for the exact finite volume one point functions, which we checked numerically. In all of these comparisons excellent agreement was found.
Topological defects in two-dimensional conformal field theories (CFTs) are canonical examples of generalized symmetries in quantum field theories. They not only encode key features of the underlying theory, but their study has also enhanced our understanding of renormalization group (RG) flows in theories that contain them. Lattice models exist which discretize the states, energy-momentum tensor, and primary fields of many 2D CFTs, raising the natural question of whether topological defects can be similarly discretized. In this thesis, we demonstrate that in many cases all topological defects of minimal model CFTs can be discretized, enabling analytical and ab-initio numerical computations of their properties. In some of these constructions, we make essential use of the integrable structure of the underlying lattice models. The lattice realization of these operators also permits the study of defect RG flows between CFTs with different topological defects inserted. These topological defects and defect RG flows are realized on the lattice not only for Hermitian but also for non-Hermitian lattice Hamiltonians. Studying topological defects in minimal model CFTs on the lattice has also opened the door to a deeper understanding of topological defects in CFTs where they remain poorly understood.