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arXiv:hep-th/0305091 · Journal of Physics A Mathematical and General

The diluteALmodels and the integrable perturbations of unitary minimal CFTs

J Suzuki

Published Dec 15, 2003• 3 citations
arXivHTMLPDF

Abstract

Recently, a set of thermodynamic Bethe ansatz (TBA) equations has been proposed by Dorey, Pocklington and Tateo for unitary minimal models perturbed by the ϕ 1,2 or ϕ 2,1 operator. We examine their results in view of the lattice analogues, dilute A L models in regimes 1 and 2. Taking M 5,6 + ϕ 1,2 and M 3,4 + ϕ 2,1 as the simplest examples, we will explicitly show that the conjectured TBA equations can be recovered from the lattice model in a scaling limit.

DOI
10.1088/0305-4470/37/2/018

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Recently, a set of thermodynamic Bethe ansatz (TBA) equations has been proposed by Dorey, Pocklington and Tateo for unitary minimal models perturbed by the ϕ 1,2 or ϕ 2,1 operator. We examine their results in view of the lattice analogues, dilute A L models in regimes 1 and 2. Taking M 5,6 + ϕ 1,2 and M 3,4 + ϕ 2,1 as the simplest examples, we will explicitly show that the conjectured TBA equations can be recovered from the lattice model in a scaling limit.
Motivated by recent studies by Dorey, Pocklington and Tateo for unitary minimal models perturbed by ϕ 1,2 , we examine the thermodynamics of one-dimensional quantum systems, whose counterparts in the two-dimensional classical model are the dilute A L models in regime 2. The functional relations for arbitrary values of L are established. Guided by numerical evidence, we obtain a set of coupled integral equations from the established relations, which yields the evaluation of the free energy at arbitrary temperature. In the scaling limit, the integral equations coincide with the thermodynamic Bethe ansatz equations (TBA) proposed by Dorey, Pocklington and Tateo, thereby supporting their results. The new fermionic representations of the Virasoro characters are remarked upon briefly.
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.
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.
By using the thermodynamic Bethe ansatz approach, we give evidence of the existence of both massive and massless behaviors for the ϕ2,1 perturbation of the M3,5 nonunitary minimal model, thus resolving apparent contradictions in the previous literature. The two behaviors correspond to changing the perturbing bare coupling constant from real values to imaginary ones. Generalizations of this picture to the whole class of nonunitary minimal models Mp,2p±1, perturbed by their least relevant operator, lead to a cascade of flows similar to that of unitary minimal models perturbed by ϕ1,3. Various aspects and generalizations of this phenomenon and the links with the Izergin–Korepin model are discussed.
We derive the fusion hierarchy of functional equations for critical A-D-E lattice models related to the sℓ ( 2 ) unitary minimal models, the parafermionic models and the supersymmetric models of conformal field theory and deduce the related TBA functional equations. The derivation uses fusion projectors and applies in the presence of all known integrable boundary conditions on the torus and cylinder. The resulting TBA functional equations are universal in the sense that they depend only on the Coxeter number of the A-D-E graph and are independent of the particular integrable boundary conditions. We conjecture generally that TBA functional equations are universal for all integrable lattice models associated with rational CFTs and their integrable perturbations. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We consider the critical nonunitary minimal model M (3, 5) with integrable boundaries and analyze the patterns of zeros of the eigenvalues of the transfer matrix and then determine the spectrum of the critical theory using the thermodynamic Bethe ansatz ( TBA ) equations. Solving the TBA functional equation satisfied by the transfer matrices of the associated A 4 restricted solid-on-solid Forrester–Baxter lattice model in regime III in the continuum scaling limit, we derive the integral TBA equations for all excitations in the ( r , s ) = (1, 1) sector and then determine their corresponding energies. We classify the excitations in terms of ( m , n ) systems .