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arXiv:hep-th/0312201 · Journal of Statistical Mechanics Theory and Experiment

Exact (d) (+)&( ) boundary flow in the tricritical Ising model

Giovanni Feverati

Published Mar 17, 2004• 4 citations
arXivHTMLPDF

Abstract

The integrable perturbation of the degenerate boundary condition ( d ) by the φ 1,3 boundary field generates a renormalization group flow down to the superposition of Cardy boundary states (+)&(−). Exact thermodynamic Bethe ansatz (TBA) equations for all the excited states are derived here extending the results of Feverati et al (2003 Nucl. Phys. B 675 469) to this case. As an intermediate step, the non-Cardy boundary conformal sector (+)&(−) is also described as the scaling limit of an A 4 lattice model with appropriate integrable boundary conditions and produces the first example of superposition of finitized Virasoro characters.

DOI
10.1088/1742-5468/2004/03/p001

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The integrable perturbation of the degenerate boundary condition ( d ) by the φ 1,3 boundary field generates a renormalization group flow down to the superposition of Cardy boundary states (+)&(−). Exact thermodynamic Bethe ansatz (TBA) equations for all the excited states are derived here extending the results of Feverati et al (2003 Nucl. Phys. B 675 469) to this case. As an intermediate step, the non-Cardy boundary conformal sector (+)&(−) is also described as the scaling limit of an A 4 lattice model with appropriate integrable boundary conditions and produces the first example of superposition of finitized Virasoro characters.
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.
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 ξ and η which play the role of the perturbing boundary fields ϕ1,3 and ϕ1,2 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.
By considering the continuum scaling limit of the A4 RSOS lattice model of Andrews-Baxter-Forrester with integrable boundaries, we derive excited state TBA equations describing the boundary flows of the tricritical Ising model. Fixing the bulk weights to their critical values, the integrable boundary weights admit two parameters ξ and η which play the role of the perturbing boundary fields φ1,3 and φ1,2 and induce the renormalization group flow between boundary fixed points. The boundary TBA equations determining the RG flows are derived in the [Formula: see text] example. The induced map between distinct Virasoro characters of the theory are specified in terms of distribution of zeros of the double row transfer matrix.
Starting from the lattice A 3 realization of the Ising model defined on a strip with integrable boundary conditions, the exact spectrum (including excited states) of all the local integrals of motion is derived in the continuum limit by means of thermodynamic Bethe ansatz techniques. It is also possible to follow the massive flow of this spectrum between the UV c = 1/2 conformal fixed point and the massive IR theory. The UV expression of the eigenstates of such integrals of motion in terms of Virasoro modes is found to have only rational coefficients and their fermionic representation turns out to be simply related to the quantum numbers describing the spectrum.
Abstract We consider perturbations of 2D CFTs by multiple relevant operators. The massive phases of such perturbations can be labeled by conformal boundary conditions. Cardy’s variational ansatz approximates the vacuum state of the perturbed theory by a smeared conformal boundary state. In this paper we study the limitations and propose generalisations of this ansatz using both analytic and numerical insights based on TCSA. In particular we analyse the stability of Cardy’s ansatz states with respect to boundary relevant perturbations using bulk-boundary OPE coefficients. We show that certain transitions between the massive phases arise from a pair of boundary RG flows. The RG flows start from the conformal boundary on the transition surface and end on those that lie on the two sides of it. As an example we work out the details of the phase diagram for the Ising field theory and for the tricritical Ising model perturbed by the leading thermal and magnetic fields. For the latter we find a pair of novel transition lines that correspond to pairs of RG flows. Although the mass gap remains finite at the transition lines, several one-point functions change their behaviour. We discuss how these lines fit into the standard phase diagram of the tricritical Ising model. We show that each line extends to a two-dimensional surface ξσ,c in a three coupling space when we add perturbations by the subleading magnetic field. Close to this surface we locate symmetry breaking critical lines leading to the critical Ising model. Near the critical lines we find first order phase transition lines describing two-phase coexistence regions as predicted in Landau theory. The surface ξσ,c is determined from the CFT data using Cardy’s ansatz and its properties are checked using TCSA numerics.
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 .