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arXiv:hep-th/0306196 · International Journal of Modern Physics A

EXCITED BOUNDARY TBA IN THE TRICRITICAL ISING MODEL

Giovanni Feverati, Paul A. Pearce, Francesco Ravanini

Published May 1, 2004• 4 citations
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Abstract

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.

DOI
10.1142/s0217751x04020373

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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.
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.
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.
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 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 .
The tricritical Ising model perturbed by the subleading energy operator [Formula: see text] was known to be an integrable scattering theory of massive kinks,14 and in fact it preserves supersymmetry. We consider here the model defined on the half-plane with a boundary and compute the associated factorizable boundary S matrix. The conformal boundary conditions of this model are identified and the corresponding S matrices are found. We also show how some of these S matrices can be perturbed and generate “flows” between different boundary conditions.
Exact renormalization-group recursion relations are derived in closed form for a one-dimensional spin-1 Ising model. The recursion relations possess tricritical and critical fixed points. We have studied the flow diagram corresponding to the renormalization-group transformations, and we have linearized about the fixed points. The predicted asymptotic homogeneity is shown to be satisfied by the true free energy, which has been computed using the transfer matrix. The pseudocritical singularities, existing in the zero-temperature limit, are marked by a double degeneracy of the largest eigenvalue of the transfer matrix, and the pseudotricritical point is marked by a triple degeneracy. The increased instability of the tricritical fixed point, relative to the critical fixed points, is shown to be directly related to the eigenvalue degeneracy of the transfer matrix being greater at a tricritical point than at a critical point.