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arXiv:math-ph/0110021 · Journal of Mathematical Physics

The dilute A4 model, the E7 mass spectrum and the tricritical Ising model

Katherine Seaton, Murray T. Batchelor

Published May 1, 2002• 2 citations
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Abstract

The exact perturbation approach is used to derive the (seven) elementary correlation lengths and related mass gaps of the two-dimensional dilute A4 lattice model in regime 2− from the Bethe Ansatz solution. This model provides a realization of the integrable φ(1,2) perturbation of the c=710 conformal field theory, which is known to describe the off-critical thermal behavior of the tricritical Ising model. The E7 masses predicted from purely elastic scattering theory follow in the approach to criticality. Universal amplitudes for the tricritical Ising model are calculated.

DOI
10.1063/1.1465515

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The exact perturbation approach is used to derive the (seven) elementary correlation lengths and related mass gaps of the two-dimensional dilute A4 lattice model in regime 2− from the Bethe Ansatz solution. This model provides a realization of the integrable φ(1,2) perturbation of the c=710 conformal field theory, which is known to describe the off-critical thermal behavior of the tricritical Ising model. The E7 masses predicted from purely elastic scattering theory follow in the approach to criticality. Universal amplitudes for the tricritical Ising model are calculated.
The dilute A3{\mathrm{A}}_{3}A3​ model is a solvable interaction round a face model with three local states and adjacency conditions encoded by the Dynkin diagram of the Lie algebra A3{\mathrm{A}}_{3}A3​. It can be regarded as a solvable spin-1 Ising model at the critical temperature in a magnetic field. One therefore expects the scaling limit to be governed by Zamolodchikov's integrable perturbation of the c=1/2 conformal field theory. Indeed, a recent thermodynamic Bethe ansatz approach succeeded in unveiling the corresponding E8{\mathrm{E}}_{8}E8​ structure under certain assumptions on the nature of the Bethe ansatz solutions. In order to check these conjectures, we perform a detailed numerical investigation of the solutions of the Bethe ansatz equations for the critical and off-critical models. Scaling functions for the ground-state corrections and for the lowest spectral gaps are obtained, which give very precise numerical results for the lowest mass ratios in the massive scaling limit. While these agree perfectly with the E8{\mathrm{E}}_{8}E8​ mass ratios, we observe one state that seems to violate the assumptions underlying the thermodynamic Bethe ansatz calculation. We also analyze the critical spectrum of the dilute A3{\mathrm{A}}_{3}A3​ model, which exhibits excitations with a finite gap on top of the massless spectrum of the Ising conformal field theory.
The dilute A_3 model is a solvable IRF (interaction-round-a-face) model with three local states and adjacency conditions encoded by the Dynkin diagram of the Lie algebra A_3. It can be regarded as a solvable version of a critical Ising model in a magnetic field. One therefore expects the scaling limit to be governed by Zamolodchikov's integrable perturbation of the c=1/2 conformal field theory. We perform a detailed numerical investigation of the solutions of the Bethe ansatz equation for the off-critical model. Our results agree perfectly with the predicted values for the lowest masses of the stable particles and support the assumptions on the nature of the Bethe ansatz solutions which enter crucially in a recent thermodynamic Bethe ansatz calculation of the factorized scattering matrix.
We give further support to Smirnov's conjecture on the exact kink S-matrix for the massive Quantum Field Theory describing the integrable perturbation of the c=0.7 minimal Conformal Field theory (known to describe the tri-critical Ising model) by the operator phi2,1\\phi_{2,1}phi2,1​. This operator has conformal dimensions (7/16,7/16)(7/16,7/16)(7/16,7/16) and is identified with the subleading magnetic operator of the tri-critical Ising model. In this paper we apply the Thermodynamic Bethe Ansatz (TBA) approach to the kink scattering theory by explicitly utilising its relationship with the solvable lattice hard hexagon model. Analytically examining the ultraviolet scaling limit we recover the expected central charge c=0.7 of the tri-critical Ising model. We also compare numerical values for the ground state energy of the finite size system obtained from the TBA equations with the results obtained by the Truncated Conformal Space Approach and Conformal Perturbation Theory.
The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.