Magnetic Correlation Length and Universal Amplitude of the Lattice E8 Ising Model

The perturbation approach is used to derive the exact correlation length ξ of the dilute AL lattice models in regimes 1 and 2 for L odd. In regime 2 the A3 model is the E8 lattice realisation of the two-dimensional Ising model in a magnetic field h at T = Tc. When combined with the singular part fs of the free energy the result for the A3 model gives the universal amplitude fs ξ 2 = 0.061 728 . . . as h → 0 in precise agreement with the result obtained by Delfino and Mussardo via the form-factor bootstrap approach. The integrable E8 quantum field theory of Zamolodchikov [1, 2] is known to be in the same universality class as the two-dimensional Ising model in a magnetic field at T = Tc. Moreover, an integrable lattice realisation of the E8 Ising model is provided by the dilute A3 model [3, 4], upon which explicit exact and numerical calculations pertaining to the Ising model in a magnetic field can be performed [3-13]. In this letter we present the correlation length of the dilute AL lattice models in regimes 1 and 2 for L odd, for which the off-critical perturbation is magnetic-like. This includes the magnetic correlation length for L = 3, of relevance to the magnetic Ising model at T = Tc. The dilute AL model is an exactly solvable, restricted solid-on-solid model defined on the square lattice. Each site of the lattice can take one of L possible (height) values, subject to the restriction that neighbouring sites of the lattice either have the same height, or differ by ±1. The Boltzmann weights of the allowed height configurations of an elementary face of the lattice are [3, 4]