Inhomogeneous Ising models with diagonal structure on a square lattice

W. F. Wolff, P. Hoever, J. Zittartz

Ken‐Yu Lin, Shang‐Fan Lee

Daniel Mba

S. Boukraa, S. Hassani, J.-M. Maillard +2 more

We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the square lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions χ (1) d and χ (2) d of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution χ (5) d , but only modulo a given prime. We use these exact linear differential equations to show that not only the Russian-doll structure but also the direct sum structure on the linear differential operators for the n -particle contributions χ ( n ) d are quite directly inherited from the direct sum structure on the form factors f ( n ) . We show that the n th particle contributions χ ( n ) d have their singularities at roots of unity. These singularities become dense on the unit circle |sinh 2 E v / kT sinh 2 E h / kT | = 1 as n → ∞ .

Ladislav Šamaj

C J Hamer

Energy eigenvalues and order parameters are calculated by exact diagonalization for the transverse Ising model on square lattices of up to 6×6 sites. Finite-size scaling is used to estimate the critical parameters of the model, confirming universality with the three-dimensional classical Ising model. Critical amplitudes are also estimated for both the energy gap and the ground-state energy.