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Physica A Statistical Mechanics and its Applications

Small-cluster study of interacting fermions on a square lattice with next-nearest-neighbour interactions

George S. Kliros

Published Apr 1, 1992• 2 citations
Publisher
DOI
10.1016/0378-4371(92)90187-u

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We show that the square two-dimensional (2D) Ising lattice with nearest- (J) and weak next-nearest-neighbour interactions (Jd ) can be mapped on a square 2D Ising lattice that has only nearest neighbour interactions (J*). For J d /J << 1 the transformation equation has the simple form . This result can be used to derive expressions for several thermodynamic functions of the square 2D Ising lattice with weak next-nearest-neighbour interactions. As an example we consider the spontaneous magnetization and compare it with low-temperature series expansion results.
The critical behaviour of the Ising model on the square lattice with ferromagnetic nearest-neighbour interactions (J) and anti-ferromagnetic next-nearest neighbour interactions (J') is discussed for small lambda =J/ mod J' mod . The singular part of the free energy is calculated to second order in lambda by perturbation theory. Near the critical temperature of the unperturbed system (J=0), this expansion is found to have a form which may be exponentiated yielding, for non-zero lambda , a non-universal critical line along which the exponents vary continuously with lambda . For small lambda , the specific heat exponent, alpha = alpha 1 lambda 2 +O( lambda 2 ) with alpha 1 approximately=1.5.