The criticality of the (2+1)-dimensional XY model is investigated with the numerical diagonalization method. So far, it has been considered that the diagonalization method would not be very suitable for analyzing the criticality in large dimensions (
d ge 3); in fact, the tractable system size with the diagonalization method is severely restricted. In this paper, we employ Novotnys method, which enables us to treat a variety of system sizes N=6,8,...,20 (N: the number of spins constituting a cluster). For that purpose, we develop an off-diagonal version of Novotnys method to adopt the off-diagonal (quantum-mechanical XY) interaction. Moreover, in order to improve the finite-size-scaling behavior, we tune the coupling-constant parameters to a scale-invariant point. As a result, we estimate the critical indices as u=0.675(20) and gamma/ u=1.97(10).
The low-lying spectrum of the three-dimensional Ising model is investigated numerically; we made use of an equivalence between the excitation gap and the reciprocal correlation length. In the broken-symmetry phase, the magnetic excitations are attrac
tive, forming a bound state with an excitation gap m_2(<2m_1) (m_1: elementary excitation gap). It is expected that the ratio m_2/m_1 is a universal constant in the vicinity of the critical point. In order to estimate m_2/m_1, we perform the numerical diagonalization for finite clusters with N le 15 spins. In order to reduce the finite-size errors, we incorporated the extended (next-nearest-neighbor and four-spin) interactions. As a result, we estimate the mass-gap ratio as m_2/m_1=1.84(3).
Multicriticality of the gonihedric model in 2+1 dimensions is investigated numerically. The gonihedric model is a fully frustrated Ising magnet with the finely tuned plaquette-type (four-body and plaquette-diagonal) interactions, which cancel out the
domain-wall surface tension. Because the quantum-mechanical fluctuation along the imaginary-time direction is simply ferromagnetic, the criticality of the (2+1)-dimensional gonihedric model should be an anisotropic one; that is, the respective critical indices of real-space (perp) and imaginary-time (parallel) sectors do not coincide. Extending the parameter space to control the domain-wall surface tension, we analyze the criticality in terms of the crossover (multicritical) scaling theory. By means of the numerical diagonalization for the clusters with Nle 28 spins, we obtained the correlation-length critical indices ( u_perp, u_parallel)=(0.45(10),1.04(27)), and the crossover exponent phi=0.7(2). Our results are comparable to ( u_{perp}, u_{parallel})=(0.482,1.230), and phi=0.688 obtained by Diehl and Shpot for the (d,m)=(3,2) Lifshitz point with the epsilon-expansion method up to O(epsilon^2).
