The fractal dimensions and the percolation exponents of geometrical spin clusters of like sign at criticality are obtained numerically for an Ising model with temperature dependent annealed bond dilution, also known as the thermalized bond Ising model (TBIM), in two dimensions. For this purpose, a modified Wolff single-cluster Monte Carlo simulation is used to generate equilibrium spin configurations on square lattices in the critical region. A tie-breaking rule is employed to identify nonintersecting spin cluster boundaries along the edges of the dual lattice. The values obtained for the fractal dimensions of the spanning geometrical clusters D c , and their interfaces D I , are in perfect agreement with those reported for the standard two-dimensional ferromagnetic Ising model. Furthermore, the variance of the winding angles results in a diffusivity κ = 3 for the two-dimensional thermalized bond Ising model, thus placing it in the universality class of the regular Ising model. A finite-size scaling analysis of the largest geometrical clusters results in a reliable estimation of the critical percolation exponents for the geometrical clusters in the limit of an infinite lattice size. The percolation exponents thus obtained are also found to be consistent with those reported for the regular Ising model. These consistencies are explained in terms of the Fisher renormalization relations, which express the thermodynamic critical exponents of systems with annealed bond dilution in terms of those of the regular model system.
The fast-forward scheme of adiabatic quantum dynamics is applied to finite regular spin clusters with various geometries and the nature of driving interactions is elucidated. Fast forward is the quasiadiabatic dynamics guaranteed by regularization terms added to the reference Hamiltonian, followed by a rescaling of time with the use of a large scaling factor. With the help of the regularization terms consisting of pairwise and three-body interactions, we apply the proposed formula [I. Setiawan et al., Phys. Rev. A 96, 052106 (2017)] to regular triangle and open linear chains for
N=3 spin systems and to triangular pyramid, square, primary star graph, and open linear chains for
N=4 spin systems. The geometry-induced symmetry greatly decreases the rank of the coefficient matrix of the linear algebraic equation for regularization terms. Choosing a transverse Ising Hamiltonian as a reference, we find, for
N=3 spin clusters, that the driving interaction consists of only the geometry-dependent pairwise interactions and there is no need for the three-body interaction and, for
N=4 spin clusters, that the geometry-dependent pairwise interactions again constitute a major part of the driving interaction, whereas the universal three-body interaction free from the geometry is necessary but plays a subsidiary role. Our scheme predicts the practical driving interaction in accelerating the adiabatic quantum dynamics of structured regular spin clusters.
A decorated lattice is suggested and the Ising model on it with three kinds of interactions K1, K2, and K3 is studied. Using an equivalent transformation, the square decorated Ising lattice is transformed into a regular square Ising lattice with nearest-neighbor, next-nearest-neighbor, and four-spin interactions, and the critical fixed point is found at K1 = 0.5769, K2 = −0.0671, and K3 = 0.3428, which determines the critical temperature of the system. It is also found that this system and the regular square Ising lattice, and the eight-vertex model belong to the same universality class.