A Broadband, Solid State Millimeter-Wave Synthesizer

Critical behavior of the geometrical spin clusters and interfaces in the two-dimensional thermalized bond Ising model

“…Thus, despite the correlation length exponent ν G = 1, we note that the geometrical clusters are too compact to represent the critical droplets, and the exponent β G = 5/96 differs appreciably from the corresponding thermal exponent β = 1/8, associated with the vanishing of the magnetization order parameter at T c . The value of the correlation length (also known as the connectivity length) exponent ν G = 1, for the geometrical clusters of the two-dimensional TBIM, is in excellent agreement with the values obtained from a real-space renormalization group analysis [1,7], high-temperature series expansion studies [26], and precision numerical simulations of the geometrical clusters of the standard two-dimensional Ising model [2,27]. This result, however, opens a question about a near perfect collapse onto a universal function for the same data obtained for the regular Ising model, but with a different exponent 15 8 [21].…”

“…Using the finite-size scaling ansatz as given in equation (11), the correlation length exponent ν G for the emerging spanning cluster, is estimated from a scaling plot of L −Dc M(L) against L 1/ν G (β/β c − 1) as shown in figure 6. By varying ν G , and evaluating the quality of the data collapse, our best estimate of the correlation length exponent for the geometrical clusters is ν G = 1.01 (2). Finally, the slope of the log-log plot of P ∞ versus L results in β G = 0.051(3), as shown in figure 7, which fits well into the hyperscaling relation for the percolation exponents in d spatial dimensions D c = d − β G ν G .…”

“…It is well known that the geometrical spin clusters (i.e., the clusters composed of the neighboring spins of the same sign), undergo a percolation transition at the thermodynamic critical temperature T c in the two-dimensional Ising model [1]. This type of behavior is believed to be generally valid for a variety of two-dimensional critical models that undergo a continuous phase transition such as the q-state Potts model (q ≤ 4) [2], which is a q-state generalization of the Ising model (q = 2) [3]. The formation of spanning spin clusters and their characteristic percolation exponents, can also be used to characterize the universality class of the corresponding thermal phase transition.…”