Interface-roughening phase diagram of the three-dimensional Ising model for all interaction anisotropies from hard-spin mean-field theory

Abstract

The roughening phase diagram of the d=3 Ising model with uniaxially anisotropic interactions is calculated for the entire range of anisotropy, from decoupled planes to the isotropic model to the solid-on-solid model, using hard-spin mean-field theory. The phase diagram contains the line of ordering phase transitions and, at lower temperatures, the line of roughening phase transitions, where the interface between ordered domains roughens. Upon increasing the anisotropy, roughening transition temperatures settle after the isotropic case, whereas the ordering transition temperature increases to infinity. The calculation is repeated for the d=2 Ising model for the full range of anisotropy, yielding no roughening transition.

Successively thresholded domain boundary roughening driven by pinning centers and missing bonds: Hard-spin mean-field theory applied tod=3Ising magnets

“…In our current study, hard-spin mean-field theory [1,2], which has been qualitatively and quantitatively successful in frustrated and unfrustrated, equilibrium and nonequilibrium magnetic ordering problems [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], including recently the interface roughening transition [3], is used to study the roughening of an interface by quenched random pinning center sites or missing bonds. The self-consistency equations of hard-spin mean-field theory [2] are…”

“…This flip will then happen for 2J z = (q − q )J xy , where again q and q are the numbers of xy neighbors bonded to the flipping spin that are, respectively, +1 and −1. The possible values are (q,q ) = (4,0), (3,1), giving the threshold values of J z /J xy = 1,2, calculationally seen in the bottom panels of Figs. 2 and 3.…”

“…Hard-spin mean-field theory [1,2] has recently been applied to Ising magnets, correctly yielding the absence and presence of an interface roughening transition respectively in d = 2 and d = 3 dimensions and producing the ordering-roughening phase diagram for isotropic and anisotropic systems [3]. The approach is now extended to the effects of quenched random pinning centers and missing bonds on the interface of isotropic and uniaxially anisotropic Ising models in d = 3.…”

Critical temperature in 2D square Ising model with anisotropic coupling constant using Kramers-Wannier duality

“…It is why some MCS, such as the one by Preis, et al [11], demonstrated the use of better hardware, thereby significantly reducing simulation times. Here, we observe solutions that incorporate MFA along one principal direction [7] and hard-spin mean field approximation (HSMFA) [8] compared with the exact solution from KWD.…”

“…Below the curve is the ordered phase where most of the spins are parallel. As T approaches 𝑇 & , the spins become more randomized eventually reaching the disordered phase where half of the spins points We also compare the behaviour of the curve to the solutions obtained by incorporating MFA along one principal direction [7] and HSMFA [8]. The mean-field theory ignores the interaction between neighbouring edges but instead considers the average behaviour of the spins.…”