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

Tolga Çağlar and A. Nihat Berker

Phys. Rev. E 84, 051129 – Published 28 November 2011

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.

Received 24 September 2011

DOI:https://doi.org/10.1103/PhysRevE.84.051129

Authors & Affiliations

Tolga Çağlar^1,2 and A. Nihat Berker^1,3

¹Faculty of Engineering and Natural Sciences, Sabancı University, Orhanlı, Tuzla 34956, Istanbul, Turkey
²Department of Physics, Koç University, Sarıyer 34450, Istanbul, Turkey
³Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

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Vol. 84, Iss. 5 — November 2011

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Images

Figure 1
For the $d = 3$ anisotropic Ising model, magnetizations $m_{i}$ versus $x y$ layer-number $i$ curves for different temperatures $1 / J_{x y}$ . Each panel shows results for the indicated anisotropy $J_{z} / J_{x y}$ . The curves in each panel, with decreasing sharpness, are for temperatures $1 / J_{x y} = 1, 3, 5, 6$ . In the upper panels, the high-temperature curves coincide with the horizontal line $m_{i} = 0$ .Reuse & Permissions
Figure 2
Local magnetization data for the $d = 3$ anisotropic Ising model. The curves, starting from the high-temperature side, are for anisotropies $J_{z} / J_{x y} = 10, 5, 2, 1, 0.5, 0.2$ . Upper panel: Magnetization absolute values $| m_{b} |$ away from the interface as a function of temperature $1 / J_{x y}$ , for different values of the anisotropy $J_{z} / J_{x y}$ . Lower panel: The deviation $| m_{b} | - | m_{i} |$ averaged over the system versus temperature $1 / J_{x y}$ for different anisotropies $J_{z} / J_{x y}$ . This averaged deviation vanishes when the interface is smooth. Note the qualitatively different low-temperature behavior in the $d = 2$ case shown in Fig. 4.Reuse & Permissions
Figure 3
For the $d = 3$ anisotropic Ising model, the calculated phase diagram showing the disordered, ordered with rough interface, and ordered with smooth interface phases. The squares indicate the exact ordering temperatures from duality at $J_{z} / J_{x y} = 0$ and from Ref. [19] at $J_{z} / J_{x y} = 1$ . The circle indicates the roughening transition temperature for the solid-on-solid limit $J_{z} / J_{x y} \to \infty$ [2]. The roughening transition is obtained by fitting the averaged deviation curves (lower panel of Fig. 2) within the range $〈 | m_{b} | - | m_{i} | 〉 = 0.01$ to 0.04, to find the temperature at which the averaged deviation reaches zero, meaning that the interface becomes localized between two consecutive layers, reversing the sign of the magnetization $m_{b}$ with no change in magnitude.Reuse & Permissions
Figure 4
For the $d = 2$ anisotropic Ising model, the deviation $| m_{b} | - | m_{i} |$ averaged over the system versus temperature $1 / J_{x y}$ for different anisotropies $J_{z} / J_{x y}$ . The curves, starting from the high-temperature side, are for anisotropies $J_{z} / J_{x y} = 10, 5, 2, 1, 0.5, 0.2$ . It is seen that the deviation does not vanish, i.e., the interface does not localize, down to zero temperature. Thus, a qualitatively different low-temperature behavior occurs, as compared with the $d = 3$ case shown in the lower panel of Fig. 2.Reuse & Permissions
Figure 5
For the $d = 2$ anisotropic Ising model, the phase diagram showing the disordered phase and the ordered phase with rough interface. The dashed curve is the exact ordering boundary $\sinh (2 J_{x}) \sinh (2 J_{z}) = 1$ obtained from duality. No ordered phase with smooth interface is found.Reuse & Permissions

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