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Abstract. The anisotropic surface free energy, Andreev surface free energy, and equilibrium crystal shape (ECS) z = z(x, y) are calculated numerically using a transfer matrix approach with the density matrix renormalization group (DMRG) method. The adopted surface model is a restricted solid-on-solid (RSOS) model with "sticky" steps, i.e., steps with a point-contact type attraction between them (p-RSOS model). By analyzing the results, we obtain a first-order shape transition on the ECS profile around the (111) facet; and on the curved surface near the (001) facet edge, we obtain shape exponents having values different from those of the universal GruberMullins-Pokrovsky-Talapov (GMPT) class. In order to elucidate the origin of the non-universal shape exponents, we calculate the slope dependence of the mean step height of "step droplets" (bound states of steps) n(p) using the Monte Carlo method, where p = (∂z/∂x, ∂z/∂y), and · represents the thermal average. Using the result of the |p| dependence of n(p) , we derive a |p|-expanded expression for the non-universal surface free energy f eff (p), which contains quadratic terms with respect to |p|. The first-order shape transition and the non-universal shape exponents obtained by the DMRG calculations are reproduced thermodynamically from the non-universal surface free energy f eff (p).

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Faceting diagrams for the step-faceting zone, the step droplet zone, and the GruberMullins-Pokrovsky-Talapov (GMPT) zone for a crystal surface are obtained by using the density matrix renormalization group method to calculate the surface tension. The model based on these calculations is the restricted solid-on-solid (RSOS) model with a point-contact-type step-step attraction (p-RSOS model) on a square lattice. The point-contact-type step-step attraction represents the energy gain obtained by forming a bonding state with orbital overlap at the meeting point of the neighboring steps. In the step-faceting zone, disconnectedness in the surface tension leads to the formation of a faceted macrostep on a vicinal surface at equilibrium. The disconnectedness in the surface tension also causes the first-order shape transition for the equilibrium shape of a crystal droplet. The lower zone boundary line (ZBL), which separates the step-faceting zone and the step droplet zone, is obtained by the condition γ 1 = lim n→ ∞ γ n /n, where γ n is the step tension of the n-th merged step. The upper ZBL, which separates the GMPT zone and the step droplet zone, is obtained by the condition A q,eff = 0 and B q,eff = 0, where A q,eff and B q,eff represent the coefficients for the |⃗ q| 2 term and the |⃗ q| 3 term, respectively, in the |⃗ q|-expanded form of the surface free energy f eff (⃗ q). Here, ⃗ q is the surface gradient relative to the (111) surface. The reason why the vicinal surface inclined in the ⟨101⟩ direction does not exhibit stepfaceting is explained in terms of the one-dimensional spinless quasi-impenetrable attractive bosons at absolute zero. C 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license

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