SOS Model

Model

\[H_{S O S}=\sum_{\langle x, y\rangle} V\left(\left|h_{x}-h_{y}\right|\right)\] where \(h_{x}\) is integer.

• Absolute value SOS model \[ H_{A S O S}=\beta_{A S O S} \sum_{\langle x, y\rangle}\left|h_{x}-h_{y}\right| \]

• Discrete Gaussian SOS model (dual of the Villian model): \[ H_{D G S O S}=\frac{\beta_{D G S O S}}{2} \sum_{\langle x, y \rangle}\left(h_{x}-h_{y}\right)^{2} \] looks exactly like the continuous Gaussian model, the difference is that \(h_x\) is integer values introduces a nontrivial interaction.

• Body Centered Solid-On-Solid (BCSOS) or F-model \[ H_{BCSOS}=K^{BCSOS} \sum_{[x, y]} \left| h_{x}-h_{y} \right| \] a 2D lattice splits in two sublattices. spins on “odd” lattice sites take values of the form \(2n + 1/2\) , and spins on “even” sites are of the form \(2n − 1/2\), \(n\) integer. The nearest neighbour spins \(h_x\) and \(h_y\) obey the constraint \(|h_x − h_y|=1\).

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General Picture

For finite positive \(K\)

• smooth phase: \(K\) is large enough, all \(h\) tends to be same, the fluctuation is very small.
• critical value:
• rough:

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• Strong coupling are mapped to weak couplings and vice versa.

• discrete height variables : the height of a crystal surface (measured in atoms)

• 2D: BCSOS or (in yet another guise) the 6-vertexmodel

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XY model with Villain action

\[ Z_{V}=\int_{-\pi}^{\pi} \prod_{x} d \Theta_{x} \prod_{<x, y>} B\left(\Theta_{x}-\Theta_{y}\right) \] with \[ B(\Theta ) = \sum_{p=-\infty}^{\infty} \exp \left(-\frac{1}{2} \beta_{V}(\Theta-2 \pi p)^{2}\right) \]