The Pinning of a Domain Wall by Weakened Bonds in Two Dimensions
The effect of thermal fluctuations on an interface between two co-existing phases can be understood as an example of the Hohenberg, Mermin and Wagner theorem . The interface between a liquid and its vapour, in the absence of a stabilising force such as gravity, provides an instance of a two-dimensional system with a continuous symmetry, under translations Vertically’. There is therefore no long-range order in the interfacial position; the interface is roughened by capillary waves, which are the relevant Goldstone modes. These considerations do not apply to a domain wall in the three-dimensional Ising model since the presence of a lattice renders the translational symmetry discrete. In this case it has been proven  that below the critical temperature of a corresponding two-dimensional Ising system the interface is localised near the ground-state position, whilst a range of evidence (low temperature series expansion , computer simulation  and mappings to other models ) suggests that just above this temperature, and well below the bulk critical point, the interface has a roughening transition to a diffuse phase dominated by capillary waves.
KeywordsDomain Wall Ising Model Weakened Bond Translational Symmetry Capillary Wave
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