Melting in two dimensions with particle conservation
The standard analysis of two-dimensional melting assumes that the low energy states for the solid are represented by the ensemble of all configurations of a gas of dislocations. However, particle conservation is a constraint that confines the system to a subset of this ensemble. The chief implication of imposing the particle conservation constraint is that the bulk modulus has finite renormalization, and the correlation exponent at criticality is ν = 0.3527 . . . . PACS numbers: 05.70.Fh, 64.60.Cn, 82.65.Dp Typeset using REVTEX I. DISLOCATIONS AND TWO DIMENSIONAL MELTING At sufficiently low temperatures, two-dimensional matter can exhibit a solid phase, characterized by long-ranged orientational order and finite shear modulus. At higher temperatures there is a liquid phase, in which the shear modulus vanishes. Nelson and Halperin [1] and Young [2], building on the seminal work of Kosterlitz and Thouless [3], have given a theory of this transition in which dislocations play a central role [4]. A dislocation in two dimensions is a point topological defect, characterized by a location ~r and a Burgers pseudovector ~b, which is the discontinuity in the (multiplyvalued) displacement field ~u(~r) along a branch line radiating from the defect, due to one or more lines of atoms having come to an end at ~r. We can equally well characterize it by the orientation ~c = (by,−bx)/|b|, which indicates the direction of the missing line of atoms. Free dislocations can relax shear stress by moving apart. At low temperatures this is prevented by the interaction between dislocations, which has the form [5]