Dislocation interaction in an adsorbate solid near the commensurate-incommensurate transition

The free energy of interaction of a pair of point dislocations, in an adsorbed crystalline layer whose periodicity, in one direction, differs very slightly from an integer $p$ times the substrate periodicity, is related to the logarithm of a correlation function $〈0\left|Q^{†}\right(x, t\left)Q\right(x^{\prime }, t^{\prime }\left)\right|0\mathrm{}〉$, where $|0\mathrm{}〉$ is the quantum-mechanical ground state of a set of noninteracting fermions in one space dimension, and $Q^{†}(x, t)$ is an operator which inserts $p$ fermions near point $x$ and time $t$, and which multiplies the wave function by ${\mathrm{}(-1\mathrm{})\mathrm{}}^{pN(x, t)}$, where $N(x, t)$ is the number of fermions to the left of point $x$. The correlation function is evaluated at large separations, and the resulting dislocation interaction is consistent with previous calculations of the long-wavelength elastic constants of the adsorbate. In particular we confirm that the adsorbate lattice, in this almost-commensurate regime, is unstable to formation of free dislocations for $p<\mathrm{}\sqrt{8}$, according to the Kosterlitz-Thouless criterion.