Topological Phase Transitions

Abstract

Previously, we have seen that the breaking of a continuous symmetry is accompanied by the appearance of massless Goldstone modes. Fluctuations of the latter lead to the destruction of long-range order at any finite temperature in dimensions d ≤ 2 — the Mermin-Wagner theorem. However, our perturbative analysis revealed only a power-law decay of spatial correlations in precisely two-dimensions — " quasi long-range order " . Such cases admit the existence of a new type of continuous phase transition driven by the proliferation of topological defects. The aim of this section is to discuss the phenomenology of this type of transition which lies outside the usual classification scheme. In classifying states of condensed matter, we usually consider two extremes: on the one hand there are crystalline solids in which atoms form a perfectly periodic array that extends to infinity in all directions. Such phases are said to possess long-range order (LRO). On the other hand there are fluids or glasses, in which the atoms are completely disordered and the system is both orientationally and positionally isotropic — that is the materials look the same when viewed from any direction. However, an intermediate state of matter is possible. In such a state the atoms are distributed at random, as in a fluid or glass, but the system is orientationally anisotropic on a macroscopic scale, as in a crystalline solid. This means that some properties of the fluid are different in different directions. Order of this sort is known as bond-orientational order. This type of quasi long-range order is manifest in properties of superfluid and su-perconducting films (i.e. two-dimensions) and in the crystallisation properties of fluid membranes. As we have seen, according to the Mermin-Wagner theorem, fluctuations of a two-component or complex order parameter destroy LRO at all finite temperatures. However, at temperatures below T c , quasi-LRO is maintained. The nature of this topo-logical phase transition was first resolved by Berezinskii (Sov. Phys. JETP 32, 493, (1971)) and later generalised to encompass a whole class of systems by Kosterlitz and Thouless 1 (J. Phys. C 5, L124 (1972); 6, 1181 (1973)). These include the melting of a two-dimensional crystal, with dislocations taking the place of vortices (Halperin and Nelson, Phys. Rev. Lett. 41, 121 (1978)). In this chapter, we will exploit a magnetic analogy to explore this unconventional type of phase transition which is driven by the condensation of topological defects known 1