Topological phase transitions in low-dimensional systems

TL;DR

This paper develops a general theory for topological phase transitions of the BKT type in low-dimensional systems, linking conformal invariance, vacuum homotopy groups, and crystallographic symmetries to critical properties.

Contribution

It introduces necessary and sufficient conditions for BKT-type transitions in low-dimensional systems, connecting topological excitations, conformal invariance, and group symmetries to critical behavior.

Findings

Critical properties in 2D characterized by Coxeter numbers and conformal field theories.

One-dimensional models exhibit ferromagnetic Dyson chain behavior.

Conditions d ≤ 2 and nontrivial homotopy groups are essential for BKT transitions.

Abstract

A general theory of the Berezinsky-Kosterlitz-Thouless (BKT) type phase transitions in low-dimensional systems is proposed. It is shown that in d-dimensional case the necessary conditions for it can take place are 1) conformal invariance of kinetic part of model action and 2) vacuum homotopy group $\pi_{d-1}$ must be nontrivial and discrete. It means a discrete vacuum degeneracy for $1d$ systems and continuous vacuum degeneracy for higher $d$ systems. For such systems topological exitations have logariphmically divergent energy and they can be described by corresponding effective field theories. In general case the sufficient conditions for existence of the BKT type phase transition are 1) constraint $d \le 2$ and 2) $\pi_{d-1}$ must have some crystallographic symmetries. Critical properties of possible low-dimensional effective theories are determined and it is shown that in two-dimensional case they are characterized by the Coxeter numbers $h_G$ of lattices from the series ${A,D,E,Z}$ and can be interpreted as those of conformal field theories with integer central charge $c=r,$ where $r$ is a rank of groups $\pi_1$ and $G.$ In one-dimensional case analogous critical properties have ferromagnetic Dyson chains with discrete Cartan-Ising spins. In contrast, critical properties of one-dimensional models with periodic potentials have a weak dependence on group $G.$