Generalized BKT Transitions

When \(W=1\) the \(U(1)\) vortices are permitted and the \(U(1)_W\) is lost. With \(W>1\), however, we can maintain a \(\mathbb{Z}_W\) global symmetry. The mixed ‘t Hooft anomaly between this \(\mathbb{Z}_W\) and the \(U(1)_S\) shift symmetry guarantees that we must find an ordered state regardless of coupling \(\kappa\).

Because we get the ‘t Hooft anomaly correct at finite lattice spacing we know that we must have order at any \(\kappa\). At high \(\kappa\) we have a CFT while at low \(\kappa\) we have spontaneous breaking of the \(\mathbb{Z}_W\) symmetry. There is a critical \(W\)-dependent \(\kappa_c\) separating these two phases. In the \(W=1\) BKT case \(\kappa_c \approx 0.74\) [8].

Showing that the modified Villain action really achieves this can be understood as a numerical demonstation that the maintenance of the continuum symmetris and ‘t Hooft anomalies.

Generalized BKT Transitions and Persistent Order on the Lattice

Reference [9] is a prelminary discussion of the phase transition at \(W\geq1\).

The BKT transition in low-dimensional systems with a \(U(1)\) global symmetry separates a gapless conformal phase from a trivially gapped, disordered phase, and is driven by vortex proliferation. Recent developments in modified Villain discretizations provide a class of lattice models which have a \(\mathbb{Z}_W\) global symmetry that counts vortices mod W, mixed ‘t Hooft anomalies, and persistent order even at finite lattice spacing. While there is no fully-disordered phase (except in the original BKT limit \(W=1\)) there is still a phase boundary which separates gapped ordered phases from gapless phases. I’ll describe a numerical Monte Carlo exploration of these phenomena.