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Generalized BKT Transitions
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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 $\kappa_c \approx 0.74$ :cite:`Janke:1993va`.

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
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Reference :cite:`Berkowitz:2024iuv` 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.