Open-boundary transverse-field Ising model admits exact invertible duality, invertible critical symmetry, and spontaneous duality breaking from differing physical sensitivities to local perturbations.
Witten, String theory dynamics in various dimensions, Nuclear Physics B443, 85 (1995)
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Invertible Symmetry and Spontaneous Duality Breaking in the Transverse-Field Ising Model
Open-boundary transverse-field Ising model admits exact invertible duality, invertible critical symmetry, and spontaneous duality breaking from differing physical sensitivities to local perturbations.