Lattice Quantum Villain Hamiltonians: Compact scalars, $U(1)$ gauge theories, fracton models and Quantum Ising model dualities

Abstract

We construct Villain Hamiltonians for compact scalars and abelian gauge theories. The Villain integers are promoted to integral spectrum operators, whose canonical conjugates are naturally compact scalars. Further, depending on the theory, these conjugate operators can be interpreted as (higher-form) gauge fields. If a gauge symmetry is imposed on these dual gauge fields, a natural constraint on the Villain operator leads to the absence of defects (e.g. vortices, monopoles,...). These lattice models therefore have the same symmetry and anomaly structure as their corresponding continuum models. Moreover they can be formulated in a way that makes the well-know dualities look manifest, e.g. a compact scalar in 2d has a T-duality, in 3d is dual to a U(1) gauge theory, etc. We further discuss the gauged version of compact scalars on the lattice, its anomalies and solution, as well as a particular limit of the gauged XY model at strong coupling which reduces to the transverse-field Ising model. The construction for higher-form gauge theories is similar. We apply these ideas to the constructions of some models which are of interest to fracton physics, in particular the XY-plaquette model and the tensor gauge field model. The XY-plaquette model in 2+1d coupled to a tensor gauge fields at strong gauge coupling is also exactly described by a transverse field quantum $J_1-J_2$ Ising model with $J_1=2J_2$, and discuss the phase structure of such models.

Accepted for publication in JHEP. 38 pages, 4 figures

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Cited by in corpus (12)

• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• Exact emergent higher-form symmetries in bosonic lattice models
• 2+1d Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Quantized axial charge of staggered fermions and the chiral anomaly
• Lattice T-duality from non-invertible symmetries in quantum spin chains
• Modified Villain formulation of abelian Chern-Simons theory
• Duality transformations and the entanglement entropy of gauge theories
• Hamiltonian Lattice Formulation of Compact Maxwell-Chern-Simons Theory
• Exotic theta terms in 2+1d fractonic field theory