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Lattice Quantum Villain Hamiltonians: Compact scalars, $U(1)$ gauge theories, fracton models and Quantum Ising model dualities
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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.
Forward citations
Cited by 5 Pith papers
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$\mathrm{SL}(2,\mathbb{Z})$ Theta Subgroup Structure of Maxwell theory in the Lattice Villain Hamiltonian Formulation
Lattice Villain Maxwell theory realizes the theta subgroup of SL(2,Z) via Hamiltonian operators S and T2, with charge exchange, the Witten effect, and a non-invertible defect with Tambara-Yamagami fusion.
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Exactly Solvable 1+1d Chiral Lattice Gauge Theories
Anomaly-free chiral U(1) gauge theories in 1+1 dimensions can be written as quadratic, exactly solvable lattice Hamiltonians, with the 34-50 model as an explicit example.
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Bosonization versus the Nielsen-Ninomiya theorem
The 2D modified Villain model's Weyl operators reproduce free Dirac correlations in the continuum, and their no-doubler Dirac kernel is non-local, showing exactly how bosonization evades the Nielsen-Ninomiya theorem.
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2d Cardy-Rabinovici model with the modified Villain lattice: Exact dualities and symmetries
The 2d Cardy-Rabinovici model is constructed on a modified Villain lattice, making rescaled theta periodicity and strong-weak duality exact at finite spacing and giving a symmetry-based phase classification.
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U(1)-gauged 2-flavor spin system in 3-D
A U(1)-gauged two-flavor spin model on a 3D lattice shows a transition that is likely weakly first order, passing near a multi-critical point, based on preliminary Monte Carlo data.
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