Lattice quantum Villain Hamiltonians: compact scalars, U(1) gauge theories, fracton models and quantum Ising model dualities

• Published in: The journal of high energy physics Vol. 2023; no. 5; pp. 17 - 38
• Main Authors: Fazza, Lucca, Sulejmanpasic, Tin
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 02-05-2023; Springer Nature B.V; SpringerOpen
• Subjects: Anomalies; Anomalies in Field and String Theories; Classical and Quantum Gravitation; Conjugates; Continuum modeling; Coupling; Duality in Gauge Field Theories; Elementary Particles; Gauge theory; Hamiltonian functions; High energy physics; Hilbert space; Ising model; Lattice Quantum Field Theory; Mathematical analysis; Numbers; Operators (mathematics); Other Lattice Field Theories; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Scalars; Solid phases; Spacetime; String Theory; Symmetry; Tensors; Vortices; Lattice Quantum Field Theory; Duality in Gauge Field Theories; Anomalies in Field and String Theories; Other Lattice Field Theories
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP05(2023)017

A
bstract
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
= 2
J
2
, and discuss the phase structure of such models.