Fractons: gauging spin models and tensor gauge theory

Abstract: The objective of the present work -- a literature review on both gapped and gapless fractonic theories -- is to pedagogically fill in the gaps between the research on fractons, and an undergraduate physics education (particularly quantum and statistical mechanics). Some familiarity with classical field theory is assumed. We will begin this work by reviewing the gauging of the Ising model to obtain the toric code. Then, following the chronological order of fracton research, we will establish the gauged spin model picture of gapped fracton theories. Next (after introducing lattice gauge theory) we will cover the developments on the tensor gauge theory front in describing gapless fracton theories. We then explain how conservation of dipole moment is key in the development of field-theoretic descriptions of fracton models, and conclude by providing future plans for work on gauging such field theories. Finally, we point out two more future plans: one being an established theoretical advance which in fact inspired this work, and the other being exciting experimental advances in ultracold atomic physics which could lead to the physical realization of fracton physics. Highlights of this work include: a derivation of the gauge field dynamics (field strength/curvature) term in the gauging of a spin system to obtain the toric code, a reduction of the 3 terms of the Kogut-Susskind lattice QED Hamiltonian to the 2 terms of the toric code, a proof of the correspondence between the invariance of a gauge field and a particular Gauss's law constraint in a tensor gauge theory context, a demonstration that conservation of dipole moment accounts for the same immobility/fractalization phenomenology that we also show from a gauged spin model perspective, and proof of the necessity for polynomial shift symmetries and higher-order spatial derivatives in a Lagrangian in order to have dipole moment conservation.