Until recently, it was widely believed that the low-energy/long-distance description of all lattice systems with local interactions is captured by a continuum quantum field theory. This framework allows a detailed and powerful description of all phases of matter and the transitions between them. However, the recently discovered fracton phases of matter present a challenge to this belief. From our perspective, the main difficulty stems from the presence of an exotic global symmetry.
In order to address this challenge, we will review and extend the standard relation between lattice systems and their continuum field theory description and will demonstrate it in examples. We will then use this understanding to incorporate some exotic systems – systems with exotic global symmetries – in a continuum field theory language.
We will discuss in detail a model with a global U(1) subsystem symmetry in 2+1 dimensions and will then discuss the gauge theory associated with this global symmetry. It has a tensor gauge symmetry and its probe particles have restricted mobility. Next, we will Higgs this U(1) tensor gauge symmetry to a discrete cyclic group. Among other things, these systems exhibit a peculiar mixing between short-distance details and long-distance phenomena, which is reminiscent of UV/IR mixing in some string theory motivated systems.
Although the more interesting systems are in 3+1 dimensions, these 2+1 dimensional models are considerably easier to work with and they exhibit most of the subtleties of their higher dimensional siblings. We will also mention some of the highlights of these 3+1 dimensional systems.
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