Citation: McGreevy, John. "Generalized symmetries in condensed matter." Annual Review of Condensed Matter Physics 14 (2023): 57-82.
Web: https://arxiv.org/abs/2204.03045
Tags: Physical, Expository, SPT/SETs
This paper discusses generalized "q-form" symmetries in condensed matter systems. Abstractly, a q-form symmetries are summarized as global symmetries for which the charged operators are of space-time dimension q. That is, symmetries which are comprised of transformations whose domain is q-dimensional. When q=0, this means that the symmetry acts on points. Hence, a 0-form symmetry is a standard gauge symmetry. When q is top dimensional, this is a global symmetry. q-form symmetries were introduced in
> Kapustin, Anton, and Ryan Thorngren. "Higher symmetry and gapped phases of gauge theories." Algebra, Geometry, and Physics in the 21st Century: Kontsevich Festschrift (2017): 177-202.
and clarified in
> Gaiotto, Davide, et al. "Generalized global symmetries." Journal of High Energy Physics 2015.2 (2015): 1-62.
One of the key philosophical takeaways from q-form symmetry is that the Landau paradigm isn't that far off. In particular, Landau said that states of matter are classified by symmetry breaking. This doesn't work for topological phases, since there is no local operator which commutes with the Hamiltonian (no "symmetry") but yet there is still non-trivial order. One way to reckon with this is to throw away the Landau paradigm. Another is to consider more generalized symmetries, in which can we find that topological order IS still classified by symmetry breaking.
One subtelty emphasized by the author is that not every symmetry can be gauged - there can be an anomaly which prevents this from happening. Two different phases can have the exact same symmetries, but have different anomaly. Including this, we get the generalized Landau paradigm: phases of matter are classified by their symmetries and their anomalies.
These sorts of generalized symmetries have also found great importance in high energy physics circles. Are good review is found here:
> Cordova, Clay, et al. "Snowmass white paper: Generalized symmetries in quantum field theory and beyond." arXiv preprint arXiv:2205.09545 (2022).