generalized-symmetries-journal-club.github.io

Generalized Symmetries Journal Club

This is a journal club covering modern advancements in the understanding of symmetry falling under the broad moniker "generalized symmetries", and their consequences in condensed matter systems. One might say that we are in the midst of a renaissance in our understanding of symmetries. In full generality, a symmetry is characterized by its rank, invertibility, and spatial modulation. Ordinary symmetries like particle number conservation are 0-form, invertible, and unmodulated: they act on all of space, are described by an invertible group action, and act homogeneously. Higher-form symmetries act on extended objects such as strings or membranes, corresponding to conservation of fluxes; non-invertible symmetries are described by a fusion category action rather than a group; and modulated symmetries act inhomogeneously in space. Generalized symmetries may be spontaneosuly broken, can protect SPT phases, may have 't Hooft anomalies, and can enforce Lieb-Schultz-Mattis-type constraints (i.e. may have mutual anomalies with translations). As just one example of their applicability, by extending Landau's paradigm of symmetry breaking to include higher-form symmetries, many "beyond-Landau" phases (such as topological/deconfined phases) may be classified by spontaneously broken higher-form symmetries.

This club and website are organized by Kristian Tyn Kai Chung. Contact: ktchung [at] pks.mpg.de

Recordings

If you are interested in learning some of the basics, I gave a pedagogical talk for condensed matter theorists which is available here:

A YouTube playlist with all recorded past talks can be found here: .

Mailing List

If you are a member of MPI-PKS, you can sign up for the mailing list here. Otherwise, contact Kai.

Format

We meet weekly on Wednesdays at 4:00 pm Dresden local time. For those in-person we meet at MPI-PKS in Room 2A1. Meetings are streamed on Zoom for external participants. An email announcement is sent before each meeting with a Zoom link and any changes to the meeting room.

Upcoming Meetings

Past Meetings

Potential Papers and Preprints to be Discussed

This is an incomplete list of some of the interesting literature surrouding generalized/higher-form symmetries, both from a high-energy perspective and from a condensed matter perspective. These have been divided into rough topics, though there may be significant overlap between these collections. If you would like to present one of these or an unlisted paper, contact Kai.

Higher-Form Symmetries

Ordinary symmetries act on 0-dimensional pointlike object (i.e. particles), and can be described as 0-form symmetries. Higher-form symmetries are symmetries which act on extended objects, e.g. a 1-form symmetry acts on string objects. Higher-form symmetries are common as emergent symmetries, and their spontaneous breaking generally yields topological phases. For spontaneously broken U(1) higher-form symmetries, their Goldstone modes are photons.

Categorical and Non-Invertible Symmetries

Categorical symmetries are those described by a (higher) fusion category instead of a group. The invertible case is captured by higher-groups (called crossed modules in older literature), which describe the way that invertible higher-form symmetries of different rank interact, and can extend non-Abelian gauge fields to higher-form. Non-invertible symmetries act non-unitarily, they generally involve a projection operator. The basic examples are Kramers-Wannier duality symmetries and Rep(G) symmetries.

Higher-Form and Higher-Rank Berry Phases

In the band theory constext, Berry phase is an ordinary 1-form gauge field in momentum space, which encodes how the Hilbert space of a band is topologically non-trivial as encoded in the Chern number. Higher-form Berry phases are also possible which described new topological invariants. Higher-rank Berry phases generally refers to those described by rank-2 or higher symmetric tensor gauge fields, which describe gauged multipolar symmetries, and are used in the classification of higher-order topological insulators. More abstractly, Berry phases can be defined on the parameter space of a gapped family of Hamiltonians, describing how the ground state wavefunction may wind non-trivially as the system is adiabatically transported through its parameter space. These Berry phases can encode anomalies and detect so-called "diabolical loci" and "unnecessary critical points", and can be generalized to highr-form topological invariants.

Anomalies and Symmetry-Protected Topological Phases (SPTs)

('t Hooft) Anomalies are a feature of quantum field theories in which a symmetry of the action fails to be a symmetry of the quantized theory. The presence of an anomalous symmetry indicates that the symmetry cannot be realized locally on-site. The anomaly is generally cured by putting the anomalous theory on the boundary of a higher-dimensional topological theory. It may also be that two symmetries have a mutual anomaly between them, for example if the generator of the two symmetries are charged with respect to each other. Anomalies are powerful RG-flow invariants which strongly constrain the physics of the system, perhaps best exemplified by Lieb-Schultz-Mattis (LSM) theorems. These forbid a trivially symmetric ground state in systems with various types of anomalies, ensuring that the anomalous symmetry is either spontaneously broken or that it acts non-trivially (e.g. projectively) on the ground state. Symmmetry-protected topological phases (SPTs) are short-range entangled phases whose ground state would be deformable to a product state if not for the presence of a symmetry, and are classified by their anomalies.

Multipolar and Subsystem Symmetries (Fractons and Symmetric Tensor Gauge Theory)

Multipole symmetries correspond to conservation of multipole moments, e.g. dipole or quadrupole moment. Closely related are subsystem symmetries, e.g. symmetries which act on planes or lines of a system. Both are related to the presence of fractons, excitations with sub-dimensional mobility, and are described by symmetric tensor gauge fields.

Entanglement

Dualities

(Lattice) Gauge Theory

Superfluids

Reviews and Miscellaneous

Other Useful Materials

Background

The generalized symmetries concept was laid out in the following paper in 2015:

A review of generalized symmetries in condensed matter has been written by McGreevy in 2022

The formalism for studying and talking about generalized symmetries is based in the language of (quantum) field theory. It features a healthy dose of differential geometry (exterior calculus of differential forms) and topology (homotopy and (co)homology). For those without background, it would be useful to learn some elementary differential geometry, in particular about differential forms. See, for example, the beginning of the book Gauge Fields, Knots And Gravity by Baez and Muniain. Additionally, some knowledge of elementary topology is useful, in particular homotopy, homology, and cohomology. See, for example, the introductions to each of the corresponding chapters in Algebraic Topology by Hatcher. For an overview of the interplay of differential forms and topology (Hodge theory), read the introduction of Differential Forms in Algebraic Topology by Bott and Tu.

Below are some papers that are mostly redundant with topics already covered in the journal club but are nonetheless interesting.