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Geometrical aspects of topological phases of matter: spatial symmetries, fractons and beyond: April 4 – May 27, 2022

Organized by: Jennifer Cano, Dominic Else, Andrey Gromov, Siddharth Parameswaran, and Yizhi You

Topological phases of matter are a long-standing subject of interest in the condensed matter community, and increasingly relevant to issues in high-energy physics. A topological phase is traditionally defined to be one which is “non-trivial” (cannot be deformed to a trivial insulator without a phase transition), but where the non-triviality cannot be ascribed merely to spontaneous symmetry breaking. Another interpretation is that the word “topological” in “topological phase” is supposed to suggest that the low-energy physics of this phase (that is, the infrared fixed point controlling the phase in the renormalization group sense) is “topological”, meaning that it can be defined on any background space-time and is sensitive only to the background topology. This program will be devoted to exploring the connections, and tension, between these two distinct notions of “topological”. The program will focus on encouraging cross-fertilization between three rapidly developing, interconnected research areas:

The past decade witnessed an explosion of activity in research on symmetry-protected topological phases. Traditionally, the symmetry group in question is either time-reversal or an internal, “on-site’’ symmetry. However, in the past several years, topological phases protected by spatial (crystalline) symmetries have emerged. Unlike topological phases described by topological quantum field theories, these phases a priori can only be defined on the particular background space-time on which the crystalline symmetries act. Consequently, the traditional approach of topological quantum field theory has to be supplemented with additional information, such as the properties of lattice defects,
which are a kind of geometric response of the system. A generic bulk-edge correspondence for these states is absent because, unlike on-site symmetries, crystal symmetries are typically broken on boundaries. In some instances, the usual correspondence is replaced by a “higher-order’’ correspondence between the d-dimensional bulk state and symmetry-preserving (d-2)-dimensional edges. Developing a systematic theory of such phases, and discovering physical examples, is an important open question.

Several years ago, a conceptually new type of gapped phases was discovered. These phases are known as fracton phases, due to the presence of topologically non-trivial excitations that can only move on lower-dimensional submanifolds, or cannot move at all. Like spatial symmetry-protected topological phases, fracton phases challenge our notion of what “topological order” means, because the low-energy theory depends on some non-topological features of space. In fact, these phases also appear to have a very complicated relationship with the geometry of the space where they reside. Intuitively, the exotic features of the excitations in these models can be viewed as stemming from a non-trivial interplay between translation symmetry and topological order. Alternatively, these phases can be viewed as higher-rank gauge theories obtained by gauging the subsystem symmetries — the symmetries which act along lower-dimensional subspaces. Fractons have attracted a broad interdisciplinary interest due to their potential relationship to lattice gauge theory, quantum computation and memory, elasticity, glassy dynamics and emergent gravity in condensed matter.

Fractional quantum Hall states exhibit a non-trivial geometric response. Since FQH phases are liquids, with continuous rotational and translational symmetries, these geometric properties are intuitively related to those of topological phases with spatial symmetries. Developing a rigorous unified approach to the geometric properties of FQH phases on equal footing with the topological phases with crystalline symmetrie will be another focus of the program. Separately, the breaking of discrete and/or continuous crystalline symmetries underpins a remarkable class of unconventional nematic quantum Hall liquids — with unusual properties analogous to those of liquid crystals familiar from classical soft condensed matter physics. Concurrently, there has been recent progress on the FQH physics that goes beyond the topological order paradigm. This builds on pioneering work by Haldane, who has argued that certain collective modes supported by a FQH liquid can be described by a fluctuating geometry. This area of study has recently witnessed further progress simultaneously on three fronts: in terms of trial states, Matrix Models, and effective theory. Quantitative properties of these modes are related to the geometric responses.

Geometry plays a central role in these topics, but currently there is no coherent picture that unifies them. Nevertheless, there are some tantalizing hints of possible close connections. We expect that the program will lead to identification of the common themes and cross-fertilization of these fields.