Higher Categorical Tools for Quantum Phases of Matter

18 Mar 2024, 08:00 → 22 Mar 2024, 17:20 America/Toronto

PI/4-405 - Bob Room (Perimeter Institute for Theoretical Physics)

Quantum phases have become a staple of modern physics, thanks to their appearance in fields as diverse as condensed matter physics, quantum field theory, quantum information processing, and topology. The description of quantum phases of matter requires novel mathematical tools that lie beyond the old symmetry breaking perspective on phases. Techniques from topological field theory, homotopy theory, and (higher) category theory show great potential for advancing our understanding of the characterization and classification of quantum phases. The goal of this workshop is to bring together experts from across mathematics and physics to discuss recent breakthroughs in these mathematical tools and their application to physical problems. 

 

Scientific Organizers

Lukas Mueller
Alex Turzillo
Davide Gaiotto
 

Sponsored in part by the Simons Collaboration on Global Categorical Symmetries 

 

 

Monday, 18 March

08:30 Registration

1 Opening Remarks

2 Analogies between QFT and lattice systems

I discuss some analogies between the Haag-Kastler approach to QFT and quantum statistical mechanics of lattice systems. As an illustrative example, I consider the interpretation of the Hall conductance of gapped 2d lattice systems as an obstruction to gauging a global symmetry of a gapped state. I argue that in order to define a proper analog of the net of algebras of observables one needs to study a category of subsets of the lattice equipped with a natural Grothendieck topology.

Speaker: Anton Kapustin (California Institute of Technology)

10:15 Break

3 Models of anyons with symmetry: a bulk-boundary correspondence

I will describe models with on-site symmetry that permutes anyons with non-trivial mutual statistics, and show that the action of this symmetry on the boundary can effectively be that of a non-invertible symmetry such as Kramers-Wannier duality. I will sketch some implications of this for anomalies in non-invertible symmetries. Finally, I will introduce a construction (based on idempotent completion) that allows us to realize all possible anyon permuting symmetries of a given topological order in an on-site way.

Speaker: Fiona Burnell (University of Minnesota)

12:00 Lunch

13:00 Free Discussion

4 Twisted Tools for (Untwisted) Quantum Field Theory

Speaker: Justin Kulp (Stony Brook University)

15:00 Break

15:30 Gong Show

Tuesday, 19 March

5 Quantum double models and Dijkgraaf-Witten theory with defects

We use 3d defect TQFTs and state sum models with defects to give a gauge theoretical formulation of Kitaev's quantum double model (for a finite group) and (untwisted) Dijkgraaf-Witten TQFT with defects. This leads to a simple description in terms of embedding quivers, groupoids and their representations. Defect Dijkgraaf-Witten TQFTs is then formulated in terms of spans of groupoids and their representations.
This is work in progress with João Faría-Martins, University of Leeds.

Speaker: Catherine Meusburger (FAU Erlangen-Nürnberg)

10:15 Break

6 Topological sectors in quantum lattice models

Speaker: Clement Delcamp (Institut des Hautes Études Scientifiques (IHES))

11:45 Group Photo

12:00 Lunch

13:00 Free Discussion

7 Douglas-Reutter 4d TQFT as a generalised orbifold

The state-sum invariants of 4d manifolds obtained from spherical fusion 2-categories due to Douglas-Reutter offer an exciting entrypoint to the study of 4d TQFTs.
In this talk we will argue that these invariants arise from a TQFT, obtained by filling the trivial 4d TQFT with a defect foam.
Such construction is known as a generalised orbifold, the Turaev-Viro-Barrett-Westbury (i.e. 3d state-sum) models are also known to arise in this way from the defects in the trivial 3d TQFT (a result by Carqueville-Runkel-Schaumann).
Advantages of this point of view offer e.g. realisations of state-spaces, examples of domain walls and commuting-projector realisations of (3+1)-dimensional topological phases.
Based on a joint project with Nils Carqueville and Lukas Müller.

Speaker: Vincentas Mulevičius (Vilnius University)

15:00 Break

15:30 Organized Discussion

Wednesday, 20 March

8 Weak Hopf symmetric tensor networks

Speaker: Andras Molnar (University of Vienna)

10:15 Break

9 Hierarchy construction for fractional quantum hall states via condensable algebras

Speaker: Carolyn Zhang (Harvard University)

12:00 Lunch

13:00 Free Discussion

10 Zesting topological order and symmetry-enriched topological order in (2+1)D

Zesting is a construction that takes a (2+1)D topological order and produces a new one by changing the fusion rules of its anyons. We'll discuss properties of zesting from a physical and computational point of view and explain how the theory produces some closely related families of topological orders, like Kitaev's 16-fold way and modular isotopes. Time permitting we'll cover a generalization of zesting to symmetry-enriched topological order and comment on connections to fusion 2-categories.

Speaker: Colleen Delaney (University of California, Berkeley)

15:00 Break

11 Motion Groupoids

The braiding statistics of point particles in 2-dimensional topological phases are given by representations of the braid groups. One approach to the study of generalised particles in topological phases, loop particles in 3-dimensions for example, is to generalise (some of) the several different realisations of the braid group.

In this talk I will construct for each manifold M its motion groupoid $Mot_M$, whose object class is the power set of M. I will discuss several different, but equivalent, quotients on motions leading to the motion groupoid. In particular that the quotient used in the construction $Mot_M$ can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows -- worldlines (e.g. monotonic `tangles').

I will also give a construction of a mapping class groupoid $MCG_M$ associated to a manifold M with the same object class. For each manifold M I will construct a functor $F \colon Mot_M \to MCG_M$, and prove that this is an isomorphism if $\pi_0$ and $\pi_1$ of the appropriate space of self-homeomorphisms of M is trivial. In particular there is an isomorphism in the physically important case $M=[0,1]^n$ with fixed boundary, for any $n\in\mathbb{N}$.

I will discuss several examples throughout.

Speaker: Fiona Torzewska (University of Bristol)

18:00 Banquet

Thursday, 21 March

12 Categorical Aspects of Symmetry in Fermionic Systems

In this talk we explore the symmetry in 1+1d fermionic systems from a category theory perspective. We argue that It requires additional structure than a tensor category over sVect, and rather captured by a fusion category equipped with a braided central functor from a specific braided fusion category, dependent on the chiral central charge. The same data defines a 1-morphism in the 4-category of braided tensor categories, which lead us to a 4d-3d-2d picture via cobordism hypothesis.

This talk is based on an ongoing work with K. Inamura.

Speaker: Kantaro Ohmori (University of Tokyo)

10:15 Break

13 Bulk Excitations of Invertible Phases

Recent development in the study of topological defects highlights the importance of understanding the multi-dimensional structure of bulk excitations inside a quantum system. When the bulk ground state is trivial, i.e. a product state, excitations on top of it are decoupled from each other and correspond to lower-dimensional phases and their defects within. In this talk, we will expand the discussion to invertible phases and study the bulk excitations in, for example, SPT phases, majorana chain, p + ip superconductor etc. We find that there is a one-to-one correspondence between bulk excitations inside a nontrivial invertible phase and those in a product state. For SPT phases, this can be shown using the symmetric Quantum Cellular Automaton that maps from the product state to the SPT state. In this talk, we will demonstrate the correspondence for more general invertible phases, those realizable using the Symmetry Topological-Field-Theory construction. Our demonstration is built upon a key property of topological orders: certain gapped boundary conditions of a topological bulk state have only relative distinctions but no absolute ones - they cannot be distinguished by any local experiments near the boundary.

This talk is based on a work under preparation with David Stephen and Xie Chen.

Speaker: Wenjie Ji (California Institute of Technology)

12:00 Lunch

13:00 Free Discussion

14 Quantum homotopy groups

An open-closed tqft is a tqft with a choice of boundary condition. Example: the sigma model for a sufficiently finite space, with its Neumann boundary. Slogan: every open-closed tqft is (sigma model, Neumann boundary) for some “quantum space”. In this talk, I will construct homotopy groups for every such “quantum space” (and recover usual homotopy groups). More precisely, these “groups” are Hopf in some category. Given a “quantum fibre bundle” (a relative open-closed tqft), I will construct a Puppe long exact sequence. Retracts in 3-categories and a higher Beck-Chevalley condition will make appearances. This project is joint work in progress with David Reutter.

Speaker: Theo Johnson-Freyd (Perimeter Institute / Dalhousie University)

15:00 Break

15:30 Organized Discussion

Friday, 22 March

15 Once extended unitary topological field theories

Motivated by the dagger category of Hilbert spaces, we explore the mathematical axiomatization of unitary topological field theory (TFT) using dagger categories. Recent advancements have elucidated the structure of higher dagger categories, paving the way for a precise definition of extended unitary TFTs. I will present an explicit formulation of once extended TFTs utilizing various versions of dagger bicategories. This framework enables a complete classification of unitary extended two-dimensional TFTs with arbitrary symmetry groups in terms of their unitary 2-representations. This is joint work in progress with Lukas Muller.

Speaker: Luuk Stehouwer (Dalhousie University)

10:15 Break

16 Bosonization and anomalies of 3d fermionic topological orders

There are two notions of a symmetry of a group G on a 3d topological order (TO): an "algebraic" symmetry, where G acts by automorphisms on the tensor category defining the (TO), and a "field-theoretic" symmetry, where the TFT corresponding to the TO is extended to manifolds with a principal G-bundle. The "field-theoretic" notion is stronger than the "algebraic" one, and the obstruction is sometimes referred to as the anomaly of the TO. The goal of this talk is to discuss a project joint with Weicheng Ye and Matthew Yu on computing these anomalies for fermionic TOs/spin TFTs: we develop a general framework employing Gaiotto-Kapustin's bosonic shadow construction. I will discuss both the mathematical conjectures our framework rests on as well as its use in
examples. The Smith long exact sequence appears in our computations.

Speaker: Arun Debray (Purdue University)

17 Closing Remarks

12:00 Lunch

13:00 Discussion and Collaboration

16:00 Perimeter Institute Friday Social