[Paper Review] Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries

Lakshya Bhardwaj, Daniel Pajer+2 more|arXiv (Cornell University)|Mar 1, 2024

Molecular spectroscopy and chirality15 citations

TL;DR

The paper builds Hasse diagrams of condensable algebras in Drinfeld centers to classify 1+1D gapless and gapped phases with fusion-category (including non-invertible) symmetries, introducing intrinsic gapless SPT/SSB phases and providing concrete examples including the first igSPT for Rep(D8).

ABSTRACT

We discuss (1+1)d gapless phases with non-invertible global symmetries, also referred to as categorical symmetries. This includes gapless phases showing properties analogous to gapped symmetry protected topological (SPT) phases, known as gapless SPT (or gSPT) phases; and gapless phases showing properties analogous to gapped spontaneous symmetry broken (SSB) phases, that we refer to as gapless SSB (or gSSB) phases. We fit these gapless phases, along with gapped SPT and SSB phases, into a phase diagram describing possible deformations connecting them. This phase diagram is partially ordered and defines a so-called Hasse diagram. Based on these deformations, we identify gapless phases exhibiting symmetry protected criticality, that we refer to as intrinsically gapless SPT (igSPT) and intrinsically gapless SSB (igSSB) phases. This includes the first examples of igSPT and igSSB phases with non-invertible symmetries. Central to this analysis is the Symmetry Topological Field Theory (SymTFT), where each phase corresponds to a condensable algebra in the Drinfeld center of the symmetry category. On a mathematical note, gSPT phases are classified by functors between fusion categories, generalizing the fact that gapped SPT phases are classified by fiber functors; and gSSB phases are classified by functors from fusion to multi-fusion categories. Finally, our framework can be applied to understand gauging of trivially acting non-invertible symmetries, including possible patterns of decomposition arising due to such gaugings.

Motivation & Objective

• Develop a phase-diagram framework (Hasse diagram) for (1+1)d phases with categorical symmetries, including both gapped and gapless SPT/SSB phases.
• Use Symmetry Topological Field Theory (SymTFT) to encode phases as condensable algebras in the Drinfeld center Z(S).
• Generalize gSPT/igSPT and gSSB/igSSB classifications to non-invertible fusion category symmetries.
• Provide concrete computations and examples for specific symmetries such as Vec(S3), Rep(D8), and Rep(D8) with non-invertible structures.

Proposed method

• Describe SymTFT for a d-dimensional theory with symmetry S as an interval with a symmetry boundary and a physical boundary.
• Classify S-symmetric phases by condensable algebras A in the Drinfeld center Z(S) and impose a partial order A1 ≤ A2 when A1 is a subalgebra of A2.
• Construct phase charges via interval compactification of bulk topological defects; charges are labeled by objects of Z(S) and constrained by Lagrangian algebras on boundaries.
• Differentiate gapped vs gapless phases by whether the corresponding condensable algebra is maximal (Lagrangian) or non-maximal.
• Introduce and identify intrinsic gapless SPT (igSPT) and intrinsic gapless SSB (igSSB) phases where no deformation connects to a gapped counterpart.
• Provide explicit calculations for examples like Z4 gSPT and Rep(D8) gSPT/SSB to illustrate the Hasse diagram structure.

Experimental results

Research questions

• RQ1 How can one organize all S-symmetric (gapped and gapless) phases in a unified partial-order framework using condensable algebras?
• RQ2 What are the intrinsic gapless phases (igSPT/igSSB) and how can they be detected within the Hasse diagram?
• RQ3 How do non-invertible (fusion category) symmetries modify the SPT/SSB classification and the associated phase boundaries?
• RQ4 What are concrete instances of igSPT/igSSB for non-invertible symmetries such as Rep(D8) and its relatives, and how are charges realized in IR/UV?
• RQ5 How do SymTFT and condensable-algebra data encode symmetry gaps Δ_S and ordinary gaps Δ in (1+1)d systems?

Key findings

• A unified Hasse-diagram framework classifies (1+1)d gapped and gapless SPT/SSB phases for fusion-category symmetries.
• Identification of intrinsic gapless SPT (igSPT) and intrinsic gapless SSB (igSSB) phases, including the first igSPT for a non-invertible symmetry Rep(D8).
• Demonstration that phase structure can be read from condensable algebras in Z(S) and their intersections with the symmetry boundary L_S, yielding explicit phase types (gSPT, igSPT, gSSB, igSSB).
• Explicit analyses for Vec(S3), Rep(D8), and Rep(D8m) show how non-invertible symmetries admit igSPT phases and richer Hasse structures.
• The SymTFT formulation encodes charges and order parameters as endable anyons on symmetry and physical boundaries, clarifying symmetry gaps Δ_S versus spectral gaps Δ.
• The Hasse diagram remains unchanged under changing S to S' with the same SymTFT, though phase interpretations vary.