Gapless Phases in (2+1)d with Non-Invertible Symmetries

Title: Gapless Phases in (2+1)d with Non-Invertible Symmetries
Authors: Lakshya Bhardwaj, Yuhan Gai, Sheng-Jie Huang, Kansei Inamura, Sakura Schäfer-Nameki, Apoorv Tiwari, Alison Warman

Problem Statement

The classification of quantum phases of matter has traditionally relied on Landau's paradigm of symmetry breaking via group symmetries. Recent developments have extended this to phases with categorical (non-invertible) symmetries, particularly in gapped systems. However, the study of gapless phases and second-order phase transitions in the presence of such categorical symmetries remains a formidable task, especially in dimensions higher than (1+1)d. While (1+1)d gapless phases with non-invertible symmetries have been explored using specific models, a systematic framework for (2+1)d gapless phases with fusion 2-category symmetries has been lacking. The central challenge is to understand how to construct and classify critical models that arise as transitions between gapped phases when the symmetries involved are non-invertible.

Methodology

The paper employs the Symmetry Topological Field Theory (SymTFT) framework as its central tool. The SymTFT for a given symmetry category $S$ in (2+1)d is a (3+1)d topological field theory, specifically a Dijkgraaf-Witten (DW) theory $Z(2\text{Vec}^\varpi_G)$ for a finite group $G$ (possibly with a twist $\varpi$), where the symmetry defects reside on the boundaries.

The core methodology involves two complementary perspectives:

1. The "Club Sandwich" Construction: This generalizes the Kennedy-Tasaki (KT) transformation. The authors start with a known phase transition for a smaller symmetry $S_0$ (the "input," e.g., the 3d Ising CFT for $\mathbb{Z}_2$). They construct a "club sandwich" by inserting a topological interface $I$ between the SymTFT of the larger symmetry $S$ and the SymTFT of the smaller symmetry $S_0$. By compactifying the interval occupied by the SymTFT of $S$, the input transition is mapped to a new transition for the larger symmetry $S$.
2. Module 2-Category Description: The interfaces between topological orders are characterized mathematically using (bi)module 2-categories. This approach generalizes the description of gapped boundaries via module categories to the study of interfaces, providing a rigorous algebraic classification of the topological defects and the resulting reduced topological orders.

The authors focus on "all-bosonic" fusion 2-category symmetries, where the SymTFT is a DW theory. They systematically classify the possible interfaces (condensable algebras) that map a (3+1)d DW theory $Z(2\text{Vec}^\varpi_G)$ to a reduced topological order $Z(2\text{Vec}^\pi_{H/N})$.

Key Contributions

1. Systematic Classification of Interfaces:
The paper provides a complete classification of interfaces between DW theories for finite groups. An interface is characterized by the data $(H, N, \pi, \mathcal{A})$, where:

• $H < G$ is a subgroup.
• $N \triangleleft H$ is a normal subgroup.
• $\pi \in H^4(H/N, U(1))$ is a twist in the reduced topological order.
• $\mathcal{A}$ is a $p^*\pi$-twisted $H$-graded fusion category (for non-minimal interfaces; for minimal interfaces, $\mathcal{A} = \text{Vec}^\omega_H$).
  This classification distinguishes between minimal interfaces (where all boundary lines are projections of bulk lines) and non-minimal interfaces (which host intrinsic topological lines).

2. Construction of Gapless Phases via KT Transformations:
Using the classified interfaces, the authors construct a systematic procedure to generate gapless phases with large categorical symmetries from known transitions with smaller symmetries. By inputting the 3d Ising CFT (or its gauged versions) as the transition for a $\mathbb{Z}_2$ symmetry, they derive a wide variety of new second-order phase transitions for categorical symmetries derived from $Z_4$, $S_3$, and $D_8$ DW theories.

3. Identification of Intrinsically Gapless Phases:
The paper establishes criteria for intrinsically gapless Symmetry Protected Topological (igSPT) and intrinsically gapless Spontaneous Symmetry Breaking (igSSB) phases.

• igSPTs: Gapless phases that preserve the full symmetry but cannot be gapped without breaking it.
• igSSBs: Gapless phases that spontaneously break the symmetry but cannot be gapped without further breaking it.
  The authors show that these phases arise when the reduced topological order carries a non-trivial twist $\pi$ that cannot be trivialized by any subgroup of the original symmetry group. Specifically, if the interface involves a non-trivial $\pi \in H^4(H/N, U(1))$ that does not pull back to zero on any proper subgroup $H' < H$, the resulting phase is intrinsically gapless.

4. Concrete Examples and Tables:
The framework is applied to specific groups:

• Abelian Groups ($Z_4$): Detailed analysis of minimal and non-minimal interfaces leading to transitions involving $Z_4$ and mixed $Z_2 \times Z_2$ symmetries.
• Non-Abelian Groups ($S_3$ and $D_8$): The authors provide exhaustive lists of phase transitions for symmetries including $2\text{Rep}(S_3)$, $2\text{Rep}(D_8)$, and 2-group symmetries like $Z_4^{(1)} \rtimes Z_2^{(0)}$.
• Tables: The paper includes comprehensive tables (Tables 1, 2, 3, 4, 5) summarizing the resulting CFTs, the nature of the phases (gSPT, gSSB, igSPT, igSSB), and the subgroup data determining the gapped limits.

Results

• The authors demonstrate that for any finite group $G$, the SymTFT framework allows for the construction of an infinite number of phase transitions between gapped phases, leading to a rich landscape of gapless critical points.
• They explicitly construct igSPTs for $D_8$ symmetry and igSSBs for $S_3$ and $D_8$ symmetries. For instance, in the $D_8$ case, a specific interface with a non-trivial 4-cocycle twist $\pi$ on the reduced $Z_2 \times Z_2$ order yields a phase that is intrinsically gapless because any gapping would require breaking the $D_8$ symmetry to a subgroup where the twist trivializes, which is impossible for the specific $\pi$ chosen.
• The paper clarifies the structure of the symmetry categories on the physical boundary after the KT transformation, showing they often form multifusion categories rather than simple fusion categories.
• The relationship between the physical "club sandwich" construction and the mathematical classification of condensable algebras (étale algebras) in the Drinfeld center is established, validating the physical intuition behind the construction.

Significance and Claims

The paper claims to provide the first systematic framework for studying second-order phase transitions with categorical symmetries in (2+1)d. Its significance lies in:

1. Bridging the Gap: It connects the well-understood gapped phases with non-invertible symmetries to the complex landscape of gapless critical points.
2. Predictive Power: By relying on the classification of interfaces and the input of known transitions (like the Ising CFT), it predicts a vast array of new critical theories and their symmetry properties without needing to solve specific lattice models from scratch.
3. New Phases of Matter: It identifies and characterizes intrinsically gapless phases (igSPTs and igSSBs) as distinct universality classes protected by non-invertible symmetries, which cannot be adiabatically connected to any gapped phase without symmetry breaking.
4. Lattice Realizability: The authors note that since the SymTFT interfaces are described by group-theoretical data, the resulting gapless phases are expected to be tractable for future lattice model constructions, extending the work of [44] which provided Hamiltonians for the gapped phases.

The paper concludes that while the formalism is comprehensive, the intrinsic complexity of the problem (infinite number of interfaces) means that the classification is exhaustive in principle but requires specific input transitions to generate concrete examples. The work sets the stage for future investigations into the lattice realizations and numerical studies of these newly predicted gapless phases.