Intrinsic NISPT Phases, igNISPT Phases, and Mixed Anomalies of Non-Invertible Symmetries

TL;DR

The paper develops intrinsic non-invertible NISPT phases in (1+1)D by exploiting mixed anomalies in non-group-theoretical fusion category symmetries and discrete gauging, and constructs explicit lattice realizations including a 2d igNISPT via anomaly resolution. It introduces a $Z_2 imesZ_2$-graded extension $ ext{C}_p$ of $ ext{Vec}_{Z_p imesZ_p}$ whose mixed anomaly becomes removable after gauging a $Z_2^t$ to obtain an anomaly-free dual $ ilde{ ext{C}}_p$, enabling a complete SPT classification that depends on $pmod 4$; the work also provides a 2d lattice model for $p=3$ and analyzes the associated igNISPT via anomaly resolution. Beyond (1+1)D, the authors outline a (3+1)D generalization using a $Z_4 imesZ_2$ extension of $Z_3^{[1]} imesZ_3^{[1]}$ 1-form symmetry, described by a 5D SymTFT, to realize intrinsic higher-dimensional NISPT phases. Together, these results connect TY fusion categories, SymTFT, and discrete gauging to construct and classify intrinsic non-invertible SPTs and their gapless counterparts, offering a unifying framework for higher-dimensional generalizations and anomaly-resolution mechanisms.

Abstract

A bosonic non-invertible Symmetry Protected Topological (NISPT) phase in (1+1)-dim is referred to as $\textit{intrinsic}$ if it cannot be mapped, under discrete gauging, to a gapped phase with any invertible symmetry, that is, if it is protected by a non-group-theoretical fusion category symmetry. We construct the intrinsic NISPT phases by performing discrete gauging in a partial SSB phase with a fusion category symmetry that has a certain mixed anomaly. Sometimes, the anomaly of that symmetry category can be alternatively understood as a self-anomaly of a proper categorical sub-symmetry; when this is the case, the same gauging provides an anomaly resolution of this anomalous categorical sub-symmetry. This allows us to construct intrinsic gapless SPT (igSPT) phases, where the anomalous faithfully acting symmetry is non-invertible; and we refer to such igSPT phases as igNISPT phases. We provide two concrete lattice models realizing an intrinsic NISPT phase and an igNISPT phase, respectively. We also generalize the construction of intrinsic NISPT phases to (3+1)-dim.

Figures (4)

• Figure 1: The fact that there is a unique enrichment from the $\mathbb{Z}_p\times \mathbb{Z}_p$-SPT $\operatorname{SPT}_{+1}$ to the $\mathrm{TY}_{p,o}$-SPT $\mathcal{F}_{o,+}$ can be established by considering the discrete gauging \ref{['eq:o_dg1']} under which the non-invertible $\mathcal{N}_o$ becomes the invertible swap symmetry $t$ exchanging $\mathbb{Z}_p^r$ and $\mathbb{Z}_p^s$. The $\operatorname{SPT}_{+1}$ under \ref{['eq:o_dg1']} is mapped to a $\mathbb{Z}_p^r$ and $\mathbb{Z}_p^s$-partial SSB phase with $\langle rs^{-1}\rangle$ unbroken. There exists a unique enrichment on the invertible symmetry frame turning it into the $(\mathbb{Z}_p^r \times \mathbb{Z}_p^s)\rtimes \mathbb{Z}_2^t$-partial SSB phase with $\langle rs^{-1},t\rangle$ unbroken. This means there exists a unique enrichment in the $\mathrm{TY}_{p,o}$ frame.
• Figure 2: SymTFT for $\mathbb{Z}_p \times \mathbb{Z}_p$ 0-form symmetry in 2d QFT $\mathcal{X}$. In Figure \ref{['fig:SymTFT1']}, choosing the topological boundary $\mathcal{B}_{\langle e_1,e_2\rangle}$ and reducing along the interval recovers the original 2d theory $\mathcal{X}$. In Figure \ref{['fig:SymTFT2']}, replacing $\mathcal{B}_{\langle e_1,e_2\rangle}$ with $\mathcal{B}_{\langle m_1, m_2\rangle}$ and then reducing along the interval leads to gauging $\mathbb{Z}_p\times \mathbb{Z}_p$ with trivial discrete torsion in $\mathcal{X}$.
• Figure 3: Relation between the bulk symmetry $U_d$ and the duality defect $\mathcal{N}_d$ in $\mathcal{X}$. In Figure \ref{['fig:SymTFT_td1']}, the interval reduction with $U_d$ insertion leads to the theory $\mathcal{X}/\mathbb{Z}_p\times \mathbb{Z}_p$ generically. In Figure \ref{['fig:SymTFT_td2']}, the bulk symmetry $U_d$ may end on a topological line operator in the bulk (known as the twist defect). Interval reduction with the $U_d$ twist defect insertion leads to an topological interface between theory $\mathcal{X}$ and $\mathcal{X}/\mathbb{Z}_p\times \mathbb{Z}_p$. If $\mathcal{B}_\mathcal{X}$ is invariant under $U_d$, in other words $\mathcal{X} \simeq \mathcal{X}/\mathbb{Z}_p\times \mathbb{Z}_p$ via the identification of the symmetry specified by $U_d$, then this topological interface becomes the duality defect $\mathcal{N}_d$.
• Figure 4: The first excited state gap scales with $L$ of \ref{['eq:igNISPT_IR']}, which is presented using $\log(\Delta E)$ vs $\log(L)$ up to $L=15$. The fitting gives $\Delta E\sim1/L^{1.767}$, which indicates no emergent conformal symmetry but it is scaling invariant. The first excited state is identified with momentum $k=2\times 2\pi/3$.