Emergent generalized symmetry and maximal symmetry-topological-order

TL;DR

<3-5 sentence high-level summary> The paper develops the maximal symmetry-topological-order (maximal symTO) framework to characterize gapless quantum liquids by treating emergent symmetry and dual symmetry on equal footing via a bulk symTO in one higher dimension. It builds on the Symm/TO correspondence and holo-equivalence to define maximal symTO, showing how boundary gapped sectors reveal emergent symmetries and how the total quantum dimension determines the maximal bulk order. Through concrete 1+1D examples (Ising, 3-state Potts, Fibonacci chains) and symmetry-twist computations, the authors demonstrate that maximal symTOs (e.g., double-Ising, double-(6,5), and double-Fibonacci related orders) encode the complete local low-energy data of gapless states. They also provide a practical method to compute symTO via symmetry twists and patch operators, linking modular-invariant partition functions to gapped boundaries and bulk orders, and propose translation symmetry can stabilize certain gapless phases protected by the emergent symTO. This framework offers a unifying perspective to classify and understand gaplessness in strongly correlated systems and suggests routes for constructing lattice realizations of generalized symmetries.

Abstract

A characteristic property of a gapless liquid state is its emergent symmetry and dual symmetry, associated with the conservation laws of symmetry charges and symmetry defects respectively. These conservation laws, considered on an equal footing, can't be described simply by the representation theory of a group (or a higher group). They are best described in terms of a topological order (TO) with gappable boundary in one higher dimension; we call this the symTO of the gapless state. The symTO can thus be considered a fingerprint of the gapless state. We propose that a largely complete characterization of a gapless state, up to local-low-energy equivalence, can be obtained in terms of its maximal emergent symTO. In this paper, we review the symmetry/topological-order (Symm/TO) correspondence and propose a precise definition of maximal symTO. We discuss various examples to illustrate these ideas. We find that the 1+1D Ising critical point has a maximal symTO described by the 2+1D double-Ising topological order. We provide a derivation of this result using symmetry twists in an exactly solvable model of the Ising critical point. The critical point in the 3-state Potts model has a maximal symTO of double (6,5)-minimal-model topological order. As an example of a noninvertible symmetry in 1+1D, we study the possible gapless states of a Fibonacci anyon chain with emergent double-Fibonacci symTO. We find the Fibonacci-anyon chain without translation symmetry has a critical point with unbroken double-Fibonacci symTO. In fact, such a critical theory has a maximal symTO of double (5,4)-minimal-model topological order. We argue that, in the presence of translation symmetry, the above critical point becomes a stable gapless phase with no symmetric relevant operator.

Figures (21)

• Figure 1: An isomorphism $f_{n-1}^{(1)}$ ( a transparent domain wall in spacetime) between two anomalous $n+1$D (gapped or gapless) quantum field theories, $\cD_n$ and $\cC_n\boxtimes_{\eZ(\cC_n)} f^{(0)}_n$ (cf. eqn. (4.3) of KZ150201690), describes a local low energy equivalence (holo-equivalence) of the two quantum field theories. Here $\eZ$ is the boundary to bulk function defined in KW1458,KZ150201690. Such an equivalence exposes the emergent symmetry described by the symTO $\eZ(\cC_n)$ in quantum field theory $\cD_n$. Note that the anomaly is given by the topological order $\eZ(\cD_n)$ in one higher dimension.
• Figure 2: A special case of Fig. \ref{['CDiso']}, where $f^{(0)}_n =\tl \cR$, $\cD_n = \underline{\cC}$, $\cC_n = \cC$, and $\eZ(\cC_n) = \text{bulk}(\cC)$. $f^{(1)}_{n-1} =\veps$ is an isomorphism, a transparent domain wall (cf. Fig. 24 and Fig. 29 in KZ200514178). Here, $\tl\cR$ is a fusion higher category describing the gapped excitations on a gapped boundary of the symTO. It describes the emergent symmetry in $\underline{\cC}$. We will refer to such a symmetry as $\tl\cR$-symmetry. Also, $\text{bulk}(\cC)$ is the symTO $\eM$ describing the holo-equivalence class of emergent symmetry $\tl\cR$.
• Figure 3: Consider two systems, $\underline{\cC}$ and $\underline{\cC'}$, with holo-equivalent symmetries, $\cR$ and $\cR'$. After restricting to the respected symmetric sub-Hilbert spaces, the two systems become identical $\cC =\cC'$ and are described by the same boundary of the symTO $\eM$. However, $\underline{\cC}$ and $\underline{\cC'}$ may have different global low energy properties from different charged sectors: $\cC\boxtimes_\eM \cR \neq \cC\boxtimes_\eM \cR'$.
• Figure 4: An $n+1$D $\tl\cR$-symmetry is anomaly-free if there exists a fusion $n$-category $\cR$ such that $\cR \boxtimes_{\eM}\tl\cR \stackrel{\veps}{\cong} n\mathcal{V}\mathrm{ec}$ where $\eM=\eZ(\tl\cR)$.
• Figure 5: $n+1$D gapless (or gapped) liquid phases, with a gravitational anomaly described by a bulk topological order $\eC$, are described and classified by the data $(W, ^\eB \cX)$. Based on the isomorphic holographic decomposition $\veps$, $W$ can be viewed as an anomalous quantum field theory, $\eC$ and $\eB$ be viewed as bulk topological orders ( braided fusion $n$-categories) and $\cX$ be viewed as a gapped domain wall ( a fusion $n$-category) between $\eC$ and $\eB$.