SciPost Phys. 17, 080 (2024) ·
published 13 September 2024

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We explore the rich landscape of higherform and noninvertible symmetries that emerge at low energies in generic ordered phases. Using that their charge is carried by homotopy defects (i.e., domain walls, vortices, hedgehogs, etc.), in the absence of domain walls we find that their symmetry defects in $D$dimensional spacetime are described by $(D1)$representations of a $(D1)$group that depends only on the spontaneous symmetrybreaking (SSB) pattern of the ordered phase. These emergent symmetries are not spontaneously broken in the ordered phase. We show that spontaneously breaking them induces a phase transition into a nontrivial disordered phase that can have symmetryenriched (non)Abelian topological orders, photons, and even more emergent symmetries. This SSB transition is between two distinct SSB phasesan ordinary and a generalized onemaking it a possible generalized deconfined quantum critical point. We also investigate the 't Hooft anomalies of these emergent symmetries and conjecture that there is always a mixed anomaly between them and the microscopic symmetry spontaneously broken in the ordered phase. One way this anomaly can manifest is through the fractionalization of the microscopic symmetry's quantum numbers. Our results demonstrate that even the most exotic generalized symmetries emerge in ordinary phases and provide a valuable framework for characterizing them and their transitions.
SciPost Phys. 16, 128 (2024) ·
published 21 May 2024

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In this note, we classify topological solitons of $n$brane fields, which are nonlocal fields that describe $n$dimensional extended objects. We consider a class of $n$brane fields that formally define a homomorphism from the $n$fold loop space $\Omega^n X_D$ of spacetime $X_D$ to a space $\mathcal{E}_n$. Examples of such $n$brane fields are Wilson operators in $n$form gauge theories. The solitons are singularities of the $n$brane field, and we classify them using the homotopy theory of ${\mathbb{E}_n}$algebras. We find that the classification of codimension ${k+1}$ topological solitons with ${k≥ n}$ can be understood using homotopy groups of $\mathcal{E}_n$. In particular, they are classified by ${\pi_{kn}(\mathcal{E}_n)}$ when ${n>1}$ and by ${\pi_{kn}(\mathcal{E}_n)}$ modulo a ${\pi_{1n}(\mathcal{E}_n)}$ action when ${n=0}$ or ${1}$. However, for ${n>2}$, their classification goes beyond the homotopy groups of $\mathcal{E}_n$ when ${k< n}$, which we explore through examples. We compare this classification to $n$form $\mathcal{E}_n$ gauge theory. We then apply this classification and consider an ${n}$form symmetry described by the abelian group ${G^{(n)}}$ that is spontaneously broken to ${H^{(n)}\subset G^{(n)}}$, for which the order parameter characterizing this symmetry breaking pattern is an ${n}$brane field with target space ${\mathcal{E}_n = G^{(n)}/H^{(n)}}$. We discuss this classification in the context of many examples, both with and without 't Hooft anomalies.
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