SciPost Phys. 18, 028 (2025) ·
published 22 January 2025
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Projective symmetries are ubiquitous in quantum lattice models and can be leveraged to constrain their phase diagram and entanglement structure. In this paper, we investigate the consequences of projective algebras formed by non-invertible symmetries and lattice translations in a generalized $1+1$D quantum XY model based on group-valued qudits. This model is specified by a finite group $G$ and enjoys a projective $\mathsf{Rep}(G)× Z(G)$ and translation symmetry, where symmetry operators obey a projective algebra in the presence of symmetry defects. For invertible symmetries, such projective algebras imply Lieb-Schultz-Mattis (LSM) anomalies. However, this is not generally true for non-invertible symmetries, and we derive a condition on $G$ for the existence of an LSM anomaly. When this condition is not met, we prove an SPT-LSM theorem: any unique and gapped ground state is necessarily a non-invertible weak symmetry protected topological (SPT) state with non-trivial entanglement, for which we construct an example fixed-point Hamiltonian. The projectivity also affects the dual symmetries after gauging $\mathsf{Rep}(G)× Z(G)$ sub-symmetries, giving rise to non-Abelian and non-invertible dipole symmetries, as well as non-invertible translations. We complement our analysis with the SymTFT, where the projectivity causes it to be a topological order non-trivially enriched by translations. Throughout the paper, we develop techniques for gauging $\mathsf{Rep}(G)$ symmetry and inserting its symmetry defects on the lattice, which are applicable to other non-invertible symmetries.
Salvatore D. Pace, Guilherme Delfino, Ho Tat Lam, Ömer M. Aksoy
SciPost Phys. 18, 021 (2025) ·
published 17 January 2025
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Modulated symmetries are internal symmetries that act in a non-uniform, spatially modulated way and are generalizations of, for example, dipole symmetries. In this paper, we systematically study the gauging of finite Abelian modulated symmetries in ${1+1}$ dimensions. Working with local Hamiltonians of spin chains, we explore the dual symmetries after gauging and their potential new spatial modulations. We establish sufficient conditions for the existence of an isomorphism between the modulated symmetries and their dual, naturally implemented by lattice reflections. For instance, in systems of prime qudits, translation invariance guarantees this isomorphism. For non-prime qudits, we show using techniques from ring theory that this isomorphism can also exist, although it is not guaranteed by lattice translation symmetry alone. From this isomorphism, we identify new Kramers-Wannier dualities and construct related non-invertible reflection symmetry operators using sequential quantum circuits. Notably, this non-invertible reflection symmetry exists even when the system lacks ordinary reflection symmetry. Throughout the paper, we illustrate these results using various simple toy models.
SciPost Phys. 17, 080 (2024) ·
published 13 September 2024
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We explore the rich landscape of higher-form and non-invertible 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 $(D-1)$-representations of a $(D-1)$-group that depends only on the spontaneous symmetry-breaking (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 symmetry-enriched (non-)Abelian topological orders, photons, and even more emergent symmetries. This SSB transition is between two distinct SSB phases--an ordinary and a generalized one--making 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 ${\pi_{k-n}(\mathcal{E}_n)}$ when ${n>1}$ and by ${\pi_{k-n}(\mathcal{E}_n)}$ modulo a ${\pi_{1-n}(\mathcal{E}_n)}$ 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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