[Paper Review] Emergent generalized symmetries in ordered phases and applications to quantum disordering

Salvatore D. Pace|arXiv (Cornell University)|Aug 10, 2023

Quantum, superfluid, helium dynamics10 citations

TL;DR

The paper shows that ordinary ordered phases with spontaneous symmetry breaking host emergent generalized symmetries, described by higher-form and non-invertible structures, and analyzes their anomalies and phase transitions.

ABSTRACT

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$\unicode{x2013}$an ordinary and a generalized one$\unicode{x2013}$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.

Motivation & Objective

• Motivate and formalize how generalized symmetries emerge at low energies in generic ordered phases.
• Classify homotopy defects via Postnikov stages and relate them to higher gauge theory with a D-1 group.
• Show that emergent symmetries are not spontaneously broken in the ordered phase and study consequences of SSB of these symmetries.
• Demonstrate phase transitions to disordered phases with topological order and emergent phenomena, potentially constituting generalized deconfined quantum critical points.
• Explore ’t Hooft anomalies between emergent and microscopic symmetries and provide evidence for a mixed anomaly.

Proposed method

• Describe homotopy defects in terms of order parameter manifolds and maps from spacetime to the order parameter manifold.
• Represent homotopy defects as magnetic defects of a D-1 group G_{\upi}^{(D-1)} higher gauge theory.
• Characterize emergent symmetries by the (D-1)-category S = (D-1)-Rep(G_{\upi}^{(D-1)}) when finite defect classes exist.
• Use the SymTFT framework to verify the symmetry category via topological boundary conditions and the Drinfeld center.
• Provide explicit examples and lattice-model illustrations to support the general construction.
• Discuss the interplay between codimension 1, 2, and higher homotopy defects and their encoding in G_{\upi}^{(D-1)}.

Experimental results

Research questions

• RQ1 What is the emergent symmetry structure carried by homotopy defects in the low-energy limit of ordered phases?
• RQ2 How are these emergent symmetries classified when the number of defect classes is finite?
• RQ3 What are the ’t Hooft anomalies involving emergent generalized symmetries and the microscopic broken symmetries?
• RQ4 How does spontaneous breaking of emergent generalized symmetries drive phase transitions to disordered phases with topological order and emergent phenomena?
• RQ5 Can the framework identify a generalized deconfined quantum critical point between ordinary and generalized SSB patterns?

Key findings

• In D dimensions, the charges of emergent generalized symmetries are carried by homotopy defects and are described by D-1 representations of a D-1 group determined by the SSB pattern.
• When there are finitely many homotopy defect classes, the emergent symmetry category is S = (D-1)-Rep(G_{\upi}^{(D-1)}).
• The emergent generalized symmetries are exact at low energies below the gap to the relevant degrees of freedom and are not spontaneously broken in the ordered phase.
• Spontaneous breaking of these emergent symmetries can induce a phase transition to a nontrivial disordered phase with topological order, emergent photons, and possibly more emergent symmetries, potentially realizing a generalized deconfined quantum critical point.
• The paper argues for a mixed ’t Hooft anomaly between the emergent generalized symmetry and the microscopic broken symmetry, evidenced by fractionalization in examples such as Z_N ferromagnets, superfluids, and isotropic ferromagnets.
• The SymTFT construction confirms the emergent symmetry structure and provides a systematic way to extract the symmetry category from topological boundary conditions.