$E_\infty^{1,2}$-type Lieb-Schultz-Mattis anomalies, deconfined quantum critical points, and non-invertible symmetry breaking
We study deconfined quantum critical points (DQCP) associated with Lieb-Schultz-Mattis (LSM) anomalies in one-dimensional spin chains. Our starting point is a structural characterization of the LSM anomaly in the Lyndon-Hochschild-Serre spectral sequence: $\omega_{\mathrm{LSM}}\in E_\infty^{1,2}= H^1(\mathbb Z_{\mathrm{trans}},H^2(G_{\mathrm{int}},\mathrm{U}(1)))\subseteq H^3(G_{\mathrm{int}}\rtimes_{\rho}\mathbb Z_{\mathrm{trans}},\mathrm{U}(1))$. Physically, this class decorates a translation defect with a projective representation of the internal symmetry $G_\mathrm{int}$. We show that gauging the internal symmetry in the presence of an $E_\infty^{1,2}$-type anomaly necessarily produces a non-invertible dual symmetry. This gives a general mechanism for type-II DQCP: in contrast to type-I examples with $E_\infty^{2,1}$-type anomalies which are dual to ordinary group-like symmetry breaking, type-II transitions are dual to spontaneous breaking of a non-invertible symmetry. We illustrate the mechanism using a spin-$1/2$ chain with an anomalous $D_8$ LSM symmetry. We construct a dimer-to-ferromagnet DQCP candidate, provide numerical evidence for a critical theory with central charge $c\approx 1$, and show, using both category theory and explicit lattice constructions, that gauging the internal symmetry yields the non-invertible $\mathrm{Rep}(H_8)$ dual symmetry.