has a deconfined U(1) gauge field coupled to a critical charged scalar. Note that the critical theory is not expressed in terms of the VBS order as would be required by the LGW paradigm (which ignores the Z2 topological order in the deconfined phase). Instead, a dual version of the U(1) gauge theory is written in terms of a "square root" of the VBS order. Onset of non-collinear antiferromagnetism: Another example of deconfined criticality in two dimensional antiferromagnets appears in the condensation of particles with electric charges (the e particle, or the spinon) from the deconfined phase of the Z2 gauge theory. As the spinon also carries quantum numbers of global spin rotations, this leads to a "Higgs" phase of the Z2 gauge theory with antiferromagnetic order and broken spin rotation symmetry; here the antiferromagnetic order parameter has SO(3) symmetry, and so should the LGW critical theory; but the deconfined critical theory for the spinons has an exact SU(2) symmetry (which is further enlarged to O(4) after neglecting irrelevant terms). Néel-VBS transition: A more subtle class of deconfined critical points has confining phases on both sides, and the fractionalized excitations present only at the critical point. The best studied examples of this class are quantum antiferromagnets with SU(N) symmetry on the square lattice. These exhibit a phase transition from a state with collinear antiferromagnetic order to a valence bond solid, but the critical theory is expressed in terms of spinons coupled to an emergent U(1) gauge field. The study of this transition involved the first computation of the scaling dimension of a monopole operator in a conformal field theory in 2+1 dimensions; more precise computations to order 1/N are in good accord with numerical studies of the Néel-VBS transition. SYK model of non-Fermi liquids and black holes Sachdev, and his first graduate student Jinwu Ye, proposed an exactly solvable model of a non-Fermi liquid, a variant of which is now called the Sachdev–Ye–Kitaev model. Its fermion correlators have a power-law decay, which was found to extend to a conformally invariant form at non-zero temperatures. The SYK model was also found to have a non-zero entropy per site in the limit of vanishing temperature (this is not equivalent to an exponentially large ground state degeneracy: instead, it is due to an exponentially small many-body level spacing, which extends across the spectrum down to the lowest energies). Based on these observations, Sachdev