transition. An important consequence is that half-integer-spin antiferromagnets (and odd Ising gauge theories) do not have a trivial confining phase, as is required by extensions of the Lieb–Schultz–Mattis theorems. These results apply also to quantum dimer models and closely related models of bosons on the square lattice. This work is now the starting point of research in symmetry enriched topological (SET) order. These results agree with numerous numerical studies of model quantum spin systems in two dimensions. Turning to experiments, VBS order was predicted by this mechanism in SrCu2(BO3)2, and has been observed by neutron scattering. A particular Z2 spin liquid state proposed for the kagome lattice antiferromagnet agrees well with a tensor network analysis, and has been proposed to describe neutron scattering and NMR experiments on herbertsmithite. A gapped spin liquid state has also been observed in the kagome lattice compound Cu3Zn(OH)6FBr, and is likely to be a Z2 spin liquid. Quantum criticality Sachdev proposed that the anomalous dynamic properties of the cuprate superconductors, and other correlated electron compounds, could be understood by proximity to a quantum critical fixed point. In the quantum critical regime of a non-trivial renormalization group fixed point (in higher than one spatial dimension) the dynamics is characterized by the absence of quasiparticles, and a local equilibration time of order ħ/(kBT). This time was proposed to be the shortest possible such time in all quantum systems. Transport measurements have since shown that this bound is close to saturation in many correlated metals. Sachdev has made numerous contributions to quantum field theories of quantum criticality in insulators, superconductors, and metals. Confinement transitions of gauge theories, and deconfined criticality Traditionally, classical and quantum phase transitions, have been described in terms of the Landau–Ginzburg–Wilson paradigm. The broken symmetry in one of the phases identifies as order parameter; the action for the order parameter is expressed as a field theory which controls fluctuations at and across the critical point. Deconfined critical points describe a new class of phase transitions in which the field theory is not expressed in terms of the order parameter. Broken symmetry and order parameters, or topological order, are present in one or both of the adjacent phases. The critical field theory is expressed in terms of deconfined fractionalized degrees of freedom that cannot exist in isolation outside the sample. Ising gauge theories: Franz Wegner introduced Ising lattice gauge theories, and their transition between confining and