Insulating Z2 spin liquids are a phase of matter with bulk anyonic quasiparticle excitations and ground-state degeneracies on manifolds with nontrivial topology. We construct a time-reversal symmetric Z2 spin liquid in two spatial dimensions using an array of quantum wires. We identify the anyons as kinks in the appropriate Luttinger-liquid description, compute their mutual statistics, and construct local operators that transport these quasiparticles.

Insulating quantum spin liquids can undergo a confinement transition to a valence bond solid via the condensation of topological excitations of the associated gauge theory. We extend the theory of such transitions to fractionalized Fermi liquids (FL∗): These are metallic doped spin liquids in which the Fermi surfaces only have gauge neutral quasiparticles.

We analyze the form of static charge susceptibility χ(q) in underdoped cuprates near axial momenta (Q,0) and (0,Q) at which short-range static charge order has been observed. We show that the momentum dependence of χ(q) is anisotropic, and the correlation length in the longitudinal direction is larger than in the transverse direction. We show that correlation lengths in both directions decrease once the system evolves into a superconductor, as a result of the competition between superconductivity and charge order. These results are in agreement with resonant x-ray scattering data [R.

We present a theory for the large suppression of the superfluid density ρs in BaFe2(As1-xPx)2 in the vicinity of a putative spin-density wave quantum critical point at a P doping, x=xc. We argue that the transition becomes weakly first order in the vicinity of xc, and disorder induces puddles of superconducting and antiferromagnetic regions at short length scales; thus, the system becomes an electronic microemulsion. We propose that frustrated Josephson couplings between the superconducting grains suppress ρs.

The last few years have seen significant experimental progress in characterizing the copper-based hole-doped high temperature superconductors in the regime of low hole density, p. Quantum oscillations, NMR, X-ray, and STM experiments have shed much light on the nature of the ordering at low temperatures. We review evidence that the order parameter in the non-Lanthanum-based cuprates is a d-form factor density-wave.

We analyze a candidate theory for the strange metal near optimal hole doping in the cuprate superconductors. The theory contains a quantum phase transition between metals with large and small Fermi surfaces of spinless fermions carrying the electromagnetic charge of the electron, but the transition does not directly involve any broken global symmetries. The two metals have emergent SU(2) and U(1) gauge fields respectively, and the transition is driven by the condensation of a real Higgs field, carrying a finite lattice momentum and an adjoint SU(2) gauge charge.

Recent experiments in the underdoped regime of the hole-doped cuprates have found evidence for an incommensurate charge density-wave state. We present an analysis of the charge ordering instabilities in a metal with antiferromagnetic correlations, where the electronic excitations are coupled to the fractionalized excitations of a quantum fluctuating antiferromagnet on the square lattice.

The nature of the pseudogap regime of cuprate superconductors at low hole density remains unresolved. It has a number of seemingly distinct experimental signatures: a suppression of the paramagnetic spin susceptibility at high temperatures, low-energy electronic excitations that extend over arcs in the Brillouin zone, X-ray detection of charge-density wave order at intermediate temperatures and quantum oscillations at high magnetic fields and low temperatures.

Metals interacting via short-range antiferromagnetic fluctuations are unstable to sign-changing superconductivity at low temperatures. For the cuprates, this leading instability leads to the well-known d-wave superconducting state. However, there is also a secondary instability to an incommensurate charge-density wave, with a predominantly d-wave form factor, arising from the same antiferromagnetic fluctuations.