After his second pedagogical lecture, Cenke Xu told me there were many questions about spin liquids and deconfined quantum criticality. To guide my last pedagogical lecture, I'd like to ask for your questions on these topics - or others if you think we could answer them. So: please add your questions here.

• Is there a simple definition of a spin liquid?
• Can quantum or thermal order by disorder split an accidental degeneracy and stabilize a gapped spin liquid?
• Relation between spin-liquids and topological states?
• How important is it to find a spin-liquid ground state (as opposed to, say, an intermediate-temperature regime that's disordered because of spin-liquid physics, followed by more conventional ordering at very low T)? Could one have exotic finite-temperature excitations in such a system?
• Are spinon Fermi surfaces unstable to BCS pairing via the Kohn-Luttinger effect?
• How easy is it to make mesoscopic samples of potential spin-liquid materials? In principle, could one force a mesoscopic system into a spin-liquid state by playing with boundary conditions to maximize frustration?
• How to distinguish a quantum spin-liquid (either gapped or gapless) from a classical one at finite temperatures (theoretically and experimentally)? How low temperature is low enough to make the distinction?
• [Question emailed by S. Sorella, entered by L. Balents] We have now a beatiful classification of spin liquids in spin models, SU(2), U(1), Z2, based on the SU(2) invariance of the electronic degrees of freedom in any spin model. My question is , is there any stability argument why this classification should hold when charge fluctuations are allowed? In particular I am referring to the Z2 SPS (sublattice pairing state) spin liquid recently supported by PSG classification of Heisenberg model on the honeycomb lattice. In this case before projection the mean field state breaks sublattice symmetry and time reversal, that is fully recovered after projection (due to PSG). In Hubbard like models, such symmetries should be really broken and a simple dimer state should be left. Or there is some field theory stability argument that SU(2) is somehow a well defined symmetry also in Hubbard model for large U/t? I spoke very long time ago with X. G. Wen, and, as far as I remember, he replied ''yes it should be...'', but I have not understood why. In the framework of variational wavefunctions, it is very difficult to define a Jastrow factor that allows to recover a spin liquid starting from a broken symmetry mean field state, as it happens in spin models for the exact Gutzwiller projection.
• [Question emailed by C. Lhuillier, entered by S. Trebst] About deconfined quantum criticality. Could it be explained in rather pedestrian terms what makes the transition from the colinear Ne'el state to the columnar VBC so different from the transition from the same colinear state to the staggered VBC?
• Would the transition from a Ne'el state, in which SU(2) is broken to Z2, to a phase with an U(1) invariant order parameter be an exotic phase transition? ( I have in mind the phase diagram of the J-K model on the square lattice)
• Would spin liquids have technological applications?