Novel paradigm for phase transitions: deconfinement, spin-quarks, and conformal phase transitions

I will review the notion of deconfined quantum criticality as a novel paradigm of quantum phase transitions. Central to this is the notion of spin-quarks (spinons) and their deconfinment across a quantum phase-transition. Numerical evidence will be presented for deconfinement of spinons as a Neel states disorders, with an emergent massless photon in the problem. However, to date no firm conclusions have been reached oin whether these deconfinement transitions are continuous or discontinuous. The central ideas of recent analytical work will be presented where these sorts of novel phase transitions may be characterised as conformal, which in a broad sense represent generalizations of the well-known Kosterlitz-Thouless phase-transition in two dimensions.