Resonating valence bond liquid physics on the triangular lattice

We give an account of the short-range RVB liquid phase on the triangular lattice, starting from an elementary introduction to quantum dimer models including details of the overlap expansion used to generate them. The fate of the topological degeneracy of the state under duality is discussed, as well as recent developments including its possible relevance for quantum computing. §1. Short-range RVB physics — some history The question posed by high-temperature superconductors is what happens to an antiferromagnetic Mott insulator upon doping with mobile charge carriers. The basic problem arises from the fact that a hole hopping through an ordered Néel background is frustrated: in the absence of spin flips, the hole leaves behind a trail of broken bonds; optimising kinetic and exchange energy at the same time appears impossible. Early on, Anderson 1) suggested that one way of resolving this dilemma would be for the magnet to enter a state which has no long-range order but is nonetheless energetically competitive. Such a quantum state could be based on singlet bonds (valence bonds), as, for a given pair of spins, these optimise the exchange energy. However, this is again a frustrated problem as each spin can only form a singlet bond with one of its neighbours. Hence, a product of singlet wavefunctions has a hard time competing against the Néel state, where each site gains energy from its four neighbours. This energy can be retrieved, at least in part, by resonance processes between different pairings of the spins into singlet bonds, as quantum mechanically a spin can after all be in a superposition of singlet bonds with different neighbours. One would then hope that such a sea of singlet pairs would present less of an obstacle to hole motion than the Néel state. Following this route, Kivelson, Rokhsar and Sethna developed what has become known as short-range resonating valence bond (SR-RVB) physics. 2) In particular, the Rokhsar-Kivelson (RK) quantum dimer model (QDM), 3) which this account is in large part devoted to, restricts the wavefunctions to contain products of valence bonds between nearest neighbours only, as opposed to the longer-range valence bonds included in more general approaches. Somewhat disappointingly, it was found that such a model would again lead to a ground state with broken translational symmetry, 3), 4) except at an isolated critical point, with order not in the spin-spin but rather typeset using PTPT E X.sty