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Arnaud Ralko, Frédéric Mila and Didier Poilblanc

Understanding electron pairing in high temperature superconductors is a major challenge in strongly correlated systems. In his milestone paper, Anderson proposed a simple connection between high temperature superconductors and Mott insulators [1]. Electron pairs "hidden" in the strongly correlated insulating parent state as Valence Bond (VB) singlets lead, once fried to move at finite doping, to a superconducting behavior. A very good candidate of the insulating parent state is the resonating VB state (RVB), a state with only exponentially decaying correlations and no lattice symmetry breaking. A simple realization of RVB state has been proposed by Rokhsar and Kivelson (RK) in the framework of an effective quantum dimer model (QDM) with only local processes and orthogonal dimer coverings [2].

In this work, we investigate in details the properties of quantum dimer model on the square and triangular lattices at finite doping. Building on the differences between the two lattices in the undoped case, we investigate to which extent the properties of the doped system are governed by the nature of the insulating parent state. This investigation is based on exact Diagonalizations and extensive Green’s Function Monte-Carlo (GFMC) simulations essentially free of the usual finite-size limitations.

**Fig.1 : Schematic phase diagram for the square and the triangular lattice at zero doping**The bipartiteness leads to different structures in the phase diagram. Out of the RK point, the square lattice is always ordered whereas the triangular lattice exhibits a large domain of RVB liquid.

In the undoped case, remarkably, these lattices lead to quite different insulating states (see Fig.1) : an ordered plaquette phase appears on the square lattice immediately away from the special RK point (the point where the potential and the kinetic term are equal), whereas a RVB liquid phase is present in the triangular lattice (no dimer ordering).

The numerical investigation with Green’s function Quantum Monte Carlo and exact diagonalizations of the doped two-dimensional quantum hard-core dimer model on the square and triangular lattices leads to a number of interesting conclusions regarding hole motion in such a dimer backgrounds.

**Fig.2 : Schematic phase diagram for the square and the triangular lattice for finite doping**The nature of the insulating parent phases leads to different behaviors once the system doped. Dashed lines correspond to the phase transitions of the zero doping phase diagram as depicted in Fig.1.

Phase separation (PS), macroscopic segregation into two phases with different hole concentrations, is often present at low doping, as suggested by earlier investigations, but our results indicate that it is related to the presence of valence bond order : In the RVB phase of the triangular lattice, PS only occurs close to the plaquette phase, where short-range dimer correlations are already strong enough. Close to the RK point, doping the RVB phase leads directly to a superfluid phase as shown from its response to an Aharonov-Bohm flux (existence of an energy barrier under a varying flux, see Fig.3).

**Fig.3 : Aharonov-Bohm flux on the torus**Upper panel : Aharonov-Bohm flux on the square lattice with periodic boundary conditions (torus). Lower panel : Half quantum flux quantization (in unit of 2.pi) of the ground state energy.

Moreover, under this Aharonov-Bhom flux, we observe that the system presents a flux quantization in units of half a flux quantum (see Fig.3), consistent with the idea that the dimer background leads to effective particles of charge 2e. All these results are in qualitative agreement with the gauge theories of high Tc superconductivity in strongly correlated systems [20].

Post-scriptum :

Details and references are given in the corresponding paper "*Phase separation
and flux quantization in the doped quantum dimer model on the square and
triangular lattices*" published in : A. Ralko, F. Mila and D. Poilblanc, Physical
Review Letters 99, 127202 (2007).

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