$d$-wave Metal phase of itinerant electrons with ring exchange on a 2-leg ladder

I will present recent results in the search for theoretical insights into non-Fermi liquid phases in two dimensions. In particular, we propose a novel conducting phase for itinerant electrons on the square lattice with strong $d$-wave correlations (i.e. a ``$d$-wave metal''; see [1] for the bosonic analog of this phase) and a candidate Hamiltonian (hopping plus four-site electron ring exchange), which we examined in search of this phase (at $T=0$) over some portion of the phase diagram. For numerical tractability, we specialize to the 2-leg ladder and study this model using variational Monte Carlo (VMC) and the density matrix renormalization group (DMRG). For the VMC, we construct trial wavefunctions corresponding to a particular slave particle decomposition of the electrons and consistent with the properties of the proposed $d$-wave metal as well as for the ``normal'' Fermi liquid phase to map out a VMC phase diagram for the candidate Hamiltonian. Meanwhile, DMRG is employed as a quasi-exact probe of this Hamiltonian and the successes and failures of the trial wavefunctions, relative to the unbiased DMRG results, will be presented. [1] O. I. Motrunich and M. P. A. Fisher, Phys. Rev. B, {\bf 75}, 235116 (2007).