Stripping the planar quantum compass model to its basics

We introduce a novel mean field theory (MFT) around the exactly soluble two leg ladder limit for the planar quantum compass model (QCM). In contrast to usual MFT, our construction respects the stringent constraints imposed by emergent, lower (here $d=1$) dimensional gauge like symmetries of the QCM. Specializing our construction to the QCM on a periodic 4-leg ladder, we find that a first order transition separates two mutually dual Ising nematic phases, in good accord with state-of-the-art numerics for the planar QCM. One pseudo spin-flip excitations in the ordered phase turn out to be two (Jordan-Wigner) fermion bound states, reminiscent of spin waves in spin-$1/2$ Heisenberg chains. We discuss the novel implications of our work on (1) emergence of coupled orbital and magnetic ordered and liquid like disordered phases, and (2) a rare instance of orbital-spin separation in $d>1$, in the context of a Kugel-Khomskii view of multi-orbital Mott insulators.