Density-matrix based numerical methods for discovering order and correlations in interacting systems

We review recently introduced numerical methods for the unbiased detection of the order parameter and/or dominant correlations, in many-body interacting systems, by using reduced density matrices. Most of the paper is devoted to the "quasi-degenerate density matrix" (QDDM) which is rooted in Anderson's observation that the degenerate symmetry-broken states valid in the thermodynamic limit, are manifested in finite systems as a set of low-energy "quasi-degenerate" states (in addition to the ground state). This method, its original form due to Furukawa et al.[Phys. Rev. Lett. 96, 047211 (2006)], is given a number of improvements here, above all the extension from two-fold symmetry breaking to arbitrary cases. This is applied to two test cases (1) interacting spinless hardcore bosons on the triangular lattice and (2) a spin-1/2 antiferromagnetic system at the percolation threshold. In addition, we survey a different method called the "correlation density matrix", which detects (possibly long-range) correlations only from the ground state, but using the reduced density matrix from a cluster consisting of two spatially separated regions.