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

Abstract

Abstract. 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 symmetrybroken 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. arXiv:1407.4189v1 [cond-mat.str-el] 16 Jul 2014Density-matrix based numerical methods for discovering order and correlations 2

Symmetry breaking and the geometry of reduced density matrices

“…It provides an explicit construction of the famous thermodynamic surface of Maxwell [3] for the case of classical and quantum spin systems, and illustrates very concisely the mathematical physics point of view of symmetry breaking as a breakdown of ergodicity [41]. It also complements the ideas developed in [52,53], where a systematic procedure for finding order parameters was developed by contrasting the RDMs of the low-lying excited states in finite size quantum many body systems. Note however that the starting point of our work is very different: we make no a priori reference to a Hamiltonian, and just consider scatter plots with respect to all possible many body wavefunctions and/or probability distributions.…”

Determination of Tomonaga-Luttinger parameters for a two-component liquid

“…First, we assumed that the degeneracy (denoted by g i ) of the first two distinct scaling dimensions to be (g 1 , g 2 ) = (6, 12) and in the second case (g 1 , g 2 ) = (4,12). In all cases, for q x < 0.5, we found the former gave a significantly better fit to the CFT formulae (35).…”

“…For example, the finite-size scaling of the EE in one-dimensional critical systems provides a precise estimate of the central charge of the corresponding CFT. More sophisticated ways of using reduced density matrices also reveal information about the low-energy scaling dimensions and operators [10][11][12][13][14][15].…”

Reduced density matrix and order parameters of a topological insulator

“…We may now use the relation between the d and f operators and the original operators in Eq. (13). This is a non-local transformation since it couples site j with the nearest-neighbor site j +1.…”