Abstract
A useful concept for finding numerically the dominant correlations of a given ground state in an interacting quantum lattice system in an unbiased way is the correlation density matrix (CDM). For two disjoint, separated clusters, it is defined to be the density matrix of their union minus the direct product of their individual density matrices and contains all the correlations between the two clusters. We show how to extract from the CDM a survey of the relative strengths of the system's correlations in different symmetry sectors and the nature of their decay with distance (power law or exponential), as well as detailed information on the operators carrying long-range correlations and the spatial dependence of their correlation functions. To achieve this goal, we introduce a new method of analysing the CDM, termed the dominant operator basis (DOB) method, which identifies in an unbiased fashion a small set of operators for each cluster that serve as a basis for the dominant correlations of the system. We illustrate this method by analysing the CDM for a spinless extended Hubbard model that features a competition between charge density correlations and pairing correlations, and show that the DOB method successfully identifies their relative strengths and dominant correlators. To calculate the ground state of this model, we use the density matrix renormalization group, formulated in terms of a variational matrix product state (MPS) approach within which subsequent determination of the CDM is very straightforward. In an extended appendix, we give a detailed tutorial introduction to our variational MPS approach for ground state calculations for one-dimensional quantum chain models. We present in detail how MPSs overcome the problem of large Hilbert space dimensions in these models and describe all the techniques needed for handling them in practice.
Highlights
In an interacting quantum lattice model the ground state may have several kinds of correlations, such as long-range order, power-law, or exponentially decaying correlations
This decomposition is possible for any two complete, r-independent operator sets OA,μ and OB,μ acting on the part of the Hilbert space of clusters A and B, respectively, which correspond to the symmetry sector Si
Summarizing, we found that the correlation density matrix (CDM) is a useful tool to detect dominant correlations in a quantum lattice system
Summary
Quantum many-body systems deal with very large Hilbert spaces even for relatively small system sizes. To be efficient the method needs a local Hilbert space with finite and small dimension, limiting its applicability to cases where the local Hilbert space is finite dimensional a priori (e.g. fermions or hard-core bosons) or effectively reduced to a finite dimension, e.g. by interactions Such a reduction is possible if there is a large repulsion between bosons on the same site such that only a few states with small occupation number will take part in the ground state. Before outlining in more detail the above-mentioned optimization scheme for determining the ground state (see section A.3), we present in section A.2 various technical ingredients needed when working with MPS
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