Physics behind the minimum of relative entropy measures for correlations

Abstract

Abstract. The relative entropy of a correlated state and an uncorrelated reference state is a reasonable measure for the degree of correlations. A key question is however which uncorrelated state to compare to. The relative entropy becomes minimal for the uncorrelated reference state that has the same oneparticle density matrix as the correlated state. Hence, this particular measure, coined nonfreeness, is unique and reasonable. We demonstrate that for relevant physical situations, such as finite temperatures or a correlation enhanced orbital splitting, other choices of the uncorrelated state, even educated guesses, overestimate correlations. e.g. in high-temperature superconductors. However, correlations are particularly difficult to deal with in theory and even a universally agreed definition, or measure, of "correlation" is hitherto lacking. There is a general agreement that a Hartree-Fock Slater determinant represents an uncorrelated state, even though such a wave function includes formally something one might call "correlations", which originate from the antisymmetrization of the wave function. Hence a correlation measure typically considers the difference of the correlated state vs. an uncorrelated Hartree-Fock calculation. In this situation, the questions are: For what quantity should one consider the difference? To which uncorrelated (possibly mixed) state should one compare to?One possibility to quantify correlation is to look at the energy differencebetween the (correlated) state investigated (E corr ) and an uncorrelated (or free) state E free , which is also coined correlation energy. This is e.g. the typical quantity considered in quantum chemistry or density functional theory (DFT) [4,5,6]. The exchange-correlation energy E xc is, as the difference to the Hartree energy, also readily accessible in DFT, at least within e.g. the local density approximation (LDA). This requires, however, still a separation into exchange and correlation part. In many-body theory on the other hand, one often considers two particle correlation functions of the type [7]

Entanglement driven phase transitions in spin–orbital models

“…The subject is rather general and it has become clear that entanglement and other concepts from quantum information provide a useful perspective for the understanding of electronic matter [19][20][21][22][23]. Other examples of entangled systems are: topologically nontrivial states [24], relativistic Mott insulators with 5d ions [25], ultracold alkaline-Earth atoms [26], and skyrmion lattices in the chiral metal MnSi [27].…”

Variational occupation numbers to a Müller-type pair-density

“…However, we got L p [ξ p (q = 0.5, Λ)] ≤ L[ξ(Λ)] at high enough Λ. Therefore, our consideration of an information-theoretic measure [27,28] of the minimum entropy deficiency principle, i.e., minimum missing information principle, demonstrates that such quantity alone is not applicable as good measure of how correlated a Hamiltonian is, in complete agreement with the forecast [28]. We stress that our conclusion is based on the simultaneous consideration of the kinetic and potential energy components of an expectation value.…”

Nonfreeness and related functionals for measuring correlation in many-fermion states

“…Indeed, nonfreeness has been used to quantify local correlation in a realistic tight-binding model of a transition metal oxide heterostructure [9].…”

“…Shortly afterward, Held and Mauser [9] pointed out that the minimizer is Γ ∆ , and and argued that (2) means that other choices of Γ necessarily overestimate correlation. In this review we will present a very thorough proof of (2).…”