Quantification of Correlations in Quantum Many-Particle Systems

Abstract

We introduce a well-defined and unbiased measure of the strength of correlations in quantum many-particle systems which is based on the relative von Neumann entropy computed from the density operator of correlated and uncorrelated states. The usefulness of this general concept is demonstrated by quantifying correlations of interacting electrons in the Hubbard model and in a series of transition-metal oxides using dynamical mean-field theory.PACS numbers: 74.70. Tx, 71.10.Fd, 71.30.+h Correlations in solids are the origin of many surprising phenomena such as Mott-insulating behavior and hightemperature superconductivity [1][2][3]. During the last few years the investigation of correlations between ultracold atoms in optical lattices has opened another fascinating field of research [4].In many-body physics correlations are conventionally defined as the effects which go beyond factorization approximations such as Hartree-Fock theory [5]. The actual strength of correlations in a given system is usually quantified by an interaction strength U relative to an energy unit such as the bandwidth W , or by comparing expectation values of particular operators, e.g., the interaction or the total energy [6][7][8][9][10]. However, there are many other quantities which can in principle be used to measure the correlation strength. Indeed "correlations" are by definition a relative concept since they always require the comparison with some reference system. Any approach which employs the expectation value of a particular set of operators for comparison will be biased. This raises a fundamental question: Is there an objective way to quantify the correlations of a system, which even allows one to compare the correlation strength of different systems?Maximal information about a general quantum state is provided by the corresponding density operator ("statistical operator")ρ = i p i |ψ i ψ i |, where p i is the probability for the quantum state |ψ i to be present in the mixture [11]. If there exists a basis in whichρ can be completely factorized (ρ →ρ PS , where PS refers to "product state"), the corresponding state is by definition uncorrelated [12]. To quantify the correlation strength of a quantum state one has to compareρ withρ PS in a suitable way.In this Letter we propose to quantify correlations within a statistical approach which is based on the concept of the von Neumann entropy [11,13]. We will show that the relative entropy [14] of a quantum state with respect to an uncorrelated product state provides a well-defined, unbiased and useful measure of the correlation strength in that system. This concept not only allows one to quantify correlations, but even to compare the correlation strength of different quantum states. In quantum theory the von Neumann entropy is defined as S(ρ) = − lnρ ρ = −Tr(ρ lnρ). It quantifies the degree to which a quantum system is in a mixed state. In this way one can also define a relative entropy [14]provided the support [15] ofρ is contained in the support ofσ; otherwise ∆S(ρ||σ) is infinite. ...

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