Two-site Hubbard model, the Bardeen-Cooper-Schrieffer model, and the concept of correlation entropy

Ziesche, P.; Gunnarsson, O.; John, W.; Beck, H. P.

doi:10.1103/physrevb.55.10270

Cited by 56 publications

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“…The simplest solvable and nontrivial system, which can still capture some of the main relevant properties of larger clusters and of the infinite chain, is provided by the two-site Hubbard Hamiltonian (see, for instance, [16]). The two-site HM, being exactly solvable, has been considered by several authors as a simplified theoretical model [16][17][18][19][20][21][22]. In particular, it is useful as a toy model for understanding the binding of molecules like H 2 [17][18][19], and it was proposed to model electron-molecular vibration coupling in organic charge-transfer salts [20] and the electronic structure in π systems [21].…”

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confidence: 99%

“…The two-site HM has been investigated by several authors [16][17][18][19][20][21][22] and proposed as a toy model for understanding the binding of molecules like H 2 [17][18][19], to model electron-molecular vibration coupling in organic charge-transfer salts [20], and to describe the electronic structure in π systems [21]. Here we propose an optical realization of the twosite Hubbard Hamiltonian based on light transport in evanescently-coupled optical waveguides which is capable of mimicking the temporal dynamics of the Hubbard system in the Wannier basis representation (3).…”

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confidence: 99%

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Classical realization of two-site Fermi-Hubbard systems

2011

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A classical wave optics realization of the two-site Hubbard model, describing the dynamics of interacting fermions in a double-well potential, is proposed based on light transport in evanescentlycoupled optical waveguides.PACS numbers: 71.10. Fd, 42.82.Et Quantum-classical analogies have been explored on many occasions to mimic and visualize in a purely classical setting the dynamical aspects embodied in a wide variety of quantum systems [1,2]. In particular, in the past two decades engineered photonic lattices have provided a useful model system to investigate wave optics analogous of solid state phenomena [2][3][4][5] [11], refer to single-particle phenomena and are based on the formal similarity between the paraxial optical wave equation in photonic lattices and the nonrelativistic Schrödinger equation of a single particle in periodic potentials [2]. However, much of the richer physics in condensed-matter comes from many-body phenomena and electron correlations. The simplest and paradigmatic model which describes correlation effects of electrons in a lattice, arising from the the competition among chemical bonding, Coulomb repulsion and Pauli exclusion principle, is perhaps provided by the Hubbard model (HM) [12]. This model is capable of capturing some many-body aspects of the electronic properties of condensed matter, such as metal-insulator transitions, itinerant magnetism, and electronic superconductivity (see, e.g., [13,14] and reference therein). In spite of the simplicity of its Hamiltonian structure, very few exact results are known for the HM, mainly for finite clusters or for the infinite onedimensional chain [13][14][15]. The simplest solvable and nontrivial system, which can still capture some of the main relevant properties of larger clusters and of the infinite chain, is provided by the two-site Hubbard Hamiltonian (see, for instance, [16]). The two-site HM, being exactly solvable, has been considered by several authors as a simplified theoretical model [16][17][18][19][20][21][22]. In particular, it is useful as a toy model for understanding the binding of molecules like H 2 [17][18][19], and it was proposed to model electron-molecular vibration coupling in organic charge-transfer salts [20] and the electronic structure in π systems [21]. Since photons are bosons and they do not interact when propagating in linear optical structures, one would expect that photonics is not a suited system to simulate in a classical setting the physics of interacting electrons in solids. In recent works [23], it has been pointed out that photonic structures could provide a noteworthy laboratory system to simulate the physics of few interacting bosons in the framework of the BoseHubbard model. In this Brief Report it is shown that light transport in suitably engineered coupled waveguide structures can mimic the dynamics of interacting fermions as well. In particular, an optical realization of the two-site HM is proposed, in which light propagation in four evanescently-coupled waveguides reproduces the temporal ...

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Classical realization of two-site Fermi-Hubbard systems

2011

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“…[8,9,10,11,12,13,14,15] In Refs. [16,17], these concepts were extended to mixed states, which requires the full density matrix instead of the 1PDM.…”

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Physics behind the minimum of relative entropy measures for correlations

Held

Mauser²

2013

Eur. Phys. J. B

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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]

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“…C(ψ) = 0 if and only if ψ is a Slater determinant. The first sum in (2) is known as the "correlation entropy" [3,6,7].…”

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confidence: 99%

“…Several of these correlation measures are functions of the eigenvalues of the 1-particle density matrix (1PDM). This class includes the "nonidempotency" of the 1PDM [2,3], the "degree of correlation" [4], and the "correlation entropy" [3,5,6,7,8].…”

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Properties of Nonfreeness: An Entropy Measure of Electron Correlation

Gottlieb

Mauser

2007

Int. J. Quantum Inform.

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Abstract"Nonfreeness" is the (negative of the) difference between the von Neumann entropies of a given many-fermion state and the free state that has the same 1-particle statistics.It also equals the relative entropy of the two states in question, i.e., it is the entropy of the given state relative to the corresponding free state. The nonfreeness of a pure state is the same as its "particle-hole symmetric correlation entropy", a variant of an established measure of electron correlation. But nonfreeness is also defined for mixed states, and this allows one to compare the nonfreeness of subsystems to the nonfreeness of the whole. Nonfreeness of a part does not exceed that in the whole; nonfreeness is additive over independent subsystems; and nonfreeness is superadditive over subsystems that are independent on the 1-particle level.

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Two-site Hubbard model, the Bardeen-Cooper-Schrieffer model, and the concept of correlation entropy

Cited by 56 publications

References 19 publications

Classical realization of two-site Fermi-Hubbard systems

Classical realization of two-site Fermi-Hubbard systems

Physics behind the minimum of relative entropy measures for correlations

Properties of Nonfreeness: An Entropy Measure of Electron Correlation

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