Thermodynamics of a 4-site Hubbard model by analytical diagonalization

Schumann, R.

doi:10.1002/1521-3889(200201)11:1<49::aid-andp49>3.0.co;2-7

Cited by 50 publications

(77 citation statements)

References 34 publications

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“…both external sources are at the same site), can be constructed as a functional G[z; ̺ e T (ω)] of the irreducible loop-spectrum ̺ e T (ω) = − 1 π Im T (ω+iδ), and G(G, T ) has a simple expression via G and T . Therefore (26) reduces to an implicit equation for T (z) and consequently also furnishes solutions for G(z) , G k (z) and G(z).…”

Section: Cumulant Perturbation Theory Of Strongly Correlated Lattice mentioning

confidence: 99%

Conserving approximations in direct perturbation theory: new semianalytical impurity solvers and their application to general lattice problems

Grewe¹,

Schmitt²,

Jabben³

et al. 2008

J. Phys.: Condens. Matter

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Abstract. For the treatment of interacting electrons in crystal lattices approximations based on the picture of effective sites, coupled in a self-consistent fashion, have proven very useful. Particularly in the presence of strong local correlations, a local approach to the problem, combining a powerful method for the short ranged interactions with the lattice propagation part of the dynamics, determines the quality of results to a large extent. For a considerable time the non crossing approximation (NCA) in direct perturbation theory, an approach originally developed by Keiter for the Anderson impurity model, built a standard for the description of the local dynamics of interacting electrons. In the last couple of years exact methods like the numerical renormalization group (NRG) as pioneered by Wilson, have surpassed this approximation as regarding the description of the low energy regime. We present an improved approximation level of direct perturbation theory for finite Coulomb repulsion U , the crossing approximation one (CA1) and discuss its connections with other generalizations of NCA. CA1 incorporates all processes up to fourth order in the hybridization strength V in a self-consistent skeleton expansion, retaining the full energy dependence of the vertex functions. We reconstruct the local approach to the lattice problem from the point of view of cumulant perturbation theory in a very general way and discuss the proper use of impurity solvers for this purpose. Their reliability can be tested in applications to e.g. the Hubbard model and the Anderson-lattice model. We point out shortcomings of existing impurity solvers and improvements gained with CA1 in this context. This paper is dedicated to the memory of Hellmut Keiter.

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Section: Cumulant Perturbation Theory Of Strongly Correlated Lattice mentioning

confidence: 99%

Conserving approximations in direct perturbation theory: new semianalytical impurity solvers and their application to general lattice problems

Grewe¹,

Schmitt²,

Jabben³

et al. 2008

J. Phys.: Condens. Matter

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show abstract

“…if the cluster is occupied with two electrons we find an U independent crossing from a twofold degenerate groundstate with spin one and spin projection one (no. 8,20) to the non-degenerate eigenstate (no. 13) with spin one and spin projection zero.…”

Section: The Spectrum Of the Trianglementioning

confidence: 99%

“…A N = 5 state is never a grand-canonical groundstate. The situation resembles closely the Hubbard square, where we also found a step from N = 2 to N = 4 and from N = 4 to N = 6 [8]. In Fig.…”

Section: The Tetrahedral Cluster Gasmentioning

confidence: 99%

“…Despite its simplicity, the only analytical solutions available up to now are the groundstate for n-site rings by help of the Bethe ansatz or transfer matrix method [5,6] and the complete solution on small clusters, i.e. the triangle, the tetrahedron [7] and the square [8]. For models containing five and more sites only approximate or numerical solutions exist.…”

Section: Introductionmentioning

confidence: 99%

“…In the present paper we study the influence of additional nearest-neighbour (nn) Coulomb interaction, exchange term in the case of both the triangle and the tetrahedron or next-nearest neighbour (nnn) hopping in the case of the square respectively, which are added to the standard Hubbard model. The latter is used in the notation of [8]. A common feature of the considered models is frustration, which is present due to geometry in the triangle and tetrahedron, even without the additional terms, and in the square due to the nnn-hopping, what becomes evident since for t ′ = t the model is equivalent to the pure Hubbard model on the tetrahedron.…”

Section: Introductionmentioning

confidence: 99%

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Rigorous solution of a Hubbard model extended by nearest-neighbour Coulomb and exchange interaction on a triangle and tetrahedron

Schumann¹

2008

Ann. Phys.

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The Hubbard model extended by either nearest-neighbour Coulomb correlation and/or nearest neighbour Heisenberg exchange is solved analytically for a triangle and tetrahedron. All eigenvalues and eigenvectors are given as functions of the model parameters in a closed form. The groundstate crossings and degeneracies are discussed both for the canonical and grand-canonical energy levels. The grand canonical potential Ω(µ, T, h) and the electron occupation N (µ, T, h) of the related cluster gases were calculated for arbitrary values (attractive and repulsive) of the three interaction constants. In the pure Hubbard model we found various steps in N (µ, T = 0, h) higher than one. It is shown that the various degeneracies of the grand-canonical energy levels are partially lifted by an antiferromagnetic exchange interaction, whereas a moderate ferromagnetic exchange modifies only slightly the results of the pure Hubbard model. A repulsive nn Coulomb correlation lifts these degeneracies completely. The relation of the cluster gas results to extended systems is discussed.

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Rigorous solution of a Hubbard model extended by nearest‐neighbour Coulomb and exchange interaction on a triangle and tetrahedron

Schumann

2008

Annalen der Physik

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The Hubbard model extended by both nearest‐neighbour (nn) Coulomb correlation and nearest‐neighbour Heisenberg exchange is solved rigorously for a triangle and tetrahedron. All eigenvalues and eigenvectors are given as functions of the model parameters in a closed analytical form. For fixed electron numbers we found a multitude of level crossings, both in the ground state and in the excited states in dependence on the various model parameters. By coupling an ensemble of clusters to an electron bath we get the cluster gas model or the cluster gas approximation, if an extended array of weak‐interacting clusters is considered. The grand‐canonical potential Ω (μ, T, h) and the electron occupation N (μ, T, h) of the related cluster gases were calculated for arbitrary values (attractive and repulsive) of the three interaction constants. For the cluster gases without the additional interactions we found various steps in N (μ, T = 0, h = 0) higher than one. The reason is the degeneration of ground states differing in their electron occupation by more than one electron. For the triangular cluster gas we have one such degeneration point. For the tetrahedral cluster gas two. As a consequence, we do not find areas with one electron in the μ‐U ground‐state phase diagram of the triangular cluster gas or with one, two and five electrons in the case of the tetrahedral cluster gas. The degeneration point of the triangular cluster gas can not be destroyed by an applied magnetic field. This holds also for the lower degeneration point of the tetrahedral cluster gas. Otherwise, the upper degeneration point breaks down at a critical magnetic field hc. The dependence of hc on U shows a maximum for strong on‐site correlation. The influence of nn‐exchange and nn‐Coulomb correlation on the ground‐state phase diagrams is calculated. Whereas antiferromagnetic nn‐exchange breaks the degeneration points of the tetrahedral cluster gas partially only, a repulsive nn‐Coulomb correlation lifts the underlying degeneracies completely. Otherwise both ferromagnetic nn‐exchange and attractive nn‐Coulomb interaction stabilise the degeneration points. The consequences of the cluster gas results for extended cluster arrays are discussed.

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Thermodynamics of a 4-site Hubbard model by analytical diagonalization

Cited by 50 publications

References 34 publications

Conserving approximations in direct perturbation theory: new semianalytical impurity solvers and their application to general lattice problems

Conserving approximations in direct perturbation theory: new semianalytical impurity solvers and their application to general lattice problems

Rigorous solution of a Hubbard model extended by nearest-neighbour Coulomb and exchange interaction on a triangle and tetrahedron

Rigorous solution of a Hubbard model extended by nearest‐neighbour Coulomb and exchange interaction on a triangle and tetrahedron

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