Thermodynamics of the extended Hubbard model in high dimensions

Dongen, P. G. J. van

doi:10.1103/physrevlett.67.757

Cited by 102 publications

(112 citation statements)

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“…As a result of the periodic boundary conditions on the cluster, this causes the near-neighbor fluctuations to be over estimated. As a result, the estimates of T N from these clusters, shown The dynamical mean-field approximation (DMFA, full circles), Staudt et al [7] (full curve), second order perturbation theory (SOPT, dotted curve) [10,11], the Heisenberg model (dashed curve) [12] and the Weiss mean-field theory for the Heisenberg model (dash-dotted curve). We took J = 4t 2 /U for both Heisenberg calculations.…”

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Efficient calculation of the antiferromagnetic phase diagram of the three-dimensional Hubbard model

et al. 2005

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The Dynamical Cluster Approximation with Betts clusters is used to calculate the antiferromagnetic phase diagram of the 3D Hubbard model at half filling. Betts clusters are a set of periodic clusters which best reflect the properties of the lattice in the thermodynamic limit and provide an optimal finite-size scaling as a function of cluster size. Using a systematic finite-size scaling as a function of cluster space-time dimensions, we calculate the antiferromagnetic phase diagram. Our results are qualitatively consistent with the results of Staudt et al. [Eur. Phys. J. B 17 411 (2000)], but require the use of much smaller clusters: 48 compared to 1000.The accurate and efficient solution of lattice Hamiltonians such as the Hubbard model is a long standing challenge in the theoretical condensed matter community. These lattice models are routinely solved on a finite periodic lattice, for example with Monte Carlo, and the calculated properties extrapolated to the infinite limit. Due to the numerical expense in solving these models for large lattices, it is imperative to choose lattices that are efficient for the estimation and extrapolation of the physical properties of interest.In this paper we use the Dynamical Cluster Approximation (DCA) [1, 2, 3] (for a review see [4]) to explore the antiferromagnetic instability in the 3D Hubbard model at half filling, withwhere c ( †) iσ (creates) annihilates an electron with spin σ on site i, n iσ is the corresponding number operator, t the hopping amplitude between nearest neighbors i, j and U the on-site Coulomb repulsion. We solve this model on a series of finite clusters chosen according to the criteria proposed by Betts et al. [5,6]. We obtain converged results extrapolating from clusters of up to only 48 sites, which are in agreement with the calculations of Staudt et al. [7], who used conventional cubic lattices of up to 1000 sites and obtained the Néel temperature via the specific heat.To solve the Hamiltonian (1) we utilized the DCA [4]. For a 3D system the DCA maps the original lattice model onto a periodic cluster of size N c = L 3 c embedded in a selfconsistent host. Thus, correlations up to a range ξ < ∼ L c are treated directly, while the longer length scale physics is described at the mean-field level. With increasing cluster size, the DCA systematically interpolates between the single-site dynamical mean-field result and the exact result, while remaining in the thermodynamic limit. We solve the cluster problem using Quantum Monte Carlo (QMC) [8]. At half-filling there is no QMC sign problem; the only systematic error in the Monte Carlo is the time step error, which can be extrapolated away.In order to calculate the phase diagram of the system in the thermodynamic limit, we employ the scalingis the Néel temperature obtained from a DCA calculation with a cluster of linear cluster size L c . This form is justified if we envision the lattice as perfectly tiled by a periodic array of non-overlapping clusters. This system becomes ordered when the antiferromagnetic ...

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Efficient calculation of the antiferromagnetic phase diagram of the three-dimensional Hubbard model

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“…Grabowski and Sham [7] carried out such an analysis for the interacting electron gas and found that vertex corrections strongly suppressed T c for plasma frequencies larger than ten percent of the fermi energy. Van Dongen [8] analyzed the Hubbard model (which can be viewed as the infinite-frequency limit of an electron-phonon model) and showed that quantum fluctuations (vertex corrections) reduced Hartree-Fock transition temperatures by more than a factor of three.…”

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“…In this limit many "direct" diagrams cancel against "exchange" diagrams (there is no interaction between electrons with the same spin) and the perturbation theory is easier to control. Quantum Monte Carlo simulations have already been performed [14], and weak-coupling expansions through second order (for transition temperatures) have also been investigated with different techniques [8,17]. Migdal's theorem states that, in the small-frequency limit, transition temperatures may be determined with a first-order approximation (using a dressed phonon propagator) because the vertex corrections vary as Ω/t * .…”

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“…The free energy must be expanded to order U 2 in order to determine the correct transition temperature in the limit |U| → 0 [8]. The necessary diagrams for the free energy in the Holstein model are presented in Fig.…”

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“…The hopping matrix elements connect the nearest neighbors of a hypercubic lattice in d-dimensions and for concreteness the infinite-dimensional limit (d → ∞) of Metzner and Vollhardt [10] is taken (1/d corrections should be less than 5% in three dimensions [8]). The unit of energy is chosen to be the rescaled matrix element t * .…”

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Weak-coupling expansions for the attractive Holstein and Hubbard models

Freericks

Scalapino

1994

Phys. Rev. B

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Weak-coupling expansions (conserving approximations) are carried out for the attractive Holstein and Hubbard models (on an infinite-dimensional hypercubic lattice) that include all bandstructure and vertex correction effects.Quantum fluctuations are found to renormalize transition temperatures by factors of order unity, but may be incorporated into the superconducting channel of Migdal-Eliashberg theory by renormalizing the phonon frequency and the interaction strength. 71.27.+a, and 71.38.+i Typeset using REVT E X 1

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Zero and finite temperature phase diagram of the spinless fermion model in infinite dimensions

Uhrig

Vlaming

1995

Annalen der Physik

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The phase diagram of the model of spinless fermions with repulsive nearest neighbour interaction is calculated analytically on a hypercubic lattice in in nite dimensions (d ! 1). In spite of its simplicity the model displays a rich phase diagram depending on the doping , the interaction U and the temperature T . The system can be in the homogeneous phase (hom), the nonsegregated AB charge density wave (ab-cdw), the AB phase separation region (ps-ab/hom; coexistence of ab-cdw and hom), the incommensurate phase (ip) or the IP phase separation region (ps-ab/ip; coexistence of ab-cdw and ip). We identify three important values of the interaction U IPL = 0:572 < U IPH = 1:914 < U IP=PS = 4:212 which distinguish four intervals of U . These imply four di erent types of phase diagrams. In all the three phase diagrams with U below U IP=PS the ip appears. We propose a new general ansatz for the order parameter of this phase. A competition between the ip, the ps-ab/ip and the ps-ab/hom is found. The relevance of our ndings for the phase scenario of the Hubbard model is shown.

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Thermodynamics of the extended Hubbard model in high dimensions

Cited by 102 publications

References 19 publications

Efficient calculation of the antiferromagnetic phase diagram of the three-dimensional Hubbard model

Efficient calculation of the antiferromagnetic phase diagram of the three-dimensional Hubbard model

Weak-coupling expansions for the attractive Holstein and Hubbard models

Zero and finite temperature phase diagram of the spinless fermion model in infinite dimensions

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