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Bulletin of the American Physical Society

Spectral properties of the one-dimensional Hubbard model: cluster dynamical mean-field approaches

Ara Go, Gun Sang Jeon

Published Mar 23, 2011• 0 citations
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Properties of the one-dimensional Hubbard model: cellular dynamical mean-field description

Ara Go, Gun Sang Jeon

The one-dimensional half-filled Hubbard model is considered at zero temperature within the cellular dynamical mean-field theory (CDMFT). By the computation of the spectral gap and the energy density with various cluster and bath sizes we examine the accuracy of the CDMFT in a systematic way, which proves the accurate description of the one-dimensional systems by the CDMFT with small clusters. We also calculate the spectral weights in a full range of the momentum for various interaction strengths. The results do not only account for the spin-charge separation, but they also reproduce all the features of the Bethe ansatz dispersions, implying that the CDMFT provides an excellent description of the spectral properties of low-dimensional interacting systems.

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Phase diagram and criticality of the three dimensional Hubbard model

G. Rohringer, A. Toschi, A. A. Katanin +1 more

Almost 50 years after the invention of the Hubbard model [1–3] and despite modern petaflop supercomputers, a precise analysis of the criticality of this most basic model for electronic correlations has not been achieved so far, at least not in three dimensions. Dynamical mean field theory (DMFT) [4–6] was a big step forward to quantitatively calculate the three dimensional Hubbard model since the major contribution of electronic correlations, i.e., the local ones, are well captured within this theory. Local correlations give rise to quasiparticle renormalization, the Mott-Hubbard transition, magnetism, and even more subtle issues such as kinks in a purely electronic models [7]. However, non-local spatial correlations are also naturally generated by a purely local Hubbard interaction, and, as it is well known, they become of essential importance in the vicinity of secondorder phase transitions. As these correlations are neglected in DMFT, this scheme can provide only for a conventional mean-field (MF) description of the critical properties. To overcome this shortcoming cluster extensions to DMFT such as the dynamical cluster approximation (DCA) and cluster-DMFT have been proposed, see Ref. 8 for a review. In these approaches non-local correlations beyond DMFT are taken into account, however only within the range of the cluster size; and due to computational limitations the actual size of the cluster is severely restricted to about 3-5 sites for each direction. Hence, short-range correlations are included by these approaches, whereas long-range ones are not. Nonetheless, Kent et al. [9] were able to extrapolate the cluster size of so-called Betts clusters to infinity, albeit assuming from the very beginning the critical exponents to be those of the Heisenberg model. This way they extrapolated the critical temperature of the paramagnetic-toantiferromagnetic phase transition which was found to be in agreement with earlier quantum Monte Carlo (QMC) simulations of Staudt et al. [10] on finite lattices.

•Apr 11, 2011•0 citations

Thermodynamic and spectral properties of quantum many-particle systems

Sebastian Fuchs

The investigation of quantum many-particle systems is a central goal of modern condensed-matter physics. Complex physical phenomena such as high-temperature superconductivity, heavy fermion behavior, and correlation-driven metal-insulator transitions are central aspects of this particular field of science. The Hubbard model is a paradigm of correlated-electron physics. It is one of the simplest quantum-mechanical lattice models capable of capturing relevant physical aspects of strongly correlated electron systems. The Hubbard model also plays an important role in the explanation of many-body phenomena observed in ultra-cold atoms trapped in optical lattices. In this thesis, we study the three-dimensional Hubbard model using quantum Monte Carlo simulations (QMC). Our aim is a precise numerical study of its properties in the thermodynamic limit. However, a direct investigation of lattice models in the thermodynamic limit is generally impossible because QMC simulations cannot treat infinite lattice sizes. Therefore, approximative schemes that allow calculations directly in the thermodynamic limit are advantageous. A widely applied approximation of this kind is the dynamical mean-field theory (DMFT) which maps the infinite-lattice problem onto a single-site impurity model embedded in a mean field. We apply cluster mean-field theories which represent a systematic extension of the DMFT by expanding the single impurity to a finite cluster. The second focus of this thesis is the calculation of spectral properties of the Hubbard model. QMC methods map the quantum-mechanical lattice model on a classical one at the expense of an additional imaginary time dimension. For this reason, QMC algorithms can only provide dynamical data on the imaginary time or frequency axis. The necessary analytic continuation to the physically relevant real axis has proven to be a difficult problem that has to be approached by advanced data analysis tools. The maximum-entropy method (MEM) is the standard technique to handle this problem. We apply the MEM to the problem of extracting momentum-resolved single particle spectra from QMC simulations of the 3D Hubbard model. Furthermore, we discuss and expand algorithms that were recently proposed as alternatives to the standard MEM approach.

•Jan 1, 2011•5 citations

Hubbard Model in Dynamical Mean-Field Theory

Vladimir I. Anisimov, Yu. A. Izyumov

Springer series in solid-state sciences•Jan 1, 2010•0 citations
The one-dimensional half-filled Hubbard model is considered at zero temperature within the cellular dynamical mean-field theory (CDMFT). By the computation of the spectral gap and the energy density with various cluster and bath sizes we examine the accuracy of the CDMFT in a systematic way, which proves the accurate description of the one-dimensional systems by the CDMFT with small clusters. We also calculate the spectral weights in a full range of the momentum for various interaction strengths. The results do not only account for the spin-charge separation, but they also reproduce all the features of the Bethe ansatz dispersions, implying that the CDMFT provides an excellent description of the spectral properties of low-dimensional interacting systems.
Almost 50 years after the invention of the Hubbard model [1–3] and despite modern petaflop supercomputers, a precise analysis of the criticality of this most basic model for electronic correlations has not been achieved so far, at least not in three dimensions. Dynamical mean field theory (DMFT) [4–6] was a big step forward to quantitatively calculate the three dimensional Hubbard model since the major contribution of electronic correlations, i.e., the local ones, are well captured within this theory. Local correlations give rise to quasiparticle renormalization, the Mott-Hubbard transition, magnetism, and even more subtle issues such as kinks in a purely electronic models [7]. However, non-local spatial correlations are also naturally generated by a purely local Hubbard interaction, and, as it is well known, they become of essential importance in the vicinity of secondorder phase transitions. As these correlations are neglected in DMFT, this scheme can provide only for a conventional mean-field (MF) description of the critical properties. To overcome this shortcoming cluster extensions to DMFT such as the dynamical cluster approximation (DCA) and cluster-DMFT have been proposed, see Ref. 8 for a review. In these approaches non-local correlations beyond DMFT are taken into account, however only within the range of the cluster size; and due to computational limitations the actual size of the cluster is severely restricted to about 3-5 sites for each direction. Hence, short-range correlations are included by these approaches, whereas long-range ones are not. Nonetheless, Kent et al. [9] were able to extrapolate the cluster size of so-called Betts clusters to infinity, albeit assuming from the very beginning the critical exponents to be those of the Heisenberg model. This way they extrapolated the critical temperature of the paramagnetic-toantiferromagnetic phase transition which was found to be in agreement with earlier quantum Monte Carlo (QMC) simulations of Staudt et al. [10] on finite lattices.
The investigation of quantum many-particle systems is a central goal of modern condensed-matter physics. Complex physical phenomena such as high-temperature superconductivity, heavy fermion behavior, and correlation-driven metal-insulator transitions are central aspects of this particular field of science. The Hubbard model is a paradigm of correlated-electron physics. It is one of the simplest quantum-mechanical lattice models capable of capturing relevant physical aspects of strongly correlated electron systems. The Hubbard model also plays an important role in the explanation of many-body phenomena observed in ultra-cold atoms trapped in optical lattices. In this thesis, we study the three-dimensional Hubbard model using quantum Monte Carlo simulations (QMC). Our aim is a precise numerical study of its properties in the thermodynamic limit. However, a direct investigation of lattice models in the thermodynamic limit is generally impossible because QMC simulations cannot treat infinite lattice sizes. Therefore, approximative schemes that allow calculations directly in the thermodynamic limit are advantageous. A widely applied approximation of this kind is the dynamical mean-field theory (DMFT) which maps the infinite-lattice problem onto a single-site impurity model embedded in a mean field. We apply cluster mean-field theories which represent a systematic extension of the DMFT by expanding the single impurity to a finite cluster. The second focus of this thesis is the calculation of spectral properties of the Hubbard model. QMC methods map the quantum-mechanical lattice model on a classical one at the expense of an additional imaginary time dimension. For this reason, QMC algorithms can only provide dynamical data on the imaginary time or frequency axis. The necessary analytic continuation to the physically relevant real axis has proven to be a difficult problem that has to be approached by advanced data analysis tools. The maximum-entropy method (MEM) is the standard technique to handle this problem. We apply the MEM to the problem of extracting momentum-resolved single particle spectra from QMC simulations of the 3D Hubbard model. Furthermore, we discuss and expand algorithms that were recently proposed as alternatives to the standard MEM approach.