We present a powerful method for calculating the thermodynamic properties of the Hubbard model in infinite dimensions, using an exact diagonalization of an Anderson model with a finite number of sites. At finite temperatures, the explicit diagonalization of the Anderson Hamiltonian allows the calculation of Green's functions with a resolution far superior to that of Quantum Monte Carlo calculations. At zero temperature, the Lanczòs method is used and yields the essentially exact zero-temperature solution of the model, except in a region of very small frequencies. Numerical results for the half-filled case in the paramagnetic phase (quasi-particle weight, self-energy, and also real-frequency spectral densities) are presented.
A detailed quantitative study of the physical properties of the infinite-dimensional Hubbard model at half filling is presented. The method makes use of an exact mapping onto a single-impurity model supplemented by a self-consistency condition. This coupled problem is solved numerically. Results for thermodynamic quantities (specific heat, entropy, . . .), one-particle spectral properties, and magnetic properties (response to a uniform magnetic field) are presented and discussed. The nature of the Mott-Hubbard metal-insulator transition found in this model is investigated. A numerical solution of the mean-field equations inside the antiferromagnetic phase is also reported.
We present a numerical solution of the infinite-dimensional Hubbard model at finite temperature in the paramagnetic phase. The problem reduces to a single-impurity Anderson model supplemented by a self-consistency condition. Using Monte Carlo methods and complete enumeration we determine the imaginary-time Green's function, the density of doubly occupied sites, and the compressibility close to half filling. All three quantities present direct evidence for a Mott insulating phase above a critical value of U.
We derive analytical expressions for the spectral moments of the dynamical response functions of the Hubbard model using the high-temperature series expansion. We consider generic dimension
d as well as the infinite-
d limit, arbitrary electron density
n, and both finite and infinite repulsion
U. We use moment-reconstruction methods to obtain the one-electron spectral function, the self-energy, and the optical conductivity. They are all smooth functions at high temperature and, at large
U, they are featureless with characteristic widths of the order of the lattice hopping parameter
t. In the infinite-
d limit, we compare the series expansion results with accurate numerical renormalization group and interaction expansion quantum Monte Carlo results. We find excellent agreement down to surprisingly low temperatures, throughout most of the bad-metal regime, which applies for $T\ensuremath{\gtrsim}(1\ensuremath{-}n)D$, the Brinkman-Rice scale. The resistivity increases linearly in
T at high temperature without saturation. This results from the
1/T behavior of the compressibility or kinetic energy, which play the role of the effective carrier number. In contrast, the scattering time (or diffusion constant) saturates at high
T. We find that $\ensuremath{\sigma}(n,T)\ensuremath{\approx}(1\ensuremath{-}n)\ensuremath{\sigma}(n=0,T)$ to a very good approximation for all
n, with $\ensuremath{\sigma}(n=0,T)\ensuremath{\propto}t/T$ at high temperatures. The saturation at small
n occurs due to a compensation between the density dependence of the effective number of carriers and that of the scattering time. The
T dependence of the resistivity displays a kneelike feature which signals a crossover to the intermediate-temperature regime where the diffusion constant (or scattering time) starts increasing with decreasing
T. At high temperatures, the thermopower obeys the Heikes formula, while the Wiedemann-Franz law is violated with the Lorenz number vanishing as
1/T2. The relevance of our calculations to experiments probing high-temperature transport in materials with strong electronic correlations or ultracold atomic gases in an optical lattice is briefly discussed.
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.