We review the dynamical mean-field theory of strongly correlated electron systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the standard mean-field construction from classical statistical mechanics to quantum problems. We discuss the physical ideas underlying this theory and its mathematical derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamical mean-field equations are reviewed and compared to each other. The method can be used for the determination of phase diagrams (by comparing the stability of various types of long-1 range order), and the calculation of thermodynamic properties, one-particle Green's functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. We also present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are also provided with this article.

We present a powerful method for calculating the thermodynamic properties of infinitedimensional Hubbard-type models using an exact diagonalization of an Anderson model with a finite number of sites. The resolution obtained for Green's functions is far superior to that of quantum Monte Carlo calculations. We apply the method to the half-filled Hubbard model for a discussion of the metal-insulator transition, and to the two-band Hubbard model where we find direct evidence for the existence of a superconducting instability at low temperatures.

Melting in two spatial dimensions, as realized in thin films or at interfaces, represents one of the most fascinating phase transitions in nature, but it remains poorly understood. Even for the fundamental hard-disk model, the melting mechanism has not been agreed on after fifty years of studies. A recent Monte Carlo algorithm allows us to thermalize systems large enough to access the thermodynamic regime. We show that melting in hard disks proceeds in two steps with a liquid phase, a hexatic phase, and a solid. The hexatic-solid transition is continuous while, surprisingly, the liquid-hexatic transition is of first-order. This melting scenario solves one of the fundamental statistical-physics models, which is at the root of a large body of theoretical, computational and experimental research. Generic two-dimensional particle systems cannot crystallize at finite temperature [1][2][3] because of the importance of fluctuations, yet they may form solids [4]. This paradox has provided the motivation for elucidating the fundamental melting transition in two spatial dimensions. A crystal is characterized by particle positions which fluctuate about the sites of an infinite regular lattice. It has long-range positional order. Bond orientations are also the same throughout the lattice. A crystal thus possesses long-range orientational order. The positional correlations of a two-dimensional solid decay to zero as a power law at large distances. Because of the absence of a scale, one speaks of "quasi-long range" order. In a two-dimensional solid, the lattice distortions preserve long-range orientational order [5], while in a liquid, both the positional and the orientational correlations decay exponentially.Besides the solid and the liquid, a third phase, called "hexatic", has been discussed but never clearly identified in particle systems. The hexatic phase is characterized by exponential positional but quasi-long range orientational correlations. It has long been discussed whether the melting transition follows a one-step first-order scenario between the liquid and the solid (without the hexatic) as in three spatial dimensions Two-dimensional melting was discovered [4] in the simplest particle system, the hard-disk model. Hard disks (of radius σ) are structureless and all configurations of nonoverlapping disks have zero potential energy. Two isolated disks only feel the hard-core repulsion, but the other disks mediate an entropic "depletion" interaction (see, e.g., [19]). Phase transitions result from an "order from disorder" phenomenon: At high density, ordered configurations can allow for larger local fluctuations, thus higher entropy, than the disordered liquid. For hard disks, no difference exists between the liquid and the gas. At fixed density η, the phase diagram is independent of temperature T = 1/k B β, and the pressure is proportional to T , as discovered by D. Bernoulli in 1738. Even for this basic model, the nature of the melting transition has not been agreed on.The hard-disk model has been simulated with the...

Ž .We discuss supersymmetric Yang-Mills theory dimensionally reduced to zero dimensions and evaluate the SU 2 and Ž .Ž . SU 3 partition functions by Monte Carlo methods. The exactly known SU 2 results are reproduced to very high precision.Ž . Our calculations for SU 3 agree closely with an extension of a conjecture due to Green and Gutperle concerning the exact Ž . value of the SU N partition functions. q

We study classical hard-core dimer models on three-dimensional lattices using analytical approaches and Monte Carlo simulations. On the bipartite cubic lattice, a local gauge field generalization of the height representation used on the square lattice predicts that the dimers are in a critical Coulomb phase with algebraic, dipolar, correlations, in excellent agreement with our large-scale Monte Carlo simulations. The non-bipartite FCC and Fisher lattices lack such a representation, and we find that these models have both confined and exponentially deconfined but no critical phases. We conjecture that extended critical phases are realized only on bipartite lattices, even in higher dimensions.The statistical mechanics of dimers on a lattice that interact with one another only via hard-core exclusion has long been of interest to mathematicians and physicists [1,2]. It is one of the simplest models describing the arrangement of anisotropic objects on a regular substrate. Applications [3] range from diatomic molecules on surfaces to spin ice in a magnetic field [4]. By Kasteleyn's theorem, on two-dimensional (2d) planar lattices, the statistical mechanics of (close-packed) dimer coverings can be computed exactly [5]. A consistent picture has emerged from this work for a large class of 2d dimer models. On bipartite 2d lattices, dimer models are in confined phases in which the free energy of two inserted test monomers (unpaired sites) increases with separation. The increase is logarithmic for phases with algebraic dimer correlations and linear for the remaining ones. An example of the former is the square lattice [5,6] and one of the latter is the exotic diamond-octagon "4-8" lattice which exhibits two confining phases as the strength of the diamond bonds is varied [3]. By contrast, non-bipartite lattices exhibit both deconfined and confined phases but always with exponentially decaying dimer correlations except at the boundary between such phases. Examples are the triangular [7,8] and kagome [9] lattices, which are deconfined with exponentially decaying correlations, and the Fisher lattice -equivalent to the 2d Ising model [10] -which exhibits a deconfinement transition [11].Dimer models on two-dimensional bipartite lattices can also be understood through their height representations [12,13]. Within this powerful framework, the two subcategories of critical and non-critical dimer correlations are described as "rough" and "flat" phases. In either case the defect interaction corresponding to the monomer free energy is long-ranged. Dimer models on non-bipartite lattices lack a similar long-wavelength description.In this paper we generalize the height representation to three dimensions. We show that dimer models on bipartite 3d lattices admit a local gauge representation which results in a "Coulomb" phase with algebraic, dipolar forms for the dimer correlations and monomer interactions that fall off inversely with their separation. We present large scale Monte Carlo simulations on 3d lattices and demonstrate that the dime...

No abstract

The phase diagram of two-dimensional continuous particle systems is studied using the event-chain Monte Carlo algorithm. For soft disks with repulsive power-law interactions ∝r^{-n} with n≳6, the recently established hard-disk melting scenario (n→∞) holds: a first-order liquid-hexatic and a continuous hexatic-solid transition are identified. Close to n=6, the coexisting liquid exhibits very long orientational correlations, and positional correlations in the hexatic are extremely short. For n≲6, the liquid-hexatic transition is continuous, with correlations consistent with the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) scenario. To illustrate the generality of these results, we demonstrate that Yukawa particles likewise may follow either the KTHNY or the hard-disk melting scenario, depending on the Debye-Hückel screening length as well as on the temperature.

We report large-scale computer simulations of the hard-disk system at high densities in the region of the melting transition. Our simulations reproduce the equation of state, previously obtained using the event-chain Monte Carlo algorithm, with a massively parallel implementation of the local Monte Carlo method and with event-driven molecular dynamics. We analyze the relative performance of these simulation methods to sample configuration space and approach equilibrium. Our results confirm the first-order nature of the melting phase transition in hard disks. Phase coexistence is visualized for individual configurations via the orientational order parameter field. The analysis of positional order confirms the existence of the hexatic phase.