Quantum cluster approaches offer new perspectives to study the complexities of macroscopic correlated fermion systems. These approaches can be understood as generalized mean-field theories. Quantum cluster approaches are non-perturbative and are always in the thermodynamic limit. Their quality can be systematically improved, and they provide complementary information to finite size simulations. They have been studied intensively in recent years and are now well established. After a brief historical review, this article comparatively discusses the nature and advantages of these cluster techniques. Applications to common models of correlated electron systems are reviewed. 1
We present a brief description of how methods of Bayesian inference are used to obtain real frequency information by the analytic continuation of imaginary-time quantum Monte Carlo data. We present the procedure we used, which is due to R. K. Bryan, and summarize several bottleneck issues.
An essentially exact solution of the infinite-dimensional Hubbard model is made possible by a new self-consistent Monte Carlo procedure. Near half filling antiferromagnetism and a pseudogap in the single-particle density of states are found for sufficiently large values of the intrasite Coulomb interaction. At half filling the antiferromagnetic transition temperature obtains its largest value when the intrasite Coulomb interaction U ~ 3. PACS numbers: 75.10.Jm, 71.10.-Hx, 75.10.Lp, 75.30.Kz The Hubbard model of strongly correlated electronic systems has been an enduring problem in condensed matter physics. It is believed to properly describe some of the properties of transition-metal oxides, and possibly high-temperature superconductors. Despite the simplicity of the model, no exact solutions exist except in one dimension [1]. Recently, a new approach [2-4] based on a dimensional expansion has been proposed to study such strongly correlated lattice models. In this paper, I present the first essentially exact numerical solution of the Hubbard model in the infinite-dimensional limit. This solution retains the physics expected in the lowdimensional model, including antiferromagnetism (Figs.
3, 4, and 5) and the formation of a correlation inducedMott-Hubbard gap in the single-particle density of states (Fig. 6).The Hamiltonian of interest is
We introduce an extension of the dynamical mean field approximation (DMFA) which retains the causal properties and generality of the DMFA, but allows for systematic inclusion of non-local corrections. Our technique maps the problem to a self-consistently embedded cluster. The DMFA (exact result) is recovered as the cluster size goes to one (infinity). As a demonstration, we study the Falicov-Kimball model using a variety of cluster sizes. We show that the sum rules are preserved, the spectra are positive definite, and the non-local correlations suppress the CDW transition temperature.Introduction. Strongly interacting electron systems have been on the forefront of theoretical and experimental interest for several decades. This interest has intensified with the discovery of a variety of Heavy Fermion and related non Fermi liquid systems and the high-T c superconductors. In all these systems strong electronic interactions play a dominant role in the selection of at least the low temperature phase. The simplest theoretical models of strongly correlated electrons, the Hubbard model (HM) and the periodic Anderson model (PAM), have remained unsolved in more than one dimension despite a multitude of sophisticated techniques introduced since the inception of the models.With the ground breaking work by Metzner and Vollhardt [1] it was realized that these models become significantly simpler in the limit of infinite dimensions, D = ∞. Namely, provided that the kinetic energy is properly rescaled as 1/ √ D, they retain only local, though nontrivial dynamics: The self energy is constant in momentum space, though it has a complicated frequency dependence. Consequently, the HM and PAM map onto a generalized single impurity Anderson model. The thermodynamics and phase diagram have been obtained numerically by quantum Monte Carlo (QMC) and other methods. [2][3][4] The name dynamical mean field approximation (DMFA) has been coined for approximations in which a purely local self energy (and vertex function) is assumed in the context of a finite dimensional electron system. While it has been shown that this approximation captures many key features of strongly correlated systems even in a finite dimensional context, the DMFA, which leads to an effective single site theory, has some obvious limitations. For example, the DMFA can not describe phases with explicitly nonlocal order parameters, such as d-wave superconductivity, nor can it describe
Maier, Jarrell, and Pruschke Reply: In his Comment [1] on our letter [2] the author claims that our numerical results for the 2D Hubbard model contradict exact results published by the author. There, it was rigorously shown that the 2D Hubbard model does not exhibit d-wave pairing long-range order at finite temperatures. This result and the Mermin-Wagner theorem precludes long-range order with a broken continuous symmetry in two dimensions [3,4]. Although within our method we do find stable d-wave superconducting solutions in the 2D model, our results do not contradict this theorem.In our Letter we presented numerical results for the 2D Hubbard model obtained with the dynamical cluster approximation (DCA). The DCA systematically incorporates causal nonlocal corrections to the dynamical mean-field approximation (DMFA) by mapping the lattice system onto an embedded cluster of size N c . Our calculation explores the initial corrections to the mean-field superconducting phase diagram of the DMFA. We used the smallest cluster (N c 4) which allows for a d-wave transition and maintains the translational and point-group symmetries of the lattice. Therefore short-ranged interactions (of the order of the cluster size) are included accurately, whereas longer-ranged interactions are treated on a mean-field level within our approach.It is well known that such methods with mean-field character yield finite transition temperatures and second order transitions even for two dimensions. However, the mean-field character of the DCA can gradually be reduced as N C tends to infinity. The author of the Comment further objected to an alleged statement in our paper that the transition we observed was identified as Kosterlitz-Thouless transition. No such statement was made. Although we did speculate about the limit of large N c and cite other such speculations [5], we did not present any scaling calculations, and this was not the thrust of our paper. Indeed, N c 4 is the largest cluster that can be treated with noncrossing approximation calculations used to obtain the results presented in our paper.The mean-field character of our calculation can be stabilized as N c !`by introducing a coupling t Ќ [6] into the third dimension t Ќ ͑k x , k y , k z ͒ 22t Ќ ͑cosk x 2 cosk y ͒ 2 cosk z (1)
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