Mark Jarrell

Bayesian inference and the analytic continuation of imaginary-time quantum Monte Carlo data

Abstract We present a way to use Bayesian statistical inference and the principle of maximum entropy to analytically continue imaginary-time quantum Monte Carlo data. We supply the details that are lacking in the seminal literature but are important for the motivated reader to understand the assumptions and approximations embodied in these methods. First, we summarize the general relations between quantum correlation functions and spectral densities. We then review the basic principles, formalism, and philosophy of Bayesian inference and discuss the application of this approach in the context of the analytic continuation problem. Next, we present a detailed case study for the symmetric, infinite-dimension Anderson Hamiltonian. We chose this Hamiltonian because the qualitative features of its spectral density are well established and because a particularly convenient algorithm exists to produce the imaginary-time Greens function data. Shown are all the intermediate steps of data and solution qualification. The importance of careful data preparation and error propagation in the analytic continuation is discussed in the context of this example. Then, we review the different physical systems and physical quantities to which these, or related, procedures have been applied. Finally, we describe other features concerning the application of our methods, their possible improvement, and areas for additional study.

Quantum Cluster Theories

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.

Anomalous normal-state properties of high-Tc superconductors : intrinsic properties of strongly correlated electron systems ?

Abstract A systematic study of optical and transport properties of the Hubbard model, based on the Metzner-Vollhardt dynamical mean-field approximation, is reviewed. This model shows interesting anomalous properties that are, in our opinion, ubiquitous to single-band strongly correlated systems (for all spatial dimensions greater than one) and also compare qualitatively with many anomalous transport features of the high-T c cuprates. This anomalous behaviour of the normal-state properties is traced to a ‘collective single-band Kondo effect’, in which a quasiparticle resonance forms at the Fermi level as the temperature is lowered, ultimately yielding a strongly renormalized Fermi liquid at zero temperature.

Nonlocal Dynamical Correlations of Strongly Interacting Electron Systems

We introduce an extension of the dynamical mean-field approximation (DMFA) that retains the causal properties and generality of the DMFA, but allows for systematic inclusion of nonlocal corrections. Our technique maps the problem to a self-consistently embedded cluster. The DMFA (exact result) is recovered as the cluster size goes to 1 (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 nonlocal correlations suppress the charge-density wave transition temperature.

Hubbard model at infinite dimensions: Thermodynamic and transport properties

We present results on the thermodynamic quantities, resistivity, and optical conductivity for the Hubbard model on a simple hypercubic lattice in infinite dimensions. Our results for the paramagnetic phase display the features expected from an intuitive analysis of the one-particle spectra and substantiate the similarity of the physics of the Hubbard model to those heavy-fermion systems. The calculations were performed using an approximate solution to the single-impurity Anderson model, wich is the key quantity entering the solution of the Hubbard model in this limit

Systematic Study of d-Wave Superconductivity in the 2D Repulsive Hubbard Model

The cluster size dependence of superconductivity in the conventional two-dimensional Hubbard model, commonly believed to describe high-temperature superconductors, is systematically studied using the dynamical cluster approximation and quantum Monte Carlo simulations as a cluster solver. Because of the nonlocality of the d-wave superconducting order parameter, the results on small clusters show large size and geometry effects. In large enough clusters, the results are independent of the cluster size and display a finite temperature instability to d-wave superconductivity.

Dynamical cluster approximation: Nonlocal dynamics of correlated electron systems

We recently introduced the dynamical cluster approximation (DCA), a technique that includes short-ranged dynamical correlations in addition to the local dynamics of the dynamical mean-field approximation while preserving causality. The technique is based on an iterative self-consistency scheme on a finite-size periodic cluster. The dynamical mean-field approximation (exact result) is obtained by taking the cluster to a single site (the thermodynamic limit). Here, we provide details of our method, explicitly show that it is causal, systematic, Phi derivable, and that it becomes conserving as the cluster size increases. We demonstrate the DCA by applying it to a quantum Monte Carlo and exact enumeration study of the two-dimensional Falicov-Kimball model. The resulting spectral functions preserve causality, and the spectra and the charge-density-wave transition temperature converge quickly and systematically to the thermodynamic limit as the cluster size increases.

d-Wave Superconductivity in the Hubbard Model

The superconducting instabilities of the doped repulsive 2D Hubbard model are studied in the intermediate to strong coupling regime with the help of the dynamical cluster approximation. To solve the effective cluster problem we employ an extended noncrossing approximation, which allows for a transition to the broken symmetry state. At sufficiently low temperatures we find stable d-wave solutions with off-diagonal long-range order. The maximal T(c) approximately 150 K occurs for a doping delta approximately 20% and the doping dependence of the transition temperatures agrees well with the generic high- T(c) phase diagram.

The two-channel Kondo route to non-Fermi-liquid metals

We present a pedagogical and critical overview of the two-channel Kondo model and its possible relevance to a number of non-Fermi-liquid alloys and compounds. We survey the properties of the model, how a magnetic two-channel Kondo effect might obtain for ions in metals, and a quadrupolar Kondo effect for ions in metals. We suggest that the incoherent metal behaviour of the two-channel Kondo-lattice model may be useful in understanding the unusual normal-state resistivity of and speculate that the residual resistivity and entropy of the two-channel lattice paramagnetic phase might be removed by either antiferromagnetic (or antiferroquadrupolar) ordering or by a superconducting transition to an odd-frequency pairing state.

Quantum Monte Carlo algorithm for nonlocal corrections to the dynamical mean-field approximation

We present the algorithmic details of the dynamical cluster approximation (DCA), with a quantum Monte Carlo (QMC) method used to solve the effective cluster problem. The DCA is a fully-causal approach which systematically restores non-local correlations to the dynamical mean field approximation (DMFA) while preserving the lattice symmetries. The DCA becomes exact for an infinite cluster size, while reducing to the DMFA for a cluster size of unity. We present a generalization of the Hirsch-Fye QMC algorithm for the solution of the embedded cluster problem. We use the two-dimensional Hubbard model to illustrate the performance of the DCA technique. At half-filling, we show that the DCA drives the spurious finite-temperature antiferromagnetic transition found in the DMFA slowly towards zero temperature as the cluster size increases, in conformity with the Mermin-Wagner theorem. Moreover, we find that there is a finite temperature metal to insulator transition which persists into the weak-coupling regime. This suggests that the magnetism of the model is Heisenberg like for all non-zero interactions. Away from half-filling, we find that the sign problem that arises in QMC simulations is significantly less severe in the context of DCA. Hence, we were able to obtain good statistics for small clusters. For these clusters, the DCA results show evidence of non-Fermi liquid behavior and superconductivity near half-filling.