An introduction to quantum cluster methods Lectures given at the CIFAR - PITP International Summer School on Numerical Methods for Correlated Systems in Condensed Matter, Sherbrooke, Canada, May 26 { June 6 2008
Classic numerical approaches to lattice models such as the Hubbard model are usually based on a solution of the model on a periodic lattice with a small number of sites. For instance, Exact Diagonalizations (ED) calculations are performed on periodic systems with no more than ∼ 20 sites and Quantum Monte Carlo (QMC) calculations are limited in practice on systems with . 100 sites. Some extrapolation is then needed to make statements about the thermodynamic (e.g. infinite-size) limit. One advantage of such approaches is their relative simplicity and lack of ambiguity. A disadvantage is that broken symmetry states need careful analysis to be identified, as they are fully revealed only in the thermodynamic limit. Quantum cluster methods are a set of closely related approaches that instead consider a finite cluster of sites embedded in the infinite lattice. The embedding is done by adding to the cluster additional fields or bath degrees of freedom such as to best represent the effect of the surrounding infinite lattice. Variational or self-consistency principles are used to set the values of these extra parameters. In these approaches, broken symmetry states can appear even for the smallest clusters used, somewhat like ordinary mean-field theory. However, unlike mean field theory, these approaches are dynamical and retain the full effect of strong correlations. These methods are usually known by their acronyms:
Classic numerical approaches to lattice models such as the Hubbard model are usually based on a solution of the model on a periodic lattice with a small number of sites. For instance, Exact Diagonalizations (ED) calculations are performed on periodic systems with no more than ∼ 20 sites and Quantum Monte Carlo (QMC) calculations are limited in practice on systems with . 100 sites. Some extrapolation is then needed to make statements about the thermodynamic (e.g. infinite-size) limit. One advantage of such approaches is their relative simplicity and lack of ambiguity. A disadvantage is that broken symmetry states need careful analysis to be identified, as they are fully revealed only in the thermodynamic limit. Quantum cluster methods are a set of closely related approaches that instead consider a finite cluster of sites embedded in the infinite lattice. The embedding is done by adding to the cluster additional fields or bath degrees of freedom such as to best represent the effect of the surrounding infinite lattice. Variational or self-consistency principles are used to set the values of these extra parameters. In these approaches, broken symmetry states can appear even for the smallest clusters used, somewhat like ordinary mean-field theory. However, unlike mean field theory, these approaches are dynamical and retain the full effect of strong correlations. These methods are usually known by their acronyms:
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.
We study the Hubbard model using the cellular dynamical mean-field theory (CDMFT) with quantum Monte Carlo (QMC) simulations. We present the algorithmic details of CDMFT with the Hirsch-Fye QMC method for the solution of the self-consistently embedded quantum cluster problem. We use the one- and two-dimensional half filled Hubbard model to gauge the performance of CDMFT+QMC particularly for small clusters by comparing with the exact results and also with other quantum cluster methods. We calculate single-particle Green's functions and self-energies on small clusters to study their size dependence in one and two dimensions. It is shown that in one dimension, CDMFT with two sites in the cluster is already able to describe with high accuracy the evolution of the density as a function of the chemical potential and the compressibility divergence at the Mott transition, in good agreement with the exact Bethe ansatz result. With increasing U the result on small clusters rapidly approaches that of the infinite size cluster. Large scattering rate and a positive slope in the real part of the self-energy in one dimension suggest that the system is a non-Fermi liquid for all the parameters studied here. In two dimensions, at intermediate to strong coupling, even the smallest cluster $({N}_{c}=2\ifmmode\times\else\texttimes\fi{}2)$ accounts for more than 95% of the correlation effect of the infinite-size cluster in the single particle spectrum, suggesting that some of the important problems in strongly correlated electron systems may be studied highly accurately with a reasonable computational effort. Finally, as an application that is sensitive to details of correlations, we show that CDMFT+QMC can describe spin-charge separated Luttinger liquid physics in one dimension. The spinon and holon branches appear only for sufficiently large system sizes.
The previous chapter is devoted to the formulation of DMFT and applications that are exact in the limit of infinite dimensions; however, in finite dimensions this is only a single-site approximation. The present chapter is devoted to clusters and other methods to treat correlation between sites. The equations are derived in a unified way, applicable to a single site or a cluster. For a cluster with more than one site there are various ways to choose the boundary conditions and the embedding procedure, and special care must be taken to satisfy causality and translation invariance. A few selected applications illustrate different techniques and results for Hubbard-type models. The essential features of DMFT are brought out in the previous chapter in the simplest form: the single-site approximation in which correlation between sites can be neglected. In this case the self-energy due to interactions can be calculated using an auxiliary system of a single site embedded in an effective medium that represents the surrounding crystal. The resulting many-body problem is equivalent to an Anderson impurity model that must be solved self-consistently. The results are exact for infinite dimensions d → ∞; however, in any finite dimension, i.e., real systems, this is just a mean-field approximation analogous to the Weiss mean field in magnetic systems or the CPA for disordered alloys. There are, however, many important properties and phenomena that require us to go beyond mean-field approximations. For example, basic questions concerning the nature of a metal–insulator transition can be established only by quantitative assessment of the correlations between different sites. Is a single site sufficient (the central idea in the arguments by Mott discussed in Sec. 3.2) or is correlation between sites (for example, an ordered antiferromagnetic state) essential for interactions to lead to an insulating state? A complete theory must establish the range of correlation needed to understand the mechanisms and to make quantitative calculations. A striking example that requires a k -dependent self-energy is the opening of a “pseudogap” in only part of the Brillouin zone in a high-temperature superconductor, as illustrated in Fig. 2.16. This chapter is devoted to methods that go beyond the single-site approximation to take into account interactions and correlations between electrons on sites.
This thesis will describe efforts to enhance our ability to simulate the 2D Hubbard model. Chapter 2 provides a background on the main computational techniques used throughout the work, including quantum Monte Carlo (QMC), dynamical mean field theory (DMFT), the dynamical cluster approximation (DCA), the continuous time auxiliary field algorithm (CTAUX), and the Maximum Entropy Method (MEM) for numeric analytic continuation. Chapter 3 presents new work on applying Twisted Boundary Conditions to the DCA framework. This method is applied in a effort to access thermodynamic (i.e. large system) information about the Hubbard model without the immense computational expense required to simulate large lattices directly. Chapter 4 describes efforts to study the extended 2D Hubbard model, which re-introduces non-local interactions between electrons that drive the formation of charge ordered phases. The work includes a thorough analysis of the phase diagram of the model away from half-filling, as well as analysis of the effect of non-local interactions on anti-ferromagnetic fluctuations and competition between the charge order and AFM states. Chapter 5 includes a derivation of the dual fermions diagrammatic expansion and the dual fermions ladder approximation used to compute corrections to single site DMFT calculations. The method is applied to the 2D Hubbard model in order to study the evolution of its spectral function, Fermi surface, and momentum dependent mass renormalization as the temperature, interaction strength, and doping are changed. The great advantage of this method is that we are able to access arbitrary momenta throughout the Brillouin zone, in contrast to the limitation to a few cluster momenta that characterize cluster methods like DCA.
These lecture notes provide an introduction to quantum cluster methods for strongly correlated systems. Cluster Perturbation Theory (CPT), the Variational Cluster Approximation (VCA) and Cellular Dynamical Mean Field Theory (CDMFT) are described, as well as the exact diagonalization solver for the cluster. Potthoff's self-energy functional formalism is reviewed. Some numerical procedures, in particular regarding the exact diagonalization method and the frequency-momentum integrals needed in VCA, are discussed in detail.
Two of the primary sources of error in the Cluster dynamical mean field theory (CDMFT) technique arise from the use of finite size clusters and finite size baths, which makes the development of impurity solvers that can treat larger systems an essential goal. In this work we introduce an impurity solver based on the recently developed dynamical variational Monte Carlo (dVMC) method. Variational Monte Carlo possesses a favorable scaling as a function of system size, which enables the treatment of systems beyond the reach of current exact diagonalization solvers. To benchmark the technique, we perform a systematic set of CDMFT calculations on the single-band one-dimensional Hubbard model. We compare to results obtained with an exact diagonalization solver for small clusters, and against the exact solution in the thermodynamic limit obtained by Lieb and Wu for larger clusters. The development of improved impurity solvers will help extend the reach of quantum cluster methods, which can be applied to a wide range of strongly correlated electron systems, promising new insights into their emergent behavior.