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arXiv:1112.5347 · Physical Review B

Extension of dynamical mean-field theory by inclusion of nonlocal two-site correlations with variable distance

Torben Jabben, N. Grewe, Sebastian Schmitt

Published Apr 13, 2012• 5 citations
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Abstract

We present an approximation scheme for the treatment of strongly correlated electrons in arbitrary crystal lattices. The approach extends the well-known dynamical mean-field theory to include nonlocal two-site correlations of arbitrary spatial extent. We extract the nonlocal correlation functions from two-impurity Anderson models where the impurity-impurity distance defines the spatial extent of the correlations included. Translational invariance is fully respected by our approach since correlation functions of any two-impurity cluster are periodically embedded to $\underline{k}$ space via a Fourier transform. As a first application, we study the two-dimensional Hubbard model on a simple-cubic lattice. We demonstrate how pseudogap formation in the many-body resonance at the Fermi level results from the inclusion of nonlocal correlations. A comparison of the spectral function with the dynamical-cluster approximation shows qualitative agreement of high- as well as low-energy features.

DOI
10.1103/physrevb.85.165122

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We present an approximation scheme for the treatment of strongly correlated electrons in arbitrary crystal lattices. The approach extends the well-known dynamical mean-field theory to include nonlocal two-site correlations of arbitrary spatial extent. We extract the nonlocal correlation functions from two-impurity Anderson models where the impurity-impurity distance defines the spatial extent of the correlations included. Translational invariance is fully respected by our approach since correlation functions of any two-impurity cluster are periodically embedded to k‾\underline{k}k​ space via a Fourier transform. As a first application, we study the two-dimensional Hubbard model on a simple-cubic lattice. We demonstrate how pseudogap formation in the many-body resonance at the Fermi level results from the inclusion of nonlocal correlations. A comparison of the spectral function with the dynamical-cluster approximation shows qualitative agreement of high- as well as low-energy features.
We study the two-dimensional Hubbard model by means of an extension of the dynamical mean-field approximation (DMFA). The extension is based on the projection technique and the DMFA, in which nonlocal contributions to the dynamics of the system are taken into account through static short-ranged correlations, while dynamical local contributions are kept by the local self-energy. The short-ranged correlations and the local self-energy are determined by mapping the lattice problem onto a self-consistently embedded cluster. We solve the cluster problem by using a noncrossing approximation. The spectral function obtained satisfies the sum rules of its first four moments. The spectra display pronounced nonlocal features including a pseudogap near the Fermi surface due to a competition between the Kondo-singlet formation and short-range spin correlations.
An extension of the dynamical mean-field approximation (DMFA), which allows inclusion of nonlocal correlations is presented. This method is based on the projection technique and the DMFA, in which nonlocal contributions are taken into account through static correlations within a cluster embedded in a self-consistent medium, and local contributions are kept by the local part of the dynamic memory function of the propagator. We examine the method by studying the spinless Falicov-Kimball model. We find that the sum rules of the spectral density and its three first moments are preserved, and new nonlocal features of the spectral function appear near the Fermi level.
We apply a recently introduced hybridization flow functional renormalization group scheme for Anderson-like impurity models as an impurity solver in a dynamical mean-field theory (DMFT) approach to lattice Hubbard models. We present how this scheme is capable of reproducing metallic and insulating solutions of the lattice model. Our setup also offers a numerically rather inexpensive method to calculate two-particle correlation functions. For the paramagnetic Hubbard model on the Bethe lattice in infinite dimensions we calculate the local two-particle vertex for the metallic and the insulating phase. Then we go to a two-site cluster DMFT scheme for the two-dimensional Hubbard model that includes short-range antiferromagnetic fluctuations and obtain the local and nonlocal two-particle vertex functions. We discuss the rich frequency structures of these vertices and compare with the vertex in the single-site solution.
Even after decades of intense research, the single band Hubbard model representing the fundamental model for interacting quantum systems and electronic correlations remains an unsolved cornerstone paradigm in theoretical solid state physics. Within the non-perturbative dynamical mean-field theory (DMFT), the lattice problem is mapped to a self-consistent auxiliary quantum impurity model leading to a local approximation of the self-energy. Nonlocal correlations can be included by the real-space cluster extension of the DMFT (CDMFT). Enlarging the unit cell makes the calculation of a numerically exact solution in CDMFT very challenging. We here use a quantum Monte-Carlo approach in the imaginary-time space in order to solve the self-consistency equations for different cluster sizes. In order to restore the broken translational symmetry re-periodization schemes for the Green function, the self-energy, or its cumulants have been introduced. However, these suffer from ambiguity and may even lead to convergence problems when attempted inside the self-consistency loop. The comparison to numerically exact diagrammatic quantum Monte Carlo calculations shows that introducing a so-called centerfocused extrapolation (CFE) to approximate the lattice self-energy yields very accurate results. Moreover, the CFE converges faster with the cluster size than previous periodization schemes. We here perform a detailed CDMFT analysis for the single-band Hubbard model on the 2D square lattice, reaching real-space cluster sizes of up to 9x9 sites. In addition to spectral properties, we also compute two-particle correlation functions which are not accessible in DMFT. Using benchmarks against diagrammatic Monte Carlo at high temperature, we show that the cluster spin susceptibility can be extrapolated to the exact result at large cluster size. In particular, the exponential decay of spin-spin correlations is very well captured by CDMFT calculations, even when correlations extend beyond the size of the cluster. We further present results at lower temperature T and larger UUU than the range currently accessible with diagrammatic Monte Carlo methods techniques. The CDMFT+CFE represents therefore a powerful computational tool to access the physics of non-local correlations beyond dynamical mean-field theory.
Strongly correlated electron systems show a rich variety of astonishing physical phenomena. However, the strong interactions make the theoretical description of these systems a highly non-trivial task. Even the simplest models cannot be solved exactly, and one has to reside to approximative solutions, which often cover only certain aspects of the physics contained in these models. In recent years progress has been made in the theoretical description of correlated lattice systems through the mapping of lattice models onto effective impurity models. The most prominent example is the mapping of the Hubbard model onto an effective single impurity Anderson model (SIAM). Within this description both models feature the prominent Kondo effect, which leads to the emergence of low energy quasiparticles. The occurrence of such excitations is a result of the dynamical screening of local magnetic moments and the corresponding formation of a low temperature Fermi liquid phase. On the other hand these theories usually fail in the description of non Fermi liquid behavior, which is for example observed in the normal state of high temperature superconductors and some Heavy-Fermion systems. Strong nonlocal fluctuations drive the dynamics of these systems. In the theoretical treatment, due to the mapping of the lattice problem to an effective single impurity problem, these fluctuations are not adequately described.\\\\ In this thesis a new self-consistent approach for the inclusion of spatial correlations is presented for the Hubbard model. In contrast to existing quantum cluster theories in this field, the approach allows for the simultaneous description of short and long range pair correlations. Due to the intimate connection of this new lattice theory with the two impurity Anderson model (TIAM) a thorough numerical investigation of the TIAM is undertaken in this thesis. The competition of the Kondo effect with the RKKY interactions is studied in great detail. Furthermore two new solvers for the TIAM are introduced, which extend existing non crossing approximations for the SIAM.
In this work, we extend the dual triply irreducible local expansion (D-TRILEX) approach for correlated electronic systems by introducing a cluster reference system for the diagrammatic expansion. This framework allows us to consistently combine the exact treatment of short-range correlation effects within the cluster, with an efficient diagrammatic description of the long-range charge and spin collective fluctuations beyond the cluster. We demonstrate the effectiveness of our approach by applying it to the one-dimensional nano-ring Hubbard model, where the low dimensionality enhances non-local correlations. Our results show that the cluster extension of D-TRILEX accurately reproduces the electronic self-energy at momenta corresponding to the Fermi energy, in good agreement with the reliable Hirsch-Fye quantum Monte Carlo solution of the problem. We further compare this method with the more computationally demanding parquet dynamical vertex approximation and find that, our method yields substantially more accurate results at momenta associated with the Fermi surface. We show that the D-TRILEX diagrammatic extension drastically reduces the periodization ambiguity of cluster quantities when mapping back to the original lattice, compared to cluster dynamical mean-field theory (CDMFT). Furthermore, we identify the CDMFT impurity problem as the main source of the translational-symmetry breaking and propose a computational scheme for improving the starting point for the cluster-diagrammatic expansion.