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A new approach to the calculation of spectra for strongly correlated systems

Theo A. Costi, Alex C. Hewson

Published Mar 21, 1991• 0 citations
Publisher

Abstract

Abstract Direct diagonalization of a small cluster embedded within a larger system gives some information about the larger system but only over the largest energy scale. Such cluster calculations may help to identify peaks in XPS and other spectra in this energy range, but are of little direct use in giving information about the lower energy excitations and thermodynamic properties. For these lower energy excitations larger range correlations have to be taken into account. Whatever the computing resources, there is a rather sharp cut-off on the size of the cluster that can be diagonalized numerically. There are various possibilities for extending these to obtain results on smaller energy scales and larger clusters.

DOI
10.1093/oso/9780198539384.003.0021

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Abstract Direct diagonalization of a small cluster embedded within a larger system gives some information about the larger system but only over the largest energy scale. Such cluster calculations may help to identify peaks in XPS and other spectra in this energy range, but are of little direct use in giving information about the lower energy excitations and thermodynamic properties. For these lower energy excitations larger range correlations have to be taken into account. Whatever the computing resources, there is a rather sharp cut-off on the size of the cluster that can be diagonalized numerically. There are various possibilities for extending these to obtain results on smaller energy scales and larger clusters.
The central problem in the modern electronic structure theory is the calculation of cor- relation energy, possibly by an approach that would account for both static and dynamic correlation in an efficient, balanced and accurate way. In this thesis, I present a collection of methods that combine the effective treatment of dynamic correlation by the coupled cluster theory with density matrix renormalization group, a well-established technique for calculations of strongly correlated systems. The connection between them is achieved via the tailored coupled clusters (TCC) ansatz, which conveniently does not impose any ad- ditional computational costs. After the successful initial assessment, we developed more efficient implementations of these methods by employing the local approaches based on pair natural orbitals. This way, we extended the range of possible applications to larger systems with thousands of basis functions. To assess the accuracy of TCC as well as its local counterparts, we performed a variety of benchmark calculations ranging from small, yet challenging systems such as the nitrogen molecule or tetramethyleneethane diradical, to larger molecules like oxo-Mn(Salen) or Fe(II)-porphyrin model. 1
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.
The cluster perturbation theory is presented in the 2D Hubbard model constructed using X operators in the Hubbard-I approximation. The short-range magnetic order is taken into account by dividing the entire lattice into individual 2 × 2 clusters and solving the eigenvalue problem in an individual cluster using exact diagonalization taking into account all excited levels. The case of half-filling taking into account jumps between nearest neighbors is considered. As a result of numerical solution, a shadow zone is discovered in the quasiparticle spectrum. It is also found that a gap in the density of states in the quasiparticle spectrum at zero temperature exists for indefinitely small values of Coulomb repulsion parameter U and increases with this parameter. It is found that the presence of this gap in the spectrum is due to the formation of a short-range antiferromagnetic order. An analysis of the temperature evolution of the density of states shows that the metal-insulator transition occurs continuously. The existence of two characteristic energy scales at finite temperatures is demonstrated, the larger scale is associated with the formation of a pseudogap in the vicinity of the Fermi level, and the smaller scale is associated with the metal-insulator transition temperature. A peak in the density of states at the Fermi level, which is predicted in the dynamic mean field theory in the vicinity of the metal-insulator transition, is not observed.