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Physical review. B, Condensed matter

Exact-diagonalization method for correlated-electron models

Canio Noce, Mario Cuoco

Published Nov 1, 1996• 22 citations
Publisher

Abstract

We present a method to compute eigenstates and eigenvalues of strongly correlated models on clusters. The calculation is performed applying the SU(2) symmetry, which allows a considerable reduction of the dimension of Hilbert space of the system. We illustrate the method classifying the states of the four-sites Hubbard model on a chain. This classification has been made using also the translational symmetry and the S\ifmmode \tilde{}\else \~{}\fi{}\ifmmode \tilde{U}\else \~{U}\fi{}(2) pseudospin symmetry. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.13047

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•Feb 5, 2012•6 citations
We present a method to compute eigenstates and eigenvalues of strongly correlated models on clusters. The calculation is performed applying the SU(2) symmetry, which allows a considerable reduction of the dimension of Hilbert space of the system. We illustrate the method classifying the states of the four-sites Hubbard model on a chain. This classification has been made using also the translational symmetry and the S\ifmmode \tilde{}\else ~{}\fi{}\ifmmode \tilde{U}\else ~{U}\fi{}(2) pseudospin symmetry. \textcopyright{} 1996 The American Physical Society.
Abstract We discuss the one-dimensional Hubbard model, on finite sites spin chain, in context of the action of the direct product of two unitary groups SU(2)×SU(2). The symmetry revealed by this group is applicable in the procedure of exact diagonalization of the Hubbard Hamiltonian. This result combined with the translational symmetry, given as the basis of wavelets of the appropriate Fourier transforms, provides, besides the energy, additional conserved quantities, which are presented in the case of a half-filled, four sites spin chain. Since we are dealing with four elementary excitations, two quasiparticles called “spinons”, which carry spin, and two other called “holon” and “antyholon”, which carry charge, the usual spin-SU(2) algebra for spinons and the so called pseudospin-SU(2) algebra for holons and antiholons, provide four additional quantum numbers.
The Lanczos algorithm has been applied to the Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 square lattice and several equal-time correlation functions have been evaluated. This computation has been made possible by a classification of all the irreducible representations of the space symmetry group, which allowed a considerable reduction of the dimension of the Hilbert space of the system. Such an analysis is quite general and might be useful in other physical applications. Our results shed light on the effects of doping on the magnetic ordering, suggest a spin-charge decoupling at large U, and show a remarkable reduction of the quasiparticle weight at strong coupling.
We present two methods to study model Hamiltonians defined on clusters. One is the symmetrization of the Hilbert space basis according to irreducible representations of the symmetry group of the Hamiltonian. The other one adapts the Lanczos diagonalization method directly to the second quantized form of the Hamiltonian studied. By the symmetrization of the basis, one takes advantage of the symmetries of the Hamiltonian, and uses the group theory to find the invariant subspaces of the Hamiltonian in the complete Hilbert space. Then one needs only to work within the invariant subspaces. This method cuts down substantially the size of the matrix representation of the Hamiltonian to be diagonalized. Therefore, one could study relatively larger systems. As a result, all the eigenvectors and eigenvalues are obtained. Consequently, one could calculate thermodynamical quantities and various correlation functions, such as, the spin correlation, the charge correlation, the superconductivity pairing correlation, and the photoemission spectra. One the other hand, the Lanczos method, mentioned here in particular, is a specially developed computer program for the study of the model Hamiltonians. This program uses the key feature of the Lanczos method in diagonalization, allows one to use the second quantized form of the Hamiltonian directly without pregeneration or storing the matrix elements of the Hamiltonian. Two methods could be combined to take the full advantage of the symmetries of the Hamiltonian by input of the symmetrized wavefunction as the starting trial wavefunction in the Lanczos method. Here we illustrate the above two methods by applying to the Hubbard, and extended Hubbard models on a simple cubic cluster, and the lattice Anderson model on the two dimensional square lattice. Other interesting results for different systems are also briefly discussed here.
A general method to construct basis functions for fermionic systems which account for the SU(2)SU(2)SU(2) symmetry and for the translational invariance of the Hamiltonian is presented. The method does not depend on the dimensionality of the system and it appears as a natural generalization of the Bethe Ansatz to the case of non integrable systems. As an example we present the block diagonalization of the Hubbard hamiltonian for finite number of sites in one and two dimensions.
Cluster perturbation theory is a technique for calculating the spectral weight of Hubbard models of strongly correlated electrons, which combines exact diagonalizations on small clusters with strong-coupling perturbation theory at leading order. It is exact in both the strong- and weak-coupling limits and provides a good approximation to the spectral function at any wave vector. Following the paper by S\'en\'echal et al. [Phys. Rev. Lett. 84,84,84, 522 (2000)], we provide a more complete description and derivation of the method. We illustrate some of its capabilities, in particular regarding the effect of doping, the calculation of ground state energy and double occupancy, the disappearance of the Fermi surface in the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model, and so on. The method is applicable to any model with on-site repulsion only.
One of the long standing problems in quantum chemistry had been the inability to exploit full spatial and spin symmetry of an electronic Hamiltonian belonging to a non-Abelian point group. Here, we present a general technique which can utilize all the symmetries of an electronic (magnetic) Hamiltonian to obtain its full eigenvalue spectrum. This is a hybrid method based on Valence Bond basis and the basis of constant z-component of the total spin. This technique is applicable to systems with any point group symmetry and is easy to implement on a computer. We illustrate the power of the method by applying it to a model icosahedral half-filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and in the largest non-Abelian point group. The C60 molecule has this symmetry and hence our calculation throw light on the higher energy excited states of the bucky ball. This method can also be utilized to study finite temperature properties of strongly correlated systems within an exact diagonalization approach. (C) 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012