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Journal of Mathematical Physics

Symmetry adapted states for Hubbard clusters

O. Tjernberg

Published Dec 1, 1998• 2 citations
Publisher

Abstract

The application of spin, pseudo spin and space group symmetry to find symmetry adapted states for the square planar Hubbard model is discussed. An approach based on pseudo spin configurations and the application of Young tableaux to the permutation group is presented. The method is illustrated for the case of a 4*4 cluster and the complete classification of the states for this cluster is given. It is shown that the linear dimension of the largest matrix block is reduced by three orders of magnitude by application of the above symmetries.

DOI
10.1063/1.532657

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The application of spin, pseudo spin and space group symmetry to find symmetry adapted states for the square planar Hubbard model is discussed. An approach based on pseudo spin configurations and the application of Young tableaux to the permutation group is presented. The method is illustrated for the case of a 4*4 cluster and the complete classification of the states for this cluster is given. It is shown that the linear dimension of the largest matrix block is reduced by three orders of magnitude by application of the above symmetries.
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.
Abstract 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. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012
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
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.
We present a spin-adapted density matrix renormalization group (DMRG) algorithm designed to target spin and spatial symmetry states that can be difficult to obtain while using a non-spin-adapted algorithm. The algorithmic modifications that have to be introduced into the usual density matrix renormalization group scheme in order to spin adapt it are discussed, and it is demonstrated that the introduced modifications do not change the overall scaling of the method. The new approach is tested on HNCO, a model system, that has a singlet-triplet curve crossing between states of the same symmetry. The advantages of the spin-adapted DMRG scheme are discussed, and it is concluded that the spin-adapted DMRG method converges better in almost all cases and gives more parallel curves to the full configuration interaction result than the non-spin-adapted method. It is shown that the spin-adapted DMRG energies can be lower than the ones obtained from the non-spin-adapted scheme. Such a counterintuitive result is explained by noting that the spin-adapted method is not a special case of the non-spin-adapted one; consequently, the spin-adapted result is not an upper bound for the non-spin-adapted energy.
We present a method for the simultaneous implementation of the spin symmetry, the pseudospin symmetry and the translational invariance in models for correlated electron systems on finite-size clusters. The application of this method leads to a significative reduction of the size of the Hamiltonian matrix blocks, thus considerably simplifying the diagonalization procedure. We illustrate the method solving exactly the Hubbard model on a 8-site chain in the Hilbert subspaces labelled by the eigenvalues of the spin, the pseudospin and the momentum operators. Some relevant thermodynamical quantities, such as the specific heat and the static magnetic susceptibility, are then evaluated and a brief discussion on the tendency of the model to exhibit ferromagnetism is also presented.