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Physical review. B, Condensed matter

Hidden symmetries of finite-size clusters with periodic boundary conditions

James K. Freericks, L. M. Falicov

Published Aug 15, 1991• 7 citations
Publisher

Abstract

Finite-size clusters with periodic boundary conditions resemble isolated clusters for a small number of sites, and infinite lattices for a large number of sites. The transition from a self-contained system to an infinite lattice passes through an intermediate region with increased (hidden) symmetry. In this high-symmetry region irreducible representations of the space group may stick together to form higher-dimensional representations of the complete symmetry group. This transition is examined for a class of simple-, body-centered-, and face-centered-cubic lattice clusters and the two-dimensional square lattice cluster. The implications of an enlarged symmetry group are also studied for a model of strongly correlated electrons interacting on eight-site clusters.

DOI
10.1103/physrevb.44.2895

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Finite-size clusters with periodic boundary conditions resemble isolated clusters for a small number of sites, and infinite lattices for a large number of sites. The transition from a self-contained system to an infinite lattice passes through an intermediate region with increased (hidden) symmetry. In this high-symmetry region irreducible representations of the space group may stick together to form higher-dimensional representations of the complete symmetry group. This transition is examined for a class of simple-, body-centered-, and face-centered-cubic lattice clusters and the two-dimensional square lattice cluster. The implications of an enlarged symmetry group are also studied for a model of strongly correlated electrons interacting on eight-site clusters.
The t-t'-J model (strongly interacting limit of a particular Hubbard model) is solved exactly on small clusters of eight sites with periodic boundary conditions for the simple-, body-centered-, and face-centered-cubic lattices and for the two-dimensional square lattice. The symmetry, k vector, and spin of the ground state are studied as functions of crystalline environment, interaction strength, and electron concentration. Phase diagrams are presented for stable solutions, and regions of parameter space that exhibit ferromagnetism and heavy-fermionic behavior are identified.
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.
The reduced density matrix of an interacting system can be used as the basis for a truncation scheme, or in an unbiased method to discover the strongest kind of correlation in the ground state. In this paper, we investigate the structure of the many-body fermion density matrix of a small cluster in a square lattice. The cluster density matrix is evaluated numerically over a set of finite systems, subject to nonsquare periodic boundary conditions given by the lattice vectors ${\mathbf{R}}_{1}\ensuremath{\equiv}({R}_{1x},{R}_{1y})$ and ${\mathbf{R}}_{2}\ensuremath{\equiv}({R}_{2x},{R}_{2y})$. We then approximate the infinite-system cluster density-matrix spectrum by averaging the finite-system cluster density matrix (i) over degeneracies in the ground state, and orientations of the system relative to the cluster, to ensure it has the proper point-group symmetry; and (ii) over various twist boundary conditions to reduce finite size effects. We then compare the eigenvalue structure of the averaged cluster density matrix for noninteracting and strongly interacting spinless fermions, as a function of the filling fraction n‾\overline{n}n, and discuss whether it can be approximated as being built up from a truncated set of single-particle operators.
We show that the thermal properties of periodic lattice systems can be approximated to that of a finite cluster with appropriate boundary conditions which include a modified Hamiltonian for the boundary variables. Imposing lattice invariance on the correlation of the local site variables is sufficient to obtain the free parameters of the boundary Hamiltonian. The degree of accuracy of the calculation depends on the interaction range allowed in the boundary Hamiltonian and the range up to which the correlation of the site variables are made lattice invariant. The Bethe approximation can be considered a trivial case of this general method for clusters of one lattice site. The reliability of the method is demonstrated with the results obtained for the two-dimensional Ising model, where a cluster of four spins and invariance conditions up to second neighbors is sufficient to reproduce some nonuniversal thermal properties of the model with an accuracy comparable or better than other more complex numerical methods.
An exact solution of an eight-site crystal model with periodic boundary conditions, a {001} thin film with face-centered-cubic crystal structure, is presented for the case of a heavy-fermion system. The object is to study modifications of the many-body electronic structure in the presence of surfaces. The model consists of (a) a single, fully symmetric orbital per site, with nearest-neighbor and second-nearest-neighbor hopping; (b) an infinite Coulomb repulsion between electrons on the same site; (c) antiferromagnetic superexchange interactions; and (d) a nearly half-filled band ((7/8 electron per site). Two structures are investigated: a two-layer and a four-layer fcc slab. The nature of the change from bulk to surface is studied by introducing a variable coupling between periodically extended slabs. The many-body spectrum, the one-electron occupancy, and the change in properties from bulk to surface are obtained and discussed.
The Drude weight of the Hubbard model on the two-dimensional square lattice is studied by the exact diagonalizations applied to clusters up to 20 sites. We carefully examine finite-size effects by consideration of the appropriate shapes of clusters and the appropriate boundary condition beyond the imitation of employing only the simple periodic boundary condition. We successfully capture the behavior of the Drude weight that is proportional to the squared hole doping concentration. Our present result gives a consistent understanding of the transition between the Mott insulator and doped metals. We also find, in the frequency dependence of the optical conductivity, that the mid-gap incoherent part emerges more quickly than the coherent part and rather insensitive to the doping concentration in accordance with the scaling of the Drude weight.