A generalized valence bond basis for the half-filled Hubbard model

Christopher Graves

Published Jan 1, 2011• 0 citations

Abstract

I present a non-orthogonal and overcomplete set of states that spans the S_tot=0 Hilbert subspace for a fermionic system with four possible spin configurations per site, on a half-filled lattice with an even number of sites. This set consists of possible pairing of the spins into three bond types: singlet bonds and positive and negative charge bonds. Although I present the specific case for the half-filled Hubbard model in one-dimension, these sets of states can be trivially extended to higher dimensions. The overlap of these generalized valence bond states form closed loops with properties that are related to operator expectation values in this basis. I verify the correctness of this basis by solving the model via exact diagonalization and comparing with known results. There is evidence of structure in the ground state that may be useful to simulations using variational wave functions.

DOI: 10.7939/r35g9n

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A generalized valence bond basis for the half-filled Hubbard model

Christopher Graves

Published Jan 1, 2011• 0 citations

Publisher

Abstract

DOI: 10.7939/r35g9n

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SO(3)\times SO(3)\times U(1)

symmetry. For rotated electrons, doubly and single occupancy are good quantum numbers for on-site repulsion

U>0

. The above states are within that description generated by occupancy configurations of charge

c

fermions and spin-singlet two-spinon

s1

bond particles. Those describe the charge and spin degrees of freedom, respectively, of the rotated electrons that singly occupy sites. While the

c

fermions have no internal structure, that of the spin-neutral

s1

bond-particle occupancy configurations is here described in terms of spinon occupancies of a well-defined effective spin lattice. In reference \cite{companion} it is confirmed that our results contribute to the further understanding of the role of electronic correlations in the spin spectrum of the parent compound La

_2

CuO

_4

. They are also of interest for studies of ultra-cold fermionic atoms on an optical lattice.

i

and

j

refer to the lattice sites and $\ensuremath{\sigma}$ to the spin. The interaction-energy functional $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ is calculated exactly for representative finite periodic systems by performing exact Lanczos diagonalizations. The relationship between $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ and the entropy $S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}]$ of independent fermions with natural-orbital occupations ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$ is analyzed. A simple approximation to the interaction energy of the half-filled Hubbard model is proposed, which takes the form $W=W(S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}])$. Using this functional we derive the ground-state energy, kinetic energy, average number of double occupations, charge distribution, magnetic susceptibility, and field-induced spin polarization in one-, two-, and three-dimensional periodic lattices. The limit of infinite dimensions is also explored. The accuracy of the method is assessed by comparison with available exact numerical or analytical results. Goals, limitations, and possible extensions of the domain of applicability of the functional are discussed.