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Physical review. B, Solid state

Estimates of the ground-state eigenvalue of the two-dimensional spin- 12 X−Y model

Robert B. Pearson

Published Aug 1, 1977• 22 citations
Publisher

Abstract

An estimate as well as upper and lower bounds are presented for the ground-state energy density per spin of the two-dimensional spin-$\frac{1}{2}$ $X\ensuremath{-}Y$ model on a square lattice. The energy density is estimated to be ${\ensuremath{\epsilon}}_{0}=\ensuremath{-}1.099\ifmmode\pm\else\textpm\fi{}0.004$ and is bounded by $\ensuremath{-}1.12498<~{\ensuremath{\epsilon}}_{0}<~\ensuremath{-}1.09440$.

DOI
10.1103/physrevb.16.1109

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An estimate as well as upper and lower bounds are presented for the ground-state energy density per spin of the two-dimensional spin-12\frac{1}{2}21​ X\ensuremath−YX\ensuremath{-}YX\ensuremath−Y model on a square lattice. The energy density is estimated to be \ensuremathϵ0=\ensuremath−1.099\ifmmode±\else\textpm\fi0.004{\ensuremath{\epsilon}}_{0}=\ensuremath{-}1.099\ifmmode\pm\else\textpm\fi{}0.004\ensuremathϵ0​=\ensuremath−1.099\ifmmode±\else\textpm\fi0.004 and is bounded by \ensuremath−1.12498< \ensuremathϵ0< \ensuremath−1.09440\ensuremath{-}1.12498<~{\ensuremath{\epsilon}}_{0}<~\ensuremath{-}1.09440\ensuremath−1.12498< \ensuremathϵ0​< \ensuremath−1.09440.
We present an improved upper bound for the ground-state energy of lattice fermion models with a sign problem. The bound can be computed by numerical simulation of a recently proposed family of deformed Hamiltonians with no sign problem. For one-dimensional models, we expect the bound to be particularly effective and practical extrapolation procedures are discussed. In particular, in a model of spinless interacting fermions and in the Hubbard model at various filling and Coulomb repulsion we show how such techniques can estimate the ground-state energies and correlation function with great accuracy.
The Heisenberg-Ising Hamiltonian $H=\ensuremath{-}\frac{1}{2}{\ensuremath{\alpha}({\ensuremath{\sigma}}_{x}\ensuremath{\sigma}_{x}^{}{}_{}{}^{\ensuremath{'}}+{\ensuremath{\sigma}}_{y}\ensuremath{\sigma}_{y}^{}{}_{}{}^{\ensuremath{'}})+\ensuremath{\epsilon}({\ensuremath{\sigma}}_{z}\ensuremath{\sigma}_{z}^{}{}_{}{}^{\ensuremath{'}})}$ for rectangular one-, two-, or three-dimensional lattices are considered. The sum is over nearest neighbors and $\ensuremath{\Delta}=\frac{\ensuremath{\epsilon}}{\ensuremath{\alpha}}$ measures the anisotropy of the coupling. Upper and lower bounds for the ground-state energy are established and these bounds apply equally well to lattices of one, two, or three dimensions. Furthermore, it is shown that the ground-state energy per nearest-neighbor pair is nondecreasing as the dimension of the lattice (one, two or three) increases.