Physical review. B, Condensed matter

Reply to ‘‘Comment on ‘t/Uexpansion for the Hubbard model’’’

Allan H. MacDonald, S. M. Girvin, D. Yoshioka

Published Feb 1, 1990• 94 citations

Abstract

It is shown that the unitary transformation used as the basis of a t/U expansion for the Hubbard model in an earlier Brief Report is not equivalent to the canonical perturbation expansion (CPE) discussed in the preceding Comment. The CPE is valid only to leading order in t/U. More evidence is offered in support of our suggestion that higher-order terms in the t/U expansion tend to favor spin-liquid states over antiferromagnetically ordered states.

DOI: 10.1103/physrevb.41.2565

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Physical review. B, Condensed matter

Reply to ‘‘Comment on ‘t/Uexpansion for the Hubbard model’’’

Allan H. MacDonald, S. M. Girvin, D. Yoshioka

Published Feb 1, 1990• 94 citations

Publisher

Abstract

DOI: 10.1103/physrevb.41.2565

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Andrzej M. Oleś

It is shown that a unitary transformation which eliminates the terms which couple states with different numbers of doubly occupied sites in the Hubbard model, proposed recently by MacDonald, Girvin, and Yoshioka, is conceptually equivalent to a canonical perturbation expansion, derived over a decade ago. In the half-filled case, the latter method gives an effective Hamiltonian in which many-body interactions result from even numbers of interconfigurational transitions. It is argued that it is hard to conclude anything about the nature of the ground state from the energies of the N\'eel state and various valence-bond states, as calculated by MacDonald et al., due to the importance of quantum fluctuations in a two-dimensional lattice.It is shown that a unitary transformation which eliminates the terms which couple states with different numbers of doubly occupied sites in the Hubbard model, proposed recently by MacDonald, Girvin, and Yoshioka, is conceptually equivalent to a canonical perturbation expansion, derived over a decade ago. In the half-filled case, the latter method gives an effective Hamiltonian in which many-body interactions result from even numbers of interconfigurational transitions. It is argued that it is hard to conclude anything about the nature of the ground state from the energies of the N\'eel state and various valence-bond states, as calculated by MacDonald et al., due to the importance of quantum fluctuations in a two-dimensional lattice.

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{H}_{0}

and the perturbation

{H}_{1}

can be expressed as ${H}_{0}=\ensuremath{\Sigma}{j}^{}{P}_{j}H{P}_{j}$ and ${H}_{1}=\ensuremath{\Sigma}{j\ensuremath{\ne}k}^{}{P}_{j}H{P}_{k}$, where

{P}_{j}

is the projection operator corresponding to a group of closely spaced effective one-electron orbital energies ${E}_{j\ensuremath{\mu}}$ with $\ensuremath{\mu}=1, 2, \dots{}, {d}_{j}$, and if $|{E}_{j\ensuremath{\mu}}\ensuremath{-}{E}_{j\ensuremath{\nu}}|\ensuremath{\ll}|{E}_{j\ensuremath{\omega}}\ensuremath{-}{E}_{k\ensuremath{\delta}}|$ with $j\ensuremath{\ne}k$. We have shown that the CPE is equivalent to the time-dependent perturbation theory. An extremely simple effective Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is obtained when the CPE is applied to the

s

-band Hubbard model at the atomic limit. An explicit form of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ to the eighth order is given, and the magnetic interaction in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is of the form of Heisenberg exchange ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}$, including far neighbors. We then use this form to compute the antiferromagnetic groundstate energy to the seventh order. Our result is compared with other works.