Néel order, ring exchange, and charge fluctuations in the half-filled Hubbard model

We investigate the ground state properties of the two-dimensional half-filled one band Hubbard model in the strong (large-$U$) to intermediate coupling limit (i.e., away from the strict Heisenberg limit) using an effective spin-only low-energy theory that includes nearest-neighbor exchange, ring exchange, and all other spin interactions to order $t{(t∕U)}^3$. We show that the operator for the staggered magnetization, transformed for use in the effective theory, differs from that for the order parameter of the spin model by a renormalization factor accounting for the increased charge fluctuations as $t∕U$ is increased from the $t∕U\to 0$ Heisenberg limit. These charge fluctuations lead to an increase of the quantum fluctuations over and above those for an $S=1∕2$ antiferromagnet. The renormalization factor ensures that the zero temperature staggered moment for the Hubbard model is a monotonously decreasing function of $t∕U$, despite the fact that the moment of the spin Hamiltonian, which depends on transverse spin fluctuations only, in an increasing function of $t∕U$. We also comment on quantitative aspects of the $t∕U$ and $1∕S$ expansions.

Figure 1

Difference between $e^{i\mathcal{S}}{\mid 0⟩}_H$ and ${\mid 0⟩}_s$ restricted to the singly occupied states. The result is compared to a ${(t∕U)}^4$ line obtained by fitting $\delta$ in the range $t∕U∊[0,0.05]$. The calculation is done on a $2\times 3$ lattice with open boundary conditions in both the $x$ and $y$ directions.

Figure 2

(Color online) $t∕U$ dependence of the staggered magnetization $M_{2,s}^{†}$, $M_{2,H}^{†}$ and ${\tilde{M}}_{2,s}$ for a $2\times 4$ lattice.

Figure 3

$(t∕U)$ dependence of $⟨M_s^{†}⟩$ and $⟨{\tilde{M}}_s^{†}⟩$ in a Holstein-Primakoff calculation of $H_s$ to order $1∕S$.