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arXiv:cond-mat/0412033 · Physical Review B

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

Jean-Yves P. Delannoy, Michel J. P. Gingras, P. C. W. Holdsworth, A.–M. S. Tremblay

Published Sep 16, 2005• 53 citations
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Abstract

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 $t∕U\ensuremath{\rightarrow}0$ 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.

DOI
10.1103/physrevb.72.115114

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We investigate the ground state properties of the two-dimensional half-filled one band Hubbard model in the strong (large-UUU) 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)3t{(t∕U)}^{3}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∕Ut∕Ut∕U is increased from the t∕U\ensuremath→0t∕U\ensuremath{\rightarrow}0t∕U\ensuremath→0 Heisenberg limit. These charge fluctuations lead to an increase of the quantum fluctuations over and above those for an S=1∕2S=1∕2S=1∕2 antiferromagnet. The renormalization factor ensures that the zero temperature staggered moment for the Hubbard model is a monotonously decreasing function of t∕Ut∕Ut∕U, despite the fact that the moment of the spin Hamiltonian, which depends on transverse spin fluctuations only, in an increasing function of t∕Ut∕Ut∕U. We also comment on quantitative aspects of the t∕Ut∕Ut∕U and 1∕S1∕S1∕S expansions.
We present an approach to the magnetic properties of the half-filled Hubbard model, based on an approximate mapping of its low-energy transverse spin excitations on to those of an effective underlying Heisenberg model, but with effective spin interactions which are self-consistently determined and not confined solely to nearest-neighbor couplings. The mapping is exact in strong-coupling and is found to be accurate over a very wide range of interaction strengths, down to weak coupling. At zero temperature, it permits ready evaluation at finite U of the one-loop effects of zero-point spin fluctuations on, e.g., the sublattice magnetization. At finite temperatures, thermodynamic properties of the system in the thermal paramagnet are studied via a physically transparent Onsager reaction field approach, which amounts to a self-consistent treatment of paramagnetic spin correlations. This is central not only in recovering the correct dimensionality dependence of antiferromagnetic long-ranged order, but also for the d=3 case of primary interest here yields a N\'eel temperature in close agreement with known strong- and weak-coupling limits. Spin correlation functions and magnetic susceptibilities also show very good agreement with quantum Monte Carlo calculations over an appreciable temperature range in which the low-lying transverse spin excitations are thermally dominant. \textcopyright{} 1996 The American Physical Society.
Spin-only descriptions of the half-filled one-band Hubbard model are relevant for a wide range of Mott insulators. In addition to the usual Heisenberg exchange, many other types of interactions, including ring exchange, appear in the effective Hamiltonian in the intermediate coupling regime. In order to improve on the quantitative description of magnetic excitations in the insulating antiferromagnetic phase of copper-oxide (cuprate) materials, and to be consistent with band-structure calculations and photoemission experiments on these systems, we include second- and third-neighbor hopping parameters, ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, into the Hubbard Hamiltonian. A unitary transformation method is used to find systematically the effective Hamiltonian and any operator in the spin-only representation. The results include all closed four-hop electronic pathways in the canonical transformation. The method generates many ring exchange terms that play an important role in the comparison with experiments on La2CuO4{\text{La}}_{2}{\text{CuO}}_{4}La2​CuO4​. Performing a spin-wave analysis, we calculate the magnon dispersion as a function of UUU, ttt, ${t}^{\ensuremath{'}}$, and ${t}^{\ensuremath{''}}$. The four parameters are estimated by fitting the magnon dispersion to the experimental results of Coldea et al. [Phys. Rev. Lett. 86, 5377 (2001)] for La2CuO4{\text{La}}_{2}{\text{CuO}}_{4}La2​CuO4​. The ring exchange terms are found essential, in particular to determine the relative sign of ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, with the values found in good agreement with independent theoretical and experimental estimates for other members of the cuprate family. The zero-temperature sublattice magnetization is calculated using these parameters and also found to be in good agreement with the experimental value estimated by Lee et al. [Phys. Rev. B 60, 3643 (1999)]. We find a value of the interaction strength $U\ensuremath{\simeq}8t$ consistent with Mott insulating behavior.
The half-filled Hubbard model on the Bethe lattice with coordination number z=3z=3z=3 is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy EEE, average local magnetization $ $, its fluctuations $ - ^2$ and various spin correlation functions $ - $ are determined as a function of the Coulomb interaction strength U/tU/tU/t. The calculated local magnetic moments $ $ increase monotonically with increasing Coulomb repulsion U/tU/tU/t forming an antiferromagnetic spin-density-wave state which matches the two sublattices of the bipartite Bethe lattice. At large U/tU/tU/t, $ $ is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins (∣∣≃0.35| |\simeq 0.35∣∣≃0.35 for U/t→+∞U/t\to +\inftyU/t→+∞). $ - ^2$ shows a maximum for U/t=2.4U/t=2.4U/t=2.4--2.9 which results from the interplay between the usual increase of $ $ with increasing U/tU/tU/t and the formation of important permanent moments $ $ at large U/tU/tU/t. NN sites show antiferromagnetic spin correlations which increase with increasing Coulomb repulsion. In contrast next NN sites are very weakly correlated over the whole range of U/tU/tU/t. The accuracy of the DMRG results is discussed by comparison with tight-binding exact results, independent DMRG calculations for the Heisenberg model and simple first-order perturbation estimates.
We analyze the density dependence of the spin stiffnesses and the stability of magnetic order with respect to quantum fluctuations in the two-dimensional Hubbard model close to half-filling. The stiffnesses are computed from the spin susceptibility obtained from a random-phase approximation in a magnetically ordered state. For a sizable next-to-nearest-neighbor hopping amplitude and a moderate Hubbard interaction, the mean-field ground state is a Néel antiferromagnet in the electron-doped regime at and above half-filling and a planar circular spiral state in the hole-doped regime below half-filling. Upon electron doping, the Néel stiffness decreases smoothly and not very steeply. By contrast, the in-plane and out-of-plane stiffnesses in the spiral state drop abruptly at half-filling. The out-of-plane stiffness even drops to zero and then increases again very slowly upon increasing hole doping. At finite temperatures, the Néel-to-spiral transition is shifted into the hole-doped regime, the stiffnesses are continuous functions of the density, and they vanish at the transition. For small hole doping, the spin stiffnesses describe the quantum fluctuations only in a small momentum range, which shrinks to zero upon approaching half-filling. Using the above results, we show that the quantum ground state of the lightly electron-doped Hubbard model remains Néel ordered, while quantum fluctuations probably destroy the spiral long-range order in the hole-doped regime, giving rise to a quantum disordered state with a spin gap.
We study electrons hopping on a kagome lattice at third filling described by an extended Hubbard Hamiltonian with on-site and nearest-neighbor repulsions in the strongly correlated limit. As a consequence of the commensurate filling and the large interactions, each triangle has precisely two electrons in the effective low-energy description, and these electrons form chains of different lengths. The effective Hamiltonian includes the ring exchange around the hexagons as well as the nearest-neighbor Heisenberg interaction. Using large-scale exact diagonalization, we find that the effective model exhibits two phases: If the charge fluctuations are small, the magnetic fluctuations confine the charges to short loops around hexagons, yielding a gapped charge-ordered phase. When the charge fluctuations dominate, the system undergoes a quantum phase transition to a resonating plaquette phase with ordered spins and gapless spin excitations. We find that a peculiar conservation law is fulfilled: the electron in the chains can be divided into two sublattices, and this division is conserved by the ring exchange term.
We study the repulsive Hubbard model at half filling in the canonical ensemble and compare its thermodynamics with the spin-12\frac1221​ antiferromagnetic Heisenberg model at superexchange coupling J0(U)=4t2/UJ_0(U)=4t^2/UJ0​(U)=4t2/U. For fixed hopping t≠0t\ne0t=0, bounded staggered fields ∣h∣≤h0|h|\le h_0∣h∣≤h0​, and inverse temperatures βJ0(U)≥ℓ0>0βJ_0(U)\ge\ell_0>0βJ0​(U)≥ℓ0​>0, we establish a volume-uniform pressure comparison with error O(U−3+∣h∣U−2)O(U^{-3}+|h|U^{-2})O(U−3+∣h∣U−2). On the Heisenberg field scale h=J0(U)xh=J_0(U)xh=J0​(U)x, the error improves to O(U−3)O(U^{-3})O(U−3) for bounded xxx. Using the quantum Lee--Yang theorem of Suzuki and Fisher, we obtain a volume-uniform O(U−1)O(U^{-1})O(U−1) comparison of staggered magnetizations on compact positive intervals of the scaled field xxx. By combining a recent fermionic Lee--Yang theorem with analytic interpolation, we improve this estimate to O(U−2log⁡U)O(U^{-2}\log U)O(U−2logU) in the thermodynamic limit. The latter analysis also establishes analyticity of the limiting Hubbard pressure at nonzero real fields. The improved comparison extends to vanishing positive fields satisfying U−3≪h(U)≪U−1U^{-3}\ll h(U)\ll U^{-1}U−3≪h(U)≪U−1, with error O((1+log⁡(U3h(U)))/(U3h(U)))O((1+\log(U^3h(U)))/(U^3h(U)))O((1+log(U3h(U)))/(U3h(U))). Consequently, sufficiently low-temperature Heisenberg Néel order yields a quantitative positive lower bound on the Hubbard staggered magnetization in this regime. Furthermore, we prove O(U−2)O(U^{-2})O(U−2) bounds on the double-occupancy and charge-defect densities and the normalized charge-density-wave second moment. At zero staggered field, this decay rate is optimal for both densities, even in the thermodynamic limit. Our approach uses finite-order Schrieffer--Wolff transformations and does not establish spontaneous Néel order at fixed finite UUU.