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Physical review. B, Condensed matter

Ground states of infinite-range spin-(1/2 quantum Heisenberg antiferromagnets

Christopher Mudry, Eduardo Fradkin

Published Dec 1, 1989• 6 citations
Publisher

Abstract

We study a class of infinite-range spin-(1/2 quantum Heisenberg antiferromagnets. We show that, in addition to ground states with macroscopic net spin in one or several sublattices (``N\'eel states''), there are valence-bond (VB) states that are local spin singlets. We show that there is a class of perturbations that stabilizes the VB state. We characterize the excitation spectrum and find an order parameter for the VB states. Finally, we extend our method to find perturbations that stabilize microscopic as well as macroscopic chiral states.

DOI
10.1103/physrevb.40.11177

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We study a class of infinite-range spin-(1/2 quantum Heisenberg antiferromagnets. We show that, in addition to ground states with macroscopic net spin in one or several sublattices (``N'eel states''), there are valence-bond (VB) states that are local spin singlets. We show that there is a class of perturbations that stabilizes the VB state. We characterize the excitation spectrum and find an order parameter for the VB states. Finally, we extend our method to find perturbations that stabilize microscopic as well as macroscopic chiral states.
We derive the singlet versions of the N\'eel states for certain n-sublattice systems, including the two-dimensional square lattice (previously considered by Carlson and Sethna) and the triangular lattice. These states are written explicitly in resonating-valence-bond form through the help of the Young-tableau formalism. Since these states are exact ground states for infinite-range models, they can be used as a test for the fermion and Schwinger boson mean-field theories (MFT). We show that the fermion MFT is incompatible with the ground state, while the boson MFT gives the exact ground state. Implications of the results with respect to the ability of the MFT's to describe antiferromagnetic long-range order are discussed.
We propose that a valence-bond-solid (VBS) order can be stabilized in certain two-dimensional antiferromagnets due to spin-lattice coupling. In contrast to the VBS state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type, the spin-lattice coupling-induced VBS state can occur when 2S2S2S is not an integer multiple of the coordination number zzz. S=2S=2S=2 spins on the triangular lattice with z=6z=6z=6 is discussed as an example. Within the Schwinger boson mean-field theory the ground state is derived as the direct product of states, one of which represents the conventional long-range ordered spins, and the other given by the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ modulation of the valence bond amplitudes. Excitation spectrum for the modulated valence bond state is worked out within the single-mode approximation. The spectrum offers a new collective mode, distinct from the spin wave excitations of the magnetically ordered ground state, and observable in neutron scattering.
A simple and efficient method for calculating the ground state for a class of antiferromagnet systems is presented. It combines the valence bond structure of the ground state for this class of systems and real space renormalization group. As an example, the entire Haldane phase for a spin-1 chain is described, from the AKLT model through to the Heisenberg model, ending at the critical WZNW SU2(2)SU_2(2)SU2​(2) model. The picture that emerges from this new method leads us to consider the relationship between the hidden topological order that characterizes the Haldane phase and potential applications in quantum communication technology.
We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling J1>0{J}_{1}>0J1​>0 and frustrating next-nearest-neighbor coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0<\ensuremath{\kappa}<1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with two-sublattice N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}=0.250(5)$, and another with four-sublattice N\'eel-II order for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}=0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}^{i}=0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range ${\ensuremath{\kappa}}_{c}^{i}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.
The ground-state properties of the spin-(1/2 Heisenberg antiferromagnet on a square lattice are studied by using a simple variational wave function that interpolates continuously between the N\'eel state and short-range resonating-valence-bond states. Exact calculations of the variational energy for small systems show that the state with the lowest energy has long-range antiferromagnetic order. The staggered magnetization in this state is approximately 70% of its maximum possible value. The variational estimate of the ground-state energy is substantially lower than the value obtained for the nearest-neighbor resonating-valence-bond wave function.