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Physical Review Letters

Extended Heisenberg models of antiferromagnetism: Analogies to the fractional quantum Hall effect

Daniel P. Arovas, Assa Auerbach

Published Feb 1, 1988• 300 citations
Publisher

Abstract

The ``valence-bond'' ground states for extended Heisenberg models discussed by Affleck et al. are shown to be of a Jastrow form. We discuss a striking analogy between these models and the ``truncated pseudopotential'' approach to the fractional quantum Hall effect. The valence-bond states are in analogy to the Laughlin ground state, and the magnon spectrum is deduced by use of a single-mode approximation.

DOI
10.1103/physrevlett.60.531

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The valence-bond'' ground states for extended Heisenberg models discussed by Affleck et al. are shown to be of a Jastrow form. We discuss a striking analogy between these models and the truncated pseudopotential'' approach to the fractional quantum Hall effect. The valence-bond states are in analogy to the Laughlin ground state, and the magnon spectrum is deduced by use of a single-mode approximation.
We study a spin-SSS ferromagnetic model with exactly-written ground states, known as the partially-magnetized valence bond solid (VBS) states with magnetization m=(S−1)/Sm=(S-1)/Sm=(S−1)/S, which is a ferromagnetic generalization of the Affleck-Kennedy-Lieb-Tasaki model. We find that the VBS state and an antiferromagnetic ground state with magnetization m=0m=0m=0 are degenerate for S=3/2S=3/2S=3/2 and S=2S=2S=2 by using the Lanczos method and the density matrix renormalization group method (DMRG). However, increasing SSS, the magnetization of the ground states is uniquely determined as the fraction m=(S−1)/Sm=(S-1)/Sm=(S−1)/S. This is not just a ferromagnet, but a quantum ferromagnet due to quantum entanglement inherent in VBS states. In the low-energy excitation spectrum, we find the coexistence of the Haldane gap and Goldstone-like ferromagnetic magnon excitation. This ``magnetic chimera'' clearly appears under a finite magnetic field. Finally, we discuss an application to the measurement-based quantum computation and an extension of the Haldane's conjecture.
In this paper, we propose a method to understand the nature of the quantum disorder phase of the two-dimensional (2D) high-spin frustrated model. The ground state and excitation properties of a fully frustrated 2D spin-1 model are studied based on a model whose ground state can be found exactly. By analogy to the pseudopotential approach in the fractional quantum Hall effect, we conclude that the ground states of the fully frustrated spin-1 model are doubly degenerate valence-bond solid (VBS) states along the horizontal or vertical direction of the square lattice. We also find that the ground state could be characterized by a nonzero string order, which rarely happened in the 2D case. The method that we used reveals the connection between the fractional quantum Hall effect and the frustrated 2D antiferromagnetism system. The VBS states capture the main character of the disordered phase in the 2D spin-1 frustrated system and can be verified by a numerical method.
We present a theory of the collective excitation spectrum in the fractional quantum Hall-effect regimes, in analogy with Feynman's theory for helium. The spectrum is in excellent quantitative agreement with the numerical results of Haldane. Within this approximation we prove that a finite gap is generic to any liquid state in the extreme quantum limit and that in this single-mode approximation gapless excitations can arise only as Goldstone modes for ground states with broken translation symmetry.