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arXiv:cond-mat/0102447 · Physical review. B, Condensed matter

Quantum disorder in the two-dimensional pyrochlore Heisenberg antiferromagnet

S. E. Palmer, J. T. Chalker

Published Aug 7, 2001• 53 citations
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Abstract

We present the results of an exact diagonalization study of the spin-$1/2$ Heisenberg antiferromagnet on a two-dimensional version of the pyrochlore lattice, also known as the square lattice with crossings or the checkerboard lattice. Examining the low-energy spectra for systems of up to 24 spins, we find that all clusters studied have nondegenerate ground states with total spin zero, and big energy gaps to states with higher total spin. We also find a large number of nonmagnetic excitations at energies within this spin gap. Spin-spin and spin-Peierls correlation functions appear to be short ranged, and we suggest that the ground state is magnetically disordered.

DOI
10.1103/physrevb.64.094412

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We present the results of an exact diagonalization study of the spin-1/21/21/2 Heisenberg antiferromagnet on a two-dimensional version of the pyrochlore lattice, also known as the square lattice with crossings or the checkerboard lattice. Examining the low-energy spectra for systems of up to 24 spins, we find that all clusters studied have nondegenerate ground states with total spin zero, and big energy gaps to states with higher total spin. We also find a large number of nonmagnetic excitations at energies within this spin gap. Spin-spin and spin-Peierls correlation functions appear to be short ranged, and we suggest that the ground state is magnetically disordered.
We consider the quantum spin-sss XXZXXZXXZ Heisenberg antiferromagnet on the two- and three-dimensional pyrochlore lattices and examine a spin-Peierls mechanism of lowering the total energy by a lattice distortion in a high magnetic field. For the exact eigenstates consisting of several independent localized magnons in a ferromagnetic environment we show the existence of a spin-Peierls instability by rigorous analytical calculations. In addition we report on exact diagonalization data for finite two-dimensional pyrochlore lattices up to N=64N=64N=64 sites. We discuss a scenario of the field-tuned changes in the ground-state properties of the frustrated spin lattices due to the coupling between spin and lattice degrees of freedom.
Exact diagonalizations of the spin-1/2 Heisenberg model on the checkerboard lattice have been performed for sizes up to N=36N=36N=36 in the full Hilbert space and N=40N=40N=40 in the restricted subspace of first neighbor dimers. This antiferromagnet does not break SU(2) symmetry and displays long-range order in four-spin S=0S=0S=0 plaquettes. Both the symmetry properties of the spectrum and various correlations functions are extensively studied. At variance with the kagom\'e antiferromagnet, the Heisenberg quantum model on a checkerboard lattice is a valence bond crystal. Some results concerning the three-dimensional spin-1/2 pyrochlore magnet (for sizes 16 and 32) are also shown: this system could behave differently from its two-dimensional analog.
Exact diagonalizations of the spin-1/2 Heisenberg model on the checkerboard lattice have been performed for sizes up to N=36 in the full Hilbert space and N=40 in the restricted subspace of first neighbor dimers. This antiferromagnet does not break SU(2) symmetry and displays long range order in 4-spin S=0 plaquettes. Both the symmetry properties of the spectrum and various correlations functions are extensively studied. At variance with the kagom{é} antiferromagnet, the Heisenberg quantum model on a checkerboard lattice is a Valence Bond Crystal. Some results concerning the 3-dimensionnal spin-1/2 pyrochlore magnet (for sizes 16 and 32) are also shown: this system could behave differently from its 2-dimensional analog.
We present exact diagonalization calculations for the spin-1/2 nearest-neighbor antiferromagnet on the pyrochlore lattice. We study a section of the lattice in the [111] direction and analyze the Hamiltonian of the breathing pyrochlore system with two coupling constants J1{J}_{1}J1​ and J2{J}_{2}J2​ for tetrahedra of different orientations and investigate the evolution of the system from the limit of disconnected tetrahedra (J2=0{J}_{2}=0J2​=0) to a correlated state at J1=J2{J}_{1}={J}_{2}J1​=J2​. We evaluate the low-energy spectrum, two and four spin correlations, and spin chirality correlations for a system size of up to 36 sites. The model shows a fast decay of spin correlations and we confirm the presence of several singlet excitations below the lowest magnetic excitation. We find chirality correlations near J1=J2{J}_{1}={J}_{2}J1​=J2​ to be small at the length scales available at this system size. Evaluation of dimer-dimer correlations and analysis of the nature of the entanglement of the tetrahedral unit shows that the triplet sector of the tetrahedron contributes significantly to the ground-state entanglement at J1=J2{J}_{1}={J}_{2}J1​=J2​.
We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone Γ(q=0)\Gamma(\mathbf q=0)Γ(q=0) and at the magnetic ordering wavevectors ±[Q=(4π/3,0)]\pm[\mathbf Q= (4\pi/3,0)]±[Q=(4π/3,0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.
Spin excitation spectra of two dimensional S=1/2S=1/2S=1/2 Heisenberg model with checkerboard structures.
We study the excitation spectra of two-dimensional spin-1/21/21/2 Heisenberg model with checkerboard structures using stochastic analytic continuation of imaginary-time correlation function obtained from quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions J1J_{1}J1​ and J2J_{2}J2​, and the tuning parameter ggg is defined as J2/J1J_{2}/J_{1}J2​/J1​. The dynamic structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, wave theory is employed to explain some features of numerical results, especially for the low-energy part. When ggg is close to 111, the features of the excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them are discussed. In contrast to the other checkerboard structures investigated in this paper, the 3×33\times 33×3 checkerboard model has distinctive excitation features, like a gap between a low-energy gapless branch and a gapped high-energy part exists when ggg is small. The gapless branch in this case can be regarded as wave in Neˊ\mathrm{\acute{e}}eˊel order formed by block spin in each 3×33\times 33×3 plaquette with effective exchange interaction originated from renormalization. One finding is that the continuum also appears in this low-energy branch, which may originated from nearly deconfined process in block spin.