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International Journal of Modern Physics B

IMPORTANCE OF LOCAL SPIN-FLIP PROCESSES IN QUANTUM ANTIFERROMAGNETS

Andrzej M. Oleś

Published Dec 1, 1989• 7 citations
Publisher

Abstract

The variational wave function proposed by Bartkowski to treat the ground state of a quantum antiferromagnet is reexamined. The fundamental equations of this method, for the energy and magnetization, are formulated in terms of a linked cluster expansion. It is found that the energy which results from the numerical minimization of the formula derived by Bartkowski is lower than originally reported. The energy is further improved by including higher order linked diagrams, which gives the energy of E0=−0.3338J per bond and magnetization of 0.356 for the square lattice. The same wave function is next used to treat the motion of a single hole in a quantum antiferromagnet. Its motion becomes possible due to the local coupling to quantum fluctuations. It is found that the spectrum is dominated by the incoherent processes, but a coherent propagating mode may be also identified. The quasiparticle dispersion is only weakly modified by quantum fluctuations present on the neighbouring bonds.

DOI
10.1142/s0217979289001159

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The variational wave function proposed by Bartkowski to treat the ground state of a quantum antiferromagnet is reexamined. The fundamental equations of this method, for the energy and magnetization, are formulated in terms of a linked cluster expansion. It is found that the energy which results from the numerical minimization of the formula derived by Bartkowski is lower than originally reported. The energy is further improved by including higher order linked diagrams, which gives the energy of E0=−0.3338J per bond and magnetization of 0.356 for the square lattice. The same wave function is next used to treat the motion of a single hole in a quantum antiferromagnet. Its motion becomes possible due to the local coupling to quantum fluctuations. It is found that the spectrum is dominated by the incoherent processes, but a coherent propagating mode may be also identified. The quasiparticle dispersion is only weakly modified by quantum fluctuations present on the neighbouring bonds.
Results for the single-particle Green's function for the strong-coupling limit of the single-band Hubbard model ($t\ensuremath{-}J$ model) obtained by numerical and analytical methods are discussed. The numerical calculation of the spectral function is performed for small systems of up to 18 sites using the Lanczos method. A detailed analysis of the spectral functions for the $t\ensuremath{-}J$ model in one and two dimensions is given. We also investigate the effect of the next-nearest-neighbor hopping terms, which appear in order t2U\frac{{t}^{2}}{U}Ut2​ and are omitted in the standard $t\ensuremath{-}J$ model. The role of quantum fluctuations on the shape of the spectra and the fundamental difference between one and two dimensions is elucidated. In 2D we find evidence for a coherent quasiparticle band of width 2J2J2J with dispersion $^{\ensuremath{\omega}}\mathrm{k}\ensuremath{\sim}(\frac{J}{2}){(cos{k}_{x}+cos{k}_{y})}^{2}$ and spin S=12S=\frac{1}{2}S=21​, which emerges at the low-energy edge of the incoherent spectrum. This form of dispersion is due to effective next-nearest-neighbor processes related to spin flips and agrees with our extension of the approach of Brinkman and Rice to finite JJJ.
A simple approach to estimation of the ground state energy of quantum antiferromagnets is developed, based on the approximation that quantum fluctuations around different bonds are independent. The ground state energy estimates are as good as spin wave theory or slightly better. A canonical transformation of the spin operators to generate bond quantum fluctuations is devised and applied to the classical ground state of the S=12S=\frac{1}{2}S=21​ Heisenberg model on the square lattice. This simple picture of quantum spin fluctuations might be useful in more complex models. The resulting nearest neighbor and next-nearest neighbor correlations can be used in an alternative derivation of spin waves in the Heisenberg model, giving an accurate dispersion as well as raising and lowering operators.
We formulate a quasiparticle theory for a single hole in a quantum antiferromagnet in the limit that the Heisenberg exchange energy is much less than the hopping matrix element, J\ensuremath{\ll}t. We consider the ground state of the spins to be either a quantum N\'eel state or a d-wave resonating-valence-bond (RVB) state. We show in a self-consistent perturbation theory that the hole spectrum is strongly renormalized by the interactions with spin excitations. The hole can be described by a narrow quasiparticle band located at an energy of order -t with a quasiparticle residue of order J/t and a bandwidth of order J. Above the quasiparticle band is an incoherent band of width of order t. Our results indicate that the energy scale for any coherent phenomenon involving the holes is \ensuremath{\delta}J, where \ensuremath{\delta} is the doping concentration. In the N\'eel state we perform a spin-wave expansion on an anisotropic Heisenberg model. In the Ising limit we reproduce previously known results and then expand perturbatively about that limit. In this expansion we find that the holes have a quasiparticle residue of Jz{J}_{z}Jz​/t and a bandwidth of ${J}_{\ensuremath{\perp}}$. In the Heisenberg limit we employ a ``dominant pole'' approximation in which we ignore contributions to the self-energy from the incoherent part of the hole spectrum. A similar technique is used to study the d-wave RVB state. The relevance of our results to recent optical experiments is discussed.
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.
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.
The spin-wave expansion is used to motivate a simple trial wave function for the N\'eel ground state of a spin-1/2 Heisenberg antiferromagnet on a square lattice; the wave function yields an upper bound on the ground-state energy per bond of (-0.3317\ifmmode\pm\else\textpm\fi{}0.0002)J, where J is the exchange constant. The wave function is easily generalizable to the case where a single hole, with hopping matrix element t, is present. The hole energy-momentum relationship is determined for J/t>0.25; the minimum hole energy is always on the zone boundary at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2). The hole bandwidth, W, has a maximum value of 1.24t at J=0.73t. The small parameter which makes all of these calculations possible is 1/(2Z-2), where Z=4 is the coordination number of the square lattice.