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arXiv:0812.3420 · Physical Review B

Exact diagonalization study of the antiferromagnetic spin- 12 Heisenberg model on the square lattice in a magnetic field

Andreas Lüscher, Andreas Martin Läuchli

Published May 5, 2009• 77 citations
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Abstract

We study the field dependence of the antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model on the square lattice by means of exact diagonalizations. In the first part, we calculate the spin-wave velocity $c$, the spin stiffness ${\ensuremath{\rho}}_{s}$, and the magnetic susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}$, and thus determine the microscopic parameters of the low-energy long-wavelength description. In the second part, we present a comprehensive study of dynamical spin-correlation functions for magnetic fields ranging from zero up to saturation. We find that, at low fields, magnons are well defined in the whole Brillouin zone but the dispersion is substantially modified by quantum fluctuations compared to the classical spectrum. At higher fields, decay channels open and magnons become unstable with respect to multimagnon scattering. Our results directly apply to inelastic neutron-scattering experiments.

DOI
10.1103/physrevb.79.195102

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We study the field dependence of the antiferromagnetic spin-12\frac{1}{2}21​ Heisenberg model on the square lattice by means of exact diagonalizations. In the first part, we calculate the spin-wave velocity ccc, the spin stiffness \ensuremathρs{\ensuremath{\rho}}_{s}\ensuremathρs​, and the magnetic susceptibility \ensuremathχ\ensuremath⊥{\ensuremath{\chi}}_{\ensuremath{\perp}}\ensuremathχ\ensuremath⊥​, and thus determine the microscopic parameters of the low-energy long-wavelength description. In the second part, we present a comprehensive study of dynamical spin-correlation functions for magnetic fields ranging from zero up to saturation. We find that, at low fields, magnons are well defined in the whole Brillouin zone but the dispersion is substantially modified by quantum fluctuations compared to the classical spectrum. At higher fields, decay channels open and magnons become unstable with respect to multimagnon scattering. Our results directly apply to inelastic neutron-scattering experiments.
We discuss spin-12\frac{1}{2}21​ Heisenberg antiferromagnet on simple square lattice in magnetic field HHH using a recently proposed bond-operator technique. It is well known that magnetically ordered phases of quantum magnets are well described at least qualitatively by the conventional spin-wave theory that only introduces quantum corrections into the classical solution of the problem. We observe that quantum fluctuations change drastically dynamical properties of the considered model at HHH close to its saturation value: the dynamical structure factor shows anomalies corresponding to Green's function poles which have no counterparts in the spin-wave theory. That is, quantum fluctuations produce multiple short-wavelength magnon modes not changing qualitatively the long-wavelength spin dynamics. Our results are in agreement with previous quantum Monte Carlo simulations and exact diagonalization of finite clusters.
Understanding the nature of the excitation spectrum in quantum spin liquids is of fundamental importance, in particular for the experimental detection of candidate materials.However, current theoretical and numerical techniques have limited capabilities, especially in obtaining the dynamical structure factor, which gives a crucial characterization of the ultimate nature of the quantum state and may be directly assessed by inelastic neutron scattering.In this work, we investigate the low-energy properties of the S ¼ 1=2 Heisenberg model on the triangular lattice, including both nearest-neighbor J 1 and next-nearestneighbor J 2 superexchanges, by a dynamical variational Monte Carlo approach that allows accurate results on spin models.For J 2 ¼ 0, our calculations are compatible with the existence of a well-defined magnon in the whole Brillouin zone, with gapless excitations at K points (i.e., at the corners of the Brillouin zone).The strong renormalization of the magnon branch (also including rotonlike minima around the M points, i.e., midpoints of the border zone) is described by our Gutzwiller-projected state, where Abrikosov fermions are subject to a nontrivial magnetic π flux threading half of the triangular plaquettes.When increasing the frustrating ratio J 2 =J 1 , we detect a progressive softening of the magnon branch at M, which eventually becomes gapless within the spin-liquid phase.This feature is captured by the band structure of the unprojected wave function (with two Dirac points for each spin component).In addition, we observe an intense signal at low energies around the K points, which cannot be understood within the unprojected picture and emerges only when the Gutzwiller projection is considered, suggesting the relevance of gauge fields for the low-energy physics of spin liquids.
The spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ model on an infinite square lattice at zero temperature has been studied and is presented here. Our methods using finite square lattices have been the exact diagonalization and quantum Monte Carlo methods. The physical properties estimated per lattice vertex include ground-state and first-excited-state energy, staggered magnetization, and susceptibility in the parallel and perpendicular directions, spin stiffness, and spin-wave velocity. In the XXZ\mathrm{XXZ}XXZ model range from the Ising model to the Heisenberg model our estimates of physical properties compare very well with estimates by several other methods published in several other papers. In the XXZ\mathrm{XXZ}XXZ range between the Heisenberg and XY\mathrm{XY}XY models there is very little in the literature to compare with our finite lattice estimates of physical properties.
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.
We study the spin-(1/2 Heisenberg antiferromagnet on an infinite square lattice. The calculational scheme known as ``paired-phonon analysis'' developed for strongly correlated quantum fluids is extended to a ``paired-magnon analysis'' to study quantum antiferromagnets. We define a complete and orthonormal set of multimagnon states and calculate the matrix elements of the Hamiltonian using a separability approximation. Our results obtained by diagonalizing the Hamiltonian matrix analytically are very similar to those obtained in spin-wave theory. We obtain -0.3290, for the ground-state energy per bond in units of the antiferromagnetic coupling and 0.303 for the ground-state staggered magnetization. These results compare well with the best-known estimates -0.334\ifmmode\pm\else\textpm\fi{}0.001 and 0.313, respectively. We derive the analytic form of the ground-state wave function in this approximation and find it to be of the same form as that assumed by Marshall in his variational studies.. The zero-point motion of long-wavelength excitations (spin waves) in the model, however, reflects a long-range tail in our wave function. We discuss the separability approximation by giving quantitative arguments which justify its validity.
We present numerical exact results for the ground state and the low-lying excitations for the spin-1/2 J1-J2 Heisenberg antiferromagnet on finite square lattices of up to N = 40 sites. Using finite-size extrapolation we determine the ground-state energy, the magnetic order parameters, the spin gap, the uniform susceptibility, as well as the spin-wave velocity and the spin stiffness as functions of the frustration parameter J2/J1. In agreement with the generally excepted scenario we find semiclassical magnetically ordered phases for J2 < J c1 2 and J2> J c2 2 separated by a gapful quantum paramagnetic phase. We estimate Jc1 2 ≈ 0.35J1 and J c2 2 ≈ 0.66J1. PACS codes: 75.10.Jm Quantized spin models 75.45.+j Macroscopic quantum phenomena in magnetic systems 1