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arXiv:1905.10689 · Physical review. B./Physical review. B

Large- S limit of the large- N theory for the triangular antiferromagnet

Shang-Shun Zhang, E. A. Ghioldi, Yoshitomo Kamiya, L. O. Manuel, A. E. Trumper, Cristian D. Batista

Published Sep 25, 2019• 19 citations
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Abstract

Large-$S$ and large-$N$ theories (spin value $S$ and spinor component number $N$) are complementary, and sometimes conflicting, approaches to quantum magnetism. While large-$S$ spin-wave theory captures the correct semiclassical behavior, large-$N$ theories, on the other hand, emphasize the quantumness of spin fluctuations. In order to evaluate the possibility of the nontrivial recovery of the semiclassical magnetic excitations within a large-$N$ approach, we compute the large-$S$ limit of the dynamical spin structure factor of the triangular lattice Heisenberg antiferromagnet within a Schwinger boson spin representation. We demonstrate that, only after the incorporation of Gaussian ($1/N$) corrections to the saddle-point ($N=\ensuremath{\infty}$) approximation, we are able to exactly reproduce the linear spin-wave theory results in the large-$S$ limit. The key observation is that the effect of $1/N$ corrections is to cancel out exactly the main contribution of the saddle-point solution, while the collective modes (magnons) consist of two spinon bound states arising from the poles of the random phase approximation propagator. This result implies that it is essential to consider the interaction of the spinons with the emergent gauge fields and that the magnon dispersion relation should not be identified with that of the saddle-point spinons.

DOI
10.1103/physrevb.100.104431

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Large-SSS and large-NNN theories (spin value SSS and spinor component number NNN) are complementary, and sometimes conflicting, approaches to quantum magnetism. While large-SSS spin-wave theory captures the correct semiclassical behavior, large-NNN theories, on the other hand, emphasize the quantumness of spin fluctuations. In order to evaluate the possibility of the nontrivial recovery of the semiclassical magnetic excitations within a large-NNN approach, we compute the large-SSS limit of the dynamical spin structure factor of the triangular lattice Heisenberg antiferromagnet within a Schwinger boson spin representation. We demonstrate that, only after the incorporation of Gaussian (1/N1/N1/N) corrections to the saddle-point (N=\ensuremath∞N=\ensuremath{\infty}N=\ensuremath∞) approximation, we are able to exactly reproduce the linear spin-wave theory results in the large-SSS limit. The key observation is that the effect of 1/N1/N1/N corrections is to cancel out exactly the main contribution of the saddle-point solution, while the collective modes (magnons) consist of two spinon bound states arising from the poles of the random phase approximation propagator. This result implies that it is essential to consider the interaction of the spinons with the emergent gauge fields and that the magnon dispersion relation should not be identified with that of the saddle-point spinons.
We compute the zero temperature dynamical structure factor S(q,ω)S({\bf q},\omega)S(q,ω) of the triangular lattice Heisenberg model using a Schwinger Boson approach that includes the Gaussian fluctuations (1/N1/N1/N correction) of the saddle point solution. While the ground state of this model exhibits a well-known 120∘^{\circ}∘ magnetic ordering, the excitation spectrum revealed by S(q,ω)S({\bf q},\omega)S(q,ω) has a strong quantum character that is not captured by low-order 1/S1/S1/S expansions. The low-energy magnons consist of two spinon bound states confined by gauge fluctuations of the auxiliary fields, which can produce internal structure of the magnon peaks. In addition, the high-energy continuum of spinon modes extends up to about three times the single-magnon bandwidth.
The Schwinger boson theory provides a natural path for treating quantum spin systems with large quantum fluctuations. In contrast to semiclassical treatments, this theory allows us to describe a continuous transition between magnetically ordered and spin liquid states, as well as the continuous evolution of the corresponding excitation spectrum. The square lattice Heisenberg antiferromagnet is one of the first models that was approached with the Schwinger boson theory. Here we revisit this problem to reveal several subtle points that were omitted in previous treatments and that are crucial to further develop this formalism. These points include the freedom for the choice of the saddle point (Hubbard-Stratonovich decoupling and choice of the condensate) and the 1/N1/N1/N expansion in the presence of a condensate. A key observation is that the spinon condensate leads to Feynman diagrams that include contributions of different order in 1/N1/N1/N, which must be accounted for to get a qualitatively correct excitation spectrum. We demonstrate that a proper treatment of these contributions leads to an exact cancellation of the single-spinon poles of the dynamical spin structure factor, as expected for a magnetically ordered state. The only surviving poles are the ones arising from the magnons (two-spinon bound states), which are the true collective modes of an ordered magnet.
We compute the dynamical structure factor of the spin-½ triangular Heisenberg model using the mean-field Schwinger boson theory. We find that a reconstructed dispersion, resulting from a non-trivial redistribution of the spectral weight, agrees quite well with the spin excitation spectrum recently found with series expansions. In particular, we recover the strong renormalization with respect to linear spin wave theory along with the appearance of roton-like minima. Furthermore, near the roton-like minima the contribution of the two spinon continuum to the static structure factor is about 40% of the total weight. By computing the density-density dynamical structure factor, we identify an unphysical weak signal of the spin excitation spectrum with the relaxation of the local constraint of the Schwinger bosons at the mean-field level. Based on the accurate description obtained for the static and dynamic ground-state properties, we argue that the bosonic spinon theory should be considered seriously as a valid alternative to interpret the physics of the triangular Heisenberg model.
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.
Schwinger boson perturbation theory for spin-SSS Kitaev-Heisenberg magnets: phase diagram and dynamical response near Kitaev spin liquids
We develop a Schwinger boson perturbative framework for the spin-SSS Kitaev-Heisenberg model. The spin-liquid saddle point of the pure Kitaev model is used as the unperturbed state, and magnetic instabilities and dynamical spin correlations are evaluated around this saddle point. We decompose the Hamiltonian exactly into two parts by exploiting the Klein duality intrinsic to the Kitaev-Heisenberg model. We perform the random-phase approximation by taking the part invariant under the duality transformation as the unperturbed term and treating the remaining term, which changes sign under it, as the perturbation. We determine the phase boundaries separating the quantum spin-liquid regimes from the adjacent magnetically ordered phases for S=1/2S=1/2S=1/2, 111, 3/23/23/2, and 222. The spin-liquid regions shrink rapidly with increasing SSS and become very narrow at S=2S=2S=2. We also compute the spin dynamics of the spin-SSS Kitaev-Heisenberg model, focusing on S=1S=1S=1, and find that the dressed dynamical spin structure factor retains a broad two-spinon continuum at finite energies, while the low-energy spectral weight softens at the ordering wave vectors of the adjacent magnetic phases. Our framework thus enables thermodynamic-limit calculations of spin dynamics near magnetic instabilities.
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.