arXiv:2501.03887 · Physical review. B./Physical review. B

Slow convergence of spin-wave expansion and magnon dispersion in the 1/3 plateau of the triangular XXZ antiferromagnet

Achille Mauri, Frédéric Mila

Published May 2, 2025• 4 citations

Abstract

Motivated by recent experiments on the quantum magnet ${\mathrm{K}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}$, we study theoretically the excitation spectrum of the nearest-neighbor triangular XXZ model in the limit of strong easy-axis anisotropy, within the up-up-down (1/3-plateau) phase. We make an expansion in $\ensuremath{\alpha}={J}_{xy}/{J}_{zz}$ instead of $1/S$ and calculate the magnon dispersion for any value of the spin $S$ at second order in $\ensuremath{\alpha}$, with two important conclusions: (i) the $1/S$ expansion converges very slowly for $S=1/2$, making spin-wave theory quantitatively inaccurate up to very large orders, and (ii) compared to the linear spin-wave predictions, our magnon dispersion presents a much better agreement with experimental results on ${\mathrm{K}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}$, for which $\ensuremath{\alpha}\ensuremath{\simeq}0.08$.

DOI: 10.1103/physrevb.111.l180402

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Nonlinear spin wave theory in the strong easy-axis limit of the triangular XXZ model

Achille Mauri, Siebe Roose, Frédéric Mila

arXiv:2511.08179Physical review. B./Physical review. BMay 1, 20262 citations

Unconventional spin dynamics and supersolid excitations in the triangular-lattice XXZ model

R. Flores-Calderón, Roderich Moessner, Frank Pollmann

arXiv:2506.15516Physical review. B./Physical review. BNov 20, 20253 citations

Magnetic excitations of the strongly-anisotropic triangular XXZ model: a projected spin-product state approach

Achille Mauri, Frédéric Mila

arXiv:2609.36140Sep 28, 20260 citations

Study of the excited states of the quantum antiferromagnets

Mohammad G. Merdan

Nov 10, 20130 citations

Magnons in a two-dimensional transverse-field XXZ model

Satyaki Kar, Keola Wierschem, Pinaki Sengupta

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

Shang-Shun Zhang, E. A. Ghioldi, Yoshitomo Kamiya +3 more

arXiv:1905.10689Physical review. B./Physical review. BSep 25, 201919 citations

arXiv:2501.03887 · Physical review. B./Physical review. B

Slow convergence of spin-wave expansion and magnon dispersion in the 1/3 plateau of the triangular XXZ antiferromagnet

Achille Mauri, Frédéric Mila

Published May 2, 2025• 4 citations

arXiv HTML PDF

Abstract

Motivated by recent experiments on the quantum magnet

{\mathrm{K}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}

, we study theoretically the excitation spectrum of the nearest-neighbor triangular XXZ model in the limit of strong easy-axis anisotropy, within the up-up-down (1/3-plateau) phase. We make an expansion in

\ensuremath{\alpha}={J}_{xy}/{J}_{zz}

instead of

1/S

and calculate the magnon dispersion for any value of the spin

S

at second order in

\ensuremath{\alpha}

, with two important conclusions: (i) the

1/S

expansion converges very slowly for

S=1/2

, making spin-wave theory quantitatively inaccurate up to very large orders, and (ii) compared to the linear spin-wave predictions, our magnon dispersion presents a much better agreement with experimental results on

{\mathrm{K}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}

, for which

\ensuremath{\alpha}\ensuremath{\simeq}0.08

DOI: 10.1103/physrevb.111.l180402

Related papers

Nonlinear spin wave theory in the strong easy-axis limit of the triangular XXZ model

Achille Mauri, Siebe Roose, Frédéric Mila

Motivated by recent experimental studies, we investigate the spectrum of the nearest-neighbour triangular XXZ model within the $1/S$ expansion, in the limit in which the exchange couplings present a strong easy-axis anisotropy $J_{xy}/J_{zz} \ll 1$ . We show that in the limit in which $1/S \to 0$ and $J_{xy} \to 0$ at fixed $V = J_{zz}/(S J_{xy})$ , the triangular spin model can be reduced to an effective boson model with quartic interactions on the honeycomb lattice. This effective model interpolates between a spin-wave ( $V \to 0$ ) and a strong-coupling limit ( $V \to \infty$ ) and encodes in a simple framework the regimes discussed by Kleine~\emph{et al.}~[Z. Phys. B Condens. Matter~{\bf 86}, 405 (1992);~{\bf 87}, 103 (1992)]. For zero field, the classical ground state of the model presents an accidental degeneracy, which can be traced to a simple symmetry of the classical energy. The model thus offers a transparent realization of a theory with quantum order-by-disorder and a pseudo-Goldstone mode. We analyze the spectrum at zero magnetic field by calculating the self-energy at one-loop order. In the calculation, we introduce a self-consistent renormalization of the energy scale and of the pseudo-Goldstone energy gap; the latter renormalization is essential to remove infrared divergences in the on-shell corrections to the energy dispersion. Finally, we discuss qualitatively the structure of the one-loop corrections in comparison with the spectrum observed experimentally in K $_{2}$ Co(SeO $_{3}$ ) $_{2}$ .Motivated by recent experimental studies, we investigate the spectrum of the nearest-neighbour triangular XXZ model within the $1/S$ expansion, in the limit in which the exchange couplings present a strong easy-axis anisotropy $J_{xy}/J_{zz} \ll 1$. We show that in the limit in which $1/S \to 0$ and $J_{xy} \to 0$ at fixed $V = J_{zz}/(S J_{xy})$, the triangular spin model can be reduced to an effective boson model with quartic interactions on the honeycomb lattice. This effective model interpolates between a spin-wave ($V \to 0$) and a strong-coupling limit ($V \to \infty$) and encodes in a simple framework the regimes discussed by Kleine~\emph{et al.}~[Z. Phys. B Condens. Matter~{\bf 86}, 405 (1992);~{\bf 87}, 103 (1992)]. For zero field, the classical ground state of the model presents an accidental degeneracy, which can be traced to a simple symmetry of the classical energy. The model thus offers a transparent realization of a theory with quantum order-by-disorder and a pseudo-Goldstone mode. We analyze the spectrum at zero magnetic field by calculating the self-energy at one-loop order. In the calculation, we introduce a self-consistent renormalization of the energy scale and of the pseudo-Goldstone energy gap; the latter renormalization is essential to remove infrared divergences in the on-shell corrections to the energy dispersion. Finally, we discuss qualitatively the structure of the one-loop corrections in comparison with the spectrum observed experimentally in K$_{2}$Co(SeO$_{3}$)$_{2}$.

arXiv:2511.08179Physical review. B./Physical review. BMay 1, 20262 citations

Unconventional spin dynamics and supersolid excitations in the triangular-lattice XXZ model

R. Flores-Calderón, Roderich Moessner, Frank Pollmann

Motivated by recent experiments, we investigate the spin- 1 2 XXZ model on the triangular lattice with strong Ising anisotropy, combining large-scale numerical simulations and analytical methods to uncover unconventional spin dynamics at T = 0 . First, we compute the dynamical spin structure factor using density matrix renormalization group (DMRG) simulations and find excellent agreement with inelastic neutron scattering data on the layered compound K 2 Co ( SeO 3 ) 2 . The low-energy spectrum reveals a rotonlike minimum at the M point, absent in linear spin-wave theory, accompanied by peak intensity and a broad continuum above it. Near the Γ point, we observe an approximately linear dispersion with vanishing spectral weight. Second, we compare two analytical frameworks that reproduce the observed features. The first is a hard-core boson approach, which includes (i) an effective staggered boson model at zero magnetic field, (ii) perturbation theory applied to the one-third magnetization plateau, and (iii) a self-consistent mean-field Schwinger boson theory (SBT). The second framework is based on a variational supersolid quantum dimer model (QDM) ansatz, combined with a single-mode approximation. The SBT captures the broad continuum, the M -point minimum, and linear dispersion at Γ , whereas the QDM reproduces the roton minimum and linear dispersion at finite momentum near Γ . Remarkably, both the QDM wave function and the DMRG ground state exhibit nearly identical structure factors with pronounced transverse photonlike excitations. Together, our comprehensive theoretical and numerical analysis elucidates the microscopic origin of supersolid excitations in the XXZ triangular-lattice model and their proximity to a spin-liquid phase observed experimentally.Motivated by recent experiments, we investigate the spin- 1 2 XXZ model on the triangular lattice with strong Ising anisotropy, combining large-scale numerical simulations and analytical methods to uncover unconventional spin dynamics at T = 0 . First, we compute the dynamical spin structure factor using density matrix renormalization group (DMRG) simulations and find excellent agreement with inelastic neutron scattering data on the layered compound K 2 Co ( SeO 3 ) 2 . The low-energy spectrum reveals a rotonlike minimum at the M point, absent in linear spin-wave theory, accompanied by peak intensity and a broad continuum above it. Near the Γ point, we observe an approximately linear dispersion with vanishing spectral weight. Second, we compare two analytical frameworks that reproduce the observed features. The first is a hard-core boson approach, which includes (i) an effective staggered boson model at zero magnetic field, (ii) perturbation theory applied to the one-third magnetization plateau, and (iii) a self-consistent mean-field Schwinger boson theory (SBT). The second framework is based on a variational supersolid quantum dimer model (QDM) ansatz, combined with a single-mode approximation. The SBT captures the broad continuum, the M -point minimum, and linear dispersion at Γ , whereas the QDM reproduces the roton minimum and linear dispersion at finite momentum near Γ . Remarkably, both the QDM wave function and the DMRG ground state exhibit nearly identical structure factors with pronounced transverse photonlike excitations. Together, our comprehensive theoretical and numerical analysis elucidates the microscopic origin of supersolid excitations in the XXZ triangular-lattice model and their proximity to a spin-liquid phase observed experimentally.

arXiv:2506.15516Physical review. B./Physical review. BNov 20, 20253 citations

Magnetic excitations of the strongly-anisotropic triangular XXZ model: a projected spin-product state approach

Achille Mauri, Frédéric Mila

Recent experiments have sparked interest on the ground-state properties and on the excitation spectrum of the spin- $1/2$ triangular XXZ model with strong Ising-like anisotropy. In this work, we introduce and analyze a ``projected'' coherent-state approximation to the model, constructed by projecting spin-product states onto the low-energy configurations of the triangular Ising antiferromagnet. Within this approximation, we discuss the structure of the ground-state ordering and the magnetization curve under a longitudinal field. We then discuss the excitation spectrum by linearizing equations of motion deriving from a time-dependent variational principle. This leads to a natural description of spin-wave-like excitations within the strongly constrained Hilbert space. At zero field, the variational approximation leads to a very flat landscape of quasi-degenerate states, which results in a low-energy pseudo-Goldstone mode in the excitation spectrum. We compare our results to experimental measurements on the cobaltite K $_{2}$ Co(SeO $_{3}$ ) $_{2}$ (KCSO). Our approximation leads to a dramatic improvement as compared to linear spin-wave theory. However, there remain deviations in the value of the pseudo-Goldstone gap and in the dispersion near $M$ points at low field.Recent experiments have sparked interest on the ground-state properties and on the excitation spectrum of the spin-$1/2$ triangular XXZ model with strong Ising-like anisotropy. In this work, we introduce and analyze a ``projected'' coherent-state approximation to the model, constructed by projecting spin-product states onto the low-energy configurations of the triangular Ising antiferromagnet. Within this approximation, we discuss the structure of the ground-state ordering and the magnetization curve under a longitudinal field. We then discuss the excitation spectrum by linearizing equations of motion deriving from a time-dependent variational principle. This leads to a natural description of spin-wave-like excitations within the strongly constrained Hilbert space. At zero field, the variational approximation leads to a very flat landscape of quasi-degenerate states, which results in a low-energy pseudo-Goldstone mode in the excitation spectrum. We compare our results to experimental measurements on the cobaltite K$_{2}$Co(SeO$_{3}$)$_{2}$ (KCSO). Our approximation leads to a dramatic improvement as compared to linear spin-wave theory. However, there remain deviations in the value of the pseudo-Goldstone gap and in the dispersion near $M$ points at low field.

arXiv:2609.36140Sep 28, 20260 citations

Study of the excited states of the quantum antiferromagnets

Mohammad G. Merdan

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 $\Gamma(\mathbf q=0)$ and at the magnetic ordering wavevectors $\pm[\mathbf Q= (4\pi/3,0)]$ , 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 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 $\Gamma(\mathbf q=0)$ and at the magnetic ordering wavevectors $\pm[\mathbf Q= (4\pi/3,0)]$, 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.

Nov 10, 20130 citations

Magnons in a two-dimensional transverse-field XXZ model

Satyaki Kar, Keola Wierschem, Pinaki Sengupta

The XXZ model on a square lattice in the presence of a transverse magnetic field is studied within the spin-wave theory to investigate the resulting canted antiferromagnet. The small- and large-field regimes are probed separately both for easy-axis and easy-plane scenarios which reveal an unentangled factorized ground state at an intermediate value of the field. Goldstone modes are obtained for the field-free XY antiferromagnet as well as for the isotropic antiferromagnet with field up to its saturation value. Moreover, for an easy-plane anisotropy, we find that there exists a nonzero field, where magnon degeneracy appears as a result of restoration of a U(1) sublattice symmetry and that, across that field, there occurs a magnon band crossing. For completeness, we then obtain the system phase diagram for $S=1/2$ via large-scale quantum Monte Carlo simulations using the stochastic series expansion technique. Our numerical method is based on a quantization of spin along the direction of the applied magnetic field and does not suffer from a sign problem, unlike comparable algorithms based on a spin quantization along the axis of anisotropy. With this formalism, we are also able to obtain powder averages of the transverse and longitudinal magnetizations, which may be useful for understanding experimental measurements on polycrystalline samples.The XXZ model on a square lattice in the presence of a transverse magnetic field is studied within the spin-wave theory to investigate the resulting canted antiferromagnet. The small- and large-field regimes are probed separately both for easy-axis and easy-plane scenarios which reveal an unentangled factorized ground state at an intermediate value of the field. Goldstone modes are obtained for the field-free XY antiferromagnet as well as for the isotropic antiferromagnet with field up to its saturation value. Moreover, for an easy-plane anisotropy, we find that there exists a nonzero field, where magnon degeneracy appears as a result of restoration of a U(1) sublattice symmetry and that, across that field, there occurs a magnon band crossing. For completeness, we then obtain the system phase diagram for $S=1/2$ via large-scale quantum Monte Carlo simulations using the stochastic series expansion technique. Our numerical method is based on a quantization of spin along the direction of the applied magnetic field and does not suffer from a sign problem, unlike comparable algorithms based on a spin quantization along the axis of anisotropy. With this formalism, we are also able to obtain powder averages of the transverse and longitudinal magnetizations, which may be useful for understanding experimental measurements on polycrystalline samples.

arXiv:1612.06608Physical review. B./Physical review. BJul 19, 20177 citations

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

Shang-Shun Zhang, E. A. Ghioldi, Yoshitomo Kamiya +3 more

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.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.

arXiv:1905.10689Physical review. B./Physical review. BSep 25, 201919 citations

Motivated by recent experiments on the quantum magnet

{\mathrm{K}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}

\ensuremath{\alpha}={J}_{xy}/{J}_{zz}

instead of

1/S

and calculate the magnon dispersion for any value of the spin

S

at second order in

\ensuremath{\alpha}

, with two important conclusions: (i) the

1/S

expansion converges very slowly for

S=1/2

{\mathrm{K}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}

, for which

\ensuremath{\alpha}\ensuremath{\simeq}0.08

Motivated by recent experimental studies, we investigate the spectrum of the nearest-neighbour triangular XXZ model within the

1/S

expansion, in the limit in which the exchange couplings present a strong easy-axis anisotropy

J_{xy}/J_{zz} \ll 1

. We show that in the limit in which

1/S \to 0

and

J_{xy} \to 0

at fixed

V = J_{zz}/(S J_{xy})

, the triangular spin model can be reduced to an effective boson model with quartic interactions on the honeycomb lattice. This effective model interpolates between a spin-wave (

V \to 0

) and a strong-coupling limit (

V \to \infty

) and encodes in a simple framework the regimes discussed by Kleine~\emph{et al.}~[Z. Phys. B Condens. Matter~{\bf 86}, 405 (1992);~{\bf 87}, 103 (1992)]. For zero field, the classical ground state of the model presents an accidental degeneracy, which can be traced to a simple symmetry of the classical energy. The model thus offers a transparent realization of a theory with quantum order-by-disorder and a pseudo-Goldstone mode. We analyze the spectrum at zero magnetic field by calculating the self-energy at one-loop order. In the calculation, we introduce a self-consistent renormalization of the energy scale and of the pseudo-Goldstone energy gap; the latter renormalization is essential to remove infrared divergences in the on-shell corrections to the energy dispersion. Finally, we discuss qualitatively the structure of the one-loop corrections in comparison with the spectrum observed experimentally in K

_{2}

Co(SeO

_{3}

)

_{2}

Recent experiments have sparked interest on the ground-state properties and on the excitation spectrum of the spin-

1/2

triangular XXZ model with strong Ising-like anisotropy. In this work, we introduce and analyze a ``projected'' coherent-state approximation to the model, constructed by projecting spin-product states onto the low-energy configurations of the triangular Ising antiferromagnet. Within this approximation, we discuss the structure of the ground-state ordering and the magnetization curve under a longitudinal field. We then discuss the excitation spectrum by linearizing equations of motion deriving from a time-dependent variational principle. This leads to a natural description of spin-wave-like excitations within the strongly constrained Hilbert space. At zero field, the variational approximation leads to a very flat landscape of quasi-degenerate states, which results in a low-energy pseudo-Goldstone mode in the excitation spectrum. We compare our results to experimental measurements on the cobaltite K

_{2}

Co(SeO

_{3}

)

_{2}

(KCSO). Our approximation leads to a dramatic improvement as compared to linear spin-wave theory. However, there remain deviations in the value of the pseudo-Goldstone gap and in the dispersion near

M

points at low field.

\Gamma(\mathbf q=0)

and at the magnetic ordering wavevectors

\pm[\mathbf Q= (4\pi/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.

S=1/2

via large-scale quantum Monte Carlo simulations using the stochastic series expansion technique. Our numerical method is based on a quantization of spin along the direction of the applied magnetic field and does not suffer from a sign problem, unlike comparable algorithms based on a spin quantization along the axis of anisotropy. With this formalism, we are also able to obtain powder averages of the transverse and longitudinal magnetizations, which may be useful for understanding experimental measurements on polycrystalline samples.

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.