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arXiv:cond-mat/9510064 · Physical review. B, Condensed matter

Spinons, solitons, and magnons in one-dimensional Heisenberg-Ising antiferromagnets

Rajiv R. P. Singh

Published May 1, 1996• 2 citations
arXivHTMLPDF

Abstract

We calculate the excitation spectra for the 1D Heisenberg-Ising antiferromagnets by expansions around the Ising limit. For S=1/2, the calculated expansion coefficients for the spinon spectra agree term by term with the solution of Johnson and McCoy. For S=1, the solitons become gapless before the Heisenberg limit is reached, signaling a transition to the Haldane phase. By applying a staggered field we calculate the one-magnon spectra for the S=1 Heisenberg chain. For S=3/2 the quantum renormalization of the spin-wave spectra is calculated to be approximately 1.16. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.11582

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We calculate the excitation spectra for the 1D Heisenberg-Ising antiferromagnets by expansions around the Ising limit. For S=1/2, the calculated expansion coefficients for the spinon spectra agree term by term with the solution of Johnson and McCoy. For S=1, the solitons become gapless before the Heisenberg limit is reached, signaling a transition to the Haldane phase. By applying a staggered field we calculate the one-magnon spectra for the S=1 Heisenberg chain. For S=3/2 the quantum renormalization of the spin-wave spectra is calculated to be approximately 1.16. \textcopyright{} 1996 The American Physical Society.
The excitation energies of magnons and solitons as well as the ground-state energy of the spin-1 antiferromagnetic Heisenberg-Ising ring are studied by the numerical method. The method of Vanden Broeck and Schwartz is used to estimate the infinite-lattice limits accurately. The result confirms Haldane's prediction that a novel singlet phase with a nonzero excitation gap appears between the gapless XY phase and the doublet ground-state N\'eel phase. It extends over the range of anisotropy (${\ensuremath{\Delta}}_{l}$\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}${\ensuremath{\Delta}}_{u}$) around the isotropic Heisenberg point \ensuremath{\Delta}=0.5. It is ascertained that at the upper transition point ${\ensuremath{\Delta}}_{u}$=0.540\ifmmode\pm\else\textpm\fi{}0.001 (1) the longitudinal magnon gap closes, (2) the soliton existing in the Ising side softens, and (3) the second derivative of the ground-state energy diverges. Around the lower transition point of ${\ensuremath{\Delta}}_{l}$=0.38\ifmmode\pm\else\textpm\fi{}0.02, the transverse magnon gap exhibits an exponential singularity accompanying with the slight anomaly of the second derivative of the ground-state energy.
We employ spin-cluster perturbation theory to investigate the evolution of the spin excitation spectra of the mixed-dimensional antiferromagnetic s=12s=\frac{1}{2}s=21​ Heisenberg models from the one-dimensional chain to the two-dimensional square and honeycomb lattices by varying the spin interactions ${J}_{\ensuremath{\perp}}$ across the chains relative to ${J}_{\ensuremath{\parallel}}$ along the chains. In addition to the well-defined magnon excitations, the spectra of the N\'eel phases for both lattices exhibit anomalous high-energy continua that evolve continuously from the continuum of one-dimensional chain through increasing ${J}_{\ensuremath{\perp}}$. Notably, the key features of the continua at the two-dimensional limit for ${J}_{\ensuremath{\perp}}/{J}_{\ensuremath{\parallel}}=1$ are well consistent with experimental observations in real materials. Our results indicate that these high-energy continua in the N\'eel phases are a result of spinon deconfinement, which is further supported by a random-phase-approximation analysis based on the spinon Hubbard models.
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.
Effective nonlinear field theory for a semiclassical long-wavelength description of one-dimensional S=1/2 Heisenberg antiferromagnets with dimerized ground states is derived within an approach using two-site coherent states. The method is applied to bond-alternating anisotropic spin chains in presence of external magnetic field and next-nearest-neighbor interactions. Linear and nonlinear excitations in singlet and magnetic phases are studied. Soliton solutions are found and their properties under semiclassical quantization are discussed. \textcopyright{} 1996 The American Physical Society.
In a previous paper (Liu et al 2012 Phys. Rev. B 85 195144 ), a variational Monte Carlo method (based on Gutzwiller projected states) was generalized to S = 1 systems. This method provided very good trial ground states for the gapped phases of an S = 1 bilinear-biquadratic (BLBQ) Heisenberg chain. In this paper, we extend the approach to study the low-lying elementary excitations in S = 1 chains. We calculate the one-magnon and two-magnon excitation spectra of the BLBQ Heisenberg chain and the results agree very well with recent data in the literature. In our approach, the difference of the excitation spectrum between the Haldane phase and the dimer phase (such as the even/odd size effect) can be understood from their different topologies of the corresponding mean field theory. We especially study the Takhtajan–Babujian critical point. Despite the fact that the 'elementary excitations' are spin-1 magnons, which are different from the spin-1/2 spinons in Bethe solution, we show that the excitation spectrum, critical exponent ( ) and central charge ( c = 1.45) calculated from our theory agree well with the Bethe ansatz solution and conformal field theory predictions.
Dynamics of one-dimensional Ising-like (strongly anisotropic) antiferromagnets is studied by the perturbation theory from the pure Ising limit and by an exact calculation for finite chains. Both calculations show that the des Cloizeaux-Gaudin (dC-G) “spin-wave” spectrum corresponds neither to a lower bound of excitation continuum nor to the peak position of the transverse response Sxx(Q, ω). A new interpretation is given of the spin-wave spectrum of CsCoCl3 without relying on the dC-G theory. It is also shown that at elevated temperatures Sxx(Q, ω) has a three-peak structure. The longitudinal response Sxx(Q, ω) at low temperatures has a weak peak at the position of spin wave peak, while with increasing temperature Sxx(Q, ω) is dominated by the central peak due to thermally activated solitons, which was predicted by Villain.