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Journal of Applied Physics

Dynamical properties of quantum spin systems in magnetically ordered product ground states

V. S. Viswanath, Joachim Stolze, Gerhard H. Muller

Published May 15, 1994• 3 citations
Publisher

Abstract

The one-dimensional spin-s XYZ model in a magnetic field of particular strength has a ferro- or antiferromagnetically ordered product ground state. The recursion method is employed to determine T=0 dynamic structure factors for systems with s=1/2, 1, 3/2. The line shapes and peak positions differ significantly from the corresponding spin-wave results, but their development for increasing values of s suggests a smooth extrapolation to the spin-wave picture.

DOI
10.1063/1.355509

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The one-dimensional spin-s XYZ model in a magnetic field of particular strength has a ferro- or antiferromagnetically ordered product ground state. The recursion method is employed to determine T=0 dynamic structure factors for systems with s=1/2, 1, 3/2. The line shapes and peak positions differ significantly from the corresponding spin-wave results, but their development for increasing values of s suggests a smooth extrapolation to the spin-wave picture.
We state the conditions under which the general spin-s quantum XYZ ferromagnet (${H}_{\mathrm{\ensuremath{-}}}$) and antiferromagnet (H+{H}_{+}H+​) with an external magnetic field along one axis, specified by the Hamiltonian ${H}_{\ifmmode\pm\else\textpm\fi{}}$=\ifmmode\pm\else\textpm\fi{} Jl=1N{\mathcal{J}}_{l=1}^{N}Jl=1N​ (Jx{J}_{x}Jx​Slx{S}_{l}^{x}Slx​Sl+1x{S}_{l+1}^{x}Sl+1x​+Jy{J}_{y}Jy​Sly{S}_{l}^{y}Sly​S l+1y_{l+1}^{y}l+1y​+Jz{J}_{z}Jz​Slz{S}_{l}^{z}Slz​Sl+1z{S}_{l+1}^{z}Sl+1z​)-h Jl=1N{\mathcal{J}}_{l=1}^{N}Jl=1N​Slz{S}_{l}^{z}Slz​ exhibits a fully ordered ground state described by a wave function which is a direct product of single-site wave functions. We present a detailed analysis of the implications for the zero-temperature dynamical properties of this model. In particular, we derive a rigorous relation between the three dynamic structure factors ${S}_{\ensuremath{\mu}\ensuremath{\mu}}$(q,\ensuremath{\omega}), \ensuremath{\mu}=x,y,z at T=0. For the special case of the s=(1/2) anisotropic XY model (Jz{J}_{z}Jz​=0), these relations are used to determine the dynamic structure factors Sxx{S}_{\mathrm{xx}}Sxx​(q,\ensuremath{\omega}) and Syy{S}_{\mathrm{yy}}Syy​(q,\ensuremath{\omega}) at T=0 and h=(Jx{J}_{x}Jx​Jy{J}_{y}Jy​)1/2{)}^{1/2})1/2 in terms of the known dynamic structure factor Szz{S}_{\mathrm{zz}}Szz​(q,\ensuremath{\omega}).
The recursion method is applied to the T=0 dynamics of the S=1/2 XXZ model on a linear chain and a square lattice. By means of new calculational techniques for the analysis of the continued-fraction coefficients pertaining to specific dynamical quantities, we obtain reliable information on the type of ordering in the ground state, on the size of gaps in the dynamically relevant excitation spectrum, on the bandwidths of dominant structures in spectral densities, on the exponents of infrared singularities, and on the detailed shape of spectral-weight distributions. We investigate some characteristic properties of the dynamic structure factors ${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}) and the spin autocorrelation functions ${\mathit{S}}^{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(\ensuremath{\omega})=${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$ tsumq{sum}_{\mathbf{q}}sumq​${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}), specifically their dependence on the uniaxial anisotropy, i.e., on the parameter which controls the type of ordering and the amount of quantum fluctuations in the ground state. We find, for example, that the different degrees of ordering in the planar regime of the one-dimensional and two-dimensional systems (criticality versus antiferromagnetic long-range order) have characteristic signatures in the dynamical properties which are conspicuously displayed in our results.
For special coupling ratios, the one-dimensional (1D) s=1/2 Heisenberg model with antiferromagnetic nearest and next-nearest neighbor interactions has a pure dimer ground state, and the 1D s=1 Heisenberg model with antiferromagnetic bilinear and biquadratic interactions has an exact valence-bond-solid ground state. The recursion method is used to calculate the T=0 spin dynamic structure factor for both models and, for the s=1/2 model, also the dimer dynamic structure factor. New results for line shapes and dynamically relevant dispersions are obtained.
We outline the recent results on the ground state for a class of one- and two-dimensional frustrated quantum spin models with competing ferro(F)- and antiferromagnetic (AF) interactions. Frustrated spin systems are known to have many interesting properties due to large quantum fluctuations. As a result of these fluctuations usual mean-field approach gives quite crude (if not false) description of these systems. Therefore, exactly solvable models are very instructive at investigations of such systems. The exact ground state wave function of proposed models has a structure of the valence-bond state (VBS) type. One of the 1D model describes the transition line between the F and AF phase. The exact singlet ground state on this line has a double-spiral ordering. Using different approximation methods we study the magnetization curve in the AF phase. The second considered set of the 1D and 2D models has an exact non-degenerate ground state with exponentially decaying spin correlations. We also proposed the 1D and 2D electronic models with exact ground state represented in terms of singlet bond functions which are the generalization of the RVB functions including ionic states.
We have found the exact (factorized) ground state of a general class of ferrimagnets in the presence of a magnetic field which includes the anisotropic and long-range interactions for arbitrary dimensional space. In particular cases our model represents many spin chains with bond alternation of antiferromagnetic-ferromagnetic coupling, ferrimagnetic spin ladders, and also homogeneous spin-sss models. The factorized ground state is a product of single-particle kets on a bipartite lattice composed of two different spins $(\ensuremath{\rho},\ensuremath{\sigma})$ which is characterized by two angles, a biangle state. The spin-wave analysis around the exact ground state shows two branch of excitations which are the origin of two dynamics of the model. The signature of these dynamics is addressed as a peak and a broaden bump in the specific heat.
A combination of world-line quantum Monte Carlo and maximum-entropy methods is used to calculate the dynamic structure factor ${\mathit{S}}_{\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{\alpha}}}$(q,\ensuremath{\omega}) for one-dimensional antiferromagnets. A comparison to exact results demonstrates that qualitative features of spectra are reproduced, but are broadened. The broadening is frequency dependent with low-frequency features being reproduced most accurately. Peak positions are well described. Results for the Heisenberg model with S\ensuremath{\le}2 demonstrate the qualitative difference between the dynamics of half-integer and integer spins predicted by Haldane. From the positions of the low-temperature q=\ensuremath{\pi} peaks quantitative estimates are produced for the S=1 and 2 gaps. The q=\ensuremath{\pi} peak position for S=1 increases with temperature in agreement with experiment. When on-site anisotropy is included for S=1 a gap splitting results that resembles that found for NENP [Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​ClO4{\mathrm{ClO}}_{4}ClO4​].