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arXiv:cond-mat/0001441 · Physical review. B, Condensed matter

Magnon dispersions in quantum Heisenberg ferrimagnetic chains at zero temperature

Nedko B. Ivanov

Published Aug 1, 2000• 25 citations
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Abstract

Within the Dyson-Maleev boson formalism, we study the zero-temperature magnon dispersions in a family of one-dimensional quantum Heisenberg ferrimagnets composed of two different spins ${(S}_{1}{,S}_{2})$ in the elementary cell. It is shown that the spin-wave theory can produce precise quantitative results for the low-energy excitations. The spin-stiffness constant ${\ensuremath{\rho}}_{s}$ and the optical magnon gap $\ensuremath{\Delta}$ of different ${(S}_{1}{,S}_{2})$ ferrimagnetic systems are calculated, respectively, to second and third order in the quasiparticle interaction. The spin-wave results are compared with available numerical estimates.

DOI
10.1103/physrevb.62.3271

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Within the Dyson-Maleev boson formalism, we study the zero-temperature magnon dispersions in a family of one-dimensional quantum Heisenberg ferrimagnets composed of two different spins (S1,S2){(S}_{1}{,S}_{2})(S1​,S2​) in the elementary cell. It is shown that the spin-wave theory can produce precise quantitative results for the low-energy excitations. The spin-stiffness constant \ensuremathρs{\ensuremath{\rho}}_{s}\ensuremathρs​ and the optical magnon gap \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ of different (S1,S2){(S}_{1}{,S}_{2})(S1​,S2​) ferrimagnetic systems are calculated, respectively, to second and third order in the quasiparticle interaction. The spin-wave results are compared with available numerical estimates.
Second-order spin-wave expansions are used to compute the ground-state energy and sublattice magnetizations of the quantum one-dimensional Heisenberg ferrimagnet with nearest-neighbor antiferromagnetic interactions and two types of alternating sublattice spins S1>S2.{S}_{1}>{S}_{2}.S1​>S2​. It is found that in the extreme quantum cases (S1,S2){(S}_{1}{,S}_{2})(S1​,S2​)=(1,1/2), (3/2,1), and (3/2,1/2), the estimates for the ground-state energy and sublattice magnetizations differ less than 0.03% for the energy and 0.2% for the sublattice magnetizations from the recently published density-matrix renormalization-group numerical calculations. The reported results strongly suggest that the quantum Heisenberg ferrimagnetic chains give another example of a low-dimensional quantum spin system where the spin-wave approach demonstrates a surprising efficiency.
We study quantum ferrimagnets in one, two, and three dimensions by using a variety of methods and approximations. These include: (i) a treatment based on the spin coherent state path-integral formulation of quantum ferrimagnets by taking into account the leading order quantum and thermal fluctuations (ii) a field-theoretical (non-linear σσσ-model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h (hc1<h<hc2h_{c1} < h < h_{c2}hc1​<h<hc2​) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.
The Schwinger-boson mean-field theory is applied to the quantum ferrimagnetic Heisenberg chain. There is a ferrimagnetic long-range order in the ground state. We observe two branches of the low-lying excitation and calculate the spin reduction, the gap of the antiferromagnetic branch, and the spin fluctuation at T=0K.T=0 \mathrm{K}.T=0K. These results agree with the established numerical results quite well. At finite temperatures, the long-range order is destroyed because of the disappearance of the Bose condensation. The thermodynamic observables, such as the free energy, magnetic susceptibility, specific heat, and the spin correlation at T>0K,T>0 \mathrm{K},T>0K, are calculated. The $T{\ensuremath{\chi}}_{\mathrm{uni}}$ has a minimum at intermediate temperatures and the spin-correlation length behaves as ${T}^{\ensuremath{-}1}$ at low temperatures. These qualitatively agree with the numerical results and the difference is small at low temperatures.
We study ground-state properties and the low-lying excitations of Heisenberg spin ladders composed of two ferrimagnetic chains with alternating site spins (S1>S2){(S}_{1}>{S}_{2})(S1​>S2​) by using the bosonic Dyson-Maleev formalism and Lanczos numerical techniques. The emphasis is on properties of the ferrimagnetic phase, which is stable for antiferromagnetic interchain couplings ${J}_{\ensuremath{\perp}}>~0$. There are two basic implications of the underlying lattice structure: (i) the spin-wave excitations form folded acoustic and optical branches in the extended Brillouin zone and (ii) the ground-state parameters (such as the on-site magnetizations and spin-stiffness constant) show a crossover behavior in the weak-coupling region $0<{J}_{\ensuremath{\perp}}<1$. The above peculiarities of the ladder ferrimagnetic state are studied up to second order in the quasiparticle interaction and by a numerical diagonalization of ladders containing up to N=12N=12N=12 rungs. The presented results for the ground-state parameters and the excitation spectrum can be used in studies on the low-temperature thermodynamics of ferrimagnetic ladders.
The low-energy structures of the quantum ferrimagnetic Heisenberg chain consisting of (5/2, 1/2, 1/2) trimers are investigated theoretically. The results of the linear spin-wave theory are compared with those from the numerical exact diagonalization calculation and the matrix product method for the lowest optical mode. The temperature behaviour of thermodynamical properties such as magnetic susceptibility, specific heat and entropy are analysed in the framework of the modified spin-wave theory. The results of the calculations are used to explain the experimental data obtained for the molecule-based heterospin magnets [Mn(hfac) 2 BNO R ] (R = H, F, Cl, Br) with one-dimensional chain structure.
In this thesis the elementary excitations of the spin-1=2 Heisenberg antiferromagnet on the square lattice are studied. This fundamental model is a paradigmatic example for long range ordered quantum phases and the spontaneous breaking of a continuous symmetry in condensed matter. The elementary excitations at long wavelengths are well understood by quantized spin waves, i.e., magnons, which are the gapless Goldstone bosons of the broken SU(2) symmetry. However, recent findings reveal an anomalous energy dip at short wavelengths, which cannot be explained by the conventional spin wave theory. The nature of spin waves at short wavelengths remains unclear to this day. A central aim of this thesis is the derivation of an effective magnon description for the square lattice antiferromagnet which is quantitatively valid for all length scales. For this purpose, the method of continuous unitary transformations (CUT) is extended to its use for gapless excitations. CUT methods provide a powerful tool for the derivation of effective quasi-particle descriptions by means of flow equations. However, typical approaches are truncated in real space restricting their scope of application to gapped quantum phases. In order to circumvent these limitations a CUT approach in momentum space is developed. The scaling dimension of operator terms is established as a suitable truncation criterion for gapless quasi-particles. Moreover, a diagrammatic representation of operator terms is introduced which simplifies the derivation and interpretation of flow equations in momentum space. The developed approach is used to transform the Heisenberg Hamiltonian into an effective Hamiltonian which conserves the number of magnons. In addition, effective observables required for the theoretical description of inelastic neutron scattering are derived. The initial magnon operators before the transformation are defined by the Dyson-Maleev representation which leads to a superficially non-hermitian Hamiltonian. As a result, the transformation constitutes a continuous similarity transformation (CST). This formalism enables the formulation of spin wave interactions by means of quartic boson operators. A crucial step of the transformation is the proper renormalization of the spin wave interaction which is achieved by a selfsimilar CST for scaling dimension d = 2. The resulting effective magnon description is consistent at all length scales. In particular, the magnon dispersion exhibits an energy dip at short wavelengths which is in quantitative agreement with recent numerical and experimental data. Further analysis of the spectral properties indicates a strong attraction between spin waves giving rise to a resonance in the longitudinal two-magnon channel. This resonance can be interpreted as the amplitude or Higgs mode of the continuously broken SU(2) symmetry with finite lifetime. The origin of the anomalous dispersion is ascribed to a scattering between the Higgs resonance and single magnon states. The theoretical results are directly compared with experimental data obtained by inelastic neutron scattering and various numerical results which support this view. The quantitative agreement between the effective spin wave approach and numerous numerical and experimental findings shows that magnons are a valid description of the elementary excitations at all length scales. Therefore, this thesis provides a final settlement of the question about the nature of the elementary excitations in the square lattice quantum antiferromagnet.