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arXiv:cond-mat/9705207 · Journal of the Physical Society of Japan

Spin Dynamics of the One-DimensionalJ-J' Model and Spin-Peierls Transition in CuGeO3

Hisatoshi Yokoyama, Yasuhiro Saiga

Published Nov 15, 1997• 38 citations
arXivHTMLPDF

Abstract

Spin dynamics as well as static properties of the one-dimensional J-J' model (S=1/2, J>0 and 0\le α=J'/J\le 0.5) are studied by the exact diagonalization and the recursion method of finite systems up to 26 sites. Especially, the dynamical structure factor S(q,ω) is investigated carefully for various values of α. As αincreases beyond the gapless-gapful critical value α_c=0.2411, there appear features definitely different from the Heisenberg model but the same with the Majumdar-Ghosh model. Some of these features depend only on the value of αand not on δ: a parameter introduced for the coupling alternation. By comparing these results with a recent inelastic neutron scattering spectrum of an inorganic spin-Peierls compound CuGeO_3 [M. Arai et al.: Phys. Rev. Lett. 77 (1996) 3649], it is found that the frustration by J' in CuGeO_3 is unexpectedly strong (α=0.4-0.45), and at least αmust be larger than α_c to some extent. The value of J is evaluated at \sim 180K consistent with other estimations. The coupling alternation is extremely small. This large frustration is a primary origin of the various anomalous properties CuGeO_3 possesses. For comparison we refer also to α'-NaV_2O_5.

DOI
10.1143/jpsj.66.3617

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Spin dynamics as well as static properties of the one-dimensional J-J' model (S=1/2, J>0 and 0\le α=J'/J\le 0.5) are studied by the exact diagonalization and the recursion method of finite systems up to 26 sites. Especially, the dynamical structure factor S(q,ω) is investigated carefully for various values of α. As αincreases beyond the gapless-gapful critical value α_c=0.2411, there appear features definitely different from the Heisenberg model but the same with the Majumdar-Ghosh model. Some of these features depend only on the value of αand not on δ: a parameter introduced for the coupling alternation. By comparing these results with a recent inelastic neutron scattering spectrum of an inorganic spin-Peierls compound CuGeO_3 [M. Arai et al.: Phys. Rev. Lett. 77 (1996) 3649], it is found that the frustration by J' in CuGeO_3 is unexpectedly strong (α=0.4-0.45), and at least αmust be larger than α_c to some extent. The value of J is evaluated at \sim 180K consistent with other estimations. The coupling alternation is extremely small. This large frustration is a primary origin of the various anomalous properties CuGeO_3 possesses. For comparison we refer also to α'-NaV_2O_5.
Applying numerical techniques to a model recently proposed for the one-dimensional spin-Peierls compound CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​, we calculate dynamical properties that can be directly compared with inelastic-neutron-scattering data and angle-resolved photoemission (ARPES) experiments. The momentum and energy dependence of the dynamical structure factor S(q,\ensuremath{\omega}) are discussed, as well as the static structure factor S(q). The latter is shown to be specially useful to estimate the strength of the spin interaction beyond nearest neighbors. The spectral function A(q,\ensuremath{\omega}) calculated from the one-particle Green's function at half-filling is shown at several values of the hole-hopping amplitude t. The results have some unique features characteristic of one-dimensional systems including small weight near the Fermi energy. The presence of ``shadow bands'' induced by short distance antiferromagnetic correlations is predicted to appear in ARPES experiments for CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​ and also for Sr2{\mathrm{Sr}}_{2}Sr2​CuO3{\mathrm{CuO}}_{3}CuO3​.
Dynamical properties of the inorganic spin-Peierls compound ${\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremath{-}{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}$ are investigated using a one-dimensional dimerized Heisenberg model. By exact diagonalizations of chains with up to 28 sites, supplemented by a finite-size scaling analysis, the dimerization parameter \ensuremath{\delta} is determined by requiring that the model reproduces the experimentally observed spin gap \ensuremath{\Delta}. The dynamical and static spin structure factors are calculated. As for CuGeO3,{\mathrm{CuGeO}}_{3},CuGeO3​, the existence of a low-energy magnon branch separated from the continuum is predicted. The present calculations also suggest that a large magnetic Raman scattering intensity should appear above an energy threshold of 1.9\ensuremath{\Delta}. The predicted photoemission spectrum is qualitatively similar to results for an undimerized chain due to the presence of sizable short-range antiferromagnetic correlations.
In a magnetic substance the gap in the Raman spectrum, ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{R}}$, and the neutron-scattering gap, ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{N}}$, are related by ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{R}}$/${\mathrm{\ensuremath{\Delta}}}_{\mathrm{N}}$\ensuremath{\approx}2 if the the magnetic excitations (magnons) are only weakly interacting. But for CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​ the experimentally observed ratio is of the order ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{R}}$/${\mathrm{\ensuremath{\Delta}}}_{\mathrm{N}}$\ensuremath{\sim}1.49--1.78, indicating attractive magnon-magnon interactions in the quasi-one-dimensional spin-Peierls compound CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​. We present numerical estimates of ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{R}}$/${\mathrm{\ensuremath{\Delta}}}_{\mathrm{N}}$ from exact diagonalization studies for finite chains and find agreement with experiment for intermediate values of the frustration parameter \ensuremath{\alpha}. An analysis of the numerical Raman intensity leads us to postulate a continuum of two-magnon bound states in the spin-Peierls phase. We discuss in detail the numerical method used, the dependence of the results on the model parameters, and a matrix-element effect due to the dimerization of the Raman operator in the spin-Peierls phase.
Summary: The interplay of charge, spin and lattice degrees of freedom is studied for quasi-one-dimensional electron and spin systems coupled to quantum phonons. Special emphasis is put on the influence of the lattice dynamics on the Peierls transition. Using exact diagonalization techniques the ground-state and spectral properties of the Holstein model of spinless fermions and of a frustrated Heisenberg model with magneto-elastic coupling are analyzed on finite chains. In the non-adiabatic regime a (T = 0) quantum phase transition from a gapless Luttinger-liquid/spin-fluid state to a gapped dimerized phase occurs at a nonzero critical value of the electron/spin-phonon interaction. To study the nature of the spin-Peierls transition at finite temperatures for the infinite system, an alternative Green’s function approach is applied to the magnetostrictive XY model. With increasing phonon frequency the structure factor shows a remarkable crossover from soft-mode to central-peak behaviour. The results are discussed in relation to recent experiments on CuGeO3. 1
Various forms of antiferromagnetic exchange interaction among quantized spins in one-dimensional and quasi-one-dimensional lattices are examined. Primary results are reported in two real compounds, the sodium vanadate NaV2O5 and the copper germanate CuGeO3, and in a class of compounds of real materials modeled as dimerized antiferromagnetically coupled spin-1/2 chains susceptible to spin-1 impurity doping and that include such examples as the strontium cuprate Sr14Cu24O41 and vanadium pyrophosphate (VO)2P2O7. Sodium vanadate, NaV2O5, a quarter-filled two-leg ladder compound that was originally thought to be composed of magnetic legs and nonmagnetic legs was subsequently shown to be a charge ordered system below room temperature. Initial models of the material as isolated antiferromagnetically coupled spin-1/2 chains can be mapped to the subsequently resolved magnetic system of the charge ordered state with the spins on molecular orbitals of V-O-V rungs antiferromagentically coupled between neighboring rungs either on the same ladder or on neighboring ladders. Comparing the model's dynamic structure factor to inelastic neutron scattering data could distinguish between the proposed spin coupling schemes and consequently reveal the details of the interaction between the lattice and the "zig-zag" charge density as well as the magnetic stabilization of the low temperature phase. Copper germanate, the first and only known inorganic spin-Peierls compound, exhibits a strong temperature dependent exchange coupling pattern among the spin-1/2 Cu2+ chains. Above the spin-Peierls transition temperature an unmodulated antiferromagnetic nearest neighbor exchange interaction J1 competes with an antiferromagnetic next nearest neighbor interaction J2. The next nearest neighbor exchange represents an effective coupling equivalent to the net effect of all longer range exchange interactions. This frustration to the nearest neighbor spin exchange produces a distinctive magnetic susceptibility χ (T) much different from the Bonner-Fisher susceptibility of the spin-1/2 Heisenberg chain with nearest neighbor antiferromagnetic exchange only. The ratio of the second to first nearest neighbor couplings α = J2/J1 ≈ 0.36 is sufficient to open a spontaneous gap in the spin-wave excitation spectrum at low temperatures. Below TSP = 14K, the dynamical structure factor is used to fit the dimerization δ and the exchange interactions J1 and J2 to the inelastic neutron scattering data of CuGeO3 at T = 10K. It is found that both δ and α increase significantly at lower temperatures, relative to the values obtained in the high temperature phase and at the onset of dimeration at 14 K. Static structure factor calculations how Scattering inconsistent with the δ−J1−J2 model and can be attributed primarily to the phonon degrees of freedom but possibly also to the couplings between chains in the b and a directions of the crystal lattice. Structurally dimerized compounds of antiferromagnetic spin-1/2 chains possess dynamic structure factor and magnetic susceptibility features that are very sensitive to doping with magnetic and nonmagnetic impurities. It is shown that the effects of spin-1 impurities are very similar to those of nonmagnetic (S = 0) impurities if the coupling between impurity and native spin is about the same magnitude as or larger than between native S = 1/2 spins. The microscopic origins of the similarity can be appreciated with the consideration that neighboring spins to the impurity find it more energetically favorable to couple to the impurity than the other spins of the lattice. Thus the spin-1 impurity and its neighboring S = 1/2 spins decouple from the lattice, creating a nonmagnetic break in the chain. The spins that now observe a nonmagnetic cluster on one side can couple to the S = 1/2 spins on the other side with a higher AF correlation than in the pure compound since they do not have to have a fluctuating dimer resonating between two neighbors. Such enhancements to the local correlations can stabilize the global AF order in one dimensional compounds as shown in theoretical and experimental studies of CuGeO3. ED calculations of the dynamic structure factor show that S(q, ω) developes states with ω = 0 in the gap at q = π. Magnetic susceptibilites in the presense of spin-1 impurities were obtained for 80 site chains by QMC simulations and for 16 site chains by ED calculations. An impurity concentration-dependent second peak in the low temperature region reveals an increasing second maximum with T as observed in doped copper germanates and suggests that the results of the study are applicable even to phonon mediated dimerization. This is not immediately apparent because spin-phonon coupling could distort the dimerization patterns. However, experimental and theoretical results show that elastic coupling between chains can stabilize the dimerization pattern due to the impurities and the results will still apply.