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Physical review. B, Condensed matter

Dynamic structure factor of the spin- 12 transverse Ising chain

Oleg Derzhko, Taras Krokhmalskii

Published Nov 1, 1997• 46 citations
Publisher

Abstract

The dynamic structure factor of the spin-$\frac{1}{2}$ transverse Ising chain is obtained by means of a numerical approach suggested earlier [O. Derzhko and T. Krokhmalskii, Ferroelectrics 192, 21 (1997)]. Frequency shapes of the dynamic structure factor at various values of the wave vector, transverse field, and temperature are displayed and discussed.

DOI
10.1103/physrevb.56.11659

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The dynamic structure factor of the spin-12\frac{1}{2}21​ transverse Ising chain is obtained by means of a numerical approach suggested earlier [O. Derzhko and T. Krokhmalskii, Ferroelectrics 192, 21 (1997)]. Frequency shapes of the dynamic structure factor at various values of the wave vector, transverse field, and temperature are displayed and discussed.
The transverse component of the dynamic form factor Sx(k,ω)\mathscr{S}^{x}(k, \omega )Sx(k,ω) of the Ising-like linear chain at various temperatures is studied by the perturbation theory about the transverse part of the Hamiltonian. The dynamic form factor Sx(k,ω)\mathscr{S}^{x}(k, \omega )Sx(k,ω) for both the ferromagnetic and antiferromagnetic couplings has the three-peak structure for k =0 at finite temperatures, and this structure persists even for any wave vector k and at any high temperature. Many interesting characters in Sx(k,ω)\mathscr{S}^{x}(k, \omega )Sx(k,ω) are obtained. The k -dependence of width of the absorption peak is remarkable in the antiferromagnetic case. The absorption peak has a finite width even at absolute zero except for k = π/2 in the antiferromagnetic case, while the shape of the absorption peak is reduced to a infinitely sharp one as the temperature is decreased to absolute zero in the ferromagnetic case.
Abstract The Ising model in a transverse field is treated by means of Green function technique to calculate the dynamic structure factor above and below Tc. The soft mode is overdamped in the disordered phase and near Tc, while it is nearly undamped at low temperatures. The coupling between the transverse soft mode and the relaxing longitudinal mode produces a central mode in the dynamic structure factor. The temperature dependence of the dynamic structure factor for zero wave vector is calculated numerically.
The paper presents a numerical approach for the calculation of thermodynamics and spin correlations for spin-½ XY chains. To illustrate the approach we performed exact finite-chain calculations of all time-dependent two-spin correlation functions for the spin-½ Ising chain in the transverse field. The computed correlation functions permit to evaluate the frequency-dependent susceptibilities and structure factors for this model. The obtained results are discussed in detail. They may have an application for the interpretation of dynamic experiments on quasi-one-dimensional crystals that can be treated within the frames of the spin-½ transverse Ising model.
The spin-12\frac{1}{2}21​ XXZ chain with easy-plane anisotropy in a transverse field describes well the thermodynamic properties of the material Cs2CoCl4{\mathrm{Cs}}_{2}{\mathrm{CoCl}}_{4}Cs2​CoCl4​ in a wide range of temperatures and fields including the region close to the spin-flop Ising quantum phase transition. For a comparison with prospective inelastic neutron scattering experiments on this compound, we present results of an extensive numerical study of its dynamic structure factor ${\mathcal{S}}^{\ensuremath{\alpha}\ensuremath{\beta}}(k,\ensuremath{\omega})$ using matrix-product-state (MPS) techniques. Close to criticality, the dynamic part of the correlator Sxx{\mathcal{S}}^{xx}Sxx longitudinal to the applied field is incoherent and possesses a small total weight as the ground state is already close to saturation. The transverse correlator Szz{\mathcal{S}}^{zz}Szz, on the other hand, is dominated by a coherent single-particle excitation with additional spectral weight at higher energies that we tentatively attribute to a repulsively bound pair of particles. With increasing temperature, the latter quickly fades and spectral weight instead accumulates close to zero wave vector just above the single-particle energy. On a technical level, we compare the numerical efficiency of real-time evolution to an MPS-based Chebyshev expansion in the present context, finding that both methods yield results of similar quality at comparable numerical costs.
We have considered a numerical scheme for the calculation of the equilibrium properties of spin-1/2 XY chains. Within its frames it is necessary to solve in the last resort only the 2N\times 2N eigenvalue and eigenvector problem but not the 2^N\times 2^N one as for an arbitrary system consisting of N spins 1/2. To illustrate the approach we have presented some new results. Namely, the xx dynamic structure factor for the Ising model in transverse field, the density of states for the isotropic chain with random intersite couplings and transverse fields that linearly depend on the surrounding couplings, and the zz dynamic structure factor for the Ising model in random transverse field. The results obtained are hopped to be useful for an interpretation of observable data for one-dimensional spin-1/2 XY substances.