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arXiv:1606.03294 · Physical review. B./Physical review. B

Dynamic structure factor of the spin- 12 XXZ chain in a transverse field

Benedikt Bruognolo, Andreas Weichselbaum, Jan von Delft, Markus Garst

Published Aug 22, 2016• 18 citations
arXivHTMLPDF

Abstract

The spin-$\frac{1}{2}$ XXZ chain with easy-plane anisotropy in a transverse field describes well the thermodynamic properties of the material ${\mathrm{Cs}}_{2}{\mathrm{CoCl}}_{4}$ 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 ${\mathcal{S}}^{xx}$ 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 ${\mathcal{S}}^{zz}$, 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.

DOI
10.1103/physrevb.94.085136

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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 S\ensuremathα\ensuremathβ(k,\ensuremathω){\mathcal{S}}^{\ensuremath{\alpha}\ensuremath{\beta}}(k,\ensuremath{\omega})S\ensuremathα\ensuremathβ(k,\ensuremathω) 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.
Improving matrix-product state techniques based on the purification of the density matrix, we are able to accurately calculate the finite-temperature dynamic response of the infinite spin-1 XXZ\text{XXZ}XXZ chain with single-ion anisotropy in the Haldane, large-DDD, and antiferromagnetic phases. Distinct thermally activated scattering processes make a significant contribution to the spectral weight in all cases. In the Haldane phase, intraband magnon scattering is prominent, and the on-site anisotropy causes the magnon to split into singlet and doublet branches. In the large-DDD phase response, the intraband signal is separated from an exciton-antiexciton continuum. In the antiferromagnetic phase, holons are the lowest-lying excitations, with a gap that closes at the transition to the Haldane state. At finite temperatures, scattering between domain-wall excitations becomes especially important and strongly enhances the spectral weight for momentum transfer $\ensuremath{\pi}$.
We consider the spin-1/2 Heisenberg XXZXXZXXZ chain in the regime of large Ising-type anisotropy $\ensuremath{\Delta}$. By a combination of duality and Jordan-Wigner transformations we derive a mapping to weakly interacting spinless fermions, which represent domain walls between the two degenerate ground states. We develop a perturbative expansion in $1/\ensuremath{\Delta}$ for the transverse dynamical spin structure factor at finite temperatures and in an applied transverse magnetic field. We present a unified description for both the low-energy temperature-activated response and the temperature evolution of the T=0T=0T=0 two-spinon continuum. We find that the two-spinon continuum narrows in energy with increasing temperature. At the same time spectral weight is transferred from the two-spinon continuum to the low-energy intraband scattering continuum, which is strongly peaked around the position of the (single) spinon dispersion (``Villain mode'').
We present a study of the dynamic structure factor of the antiferromagnetic spin-12\frac{1}{2}21​ Heisenberg chain at finite temperatures and finite magnetic fields. Using quantum Monte Carlo based on the stochastic series expansion and maximum entropy methods, we evaluate the longitudinal and the transverse dynamic structure factors from vanishing magnetic fields up to and above the threshold Bc{B}_{c}Bc​ for ferromagnetic saturation, as well as for high and for intermediate temperatures. We study the field-induced redistribution of spectral weight contrasting longitudinal versus transverse excitations. At finite fields below saturation incommensurate low-energy modes are found consistent with zero-temperature Bethe ansatz. The crossover between the field-induced ferromagnet above Bc{B}_{c}Bc​ and the Luttinger liquid below Bc{B}_{c}Bc​ is analyzed in terms of the transverse spin dynamics. Evaluating sum rules we assess the quality of the analytic continuation and demonstrate excellent consistency of the maximum entropy results.
Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.
Dynamical correlation functions contain important physical information on correlated spin models. Here a dynamical theory suitable to the Δ=1 isotropic spin-1/2 Heisenberg chain in a longitudinal magnetic field is extended to anisotropy Δ>1. The aim of this paper is the study of the (k,ω)-plane line shape of the spin dynamical structure factor components S+−(k,ω), S−+(k,ω), and Szz(k,ω) of the Δ>1 spin-1/2 Heisenberg-Ising chain in a longitudinal magnetic field near their (k,ω)-plane sharp peaks. However, the extension of the theory to anisotropy Δ>1 requires as a first step the clarification of the nature of a specific type of elementary magnetic configurations in terms of physical spins 1/2. To reach that goal, that the spin SU(2) symmetry of the isotropic Δ=1 point is in the case of anisotropy Δ>1 replaced by a continuous quantum group deformed q-spin SUq(2) symmetry plays a key role. For Δ>1, spin projection Sz remains a good quantum number whereas spin S is not, being replaced by the q-spin Sq in the eigenvalue of the Casimir generator of the continuous SUq(2) symmetry. Based on the isomorphism between the irreducible representations of the Δ=1 spin SU(2) symmetry and Δ>1 continuous SUq(2) symmetry and on their relation to the occupancy configurations of the Bethe-ansatz quantum numbers one finds that the elementary magnetic configurations under study are q-spin neutral. This determines the form of S matrices on which the extended dynamical theory relies. They are found in this paper to describe the scattering of n-particles whose relation to configurations of physical spins 1/2 is established. Specifically, their internal degrees of freedom refer to unbound q-spin singlet pairs of physical spins 1/2 described by n=1 real single Bethe rapidities and to n=2,3,... bound such q-spin singlet pairs described by Bethe n-strings for n>1. There is a relationship between the negativity and the length of the momentum k interval of the k dependent exponents that control the power-law line shape of the spin dynamical structure factor components near the lower threshold of a given (k,ω)-plane continuum and the amount of spectral weight over the latter. Using such a relationship, one finds that in the thermodynamic limit the significant spectral weight contributions from Bethe n-strings at a finite longitudinal magnetic field refer to specific two-parametric (k,ω)-plane gapped continua in the spectra of the spin dynamical structure factor components S+−(k,ω) and Szz(k,ω). In contrast to the Δ=1 isotropic chain, excited energy eigenstates including up to n=3 Bethe n-strings lead to finite spectral-weight contributions to S+−(k,ω). Most spectral weight stems though from excited energy eigenstates whose q-spin singlet Sz=Sq=0 pairs of physical spins 1/2 are all unbound. It is associated with (k,ω)-plane continua in the spectra of the spin dynamical structure factor components that are gapless at some specific momentum values. We derive analytical expressions for the line shapes of S+−(k,ω), S−+(k,ω), and Szz(k,ω) valid in the vicinity of (k,ω)-plane lines of sharp peaks. Those are mostly located at and just above lower thresholds of (k,ω)-plane continua associated with both states with only unbound q-spin singlet pairs and states populated by such pairs and a single n=2 or n=3 n-particle. Our results provide physically interesting and important information on the microscopic processes that determine the dynamical properties of the non-perturbative spin-1/2 Heisenberg-Ising chain in a longitudinal magnetic field.
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