We analyze the absorption of microwaves by the Heisenberg-Ising chain combining exact calculations, based on the integrability of the model, with numerical calculations. Within linear response theory, the absorbed intensity is determined by the imaginary part of the dynamical susceptibility. The moments of the normalized intensity can be used to define the shift of the resonance frequency induced by the interactions and the linewidth independently of the shape of the spectral line. These moments can be calculated exactly as functions of temperature and strength of an external magnetic field, as long as the field is directed along the symmetry axis of the chain. This allows us to discuss the linewidth and the resonance shift for a given magnetic field in the full range of possible anisotropy parameters. For the interpretation of these data, we need a qualitative knowledge of the line shape, which we obtain from fully numerical calculations for finite chains. Exact analytical results on the line shape are out of reach of current theories. From our numerical work, we could extract, however, an empirical parameter-free model of the line shape at high temperatures, which is extremely accurate over a wide range of anisotropy parameters, and is exact at the free-fermion and isotropic points. Another prediction of the line shape can be made in the zero-temperature and zero magnetic field limits, where the sufficiently anisotropic model shows strong absorption. For anisotropy parameters in the massive phase, we derive the exact two-spinon contribution to the spectral line. From the intensity sum rule, it can be estimated that this contribution accounts for more than 80$%$ of the spectral weight if the anisotropy parameter is moderately above its value at the isotropic point.
We analyze the absorption of microwaves by the Heisenberg-Ising chain combining exact calculations, based on the integrability of the model, with numerical calculations. Within linear response theory, the absorbed intensity is determined by the imaginary part of the dynamical susceptibility. The moments of the normalized intensity can be used to define the shift of the resonance frequency induced by the interactions and the linewidth independently of the shape of the spectral line. These moments can be calculated exactly as functions of temperature and strength of an external magnetic field, as long as the field is directed along the symmetry axis of the chain. This allows us to discuss the linewidth and the resonance shift for a given magnetic field in the full range of possible anisotropy parameters. For the interpretation of these data, we need a qualitative knowledge of the line shape, which we obtain from fully numerical calculations for finite chains. Exact analytical results on the line shape are out of reach of current theories. From our numerical work, we could extract, however, an empirical parameter-free model of the line shape at high temperatures, which is extremely accurate over a wide range of anisotropy parameters, and is exact at the free-fermion and isotropic points. Another prediction of the line shape can be made in the zero-temperature and zero magnetic field limits, where the sufficiently anisotropic model shows strong absorption. For anisotropy parameters in the massive phase, we derive the exact two-spinon contribution to the spectral line. From the intensity sum rule, it can be estimated that this contribution accounts for more than 80 of the spectral weight if the anisotropy parameter is moderately above its value at the isotropic point.
The exact expression derived by Bougourzi, Couture, and Kacir for the two-spinon contribution to the dynamic spin structure factor S(q,\ensuremath{\omega}) of the one-dimensional s=1/2 Heisenberg antiferromagnet at T=0 is evaluated for direct comparison with finite-chain transition rates (N\ensuremath{\leqslant}28) and an approximate analytical result previously inferred from finite-N data, sum rules, and Bethe ansatz calculations. The two-spinon excitations account for 72.89% of the total intensity in S(q,\ensuremath{\omega}). The singularity structure of the exact result is determined analytically and its spectral-weight distribution evaluated numerically over the entire range of the two-spinon continuum. The leading singularities of the frequency-dependent spin autocorrelation function, static spin structure factor, and q dependent susceptibility are determined via sum rules. The impact of the non-two-spinon excitations on the integrated intensity, the susceptibility, the frequency moments, and the Euclidian time representation of S(q,\ensuremath{\omega}) is studied on the basis of finite-size data.
The spin fluctuations parallel to the external magnetic field in the ground state of the one-dimensional (1D) s=21 Heisenberg antiferromagnet are dominated by a two-parameter set of collective excitations. In a cyclic chain of N sites and magnetization 0<Mz<N/2, the ground state, which contains 2Mz spinons, is reconfigured as the physical vacuum for a different species of quasiparticles, identifiable in the framework of the coordinate Bethe ansatz by characteristic configurations of Bethe quantum numbers. The dynamically dominant excitations are found to be scattering states of two such quasiparticles. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty},$ these collective excitations form a continuum in $(q,\ensuremath{\omega})$ space with an incommensurate soft mode. Their matrix elements in the dynamic spin structure factor ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ are calculated directly from the Bethe wave functions for finite N. The resulting line- shape predictions for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ complement the exact results previously derived via algebraic analysis for the exact two-spinon part of ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ in the zero-field limit. They are relevant for neutron-scattering experiments on quasi-1D antiferromagnetic compounds in a strong magnetic field.
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.
We compute the exact four-spinon contribution to the zero-temperature dynamical structure factor of the spin-1/2 Heisenberg isotropic antiferromagnet in zero magnetic field, directly in the thermodynamic limit. We make use of the expressions for matrix elements of local spin operators obtained by Jimbo and Miwa using the quantum affine symmetry of the model, and of their adaptation to the isotropic case by Abada, Bougourzi and Si-Lakhal (correcting some overall factors). The four-spinon contribution to the first frequency moment sum rule at fixed momentum is calculated. This shows, as expected, that most of the remaining correlation weight above the known two-spinon part is carried by four-spinon states. Our results therefore provide an extremely accurate description of the exact structure factor.
The exact two-spinon part of the dynamic spin structure factor ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ for the one-dimensional s=1/2,XXZ model at T=0 in the antiferromagnetically ordered phase is calculated using recent advances in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The two-spinon excitations form a two-parameter continuum consisting of two partly overlapping sheets in $(Q,\ensuremath{\omega})$ space. The spectral threshold has a smooth maximum at the Brillouin zone boundary $(Q=\ensuremath{\pi}/2)$ and a smooth minimum with a gap at the zone center (Q=0). The two-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the two-spinon transition rates, the two regimes $0<~Q<{Q}_{\ensuremath{\kappa}}$ (near the zone center) and ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2$ (near the zone boundary) must be distinguished, where ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}0$ in the Heisenberg limit and ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}\ensuremath{\pi}/2$ in the Ising limit. In the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2,$ the two-spinon transition rates relevant for ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ are finite at the lower boundary of each sheet, decrease monotonically with increasing \ensuremath{\omega}, and approach zero linearly at the upper boundary. The resulting two-spinon part of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime $0<{Q}_{\ensuremath{\kappa}}<~\ensuremath{\pi}/2,$ in contrast, the two-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. In the associated two-spinon line shapes of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega}),$ the linear cusps at the continuum boundaries are replaced by square-root cusps. Existing perturbation studies have been unable to capture the physics of the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2.$ However, their line-shape predictions for the regime $0<~Q<{Q}_{\ensuremath{\kappa}}$ are in good agreement with the exact results if the anisotropy is very strong. For weak anisotropies, the exact line shapes are more asymmetric.
Spin-dynamical calculations on one-dimensional systems have relied heavily on classical ($s=\ensuremath{\infty}$) theories, despite abundant evidence that quantum effects can be extremely important at low temperatures. We present a new approach to the spin dynamics of the one-dimensional isotropic s=21 Heisenberg antiferromagnetic (HB AF) which does not involve the many-body techniques usually employed. It is based on analytic Bethe ansatz calculations of excitation energies and densities of states combined with finite-chain calculations of matrix elements. An important feature of our method is the use of rigorous selection rules and the introduction of new selection rules, which are valid for macroscopic systems in a magnetic field. We show that in zero field the dynamical two-spin correlation function ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q,\ensuremath{\omega})$ at T=0 is governed by a two-parameter continuum of spin-wave-type excitations. In nonzero field, the longitudinal component ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ and the transverse components ${S}_{\mathrm{xx}}(q,\ensuremath{\omega})\ensuremath{\equiv}{S}_{\mathrm{yy}}(q,\ensuremath{\omega})$ behave quite differently because they are dominated by different continua of excitations. The former is characterized by a lowest excitation branch with a zero-frequency mode moving from the zone boundary ($q=\ensuremath{\pi}$) towards the zone center (q=0) as the field increases, whereas the latter is characterized by a lowest branch with a zero frequency mode moving from $q=0 \mathrm{to} \ensuremath{\pi}$ with increasing field. The first part of our work features an approximate analytic expression for ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q,\ensuremath{\omega})$ at zero temperature and in zero field. Although our expression is not rigorous, exact sum rules are violated only by a small amount, and good agreement exists with the few known exact results. Our studies are extended to nonzero temperatures by placing major reliance on exact finite-chain calculations. Our work was stimulated by recent neutron scattering experiments and is oriented towards experimental comparisons. Our result for the s=21 integrated intensity is in much better agreement with neutron scattering data on CuCl2\ifmmode\cdot\else\textperiodcentered\fi{}2N(C5D5) (CPC) than the corresponding semiclassical result. Moreover, the spectral-weight distribution in ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q,\ensuremath{\omega})$ shows increasing asymmetry as $q\ensuremath{\rightarrow}\ensuremath{\pi}$, a quantum effect, again in agreement with more recent neutron scattering data. The second part of our work deals with the effects of an applied magnetic field. We extend the analytic work of Ishimura and Shiba to obtain expressions for the energies and densities of states of the various excitation continua. It is shown that these continua are expected to give rise to multiple structures in the scattering intensity. Our results appear to be in quantitative agreement with preliminary results of a neutron study in CPC in a field of 70 kOe, revealing anomalous scattering intensity peaks. Our results repeatedly demonstrate the inadequacy of classical spin-wave theory for this problem. They call for additional experimental studies on quasi-one-dimensional antiferromagnets to examine other unusual and interesting phenomena predicted by our approach.