An exact diagonalization of the XXX one-dimensional Heisenberg model a magnet with the single-node 1/2 reveals a specific degeneracy in the sector of r = 3 reversed spins at the centre k = 0 of the Brillouin zone. The degeneracy manifests itself by the second power (i.e. the double degeneracy) of a factor of the characteristic polynomial of the related secular matrix. The factor, treated as a polynomial in the variable x being an eigenenergy, is indecomposable over the field Q of rationals, and its roots correspond to appropriate rigged configurations of strings. The degeneracy is associated with the fact that quasimomenta of the two-string and the one-string are conserved independently each of the other, which is a particular consequence of integrability of the system.
An exact diagonalization of the XXX one-dimensional Heisenberg model a magnet with the single-node 1/2 reveals a specific degeneracy in the sector of r = 3 reversed spins at the centre k = 0 of the Brillouin zone. The degeneracy manifests itself by the second power (i.e. the double degeneracy) of a factor of the characteristic polynomial of the related secular matrix. The factor, treated as a polynomial in the variable x being an eigenenergy, is indecomposable over the field Q of rationals, and its roots correspond to appropriate rigged configurations of strings. The degeneracy is associated with the fact that quasimomenta of the two-string and the one-string are conserved independently each of the other, which is a particular consequence of integrability of the system.
We perform an exact diagonalization of the Heisenberg Hamiltonian for the magnetic ring consisting of N=6 nodes, compare the result with the Bethe Ansatz (BA) approach and verify the hypothesis of strings. We find and explain the failure of BA method in a single case of the highest weight eigenvector at the boundary of the Brillouin zone for the case of two spin deviations. The explanation involves the orbit structure of the corresponding set of magnetic configurations. We demonstrate that the finite size effects for N=6 only slightly deform the shape of asymptotic strings.
XXX Heisenberg s1{2 model has been examined in detail during last decades, however, recently one may find some new insights into that issue.Among several approaches describing the eigenproblem for the finite case, a close look into the structure of Bethe equations (BE) for the two-magnon sector case seems to be particularly interesting.BE enable us to evaluate parameters labeling eigenstates of a magnet, however to find appropriate sets of winding numbers, which parametrize BE, one has to apply the Inverse Bethe Ansatz method.On the other hand, one may choose a different -combinatoric approach -which also parametrizes Bethe eigenstates, with the use of rigging numbers describing string configurations.We present an idea of comparison of the concepts mentioned above for the particular case of two-spin deviations sector.
We study the field dependence of the antiferromagnetic spin-21 Heisenberg model on the square lattice by means of exact diagonalizations. In the first part, we calculate the spin-wave velocity c, the spin stiffness ${\ensuremath{\rho}}_{s}$, and the magnetic susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}$, and thus determine the microscopic parameters of the low-energy long-wavelength description. In the second part, we present a comprehensive study of dynamical spin-correlation functions for magnetic fields ranging from zero up to saturation. We find that, at low fields, magnons are well defined in the whole Brillouin zone but the dispersion is substantially modified by quantum fluctuations compared to the classical spectrum. At higher fields, decay channels open and magnons become unstable with respect to multimagnon scattering. Our results directly apply to inelastic neutron-scattering experiments.
We compare the excitation spectra in the presence of a magnetic field of a number of integrable (exactly solvable) and nonintegrable quantum spin chains of various spin value s. The archetypal Bethe-ansatz integrable model is the s= 1/2 Heisenberg antiferromagnet (HB AFM). The excitation spectra are characterized by a soft mode which tracks across the Brillouin zone as the field increases to its saturation value. A class of Bethe-ansatz integrable models with SU(2) symmetry and the general spin s display excitation spectra qualitatively similar to the spin- 1/2 model above, for all s. A second class of Bethe-ansatz integrable models has SU(n) symmetry, where n=2s+1. Like the SU(2) integrable chains, these models have gapless excitation spectra, but the basic Brillouin zone changes from k=±2π/(2s+1)a. Studies show that periodicity of the SU(3) member of the class changes (increases) as the field increases to saturation. For both classes of integrable models, there is a single type of excitation pattern which is generically similar for all s. In the case of the other models, on the other hand, numerical studies show that the excitations divide into at least two distinct classes. In the case of the s=1 HB AFM, at high fields (corresponding to SzT=N,N−1, . . .,N/2) the excitations map approximately onto the complete set of excitations for s= 1/2 , whereas at low fields (SzT=N/2,N/2−1,. . .,0) the excitations have notable classical character. In the case of the s=1 model with pure biquadratic exchange, one set of excitations, corresponding to SzT even (SzT=N,N−2,. . .,2,0), again shows an approximate mapping to the complete excitation set for s= 1/2 . The second class of excitations, corresponding to SzT odd, are very different. They are symmetric about k=±π/2a for all SzT, i.e., correspond to a basic Brillouin zone of ±π/2a.
Bethe Ansatz is the famous method of determination of eigenstates and eigenenergies for a wide range of quantum problems, e.g. for the Heisenberg XXX s = 1/2 model.The Bethe equations applied to solve the problem of N nodes and r overturned spins on a magnetic chain are labeled by sets of winding numbers {ni}, however the condition for admissible sets give an overcomplete number of results.On the other hand, combinatorial objects, so called paths, give the exact number of eigenvectors for the problem described by (N, r) values.The paper presents the method of determining the set of winding numbers from the appropriate path for the sector of r = 2 spin deviations.
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.