Quantum spin chains of higher spin are interesting in the case of both ferromagnets and antiferromagnets. They comprise a wide range of model types—Bethe–Ansatz integrable, nonintegrable, and with limited integrability. It is not well appreciated that the two-magnon (two-body) problems is always solvable, by Bethe–Ansatz and Green’s-function techniques, regardless of the integrability character of the entire model, with interesting implications in the context of solitons and bound solitons. Chubukov and Kveschenko have suggested using the two-magnon problem in a test for integrability and claim to find additional classes of integrable spin chains for s=1 and s= (3)/(2) . Here we investigate this situation.
Quantum spin chains of higher spin are interesting in the case of both ferromagnets and antiferromagnets. They comprise a wide range of model types—Bethe–Ansatz integrable, nonintegrable, and with limited integrability. It is not well appreciated that the two-magnon (two-body) problems is always solvable, by Bethe–Ansatz and Green’s-function techniques, regardless of the integrability character of the entire model, with interesting implications in the context of solitons and bound solitons. Chubukov and Kveschenko have suggested using the two-magnon problem in a test for integrability and claim to find additional classes of integrable spin chains for s=1 and s= (3)/(2) . Here we investigate this situation.
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.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.
This study concerns the concept of nonintegrability in quantum many-body systems, which is related to the important and unresolved problem of quantum chaos. Our findings strongly indicate that nonintegrability affects the reliability of many approximation techniques which have proved to be successful in the study of integrable models. This report is based on finite-size studies of the low-lying spectral excitations of both integrable and nonintegrable 1D quantum spin models. In integrable cases, the characteristic excitation pattern of the infinite system is apparent even in relatively short chains. This is generally not the case in nonintegrable systems where we observe several classes of excitations with qualitatively different character. In some situations, the nature of the lowest-lying excitations actually changes with increasing system size, which makes finite-size studies very vulnerable to misleading conclusions if care is not taken.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.
An anisotropic integrable spin chain, consisting of spins s = 1 and , is investigated [1]. It is characterized by two real parameters and , the coupling constants of the spin interactions. For the case and the ground-state configuration is obtained by means of thermodynamic Bethe ansatz. Furthermore, the low excitations are calculated. It turns out that apart from free magnon states being the holes in the ground-state rapidity distribution, there exist bound states given by special string solutions of Bethe ansatz equations (BAE) in analogy to [13]. The dispersion law of these excitations is calculated numerically.
Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...