We comment on a recent paper by Kennedy (J. Phys. A: Math. Gen. 25 (1992) 2809) in which a systematic search for integrable spin-S su(2)-invariant quantum chains for S<or=6 revealed four spin-S families of integrable chains along with an additional integrable chain at S=3. We identify these su(2)-invariant chains with known G-invariant R-matrices, where G is a simple Lie algebra, and give arguments that Kennedy's results may well constitute the complete classification of integrable spin-S su(2)-invariant chains.
In this thesis we theoretically investigate the close relation between spin chains and Yangians. First, we investigate the Haldane-Shastry model, which symmetry algebra is a representation of the Yangian. We show this in full detail for the spin-1/2 model. We also give an argument why a spin-1 chain with Yangian symmetry cannot exist. \nAfter that, we turn to integrable spin chains which are constructed using the quantum inverse scattering method. Using this method, an integrable spin chain can be constructed from each so called R-matrix, which is a solution of the Yang-Baxter equation. Our main result is the construction of an R-matrix that is invariant under the adjoint representation of SU(n), which is found using an intertwining operator for the corresponding Yangian representation. The integrable\nspin chain that corresponds to this R-matrix is non-Hermitian, making its physical interpretation unclear. These results have been published recently as a preprint under arXiv:1606.02516. \nFinally, integrable spin chains with defects are studied. We present the Hamiltonian for the Heisenberg model and the Babujian-Takhtajan model with defects, such that these models keep there integrability while staying Lie algebra invariant.
Abstract We introduce a new class of open, translationally invariant spin chains with long-range interactions depending on both spin permutation and (polarized) spin reversal operators, which includes the Haldane-Shastry chain as a particular degenerate case. The new class is characterized by the fact that the Hamiltonian is invariant under “twisted” translations, combining an ordinary translation with a spin flip at one end of the chain. It includes a remarkable model with elliptic spin-spin interactions, smoothly interpolating between the XXX Heisenberg model with anti-periodic boundary conditions and a new open chain with sites uniformly spaced on a half-circle and interactions inversely proportional to the square of the distance between the spins. We are able to compute in closed form the partition function of the latter chain, thereby obtaining a complete description of its spectrum in terms of a pair of independent su(1|1) and su(m/2) motifs when the number m of internal degrees of freedom is even. This implies that the even m model is invariant under the direct sum of the Yangians Y (gl(1|1)) and Y (gl(0|m/2)). We also analyze several statistical properties of the new chain’s spectrum. In particular, we show that it is highly degenerate, which strongly suggests the existence of an underlying (twisted) Yangian symmetry also for odd m.
We construct an open quantum spin chain from the “twisted” [Formula: see text]R matrix in the fundamental representation which has the quantum algebra symmetry Uq[ su (2)]. This anisotropic spin-1 chain is different from the Uq[ su (2)]-invariant chain constructed from the “untwisted” [Formula: see text] spin-1 R matrix (namely, the spin-1 XXZ chain of Fateev-Zamolodchikov with boundary terms) but, nevertheless, is also completely integrable. We discuss the general case of an R matrix of the type g(k), where k∈{1, 2, 3}, and g is any simple Lie algebra.
To begin with, we introduce several exact models for SU(3) spin chains: First is a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation
3 of SU(3) if the original spins of the model transform under representation 3. Second is a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement and, hence, a Haldane gap in the excitation spectrum. We generalize some of our models to
SU(n). Finally, we use the emerging rules for the construction of valence bond solid states to argue that models of antiferromagnetic chains of
SU(n) spins, in general, possess a Haldane gap if the spins transform under a representation corresponding to a Young tableau consisting of a number of boxes $\ensuremath{\lambda}$ which is divisible by
n. If $\ensuremath{\lambda}$ and
n have no common divisor, the spin chain will support deconfined spinons and not exhibit a Haldane gap. If $\ensuremath{\lambda}$ and
n have a common divisor different from
n, it will depend on the specifics of the model including the range of the interaction.
We use extensive density-matrix renormalization-group calculations to show that a classification of
SU(n) spin chains with regard to the existence of spinon confinement and hence a Haldane gap obtained previously for valence bond solid models applies to
SU(n) Heisenberg chains as well. In particular, we observe spinon confinement due to a next-nearest-neighbor interaction in the SU(4) representation 10 chain.