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arXiv:0704.3046 · Nuclear Physics B

A novel quasi-exactly solvable spin chain with nearest-neighbors interactions

Alberto Enciso B, Federico Finkel, Artemio González-López, Miguel A. Rodrı́guez

Published Jul 12, 2007• 5 citations
arXivHTMLPDF
DOI
10.1016/j.nuclphysb.2007.07.001

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Most of the existing dynamical studies in one dimension on magnetic insulators have considered the simplest spin models with nearest-neighbor interactions. In real systems, however, it is possible that longer range interactions are not entirely negligible. It is expected that the inclusion of next-nearest-neighbor interactions between spins in one-dimensional spin models will introduce a multitude of new frequencies in addition to the ones already present in the dynamics that arises due to nearest-neighbor interactions. We first present an exact solution for the dynamical xx-spin-pair correlations in an Ising chain with both nearest- and next-nearest-neighbor interactions to confirm our expectation. We next show, via an approximate analytical calculation, that the dynamical zz-spin-pair correlations in the next-nearest-neighbor transverse Ising chain when plotted as a function of time is noticeably different with respect to the exactly solvable nearest-neighbor transverse Ising chain at T\ensuremath{\rightarrow}\ensuremath{\infty} when the next-nearest-neighbor interaction is \ensuremath{\gtrsim}1/2 of the magnitude of the nearest-neighbor interaction. The effects could be fairly subtle in the time domain representation and in the spectral function when these additional interactions are weak (i.e., 1/2 of the nearest-neighbor interaction magnitude). The general conclusions reached in this work are expected to be valid for other simple quantum spin models such as the XY and XXZ models in one dimension. \textcopyright{} 1996 The American Physical Society.
Motivated by frustrated magnets and quasi-one-dimensional magnetic materials, we study the magnetic properties of one-dimensional (1D) Ising chains with nearest-neighbor (NN) and weaker next-to-nearest-neighbor (NNN) interactions in the presence of vacancy defects. The effect of a vacancy on the magnetic susceptibility of a spin chain is twofold: it reduces the length of the chain by an effective ``vacancy size'' and may also act as a free spin, a ``quasispin,'' with a Curie-type ${\ensuremath{\chi}}_{\text{quasi}}=\ensuremath{\langle}{S}^{2}\ensuremath{\rangle}/T$ contribution to the susceptibility. In chains with antiferromagnetic short-range order, the susceptibility of vacancy-free chains is exponentially suppressed at low temperatures, and quasispins dominate the effect of impurities on the chains' magnetic properties. For chains with antiferromagnetic NN interactions, the quasispin matches the value $\ensuremath{\langle}{S}^{2}\ensuremath{\rangle}=1$ of the Ising spins in the chain for ferromagnetic NNN interactions and vanishes for antiferromagnetic NNN interactions. For chains with ferromagnetic short-range order, quasispin effects are insignificant due to exponentially large low-temperature susceptibilities, and the dominant effect of a vacancy is effectively changing the length of the chain.
Motivated by recent experiments on quasi-one-dimensional spin-frustrated quantum spin systems, we study the dynamic properties of exactly solvable quantum spin models with spin frustration. The ground-state dynamic structure factor, static structure factor (important for the neutron scattering), inhomogeneous static magnetic susceptibility, etc., are calculated. The results differ strongly from the behavior of similar characteristics of a standard Heisenberg quantum spin chain. We believe that the obtained behavior of the mentioned characteristics is generic for quantum spin chains with nearest- and next-nearest-neighbor interactions.