The analytical and numerical (Monte Carlo and exact diagonalisation) estimates of phase diagrams of frustrated Ising models in transverse fields are discussed here. Specifically we discuss the Sherrington–Kirkpatrick model in transverse field and the Axial Next-Nearest Neighbour Ising (ANNNI) model in transverse field. The effects of quantum fluctuations (induced by the transverse field) on the ground and excited states of such systems with competing interactions (frustration) are also discussed. The results are compared to those available for other frustrated quantum systems.
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.
To study the ground state of an axial next-nearest-neighbor Ising chain under transverse field as a function of frustration parameter kappa and field strength Gamma, we present here two different perturbative analyses. In one, we consider the (known) ground state at kappa=0.5 and Gamma=0 as the unperturbed state and treat an increase of the field from 0 to Gamma coupled with an increase of kappa from 0.5 to 0.5+rGamma/J as perturbation. The first-order perturbation correction to eigenvalue can be calculated exactly and we could conclude that there are only two phase-transition lines emanating from the point kappa=0.5, Gamma=0. In the second perturbation scheme, we consider the number of domains of length 1 as the perturbation and obtain the zeroth-order eigenfunction for the perturbed ground state. From the longitudinal spin-spin correlation, we conclude that floating phase exists for small values of transverse field over the entire region intermediate between the ferromagnetic phase and antiphase.
The correlation functions ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$ and ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)$ are calculated for the one-dimensional Ising model with a transverse field
h at
T=0. This model corresponds to the
XY model with $\ensuremath{\gamma}=1$, and is equivalent to the two-dimensional anisotropic (${J}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$, ${J}_{2}\ensuremath{\rightarrow}0$) Ising model. The additional dimension in the classical model is the imaginary time of the quantum model. For all values of
h, ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)=\ensuremath{-}(\frac{1}{{h}^{2}})(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{t}^{2}}){\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$. At the critical field
h=hc=1, ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)\ensuremath{\sim}{({R}^{2}\ensuremath{-}{t}^{2})}^{\frac{\ensuremath{-}1}{8}}$. For
h<1,h>1 the results already obtained for the
XY model are recovered. We give some consequences from this equivalence in higher dimension, concerning the behavior of the correlation function at the critical field at
T=0.
The Ising model with a transverse field has been investigated for both positive and negative exchange interactions. Effective Hamiltonians are used for a pair of nearest-neighbour spins. The transition temperature and susceptibility above the transition temperature are calculated. The critical magnitude of the transverse field for which the transition temperature becomes 0 K is also evaluated. When the transverse field is absent, the result of the present method is reduced to that of the Bethe approximation.