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arXiv:cond-mat/0206055 · Physical review. B, Condensed matter

Quantum phase transitions in alternating transverse Ising chains: Analytical and numerical results

Oleg Derzhko, Johannes Richter, Taras Krokhmalskii, O. Zaburannyi

Published Oct 2, 2002• 14 citations
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Abstract

We examine the ground state properties of the $s=\frac{1}{2}$ transverse Ising chain with regularly alternating bonds and fields, using exact analytical results and exact numerical data for long (up to $N=900)$ and short $(N=20)$ chains. For a given period of alternation the system may exhibit a series of quantum phase transitions, where the number of transitions depends on the concrete set of the parameters of the Hamiltonian. The critical behavior for the nonuniform and uniform chains is the same.

DOI
10.1103/physrevb.66.144401

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The ground state phase diagram of the alternating spin-1/2 chains with anisotropic ferromagnetic coupling under the influence of a symmetry breaking transverse magnetic field is studied. We have used the exact diagonalization technique. In the limit where the antiferromagnetic coupling is dominant, we have identified two Ising-type quantum phase transitions. We have calculated two critical fields hc1h_{c_{1}}hc1​​ and hc2h_{c_{2}}hc2​​, corresponding to the transition between different magnetic phases of the system. It is found that the intermediate state ($h_{c_{1}}
Using continued fractions we study the ground state properties of the spin-½ Ising chain in a transverse field with periodically varying interaction strengths and external fields. We consider in detail the chain having a period-2 modulation of the interaction strengths and compare the results obtained with those corresponding to the spin-½ isotropic XY chain in a transverse field. In contrast to the behaviour of the transverse XY chain, the transverse Ising chain does not exhibit a step-like magnetization versus field dependence caused by the alternation of bonds. Its susceptibility exhibits a logarithmic singularity at a field determined by interaction strengths, and it is stable with respect to spin-Peierls dimerization.
The ground-state behaviour of the frustrated quantum spin-1/2 two-leg ladder with the Heisenberg intra-rung and Ising inter-rung interactions is examined in detail. The investigated model is transformed into the quantum Ising chain with composite spins in an effective transverse and longitudinal field by employing either the bond-state representation or the unitary transformation. It is shown that the ground state of the Heisenberg–Ising ladder can be descended from three exactly solvable models: the quantum Ising chain in a transverse field, the ‘classical’ Ising chain in a longitudinal field or the spin-chain model in a staggered longitudinal–transverse field. The last model serves in evidence of the staggered bond phase with alternating singlet and triplet bonds on the rungs of a two-leg ladder, which appears at moderate values of the external magnetic field and consequently leads to a fractional plateau at a half of the saturation magnetization. The ground-state phase diagram totally consists of five ordered and one quantum paramagnetic phase, which are separated from each other either by the lines of discontinuous or continuous quantum phase transitions. The order parameters are exactly calculated for all five ordered phases and the quantum paramagnetic phase is characterized through different short-range spin–spin correlations.