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Physica Scripta

The ground state properties of theS= 1/2 antiferromagneticXYmodel with quasiperiodic modulation

Li Peng-Fei, Chen Yu-Guang

Published Apr 29, 2009• 1 citation
Publisher

Abstract

The low-energy properties of the S = 1/2 antiferromagnetic XY model with quasiperiodic modulation are investigated for the consideration of spin–phonon coupling. It is found that the lattice distortion at every site displays a self-similar character, which is induced by quasiperiodic modulation. As quasiperiodic modulation is strengthened, the average lattice distortion may decrease and increase for strong and weak enough spin–phonon coupling, respectively. Correspondingly, the scaling behavior of the energy gap in this system is also strongly dependent on the spin–phonon coupling.

DOI
10.1088/0031-8949/79/05/055701

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The low-energy properties of the S = 1/2 antiferromagnetic XY model with quasiperiodic modulation are investigated for the consideration of spin–phonon coupling. It is found that the lattice distortion at every site displays a self-similar character, which is induced by quasiperiodic modulation. As quasiperiodic modulation is strengthened, the average lattice distortion may decrease and increase for strong and weak enough spin–phonon coupling, respectively. Correspondingly, the scaling behavior of the energy gap in this system is also strongly dependent on the spin–phonon coupling.
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The low energy behavior of the S=1/2 antiferromagnetic XY-like XXZ chains with precious mean quasiperiodic exchange modulation is studied by the density matrix renormalization group method. It is found that the energy gap of the chain with length N scales as exp⁡(−cNω)\exp (-cN^ω)exp(−cNω) with nonuniversal exponent ωωω if the Ising component of the exhange coupling is antiferromagnetic. This behavior is expected to be the characteristic feature of the quantum spin chains with relevant aperiodicity. This is in contrast to the XY chain for which the precious mean exchange modulation is marginal and the gap scales as N−zN^{-z}N−z. On the contrary, it is also verified that the energy gap scales as N−1N^{-1}N−1 if the Ising component of the exhange coupling is ferromagnetic. Our results are not only consistent with the recent bosonization analysis of Vidal, Mouhanna and Giamarchi but also clarify the nature of the strong coupling regime which is inaccesssible by the bosonization approach.
Low energy behavior of the S = 1/2 antiferromagnetic XXZ chains with Fibonacci exchange/field modulation is studied by the density matrix renormalization group method. For both types of modulation, the energy gap of the chain with length N scales as exp(-cNω ) with nonuniversal exponent ω if the z-component of the exchange coupling is antiferromagnetic. For the field modulation case, the possibility of fully localized ground state is suggested. This is in contrast to the XY chains for which the gap scales as N−z .
We investigate the low-energy properties of antiferromagnetic quantum XXZ spin chains with couplings following two-letter aperiodic sequences, by an adaptation of the Ma-Dasgupta-Hu renormalization-group method. For a given aperiodic sequence, we argue that, in the easy-plane anisotropy regime, intermediate between the XX and Heisenberg limits, the general scaling form of the thermodynamic properties is essentially given by the exactly known XX behavior, providing a classification of the effects of aperiodicity on XXZ chains. As representative illustrations, we present analytical and numerical results for the low-temperature thermodynamics and the ground-state correlations for couplings following the Fibonacci quasiperiodic structure and a binary Rudin-Shapiro sequence, whose geometrical fluctuations are similar to those induced by randomness.
An analytical approach, based on the unitary transformation method, has been developed to study the effect of quantum lattice fluctuations on the ground state of an XY\mathrm{XY}XY spin-Peierls chain, which is equivalent to the spinless Su-Schrieffer-Heeger model in half-filling after the Jordan-Wigner transformation. We show that when the spin-phonon coupling constant ${\ensuremath{\alpha}}^{2}/4K$ decreases or the phonon frequency ${\ensuremath{\omega}}_{\ensuremath{\pi}}$ increases, the lattice dimerization and the gap in the fermion spectrum decrease gradually. At some critical value $({\ensuremath{\alpha}}^{2}{/4K)}_{c}$ or ${\ensuremath{\omega}}_{\ensuremath{\pi}c},$ the system becomes gapless and the lattice dimerization disappears. This can be attributed to the fact that the ground state fails to develop the spin-Peierls long-range order because of the quantum lattice fluctuations.
We consider the antiferromagnetic spin-1∕21∕21∕2 Heisenberg model on a two-dimensional bipartite quasiperiodic tiling. The broken symmetry ground state in this model is inhomogeneous, reflecting the fact that there are a variety of local environments in such a structure. An important symmetry of the quasicrystal, namely that of invariance under discrete scale transformations is used to define an approximate real space renormalization scheme for the octagonal tiling. We solve for some of the fixed point properties of this quasiperiodic antiferromagnet. The ground state energy and local order parameters are calculated, and the results compare favorably with numerical values obtained by quantum Monte Carlo calculation. Despite the novel features of the ground state in this type of antiferromagnet, there are some interesting similarities with the well-known square lattice antiferromagnet. The most striking of these is the proximity of the values of the ground state energies of these two paradigms for two very dissimilar classes of solids.