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Journal of the Physical Society of Japan

Ground-State Properties of the One-Dimensional Isotropic Spin-1 Heisenberg Antiferromagnet with Competing Interactions

Takashi Tonegawa, Makoto Kaburagi, Naoko Ichikawa, Isao Harada

Published Aug 15, 1992• 150 citations
Publisher

Abstract

Ground-state properties are studied for the one-dimensional isotropic spin-1 Heisenberg magnet with antiferromagnetic nearest-neighbor (nn) and next-nearest-neighbor (nnn) interactions. The energy, the singlet-triplet energy gap, the two-spin correlation function, and the den Nijs and Rommelse string correlation function in the ground states of finite-size systems of up to 16 spins are calculated exactly. By extrapolating these results, the ground-state properties at the infinite-size limit are discussed. It is found, in particular, that as the ratio of the nnn interaction constant to the nn one increases, the string order parameter (the limiting value of the longest-distance string correlation function) as well as the Haldane gap (the limiting value of the singlet-triplet energy gap) divided by the sum of both interaction constants increases. This result implies that the competition between the antiferromagnetic nn and nnn interactions stabilizes the Haldane phase.

DOI
10.1143/jpsj.61.2890

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Physical review. B, Condensed matter•May 1, 1990•23 citations
Ground-state properties are studied for the one-dimensional isotropic spin-1 Heisenberg magnet with antiferromagnetic nearest-neighbor (nn) and next-nearest-neighbor (nnn) interactions. The energy, the singlet-triplet energy gap, the two-spin correlation function, and the den Nijs and Rommelse string correlation function in the ground states of finite-size systems of up to 16 spins are calculated exactly. By extrapolating these results, the ground-state properties at the infinite-size limit are discussed. It is found, in particular, that as the ratio of the nnn interaction constant to the nn one increases, the string order parameter (the limiting value of the longest-distance string correlation function) as well as the Haldane gap (the limiting value of the singlet-triplet energy gap) divided by the sum of both interaction constants increases. This result implies that the competition between the antiferromagnetic nn and nnn interactions stabilizes the Haldane phase.
Ground-state properties are studied for the one-dimensional anisotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions, which are assumed to have the same form as regards the anisotropy. The singlet-triplet energy gap, the magnetization, the energy, and the two-spin correlation function in the ground states of finite-size systems of up to 20 spins are calculated exactly. By extrapolating these results, the values of the above quantities at the infinite-size limit are estimated. Special attention is paid to the limiting ground-state phase diagram on the plane of the ratio of the next-nearest-neighbor interaction constant to the nearest-neighbor one versus the anisotropy parameter. It is found that, when the interactions have the XY-type anisotropy, the limiting ground state is in the spin-fiuid phase or in the dimer phase depending upon whether the ratio between the interaction constants is smaller or larger than a critical value. It is also found that, when the interactions have the Ising-type anisotropy, a novel phase with unsaturated magnetization appears between the ferromagnetic and (2, 2) antiphase phases in the limiting ground-state phase diagram.
Ground-state properties are studied for the one-dimensional anisotropic spin-1/2 Heisen­ berg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest­ neighbor interactions, which are assumed to have the same form as regards the anisotropy. The singlet-triplet energy gap, the magnetization, the energy, and the two-spin correlation function in the ground states of finite-size systems of up to 20 spins are calculated exactly. By extrapolating these results, the values of the above quantities at the infinite-size limit are estimated. Special attention is paid to the limiting ground-state phase diagram on the plane of the ratio of the next-nearest-neighbor interaction constant to the nearest-neighbor one versus the anisotropy parameter. It is found that, when the interactions have the XY-type anisotropy, the limiting ground state is in the spin-fluid phase or in the dimer phase depending upon whether the ratio between the interaction constants is smaller or larger than a critical value. It is also found that, when the interactions have the Ising-type anisotropy, a novel phase with unsaturated magnetization appears between the ferromagnetic and (2, 2) antiphase phases in the limiting ground-state phase diagram.
We study the ground-state and finite-temperature properties of the one-dimensional isotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions which compete with each other. Extrapolating the exact results for finite-size systems of up to 20 spins, we estimate the ground-state energy and the ground-state two-spin correlation function in the infinite-size limit. Using the cluster transfer matrix method, we also calculate the temperature dependence of the internal energy and the specific heat and that of the inverse correlation length and the wave number, by both of which the asymptotic behavior of the two-spin correlation function in the long-distance limit is characterized. We compare the results of these calculations with those obtained previously for the case where the nearest-neighbor interaction is antiferromagnetic.
We present low-temperature static and dynamic properties of the quantum one-dimensional isotropic integer spin Heisenberg magnet with antiferromagnetic nearest-neighbour (nn) and next-nearest-neighbour (nnn) interactions. The modified spin-wave theory is used to provide quantities such as the spin-wave dispersion relation, the ground-state energy, the gap and its dependence with temperature, and the asymptotic behaviour of the spin–spin correlation function. The ground state energy and the singlet–triplet energy gap are obtained for several values of j, defined as the ratio of the nnn interaction constant to the nn one. Our results show that the ground-state and the gap energies increase with j, in accordance with numerical results available in the literature. The calculation of the dynamic correlation function is performed using the projection operator formalism: the procedure includes up to two-magnon processes. We show that, depending on the values of the frustration parameter j, the wavevector and the temperature, a double-peak structure for the dynamical correlation function can develop.
We study the ground-state properties of the one-dimensional anisotropic spin-1/2 antiferromagnetic Heisenberg model with competing antiferromagnetic second-neighbor Ising interaction. Using the modified Lanczos method, we obtain the phase diagram in parameter space and different correlation functions, which we use to characterize the different phases. For small next-nearest-neighbor parameters we evaluate the critical exponent \ensuremath{\eta} of the spin-spin correlation function. We find good agreement between the numerical results and those obtained using the Luther and Peschel approach.