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Journal of the Physical Society of Japan

Longitudinal and Transverse Susceptibilities of the One-Dimensional Ising Model with Competing Interactions

Isao Harada

Published Dec 1, 1983• 17 citations
Publisher

Abstract

Exact expressions for the wave-number dependent longitudinal and transverse susceptibilities of the one-dimensional Ising model with competing interactions are obtained by combining the linear response theory with the transfer matrix method. The longitudinal susceptibility exhibits a characteristic wave-number dependence, while the transverse susceptibility is independent of wave-number. Special attention is paid to the condition under which the longitudinal susceptibility has a maximum value at nonzero wave-number. Furthermore, the phase diagram of a system with weakly coupled Ising chains is considered in connection with the one-dimensional susceptibility.

DOI
10.1143/jpsj.52.4099

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Exact expressions for the wave-number dependent longitudinal and transverse susceptibilities of the one-dimensional Ising model with competing interactions are obtained by combining the linear response theory with the transfer matrix method. The longitudinal susceptibility exhibits a characteristic wave-number dependence, while the transverse susceptibility is independent of wave-number. Special attention is paid to the condition under which the longitudinal susceptibility has a maximum value at nonzero wave-number. Furthermore, the phase diagram of a system with weakly coupled Ising chains is considered in connection with the one-dimensional susceptibility.
The critical properties of the one-dimensional spin-1/2 transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. The maximum of the fidelity susceptibility was used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction J_{2}, we calculated the fidelity susceptibility as a function of the transverse field B_{x} and the longitudinal field B_{z}. The phase diagram in the (B_{x},B_{z})-plane shows three phases. These findings are in contrast with the published literature that claims that the system has only two phases. For B_{x}<1, we observed an antiferromagnetic phase for small values of B_{z} and a paramagnetic phase for large values of B_{z}. For B_{x}>1 and low B_{z}, we found a disordered phase that undergoes a second-order phase transition to a paramagnetic phase for large values of B_{z}.
We calculate the zero-temperature q-dependent susceptibilities of the one-dimensional, S=1/2, transverse Ising model at the critical magnetic field and of the isotropic XY model in zero field which have not been previously determined. Our method, which is based on a rigorous method of calculating dynamic correlation functions for these models, provides precise numerical values for the susceptibilities at wave numbers q=k\ensuremath{\pi}/M for integral M and odd integral k, as well as exact analytic results for the dominant singularities at q=0 and q=\ensuremath{\pi}.