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Journal of Statistical Physics

The quantum ground state of a Heisenberg ferromagnet with an ?easy-plane? type anisotropy

Andrei A. Zvyagin, V. M. Tsukernik

Published Jan 1, 1985• 1 citation
Publisher
DOI
10.1007/bf01017870

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We consider the determination of the ground state energy and the lowest lying excitations for ferromagnetic Heisenberg spin chains with uniaxial anisotropy of the “easy plane” type in an external magnetic field directly along the anisotropy axis. The ratio of the anisotropy and exchange constants (α) is assumed small, but the size of the site spin S is arbitrary. To first order in α the exact results are shown to differ from the quasiclassical results only by the substitution α→α˜=α[1−(1/2)S]. The asymptotic behavior of the transverse spin correlation function is discussed, and a comparison is made between the theoretical results and recent experiment data.

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The spin-$1/2$ Heisenberg chain with a ferromagnetic first-neighbor exchange coupling ${J}_{1}$ and an antiferromagnetic second-neighbor ${J}_{2}$ has a Haldane dimer ground state with an extremely small spin gap. Thus the ground state is readily altered by perturbations. Here, we investigate the effects of XXZ exchange magnetic anisotropy of both the easy-axis and easy-plane types and an alternation in ${J}_{1}$ on the ground state, the spin gap, and magnetic properties of the frustrated ferromagnetic spin-$1/2$ chain. It is found that there are two distinct dimerized spin-gap phases, in one of which the spin gap and the magnetic susceptibility are extremely small around the SU(2) symmetric case and in the other they are moderately large far away from the SU(2) symmetric case. A small alternation in the amplitude of ${J}_{1}$ rapidly shortens the pitch of spin correlations towards the four-spin periodicity, as in the limit of ${J}_{1}/{J}_{2}\ensuremath{\rightarrow}0$. These effects are not sufficient to quantitatively explain overall experimentally observed magnetic properties in the quasi-one-dimensional spin-gapped magnetoelectric cuprate ${\mathrm{Rb}}_{2}{\mathrm{Cu}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{12}$ that exhibits ferroelectricity stabilized by a magnetic field. Our results are also relevant to ${\mathrm{Cs}}_{2}{\mathrm{Cu}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{12}$, where the ferromagnetic intrachain and antiferromagnetic interchain order has recently been found, in a single chain level. We also reveal the nature of symmetry-protected topological phase transitions in the model by mapping onto effective spin-1 chain models.

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Long range order in the anisotropic quantum ferromagnetic Heisenberg model

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We consider the determination of the ground state energy and the lowest lying excitations for ferromagnetic Heisenberg spin chains with uniaxial anisotropy of the “easy plane” type in an external magnetic field directly along the anisotropy axis. The ratio of the anisotropy and exchange constants (α) is assumed small, but the size of the site spin S is arbitrary. To first order in α the exact results are shown to differ from the quasiclassical results only by the substitution α→α˜=α[1−(1/2)S]. The asymptotic behavior of the transverse spin correlation function is discussed, and a comparison is made between the theoretical results and recent experiment data.
The spin-1/21/21/2 Heisenberg chain with a ferromagnetic first-neighbor exchange coupling J1{J}_{1}J1​ and an antiferromagnetic second-neighbor J2{J}_{2}J2​ has a Haldane dimer ground state with an extremely small spin gap. Thus the ground state is readily altered by perturbations. Here, we investigate the effects of XXZ exchange magnetic anisotropy of both the easy-axis and easy-plane types and an alternation in J1{J}_{1}J1​ on the ground state, the spin gap, and magnetic properties of the frustrated ferromagnetic spin-1/21/21/2 chain. It is found that there are two distinct dimerized spin-gap phases, in one of which the spin gap and the magnetic susceptibility are extremely small around the SU(2) symmetric case and in the other they are moderately large far away from the SU(2) symmetric case. A small alternation in the amplitude of J1{J}_{1}J1​ rapidly shortens the pitch of spin correlations towards the four-spin periodicity, as in the limit of ${J}_{1}/{J}_{2}\ensuremath{\rightarrow}0$. These effects are not sufficient to quantitatively explain overall experimentally observed magnetic properties in the quasi-one-dimensional spin-gapped magnetoelectric cuprate Rb2Cu2Mo3O12{\mathrm{Rb}}_{2}{\mathrm{Cu}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{12}Rb2​Cu2​Mo3​O12​ that exhibits ferroelectricity stabilized by a magnetic field. Our results are also relevant to Cs2Cu2Mo3O12{\mathrm{Cs}}_{2}{\mathrm{Cu}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{12}Cs2​Cu2​Mo3​O12​, where the ferromagnetic intrachain and antiferromagnetic interchain order has recently been found, in a single chain level. We also reveal the nature of symmetry-protected topological phase transitions in the model by mapping onto effective spin-1 chain models.