Linear chains (and rings) of $S=\frac{1}{2}$ spins with the anisotropic (Ising-Heisenberg) Hamiltonian $\mathcal{H}=\ensuremath{-}2J\ensuremath{\Sigma}\stackrel{N}{i=1}{{{S}_{i}}^{z}{{S}_{i+1}}^{z}+\ensuremath{\gamma}({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})}\ensuremath{-}g\ensuremath{\beta}\ensuremath{\Sigma}\stackrel{N}{i=1}\mathrm{H}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{i}$ have been studied by exact machine calculations for $N=2 \mathrm{to} 11$ , $\ensuremath{\gamma}=0 \mathrm{to} 1$ and for ferro- and antiferro-magnetic coupling. The results reveal the dependence on finite size and anisotropy of the spectrum and dispersion laws, of the energy, entropy, and specific heat, of the magnetization and susceptibilities, and of the pair correlations. The limiting $N\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior is accurately indicated, for all $\ensuremath{\gamma}$, in the region $\frac{\mathrm{kT}}{|J|}>~0.5$ which includes the maxima in the specific heat and susceptibility. The behavior of thermal and magnetic properties of infinite chains at lower temperatures is estimated by extrapolation. For infinite antiferromagnetic chains the ground-state degeneracy, the anisotropy gap, and the magnetization, perpendicular susceptibility, and pair correlations at $T=0$ are similarly studied. Estimates of the long-range order suggest that it vanishes only at the Heisenberg limit $\ensuremath{\gamma}=1$ and confirm the accuracy of Walker's perturbation series in $\ensuremath{\gamma}$.Linear chains (and rings) of $S=\frac{1}{2}$ spins with the anisotropic (Ising-Heisenberg) Hamiltonian $\mathcal{H}=\ensuremath{-}2J\ensuremath{\Sigma}\stackrel{N}{i=1}{{{S}_{i}}^{z}{{S}_{i+1}}^{z}+\ensuremath{\gamma}({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})}\ensuremath{-}g\ensuremath{\beta}\ensuremath{\Sigma}\stackrel{N}{i=1}\mathrm{H}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{i}$ have been studied by exact machine calculations for $N=2 \mathrm{to} 11$, $\ensuremath{\gamma}=0 \mathrm{to} 1$ and for ferro- and antiferro-magnetic coupling. The results reveal the dependence on finite size and anisotropy of the spectrum and dispersion laws, of the energy, entropy, and specific heat, of the magnetization and susceptibilities, and of the pair correlations. The limiting $N\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior is accurately indicated, for all $\ensuremath{\gamma}$, in the region $\frac{\mathrm{kT}}{|J|}>~0.5$ which includes the maxima in the specific heat and susceptibility. The behavior of thermal and magnetic properties of infinite chains at lower temperatures is estimated by extrapolation. For infinite antiferromagnetic chains the ground-state degeneracy, the anisotropy gap, and the magnetization, perpendicular susceptibility, and pair correlations at $T=0$ are similarly studied. Estimates of the long-range order suggest that it vanishes only at the Heisenberg limit $\ensuremath{\gamma}=1$ and confirm the accuracy of Walker's perturbation series in $\ensuremath{\gamma}$.

Physical ReviewAug 3, 19642,497 citations

One-dimensional anisotropic Heisenberg chain

R. J. Baxter

Annals of PhysicsApr 1, 1972413 citations

Spin-1/2 asymmetric diamond Ising-Heisenberg chain

B. M. Lisnii

The ground state and the thermodynamics of a spin-1/2 asymmetric diamond Ising--Heisenberg chain are considered. For the $XYZ$ anisotropic Heisenberg interaction, the exact calculations of the free energy, entropy, heat capacity, magnetization, and magnetic susceptibility are performed using the method of decoration-iteration transformation. In the case of antiferromagnetic interactions (Ising and $XXZ$ anisotropic Heisenberg ones), the ground state, magnetization process, temperature dependence of the magnetization, magnetic susceptibility, and heat capacity are investigated. The influence of geometric frustration and quantum fluctuations on these characteristics is studied.The ground state and the thermodynamics of a spin-1/2 asymmetric diamond Ising--Heisenberg chain are considered. For the $XYZ$ anisotropic Heisenberg interaction, the exact calculations of the free energy, entropy, heat capacity, magnetization, and magnetic susceptibility are performed using the method of decoration-iteration transformation. In the case of antiferromagnetic interactions (Ising and $XXZ$ anisotropic Heisenberg ones), the ground state, magnetization process, temperature dependence of the magnetization, magnetic susceptibility, and heat capacity are investigated. The influence of geometric frustration and quantum fluctuations on these characteristics is studied.

arXiv:1112.1846Dec 8, 20111 citation

Linear chains (and rings) of

S=\frac{1}{2}

spins with the anisotropic (Ising-Heisenberg) Hamiltonian $\mathcal{H}=\ensuremath{-}2J\ensuremath{\Sigma}\stackrel{N}{i=1}{{{S}_{i}}^{z}{{S}_{i+1}}^{z}+\ensuremath{\gamma}({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})}\ensuremath{-}g\ensuremath{\beta}\ensuremath{\Sigma}\stackrel{N}{i=1}\mathrm{H}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{i}$ have been studied by exact machine calculations for

N=2 \mathrm{to} 11

, $\ensuremath{\gamma}=0 \mathrm{to} 1$ and for ferro- and antiferro-magnetic coupling. The results reveal the dependence on finite size and anisotropy of the spectrum and dispersion laws, of the energy, entropy, and specific heat, of the magnetization and susceptibilities, and of the pair correlations. The limiting $N\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior is accurately indicated, for all $\ensuremath{\gamma}$, in the region

\frac{\mathrm{kT}}{|J|}>~0.5

which includes the maxima in the specific heat and susceptibility. The behavior of thermal and magnetic properties of infinite chains at lower temperatures is estimated by extrapolation. For infinite antiferromagnetic chains the ground-state degeneracy, the anisotropy gap, and the magnetization, perpendicular susceptibility, and pair correlations at

T=0

are similarly studied. Estimates of the long-range order suggest that it vanishes only at the Heisenberg limit $\ensuremath{\gamma}=1$ and confirm the accuracy of Walker's perturbation series in $\ensuremath{\gamma}$.

The ground state and the thermodynamics of a spin-1/2 asymmetric diamond Ising--Heisenberg chain are considered. For the

XYZ

anisotropic Heisenberg interaction, the exact calculations of the free energy, entropy, heat capacity, magnetization, and magnetic susceptibility are performed using the method of decoration-iteration transformation. In the case of antiferromagnetic interactions (Ising and

XXZ

anisotropic Heisenberg ones), the ground state, magnetization process, temperature dependence of the magnetization, magnetic susceptibility, and heat capacity are investigated. The influence of geometric frustration and quantum fluctuations on these characteristics is studied.