The Ising Model and the Heisenberg Spin Chain

Mustansir Barma

The free energy of a classical Heisenberg chain with three or four spin interactions is shown to equal that of a spin-infinity Ising model. The two-spin Heisenberg correlation function equals a many-spin Ising correlation function and can oscillate with distance.

Taras Verkholyak, Jozef Strečka

The spin-$\frac{1}{2}$ Heisenberg orthogonal-dimer chain is considered within the perturbative strong-coupling approach, which is developed from the exactly solved spin-$\frac{1}{2}$ Ising-Heisenberg orthogonal-dimer chain with the Heisenberg intradimer and the Ising interdimer couplings. Although the spin-$\frac{1}{2}$ Ising-Heisenberg orthogonal-dimer chain exhibits just intermediate plateaus at zero, one-quarter, and one-half of the saturation magnetization, the perturbative treatment up to second order stemming from this exactly solvable model additionally corroborates the fractional one-third plateau as well as the gapless Luttinger spin-liquid phase. It is evidenced that the approximate results obtained from the strong-coupling approach are in an excellent agreement with the state-of-the-art numerical data obtained for the spin-$\frac{1}{2}$ Heisenberg orthogonal-dimer chain within the exact diagonalization and density-matrix renormalization group method. The nature of individual quantum ground states is comprehensively studied within the developed perturbation theory.

Walter Greiner, Ludwig Neise, Horst Stöcker B

J. S. Valverde, Onofre Rojas, S. M. de Souza

The asymmetric tetrahedron is composed of all edges of a tetrahedron represented by Ising interactions except for one, which has a Heisenberg-type interaction. This asymmetric tetrahedron is arranged by connecting a vertex whose edges are only Ising-type interactions to another vertex with the same structure for another tetrahedron. The process is replicated and this kind of lattice we call the asymmetric Ising–Heisenberg chain. We have studied the ground-state phase diagram for this kind of model. In particular, we consider two situations in the Heisenberg-type interaction: (i) the asymmetric tetrahedral spin-(1/2, 1/2) Ising-XYZ chain and (ii) the asymmetric tetrahedral spin-(1/2, 1) Ising-XXZ chain, where we have found a rich phase diagram and a number of multicritical points. Additionally we have also studied their thermodynamical properties and the correlation function, using the decorated transformation. We have mapped the asymmetric tetrahedral Ising–Heisenberg chain in an effective Ising chain, and we have also concluded that it is possible to evaluate the partition function including a longitudinal external magnetic field.

Subir Sachdev A