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arXiv:0807.4796 · Journal of Physics Condensed Matter

Phase diagram of the asymmetric tetrahedral Ising–Heisenberg chain

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

Published Aug 1, 2008• 47 citations

Abstract

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.

DOI: 10.1088/0953-8984/20/34/345208

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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.

A mixed Ising-Heisenberg spin system consisting of triangular

XXZ

-Heisenberg spin clusters assembled into a chain by alternating with Ising spins interacting to all three spins in the triangle is considered. The exact solution of the model is given in terms of the generalized decoration-iteration map and within the transfer-matrix technique. Exact expressions for thermodynamic functions are derived. Ground-state phase diagrams and thermodynamic and magnetic properties of the system are examined.

The phase diagram of a coupled tetrahedral Heisenberg model is obtained. The quantum chain has a local gauge symmetry and its eigenspectrum is obtained by the composition of the eigenspectra of $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ XXZ chains with arbitrary distribution of $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{3}{2}$ impurities. The phase diagram is quite rich with an infinite number of phases with ferromagnetic, antiferromagnetic, or ferrimagnetic order. In some cases the ground state and the low-lying eigenlevels of the model can be exactly calculated since they coincide with the eigenlevels of the exactly integrable XXZ chain. The thermodynamical properties of the model at low temperatures are also studied through finite-size analysis.

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 frustrated spin-1/2 Ising-Heisenberg ladder with Heisenberg intra-rung and Ising inter-rung interactions is exactly solved in a longitudinal magnetic field by taking advantage of the local conservation of the total spin on each rung and the transfer-matrix method. We have rigorously calculated the ground-state phase diagram, magnetization process, magnetocaloric effect, and basic thermodynamic quantities for the model, which can be alternatively viewed as an Ising-Heisenberg tetrahedral chain. It is demonstrated that a stepwise magnetization curve with an intermediate plateau at half of the saturation magnetization is also reflected in respective stepwise changes of the concurrence serving as a measure of bipartite entanglement. The ground-state phase diagram and zero-temperature magnetization curves of the Ising-Heisenberg tetrahedral chain are contrasted with the analogous results of the purely quantum Heisenberg tetrahedral chain, which have been obtained through density-matrix renormalization group (DMRG) calculations. While both ground-state phase diagrams fully coincide in the regime of weak inter-rung interaction, the purely quantum Heisenberg tetrahedral chain develops Luttinger spin-liquid and Haldane phases for strongly coupled rungs, which are absent in the Ising-Heisenberg counterpart model.

Phase diagram of a discrete counterpart of the classical Heisenberg model, the truncated tetrahedral model, is analyzed on the square lattice, when the interaction is ferromagnetic. Each spin is represented by a unit vector that can point to one of the 12 vertices of the truncated tetrahedron, which is a continuous interpolation between the tetrahedron and the octahedron. Phase diagram of the model is determined by means of the statistical analog of the entanglement entropy, which is numerically calculated by the corner transfer matrix renormalization group method. The obtained phase diagram consists of four different phases, which are separated by five transition lines. In the parameter region, where the octahedral anisotropy is dominant, a weak first-order phase transition is observed.