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arXiv:0806.2568 · Journal of Physics Conference Series

Spontaneous order in the highly frustrated spin-1/2 Ising-Heisenberg model on the triangulated Kagomé lattice due to the Dzyaloshinskii-Moriya anisotropy

Jozef Strečka, Lucia Čanová

Published Jan 1, 2009• 15 citations
arXivHTMLPDF

Abstract

The spin-1/2 Ising-Heisenberg model on the triangulated Kagomé (triangles-in-triangles) lattice is exactly solved by establishing a precise mapping correspondence to the simple spin-1/2 Ising model on Kagomé lattice. It is shown that the disordered spin liquid state, which otherwise occurs in the ground state of this frustrated spin system on assumption that there is a sufficiently strong antiferromagnetic intra-trimer interaction, is eliminated from the ground state by arbitrary but non-zero Dzyaloshinskii-Moriya anisotropy.

DOI
10.1088/1742-6596/145/1/012012

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Physical review. E•Apr 13, 2022•1 citation
The spin-1/2 Ising-Heisenberg model on the triangulated Kagomé (triangles-in-triangles) lattice is exactly solved by establishing a precise mapping correspondence to the simple spin-1/2 Ising model on Kagomé lattice. It is shown that the disordered spin liquid state, which otherwise occurs in the ground state of this frustrated spin system on assumption that there is a sufficiently strong antiferromagnetic intra-trimer interaction, is eliminated from the ground state by arbitrary but non-zero Dzyaloshinskii-Moriya anisotropy.
The geometrically frustrated spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices is exactly solved by combining the generalized star-triangle transformation with the method of exact recursion relations. The ground-state and finite-temperature phase diagrams are rigorously calculated along with both sublattice magnetizations of the Ising and Heisenberg spins. It is evidenced that the Ising-Heisenberg model on triangulated Husimi lattices with two or three interconnected triangles-in-triangles units displays in a highly frustrated region a quantum disorder irrespective of temperature, whereas the same model on triangulated Husimi lattices with a greater connectivity of triangles-in-triangles units exhibits at low enough temperatures an outstanding quantum order due to the order-by-disorder mechanism. The quantum reduction of both sublattice magnetizations in the peculiar quantum ordered state gradually diminishes upon increasing the coordination number of the underlying Husimi lattice.
The spin-12\frac{1}{2}21​ Ising-Heisenberg model on two geometrically related triangles-in-triangles lattices is exactly solved through the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the effective spin-12\frac{1}{2}21​ Ising model on a triangular lattice. Basic thermodynamic quantities were exactly calculated within this rigorous mapping method along with the ground-state and finite-temperature phase diagrams. Apart from the classical ferromagnetic phase, both investigated models exhibit several unconventional quantum ordered and disordered ground states. A mutual competition between two ferromagnetic interactions of basically different character generically leads to the emergence of a quantum ferromagnetic phase, in which a symmetric quantum superposition of three up-up-down states of the Heisenberg trimers accompanies a perfect alignment of all Ising spins. In the highly frustrated regime, we have either found the disordered quantum paramagnetic phase with a substantial residual entropy or a similar but spontaneously ordered phase with a nontrivial criticality at finite temperatures. The latter quantum ground state exhibits a striking coexistence of imperfect spontaneous order with partial disorder, which is evidenced by a quantum reduction of the spontaneous magnetization of Heisenberg spins that indirectly causes a small reduction of the spontaneous magnetization of otherwise classical Ising spins.
A preponderance of evidence suggests that the ground state of the nearest-neighbor S=1/2S=1/2S=1/2 antiferromagnetic Heisenberg model on the kagome lattice is a gapless spin liquid. Many candidate materials for the realization of this model possess in addition a Dzyaloshinskii-Moriya (DM) interaction. We study this system by tensor-network methods and deduce that a weak but finite DM interaction is required to destabilize the gapless spin-liquid state. The critical magnitude, ${D}_{c}/J\ensuremath{\simeq}0.012(2)$, lies well below the DM strength proposed in the kagome material herbertsmithite, indicating a need to reassess the apparent spin-liquid behavior reported in this system.
A small value of the spin gap in quantum antiferromagnets with strong frustration makes them susceptible to nominally small deviations from the ideal Heisenberg model. One such perturbation, the anisotropic Dzyaloshinskii-Moriya (DM) interaction, is an important perturbation for the S=1/2S=1/2S=1/2 kagome antiferromagnet, one of the current candidates for a quantum-disordered ground state. We study the influence of the DM term in a related one-dimensional system, the sawtooth chain, which has valence-bond order in its ground state. Through a combination of analytical and numerical methods, we show that a relatively weak DM coupling, 0.115J0.115J0.115J, is sufficient to destroy the valence-bond order, close the spin gap, and turn the system into a Luttinger liquid with algebraic spin correlations. A similar mechanism may be at work in the kagome antiferromagnet.
The spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice is exactly solved in a magnetic field within the framework of the generalized star-triangle transformation and the method of exact recursion relations. The generalized star-triangle transformation establishes an exact mapping correspondence with the effective spin-1/2 Ising model on a triangular Husimi lattice with a temperature-dependent field, pair and triplet interactions, which is subsequently rigorously treated by making use of exact recursion relations. The ground-state phase diagram of a spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice, which bears a close resemblance with a triangulated kagomé lattice, involves, in total, two classical and three quantum ground states manifested in respective low-temperature magnetization curves as intermediate plateaus at 1/9, 1/3, and 5/9 of the saturation magnetization. It is verified that the fractional magnetization plateaus of quantum nature have character of either dimerized or trimerized ground states. A low-temperature magnetization curve of the spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice resembling a triangulated kagome lattice may exhibit either no intermediate plateau, a single 1/3 plateau, a single 5/9 plateau, or a sequence of 1/9, 1/3, and 5/9 plateaus depending on a character and relative size of two considered coupling constants.
The spin-1/2 Ising-Heisenberg model on martini and martini-diced lattices is exactly solved using a star-triangle transformation, which affords an exact mapping correspondence to an effective spin-1/2 Ising model on a triangular lattice. The ground-state phase diagram of both investigated quantum spin models display two spontaneously ordered ferromagnetic phases and one macroscopically degenerate disordered phase. In contrast to a classical ferromagnetic phase where the spontaneous magnetization of the Ising as well as Heisenberg spins acquire fully saturated values, the spontaneous magnetization of the Heisenberg spins is subject to a quantum reduction to one-third of its saturated value within a quantum ferromagnetic phase. The spontaneous magnetization and logarithmic divergence of the specific heat as the most essential features of both ferromagnetic phases disappear whenever the investigated quantum spin model is driven to the highly degenerate disordered phase. The disordered phase with nonzero residual entropy originates either from a geometric spin frustration caused by antiferromagnetic interactions or more strikingly it may also alternatively arise from a competition of the ferromagnetic Ising and Heisenberg interactions of easy-axis and easy-plane type, respectively. All three available ground states coexist together at a single triple point, around which anomalous magnetic and thermodynamic behavior can be detected.