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arXiv:1407.8229 · Physical Review B

Exact ground states of a spin- 12 Ising-Heisenberg model on the Shastry-Sutherland lattice in a magnetic field

Taras Verkholyak, Jozef Strečka, Frédéric Mila, Kai Phillip Schmidt

Published Oct 16, 2014• 26 citations
arXivHTMLPDF

Abstract

Exact ground states of a spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice with Heisenberg intradimer and Ising interdimer couplings are found by two independent rigorous procedures. The first method uses a unitary transformation to establish a mapping correspondence with an effective classical spin model, while the second method relies on the derivation of an effective hard-core boson model by continuous unitary transformations. Both methods lead to equivalent effective Hamiltonians providing a convincing proof that the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice exhibits a zero-temperature magnetization curve with just two intermediate plateaus at one-third and one-half of the saturation magnetization, which correspond to stripe and checkerboard orderings of singlets and polarized triplets, respectively. The nature of the remarkable stripe order relevant to the one-third plateau is thoroughly investigated with the help of the corresponding exact eigenvector. The rigorous results for the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice are compared with the analogous results for the purely classical Ising and fully quantum Heisenberg models. Finally, we discuss to what extent the critical fields of ${\mathrm{SrCu}}_{2}{(\mathrm{BO}}_{3}){}_{2}$ and ${(\mathrm{CuCl})\mathrm{Ca}}_{2}{\mathrm{Nb}}_{3}{\mathrm{O}}_{10}$ can be described within the suggested Ising-Heisenberg model.

DOI
10.1103/physrevb.90.134413

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Exact ground states of a spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice with Heisenberg intradimer and Ising interdimer couplings are found by two independent rigorous procedures. The first method uses a unitary transformation to establish a mapping correspondence with an effective classical spin model, while the second method relies on the derivation of an effective hard-core boson model by continuous unitary transformations. Both methods lead to equivalent effective Hamiltonians providing a convincing proof that the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice exhibits a zero-temperature magnetization curve with just two intermediate plateaus at one-third and one-half of the saturation magnetization, which correspond to stripe and checkerboard orderings of singlets and polarized triplets, respectively. The nature of the remarkable stripe order relevant to the one-third plateau is thoroughly investigated with the help of the corresponding exact eigenvector. The rigorous results for the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice are compared with the analogous results for the purely classical Ising and fully quantum Heisenberg models. Finally, we discuss to what extent the critical fields of SrCu2(BO3)2{\mathrm{SrCu}}_{2}{(\mathrm{BO}}_{3}){}_{2}SrCu2​(BO3​)2​ and (CuCl)Ca2Nb3O10{(\mathrm{CuCl})\mathrm{Ca}}_{2}{\mathrm{Nb}}_{3}{\mathrm{O}}_{10}(CuCl)Ca2​Nb3​O10​ can be described within the suggested Ising-Heisenberg model.
Spin-1/2 Heisenberg model on the Shastry-Sutherland lattice is considered within the many-body perturbation theory developed from the exactly solved spin-1/2 Ising-Heisenberg model with the Heisenberg intradimer and Ising interdimer interactions. The former model is widely used for a description of magnetic properties of the layered compound SrCu _2 2 (BO _3 3 ) _2 2 , which exhibits a series of fractional magnetization plateaux at sufficiently low temperatures. Using the novel type of many-body perturbation theory we have found the effective model of interacting triplet excitations with the extended hard-core repulsion, which accurately recovers 1/8, 1/6 and 1/4 magnetization plateaux for moderate values of the interdimer coupling. A possible existence of a striking quantum phase of bound triplons is also revealed at low enough magnetic fields.
We study magnetic properties of the spin-12\frac{1}{2}21​ Ising-like XXZ model on the Shastry-Sutherland lattices with long-range interactions, using the quantum Monte Carlo method. This model shows magnetization plateau phases at one-half and one-third of the saturation magnetization when additional couplings are considered. We investigate the finite temperature transition to one-half and one-third plateau phases. The obtained results suggest that the former case is of the first order and the latter case is of the second order. We also find that the system undergoes two successive transitions with the two-dimensional Ising model universality although there is a single phase transition in the Ising limit case. Finally, we estimate the coupling ratio to explain the magnetization process observed in TmB4{\text{TmB}}_{4}TmB4​.
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.
Magnetization dependence on the external magnetic field was studied in a system with the Shastry–Sutherland lattice using exact diagonalization method at zero temperature within the xxz-model with spin S = 1/2. It was found that contrary to the classical Heisenberg model, the magnetization plateaus corresponding to m* = m/msat = 1/4, 1/3, and 1/2 (here msat is the saturation magnetization) exist even in the case of isotropic exchange interactions. The effect of exchange interaction on the width of the plateaus was studied. It was shown that depending on the ratio of exchange constants, there exist three regions corresponding to different types of the behavior of the system: the regions of a Neel-like antiferromagnet, a “dimer” antiferromagnet, and an intermediate region where the unique properties of Shastry–Sutherland lattices are most pronounced. The boundaries of these regions were also determined.
The quantum spin-12\frac{1}{2}21​ orthogonal-dimer chain with the Heisenberg intradimer and Ising interdimer interactions in a magnetic field is considered by a rigorous approach. The model conserves the zzz component of total spin on vertical Heisenberg bonds, and this property is used to calculate exactly the partition function using the transfer-matrix method. We have found the ground-state phase diagram of the given model in a magnetic field as well as the macroscopic degeneracy along field-induced transitions accompanied with the magnetization jumps. The model exhibits two intermediate fractional plateaux at one quarter and one half of the saturation magnetization. We have examined the effect of the exchange anisotropy in the XXZXXZXXZ Heisenberg intradimer interaction on the ground state. It is shown that the one-quarter and one-half plateaux may disappear from the magnetization curve for the ferromagnetic Heisenberg intradimer interaction. We have also studied rigorously the effect of frustrated interactions on the thermodynamic and magnetic properties of the model and show how the macroscopic degeneracy of the ground state is reflected in the low-temperature behavior of the magnetization, entropy, and specific heat. A possibility of observing enhanced magnetocaloric effect during the adiabatic demagnetization is discussed in detail.
The ground state and zero-temperature magnetization process of the spin-1/2 Ising-Heisenberg model on two-dimensional triangles-in-triangles lattices are exactly calculated using eigenstates of the smallest commuting spin clusters. Our ground-state analysis of the investigated classical-quantum spin model reveals three unconventional dimerized or trimerized quantum ground states besides two classical ground states. It is demonstrated that the spin frustration is responsible for a variety of magnetization scenarios with up to three or four intermediate magnetization plateaus of either quantum or classical nature. The exact analytical results for the Ising-Heisenberg model are confronted with the corresponding results for the purely quantum Heisenberg model, which were obtained by numerical exact diagonalizations based on the Lanczos algorithm for finite-size spin clusters of 24 and 21 sites, respectively. It is shown that the zero-temperature magnetization process of both models is quite reminiscent, and hence, one may obtain some insight into the ground states of the quantum Heisenberg model from the rigorous results for the Ising-Heisenberg model even though exact ground states for the Ising-Heisenberg model do not represent true ground states for the pure quantum Heisenberg model.