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arXiv:cond-mat/9807075 · Physical Review Letters

Exact Dimer Ground State of the Two Dimensional Heisenberg Spin System SrCu2(BO3)2

Shin Miyahara, Kazuo Ueda

Published May 3, 1999• 408 citations
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Abstract

The two dimensional Heisenberg model for ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2}$ has an exact dimer ground state which was proven by Shastry and Sutherland almost twenty years ago. The critical value of the quantum phase transition from the dimer state to the N\'eel ordered state is determined. Analysis of the experimental data shows that $\mathrm{SrCu}{}_{2}(\mathrm{BO}{}_{3}{)}_{2}$ has the dimer ground state but its coupling constants are close to the transition point, which leads to an unusual temperature dependence of the susceptibility. The almost localized nature of the triplet excitations explains the plateaus observed in the magnetization curve.

DOI
10.1103/physrevlett.82.3701

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arXiv:cond-mat/0012396•Journal of the Physical Society of Japan•May 15, 2001•25 citations
The two dimensional Heisenberg model for SrCu2(BO3)2{\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2}SrCu2​(BO3​)2​ has an exact dimer ground state which was proven by Shastry and Sutherland almost twenty years ago. The critical value of the quantum phase transition from the dimer state to the N'eel ordered state is determined. Analysis of the experimental data shows that SrCu2(BO3)2\mathrm{SrCu}{}_{2}(\mathrm{BO}{}_{3}{)}_{2}SrCu2​(BO3​)2​ has the dimer ground state but its coupling constants are close to the transition point, which leads to an unusual temperature dependence of the susceptibility. The almost localized nature of the triplet excitations explains the plateaus observed in the magnetization curve.
The magnetic properties of SrCu 2 (BO 3 ) 2 are reviewed from a theoretical point of view. SrCu 2 (BO 3 ) 2 is a new two-dimensional spin gap system and its magnetic properties are well described by a spin-1/2 antiferromagnetic Heisenberg model of the orthogonal dimer lattice. The model has a dimer singlet ground state whose exactness was proven by Shastry and Sutherland for a topologically equivalent model more than 20 years ago. The exactness of the ground state is maintained even if interlayer couplings are introduced for SrCu 2 (BO 3 ) 2 . In the two-dimensional model, quantum phase transitions take place between different ground states for which three phases are expected: a gapped dimer singlet state, a plaquette resonating valence bond state and a gapless magnetic ordered state. Analysis of the experimental data shows that the dimer singlet ground state is realized in SrCu 2 (BO 3 ) 2 . The orthogonality of the dimer bonds, which is the underlying symmetry of the exactness of the ground state, also leads to an unusual property of elementary excitations, namely the almost localized nature of the triplet excitations. Application of an external magnetic field changes the density of the triplet excitations. In general, there is competition between kinetic energies and interaction energies between triplets. The almost localized nature of the triplets makes it easy to form regular lattices. In fact, at certain densities, where the commensurability energy is significant, the triplet excitations form superstructures and plateaux appear at 1/2, 1/3, 1/4 and 1/8 in the magnetization curve. In high-magnetic-field experiments, magnetic plateaux at magnetizations of 1/3, 1/4 and 1/8 have been observed. Translational symmetry of the lattice is spontaneously broken at the plateaux, except for the 1/2 plateau. The 1/3 and 1/4 plateaux are expected to have magnetic superstructures of stripe form while the 1/2 plateau has a square unit cell and the 1/8 plateau a rhomboid cell. Especially at the 1/8 plateau, nuclear magnetic resonance experiments indicate the presence of at least 11 distinct Cu sites with different spin polarizations, which is the first evidence of breaking of the translational symmetry at the plateau phase. The spin texture calculated on the basis of a Heisenberg model with adiabatic spin–phonon coupling is consistent with the experimental results.
We investigate quantum phase transitions in the frustrated antiferromagnetic Heisenberg model for SrCu2(BO3)2\mathrm{Sr}{\mathrm{Cu}}_{2}({\mathrm{BO}}_{3}{)}_{2}SrCu2​(BO3​)2​ by using the series expansion method. It is found that a novel spin-gap phase, adiabatically connected to the plaquette-singlet phase, exists between the dimer and the magnetically ordered phases known thus far. When the ratio of the competing exchange couplings $\ensuremath{\alpha}({\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}J}^{\ensuremath{'}}/J)$ is varied, this spin-gap phase exhibits a first- (second-) order quantum phase transition to the dimer (the magnetically ordered) phase at the critical point ${\ensuremath{\alpha}}_{c1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.677(2)$ [ ${\ensuremath{\alpha}}_{c2}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.86(1)$]. Our results shed light on some controversial arguments about the nature of quantum phase transitions in this model.
We study a generalized Shastry-Sutherland model for the material SrCu2(BO3)(2). Along a line in the parameter space, we show rigorously that the model has a first-order phase transition between dimerized and Neel-ordered ground states. Furthermore, when a magnetic field is applied in the dimerized phase, magnetization plateaus develop at commensurate values of the magnetization. We also discuss various aspects of the phase diagram and properties of this model away from this exactly soluble line, which include gap-closing continuous transitions between dimerized and magnetically ordered phases.
The specific heat of a two-dimensional spin gap system SrCu 2 (BO 3 ) 2 realizing the Shastry-Suther-land model was measured between 1.3 and 25 K under various magnetic fields up to 12 T. The analysis based on an isolated dimer model in a low temperature region revealed that the value of the spin gap at zero field is Δ = 34.4 K. It turned out that Δ decreases in proportion to H due to the Zeeman splitting of the excited triplet levels. This simplest model, however, fails to reproduce the result in a high-temperature region, suggesting rather strong spin-spin correlation of the system.
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.
Magnetization plateaus in the Shastry-Sutherland model for SrCu2(BO3)2{\rm SrCu_2(BO_3)_2}SrCu2​(BO3​)2​ are studied by the perturbation expansion method. The fourth-order effective Hamiltonian which describes the dynamics of triplet dimers (TD's) with Sztot=1S^{\rm tot}_z=1Sztot​=1 in the singlet sea is derived and then partially diagonalized for the space that consists of the TD configurations with the lowest second-order energy. The fourth-order terms are treated within a low-density approximation. Our procedure makes clear how TD interactions are responsible for the formation of magnetization plateaus. Particularly, the 1/4-plateau is obtained by the fourth-neighbor TD repulsion in the fourth-order perturbation, and a diagonal stripe arrangement of TD's appears at this plateau.