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arXiv:cond-mat/0005322 · Physical Review Letters

Dispersion and Symmetry of Bound States in the Shastry-Sutherland Model

Christian Knetter, Alexander Bühler, E. Müller‐Hartmann, Götz Silvester Uhrig

Published Oct 30, 2000• 146 citations
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Abstract

Bound states made from two triplet excitations on the Shastry-Sutherland lattice are investigated. Based on the perturbative unitary transformation by flow equations quantitative properties like dispersions and qualitative properties like symmetries are determined. The high order results [up to $({J}_{2}/{J}_{1}{)}^{14}$] permit one to fix the parameters of ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2}$ precisely: ${J}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6.16(10)\mathrm{meV}$, $x\ensuremath{\mathrel{:=}}{J}_{2}/{J}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.603(3)$, ${J}_{\ensuremath{\perp}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.3(2)\mathrm{meV}$. At the border of the magnetic Brillouin zone a general double degeneracy is derived. An unexpected instability in the triplet channel at $x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.63$ indicates a transition towards another phase. The possible nature of this phase is discussed.

DOI
10.1103/physrevlett.85.3958

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Bound states made from two triplet excitations on the Shastry-Sutherland lattice are investigated. Based on the perturbative unitary transformation by flow equations quantitative properties like dispersions and qualitative properties like symmetries are determined. The high order results [up to (J2/J1)14({J}_{2}/{J}_{1}{)}^{14}(J2​/J1​)14] permit one to fix the parameters of SrCu2(BO3)2{\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2}SrCu2​(BO3​)2​ precisely: J1=6.16(10)meV{J}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6.16(10)\mathrm{meV}J1​=6.16(10)meV, x\ensuremath:=J2/J1=0.603(3)x\ensuremath{\mathrel{:=}}{J}_{2}/{J}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.603(3)x\ensuremath:=J2​/J1​=0.603(3), J\ensuremath⊥=1.3(2)meV{J}_{\ensuremath{\perp}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.3(2)\mathrm{meV}J\ensuremath⊥​=1.3(2)meV. At the border of the magnetic Brillouin zone a general double degeneracy is derived. An unexpected instability in the triplet channel at x=0.63x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.63x=0.63 indicates a transition towards another phase. The possible nature of this phase is discussed.
We study an extended Shastry-Sutherland model for SrCu2(BO3)2 and analyze the low lying parts of the energy spectrum by means of a perturbative unitary transformation based on flow equations.The derivation of the 1-magnon dispersion (elementary triplets) is discussed.Additionally, we give a quantitative description (symmetries and energies) of bound states made from two elementary triplets.Our high order results allow to fix the model parameters for SrCu2(BO3)2 precisely: J1 = 6.16(10)meV, x := J2/J1 = 0.603(3), J ⊥ = 1.3(2)meV.To our knowledge this is the first quantitative treatment of bound states in a true 2d model.
We study an extended Shastry–Sutherland model for SrCu2(BO3)2 and analyze the low-lying parts of the energy spectrum by means of a perturbative unitary transformation based on flow equations. The derivation of the one-magnon dispersion (elementary triplets) is discussed. Additionally, we give a quantitative description (symmetries and energies) of bound states made from two elementary triplets. Our high-order results allow us to fix the model parameters for SrCu2(BO3)2 precisely: J1 = 6.16(10) meV, x := J2/J1 = 0.603(3), J[Formula: see text] =1.3(2) meV. To our knowledge this is the first quantitative treatment of bound states in a true 2D model. PACS Nos.: 75.40Gb, 75.30Kz, 75.50Ee, 75.10Jm
We investigate the one-triplet dispersion in a modified Shastry-Sutherland model for SrCu 2 (BO 3 ) 2 by means of a series expansion about the limit of strong dimerization. Our perturbative method is based on a continuous unitary transformation that maps the original Hamiltonian to an effective, energy-quanta-conserving block diagonal Hamiltonian eff . The dispersion splits into two branches which are nearly degenerate. We analyse the symmetries of the model and show that space group operations are necessary to explain the degeneracy of the dispersion at k = 0 and at the border of the magnetic Brillouin zone. Moreover, we investigate the behaviour of the dispersion for small | k | and compare our results to inelastic neutron scattering data.
Configurations of singlets and triplets on the Shastry-Sutherland lattice (SSL) have been proposed in the literature as variational ground states of the Shastry-Sutherland model at a fixed magnetization M. We prove that isolated triplet excitations on the SSL are unstable if the coupling $\ensuremath{\alpha}$ falls below a critical value ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}2.0.$ The instability should be visible in the compound SrCu2(BO3)2,{\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2},SrCu2​(BO3​)2​, where a coupling ${\ensuremath{\alpha}}^{*}=1.48$ is realized.
By using the perturbation expansion up to the fifth order, we study the two-triplet-dimer excitation spectra in the Shastry-Sutherland model, where the localized nature of a triplet-dimer, the propagation of a triplet-dimer pair by the correlated hopping and the long-range interactions between triplet-dimers play an essential role. It is found that the dispersion relations for first-neighbor triplet-dimer pair excitations with S=1 and p-type symmetry qualitatively explain the second-lowest branch observed in the neutron inelastic scattering experiment. It is also predicted that the second-lowest branch consists of two components, p_x- and p_y-states, with slightly different excitation energies. The origin of the singlet mode at 3.7meV observed in the Raman scattering experiment is also discussed.
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.