Determining ground states of correlated electron systems is fundamental to understanding novel phenomena in condensed matter physics. A difficulty, however, arises in a geometrically frustrated system in which the incompatibility between the global topology of an underlying lattice and local spin interactions gives rise to macroscopically degenerate ground states 1 , potentially prompting the emergence of quantum spin states, such as resonating valence bond (RVB) and valence bond solid (VBS). Although theoretically proposed to exist in a kagome lattice -one of the most highly frustrated lattices in two dimensions (2D) being comprised of corner-sharing triangles -such quantum-fluctuation-induced states have not been observed experimentally. Here we report the first realization of the "pinwheel" VBS ground state in the S = 1 2 deformed kagome lattice antiferromagnet Rb 2 Cu 3 SnF 12 . In this system, a lattice distortion breaks the translational symmetry of the ideal kagome lattice and stabilizes the VBS state.
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 firstneighbor 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, px-and py-states, with slightly different excitation energies. The origin of the singlet mode at 3.7meV observed in the Raman scattering experiment is also discussed. There has been a growing interest in low-dimensional quantum spin systems, since one can observe a variety of properties where classical pictures break down. In the last decade, spin gapped ground states, particularly, in two-dimensional systems have received considerable attention in connection with the high-T c superconductivity. For instance, a quasi-two-dimensional compound CaV 4 O 9 is known to have a spin gap originating in the plaquette RVB mechanism. 1, 2, 3, 4)Two years ago, another new two-dimensional spin gap system SrCu 2 (BO 3 ) 2 was found by Kageyama et al. 5) In this compound, magnetic ions Cu 2+ (S = 1/2) are arranged as shown in Fig. 1. Miyahara and Ueda 6) pointed out that SrCu 2 (BO 3 ) 2 is an experimental realization of a special class of the Heisenberg antiferromagnets which is called the Shastry-Sutherland modelwhere i, j ( i, j ) denotes a (next-)nearest-neighbor pair of spins. The strength of the interdimer coupling is now considered λ ∼ 0.63. 8, 9) The orthogonal dimer structure as seen in Fig. 1 leads to the following unique properties: the direct product of the singlet-dimer states of the nearest-neighbor pairs of spins is the ground state exactly for λ <∼ 0.7 and a triplet-dimer in the singlet sea is almost localized. 6, 9)The nature of the quantum phase transition has been one of important subjects in theoretical investigations. 6,9,10,11,12) At first stage of the investigations, it was assumed that the exact dimer singlet state is destabi- * E-mail: yfuku@ph.noda.sut.ac.jp in SrCu 2 (BO 3 ) 2 . The closed circles represent copper ions. The nearest-neighbor bonds are expressed by the bold lines and the next nearest-neighbor bonds by the gray lines. The vectors, ex and ey, are the primitive translation vectors in this lattice. Note that there exist two types of third-neighbor triplet-dimer pairs, which are indicated by 3n and 3n ′ . For brevity, we call the farmer the third-neighbor pair. Within the fifth-order calculation, the diagonal interactions between triplet-dimers appear for the first-neighbor (1n), second-neighbor (2n), third-neighbor (3n) and fourth-neighbor (4n) triplet-dimer pairs.lized against the Néel ordered state when λ is increased. A recent study, however, suggested that ...
The magnetization process in the Shastry-Sutherland system is studied by using the thirdorder perturbation expansion. It is shown that the 1/3-plateau is realized by the second-order perturbation, which is not prevented by the off-diagonal part. In each subspace whose magnetization per dimer is less than 1/3, the lowest energy state is determined by a small but finite energy-gain due to the third-order correlated flip terms and there exists no plateau originating from the third-order effect. Our results are compared with those of the exact diagonalization method to discuss the validity of truncation of states in our perturbation theory.In the last couple of years there has been a growing interest in the phenomenon of magnetization plateaus in two-dimensional spin systems. For instance, the existence of plateaus in the magnetization curve is theoretically shown in the multiple-spin exchange model on the triangular lattice 1) and the Heisenberg antiferromagnet on the 1/5-depleted square lattice. 2) As for the experimental investigations of this phenomenon, recently, Kageyama et al. measured the magnetization curve of SrCu 2 (BO 3 ) 2 up to the magnetic field H = 45 T and observed intermediate plateaus at 1/8 (for 27.9 < H < 29.8 T) and 1/4 (for 37.0 < H < 41.0 T) of the full Cu moment. 3) In this compound, magnetic ions Cu 2+ (S = 1/2) are arranged as shown in Fig. 1.As a model for this system, Miyahara and Ueda 4) proposed a Heisenberg antiferromagnet with nearest neighbor (NN) and next nearest neighbor (NNN) couplings:which is equivalent to a model with the exact dimer ground state (for J <∼ 0.7J) proposed by Shastry and Sutherland. 5) It is reported that experimental data of the uniform susceptibility are reproduced by this model with J 100 K and J 68 K, where the exact diagonalization method is employed. 4) Another estimation J 83 K and J 55 K is obtained on the basis of the theoretical calculations of the spin gap and the susceptibility by the dimer expansion, where the disconnected dimer model defined by the first term in eq. (1) is taken as an unperturbed part and the interdimer coupling in the second term is taken as a perturbation. 6, 7)The existence of the exact dimer ground state in the present model stems from a symmetric relation in the interdimer coupling as pointed out in ref. 4. This leads to other interesting properties: an almost dispersionless triplet excitation spectrum 4) and the experimental observation of the second lowest band of triplet excitations. 8) It is natural to expect that the observed magnetization plateaus in SrCu 2 (BO 3 ) 2 originate from the symmetric properties of the interdimer coupling. 4) We now turn to the study of the magnetization process. At first we present the magnetization curves for the clusters composed of 8 ( √ 8 × √ 8), 10 ( √ 10 × √ 10) and 12 ( √ 8 × √ 18) dimers in Fig. 2, where we choose J /J(≡ λ) = 0.35. We find considerable system size dependence in the magnetization curves especially for M ≤ M s /2, where M s represents the saturation value of the magnetiz...
A frustrated Ising model on a diamond hierarchical lattice is studied. We obtain the exact partition function of this model and calculate the transition temperature, specific heat, entropy, magnetization, and ferromagnetic correlation function. Depending on the magnitude of a parameter giving the frustration, there exist three types of ground states: ferromagnetic, classical spin-liquid with highly developed short-range order, and paramagnetic. The dependence of the zero-temperature entropy on the frustration parameter has an infinite number of steps. The temperature dependence of the specific heat exhibits many peaks with decreasing temperature and entropy loss. The dominant spin configurations at low temperatures are also specified.KEYWORDS: classical spin-liquid ground state, frustration, diamond hierarchical lattice, Ising model, phase transition IntroductionMany magnetic systems with competing interactions exhibit frustration that leads to multiple ground states called spin glass or spin liquid even at the classical level. These interesting properties are attributable to the delicate balance of frustrated spin-spin interactions. If we can obtain the exact partition function of a model with frustrated magnetic phenomena, we can obtain a better understanding of complicated frustrated phenomena.In statistical physics, few exactly soluble models with phase transitions are known, such as two-dimensional Ising model, 1 eight-vertex model, 2 one-dimensional van der Waals gas model, 3 and a hierarchical model. 4 Since Berker and Ostlund proposed a hierarchical model related to the renormalization group method, 4, 5 many different models on hierarchical lattices have been proposed and developed.5-8 The magnetic and thermodynamic behaviors have been studied, 9-11 and the distribution of the zeros of the partition function and the critical exponents have also been obtained.10, 12 Using a hierarchical model with competing ferro-and antiferromagnetic interactions, McKay et al. studied the spin-glass behavior 13 and Nogueria et al. investigated the local magnetization.14 However, to the best of our knowledge, no hierarchical model describes the spin-liquid ground state. In this study, we consider a hierarchical Ising model with frustrated interactions that lead to the classical spinliquid ground state. This paper is organized as follows. We introduce our diamond hierarchical lattice and frustrated Ising model in §2 and describe our recursion relations in §3. Thermodynamic and ground state properties are described in §4 and §5, respectively. In §6, the temperature evolution of dominant spin configurations at low temperatures is discussed. In §7, we summarize the results obtained in this study.
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