Spin nematic order is investigated for an S ¼ 1 spin model on a triangular lattice with bilinearbiquadratic interactions. We particularly studied an antiferro nematic order phase with a three-sublattice structure, and magnetic properties are calculated at zero temperature by bosonization. Two types of bosonic excitations are found and we calculated dynamic and static spin correlations. One is a gapless excitation with linear energy dispersion around k $ 0, and this leads to a finite spin susceptibility at T ¼ 0 and would have a specific heat CðTÞ $ T 2 at low temperatures. These behaviors can explain many of the characteristic features of a recently discovered spin liquid state in the triangular magnet, NiGa 2 S 4 .KEYWORDS: antiferromagnets, triangular lattice, spin nematics, quadrupolar ordering DOI: 10.1143/JPSJ.75.083701The concept of a spin liquid was introduced thirty years ago by Anderson, 1,2) as a quantum disordered state in which the spin-spin correlation function does not show a longrange order but only short-range correlations. This issue has been studied intensively since then, both theoretically and experimentally. Frustration and quantum fluctuations are considered to be two ingredients needed to realize a spin liquid, and a spin-1/2 Heisenberg antiferromagnet on a triangular lattice was the first candidate. Many antiferromagnetic materials with a triangular lattice structure have been studied to see if they show a spin liquid behavior, but most of them turned out to exhibit some long-range order at low temperatures. The very few exceptions are a 3 He thin layer 3) and organic -(BEDT-TTF) 2 Cu 2 (CN) 3 , 4) and recently a new spin-liquid material NiGa 2 S 4 was discovered. 5)Whereas the former two are spin-1/2 systems, NiGa 2 S 4 is a spin-1 system.In NiGa 2 S 4 , spins of Ni 2þ ions form triangular layers with undistorted regular triangular units, and the layers are stacked along the c-axis. These layers are effectively decoupled, since the Ni-Ni distance is more than three times longer between layers. This system showed various lowtemperature properties that indicate a spin liquid state. First of all, no singularity was observed in specific heat down to the lowest temperature, T ¼ 0:3 K, indicating the absence of phase transitions. Moreover, the specific heat shows a power-law behavior, C $ T 2 , below 10 K. Secondly, the magnetic susceptibility gradually increased with decreasing temperature, and approached a finite value. Thirdly, a neutron experiment revealed a peak at an incommensurate wavevector Q $ ð= ffiffi ffi 3 p ; 0Þ. However, this was not a magnetic Bragg peak, and spin correlation length did not diverge but saturated to about $ 20 Å , only seven lattice units.The absence of magnetic long-range order and the presence of critical behaviors are necessary conditions to identify a spin liquid, and these were satisfied in NiGa 2 S 4 . These may suggest a finite spin gap instead, but this contradicts a nonvanishing temperature dependence of susceptibility. In this paper, we will examine the...

Using the (dynamical) density-matrix renormalization group method, we study the low-energy physics of three-leg antiferromagnetic Heisenberg model where the periodic boundary conditions are applied in the rung direction. We confirm that the spin excitations are always gapped as long as the intra-ring couplings form a regular triangle. From precise finite-size-scaling analyses of the spin gap and dimerization order parameter, we also find that the spin gap is collapsed by very small asymmetric modulation of the intra-ring couplings. Moreover, the dynamical spin structure factors on the intra-and inter-leg correlations are calculated. It is demonstrated that the low-lying structure of the inter-leg spectra is particularly affected by the asymmetric modulation.

Motivated by the recently discovered triangular magnet NiGa 2 S 4 , we investigated a spin antiferronematic phase in the spin-1 bilinear-biquadratic model. We obtained the energy dispersion of elementary excitations by bosonizing spin operators, and the dynamic and static spin correlations are calculated. Low energy properties of NiGa 2 S 4 are consistently explained using the scenario of nematic order, including the algebraic temperature dependence of the specific heat C(T ) ∼ T 2 and a finite value of the magnetic susceptibility at zero temperature. We also calculated the rate of relaxation of the nuclear magnetic resonance, and found T −1 1 ∝ T 3 , a scaling different from that in the magnetically ordered phase. Some comments are also given on how to identify antiferronematic orders experimentally.

The electron addition spectrum A + (k, ω) is obtained analytically for the one-dimensional (1D) supersymmetric t-J model with 1/r 2 interaction. The result is obtained first for a small-sized system and its validity is checked against the numerical calculation. Then the general expression is found which is valid for arbitrary size of the system. The thermodynamic limit of A + (k, ω) has a simple analytic form with contributions from one spinon, one holon and one antiholon all of which obey fractional statistics. The upper edge of A + (k, ω) in the (k, ω) plane includes a delta-function peak which reduces to that of the single-electron band in the low-density limit.71.10. Pm, 75.10.Jm, The concept of spinons and holons, both of which obey the fractional statistics [1], has turned out to be useful in approaching to 1D electron systems. In terms of these quasi-particles one can inquire into not only the low-energy and low-wavelength limit, but the global feature of the dynamics. Hence special interest has been cherished in the global dynamics from both theoretical and experimental points of view. For example, angle resolved photoemission [2] has revealed some evidence of the spin-charge separation by resolution of holon and spinon contributions. On the theoretical side, numerical studies have been performed for the 1D t-J model for a small number of lattice sites [3] and some structures have been ascribed to spinons and holons. For deeper understanding of the overall dynamics, demand is growing for analytic theory which can go to the thermodynamic limit. Partly analytic theory is available for the single-particle spectral functions of the t-J model in the J → 0 limit [4]. A notable feature is that a satellite band is observed whose intensity is comparable to that of the main band. It is natural to ask how the finite J influences the dynamics.In the supersymmetric t-J model with 1/r 2 interaction [5], spinons and holons appear in the simplest manner. In fact exact thermodynamics for the model [6] can be interpreted in terms of free spinons and holons. Ha and Haldane [7] analyzed numerical results for dynamics in finite-sized systems, and found that only a few number of elementary excitations contribute to spectral functions. They proposed a momentum-frequency region where each spectral function takes nonzero values in the thermodynamic limit, but they did not obtain the spectral functions themselves. Recently, exact results have been derived for a particular component [8], and for a particular momentum range of the spectral weight [9].In this paper we report on the analytical result of the electron addition spectrum for the t-J model at zero temperature. The electron addition spectral function is relevant to the angle resolved inverse photoemission spectroscopy. Our result constitutes the first analytical knowledge for dynamical quantities of lattice electrons with no restriction on the system size, the density and the momentum-frequency range. Although we cannot provide the formal proof for the exactness, the analy...