Session A8: Focus Session: Frustrated Magnetism - Triangular Lattice

Spin Correlations in the quasi-triangular magnet, Cu$_{2}$(OH)$_{3}$NO$_{3}$

We have investigated the structural and magnetic properties of the spin S = $\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $ antiferromagnetic quasi-triangular lattice materials: Cu$_{2(1-x)}$Zn$_{2x}$(OH)$_{3}$NO$_{3}$ (0 $<$ x $<$ 0.65) using a.c. susceptibility, heat capacity [1,2] and neutron scattering. After a brief introduction to the bulk properties of this family of materials, we will discuss recent inelastic neutron scattering results from the pure and doped materials. The temperature dependence of the quasielastic scattering reveals an abundance of slow spin dynamics at elevated temperatures. This scattering collapses as the system is cooled through its ordering temperature (11 K) and a \textbf{Q}-independent mode is observed at finite energy. These results will be compared to those seen in other triangular systems with a Kagome motif. \\[4pt] [1] J. Wu, J.S. Wildeboer, F. Werner, A. Seidel, Z. Nussinov, and S.A. Solin, Europhysics Letters,~\textbf{93,}~67001 (2011).\\[0pt] [2] J. Wu, A. K. Gangopadhyay, P. Kanjanaboos and S. A. Solin, J. Phys.: Condens. Matter \textbf{22,} 334211 -- 334222 (2010).   

Optical spectroscopy of the Triangular Lattice Antiferromagnets CuCrO$_2$ and $\alpha$-CaCr$_2$O$_4$

We will compare and discuss our results obtained by optical spectroscopy on CuCrO$_2$ and $\alpha$-CaCr$_2$O$_4$. While CuCrO$_2$ is famous for its multiferroicity [1], in $\alpha$-CaCr$_2$O$_4$ a polarization can only be observed under the application of electric or magnetic field, despite having a closely related structure [2]. At near infrared and visible light frequencies we observe Cr$^{3+}$ crystal field absorptions and below T$_N$ excitons and exciton-magnon-transitions appear. The width of these exciton-magnon transitions is analyzed with respect to the existence of Z$_2$ vortices as proposed by Kojima et al. [3]. \\[4pt] [1] S. Seki et al., Phys. Rev. Lett. 101, 067240 (2008)\\[0pt] [2] K. Singh et al., Phys. Rev. B 84, 064129 (2011)\\[0pt] [3] N. Kojima et al., J. Phys. Soc. Jpn. 62, 4137 (1993)   

Quantum Spin Fluctuations for an Incommensurate Spiral

Quantum spin fluctuations are investigated for the incommensurate spiral state of a geometrically-frustrated triangular-lattice antiferromagnet. With increasing anisotropy, the average spin amplitude becomes larger but the spiral becomes more distorted. Quantum fluctuations enhance both the wavevector of the distorted spiral and the critical anisotropy above which it undergoes a first-order transition into a collinear state. An experimental technique is proposed to isolate the effects of quantum fluctuations from the classical distortion of the spiral. Results of this work are used to estimate the change in spin amplitude and ellipticity in the multiferroic state of CuFeO$_{2}$. Research sponsored by the Division of Materials Sciences and Engineering, U.S. Department of Energy under contract with UT-Battelle, LLC.   

Phase Diagram of the Spatially Anisotropic Heisenberg Model on a Triangular Lattice

The spatially anisotropic spin-1/2 Heisenberg model on a triangular lattice is examined using a renormalization group (RG) analysis in the limit of weak interchain coupling ($J'/J \ll 1$ or large anisotropy). In agreement with a previous similar study, a collinear antiferromagnetic (CAF) state is found for large anisotropies. Retaining a marginal term that was not previously included, elucidates the strong competition between spiral and CAF order and suggests a direct transition at a fairly large anisotropy to incommensurate spiral order with a significantly renormalized wave vector that smoothly connects to the commensurate spiral state at the isotropic point. The significant renormalization of the ordering vector of this incommensurate spiral phase is argued to explain why many numerical studies found spin liquid behavior close to the isotropic point, (i.e. $J'/J \sim 0.85$). The agreement between the predictions of our RG analysis for the correlation function of the staggered magnetization on next-nearest-neighbor chains with DMRG results for small anisotropies further supports this picture. Finally, the effect on the phase diagram of Dzyaloshinskii-Moriya interactions, relevant to Cs$_2$CuCl$_4$, will be discussed.   

Variational Monte Carlo study of quantum spin=1 liquid phases in the extended triangular-lattice Heisenberg model

Recent experiments in the compound Ba3NiSb2O9 [PRL 107, 197204] indicated that quantum-spin liquid phases in a spin S=1 anti-ferromagnet may exist. Motivated by these experiments, we construct quantum spin=1 liquid states with three flavors of fermionic spinons. We use variational Monte Carlo calculations to investigate the phase diagram of a triangular-lattice quantum Heisenberg model with single-ion anisotropy, bi-quadratic, and ring-exchange terms. We compare the energies of the spin-liquid states with conventional magnetically ordered states. We find that in some parameter ranges, an exotic gapless U(1) spin liquid is stabilized. In other parameter ranges, a BCS pairing instability with unconventional symmetry gaps out some of the spinons. We discuss our findings in relation with present and future experiments.   

Magnetic Structure and Spin Waves in Co$_{2}$(OD)$_{3}$Cl

We have examined the magnetic structure of Co$_{2}$(OD)$_{3}$Cl with magnetic Co$^{2+}$ (3$d^{7}$; s= 3/2) ions using neutron powder diffraction data. Magnetic structure that yields the best refinement factor is an ``umbrella''-type antiferromagnetic structure with ab-components of magnetic moments in the kagome plane forming a q=0 120$^{\circ}$ structure and the moments are canted out of the plane by $\sim $ 40$^{\circ}$. The magnetic moments in the triangular plane are aligned ferromagnetically along the c-axis. We have performed linear spin wave calculations considering the nearest neighbor interactions J within the kagome plane and J$_{F}$ between the kagome and the triangular plane. The effects of J$_{F}$/J and the canting angle to the spin wave dispersion was studied thoroughly, to find out an effective spin hamiltonian that explains our inelastic neutron scattering data with two prominent excitation modes centered at 3 meV and 19 meV. Single ion-type anisotropy was also included in the spin hamiltonian to study its effect to the spin wave excitation.   

Partially disordered state and spin-lattice coupling in an $S$=3/2 triangular lattice antiferromagnet Ag$_2$CrO$_2$

Ag$_2$CrO$_2$ consists of triangular lattice planes of CrO$_2$, which are well separated by the metallic Ag$_2$ layers. [1] This compound is an $S$=3/2 frustrated triangular lattice antiferromagnet without orbital degree of freedom. We performed neutron diffraction experiments on a powder sample of Ag$_2$CrO$_2$ on a neutron powder diffractometer HB-2A and a triple-axis neutron spectrometer HB-1, installed at HFIR in Oak Ridge National Laboratory. With decreasing temperature, a short-range 4-sublatice spin state develops. However, a long-range partially disordered state with 5 sublattices abruptly appears at $T\rm_N$=24 K, accompanied by a structural distortion, and persists at least down to 2 K. The spin-lattice coupling stabilizes the anomalous state, which is expected to appear only in limited ranges of further-neighbor interactions and temperature. It was found that the spin-lattice coupling is a common feature in triangular lattice antiferromagnets with multiple-sublattice spin states, since the triangular lattice is elastic. \\[4pt] [1] H. Yoshida {\it et al.}, to appear in J. Phys. Soc. Jpn.   

Quantum Critical Magnetization Behaviors of the Kagome- and Triangular-Lattice Antiferromagnets

Magnetization process of the S=1/2 isotropic Heisenberg antiferromagnets on the kagome and triangular lattices are studied. Data from numerical-diagonalization method up to 39-spin systems, are reexamined from the viewpoint of the derivative of the magnetization with respect to the magnetic field. We find that the behavior of the derivative around the 1/3 height of the magnetization saturation is quite different from the cases of typical magnetization plateaux for the kagome-lattice antiferromagnet. This new phenomenon is called the ``magnetization ramp'' [1]. We also compare it with the 1/3 magnetization plateau of the triangular antiferromagnet. The critical exponent analysis indicates a clear difference between the magnetization plateau and ramp [2]. In addition using the numerical diagonalization up to 42-spin systems we suggest that the kagome-lattice antiferromagnet has a gapless singlet-triplet excitation in the thermodynamic limit [3].\\[4pt] [1] H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 79 (2010) 053707.\\[0pt] [2] T. Sakai and H. Nakano: Phys. Rev. B 83 (2011) 100405(R).\\[0pt] [3] H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 80 (2011) 053704.   

Phase Diagram of Spin-1/2 Triangular Antiferromagnet in a Magnetic Field

We investigate the spin-1/2 quantum Heisenberg antiferromagnet on both two-dimensional triangular lattice and N-leg triangular ladder. The model describes isotropic Heisenberg chains (exchange constant $J$) coupled antiferromagnetically through interchain diagonal bonds (exchange constant $J$'). We study different regions using various controlled theoretical methods. Primarily we focus on the region slightly below saturation field. We show that the cone-coplanar state transition is absent, while commensurate-incommensurate transition emerges right below the saturation field for two-dimensional triangular lattice. We also determine the ground states in the limit $J'\ll J$, using one-dimensional bosonization, renormalization group methods and current algebra. Finally, we compare our theoretical result with DMRG result for N-leg ladder.   

Spin Correlations and Excitations in the Quasi-2D Triangular Bilayer Spin Glass LuCoGaO4

LuCoGaO4 is a layered magnetic-bilayer material wherein Co2+ magnetic moments and nonmagnetic Ga3+ ions are randomly distributed on planar triangular bilayers. This makes it an ideal case to study the interplay between geometric frustration, site disorder and low dimensionality and its influence on the magnetic ground of the system. This novel material has been grown for the first time in single crystal form at McMaster University. We have performed magnetization measurements, revealing a previously identified spin glass transition near Tf$\sim $19K, and a Curie Weiss temperature of Tcw$\sim $-96K, consistent with antiferromagnetic interactions[1]. We discuss time-of-flight neutron scattering measurements using SEQUOIA at SNS which elucidate the evolution of the static and dynamic spin correlations in LuCoGaO4 over a range of temperatures from T$<<$ Tf to T$>$Tcw. We observe quasielastic scattering at (1/3,1/3,L) positions in reciprocal space and rods of scattering along the c*-direction, consistent with short range antiferromagnetic correlations within decoupled bilayers, and which comfirm the 2-dimensional character of this system. Inelastic scattering measurements show a gapped $\sim $ 12 meV spin excitation which softens and broadens in energy, filling in the gap on a temperature scale of $\sim $ Tcw/2. [1] Cava et al., J. Solid State Chem. 140, 337 (1998).   

Emergence and lifting of frustration for dipolar molecules

Ising spins on a triangular lattice are like a ``harmonic oscillator'' of geometric frustration. We address the general question of how this frustration is lifted in a system of dipolar molecules confined to 2+epsilon dimensions. When the confinement to two dimensions is strong, the dipoles arrange in a triangular lattice. Upon reduction of the confinement, the dipoles undergo a transition out of the two dimensional plane, leading to an Ising degree of freedom. Spin--lattice coupling gives rise to an effective model in terms of quantum dimers. We discuss the resulting phases and the implications to current experiments with hetero-nuclear molecules.   

Spin dynamics in Kagom\'{e}-staircase multiferroic Ni$_3$V$_2$O$_8$

The coupling of magnetic and ferroelectric order has recently been drawing a lot of interest in condensed matter science given the fundamental interest and potential applications. Ni$_3$V$_2$O$_8$ (NVO) is a $S=1$ magnet with Ni$^{2+}$ ions arranged in a weakly coupled buckled Kagom\'{e}-staircase planes. Its complex magnetic phase diagram exhibits four different zero field incommensurate and commensurate magnetic phases below 10~K, with only one developing ferroelectric order. We present here a detailed study of low temperature magnetic dynamics in this geometrically frustrated spin system. Using single crystal inelastic neutron scattering technique we map the magnetic excitation spectra across all the magnetic phase transitions. We found that the spin-waves, well formed in the base temperature nonferroelectric phase at $T < 3$~K, are considerably damped when the system enters the low-temperature incommensurate phase with ferroelectric order ($3.9 < T < 6.3$~K). Finally, we discuss models that describe the coupling between magnetic and ferroelectric properties in the incommensurate magnets.   

Temperature-dependent Raman studies of the magnetodielectric vanadates: Ni$_{3}$V$_{2}$O$_{8}$, Co$_{3}$V$_{2}$O$_{8}$, and K$_{2}$V$_{3}$O$_{8}$

The kagome staircase compounds, Ni$_{3}$V$_{2}$O$_{8}$ and Co$_{3}$V$_{2}$O$_{8}$, are known to exhibit dielectric anomalies at the magnetic transitions; and inhomogenously mixed-valent K$_{2}$V$_{3}$O$_{8}$ has been shown to have a strong dependence of the dielectric constant on applied magnetic field. However, while strong spin-lattice coupling is generally associated with the complex phases and phenomena in these materials, there has been little microscopic information about this coupling. In this talk, we present temperature-dependent Raman scattering studies of Ni$_{3}$V$_{2}$O$_{8}$, Co$_{3}$V$_{2}$O$_{8}$, and K$_{2}$V$_{3}$O$_{8}$ that allow us to investigate the microscopic connection between the strong spin-lattice coupling and the magnetodielectric effects in these materials.   

Competing Interactions in the $S$=3/2 Kagome Staircase Co$_{3}$V$_{2}$O$_{8}$: Evolution of the Commensurate and Incommensurate Phases in a Magnetic Field

Single crystal neutron diffraction studies have been performed on the $S$~=~3/2 kagome staircase compound Co$_{3}$V$_{2}$O$_{8}$ with a magnetic field applied along the magnetization easy-axis ($\vec{H}$~$||$~$\vec{a}$). Previous zero-field measurements reported incommensurate, transversely polarized spin density wave (SDW) phases [with a temperature dependent propagation vector of $\vec{k}$~=~(0~$\delta$~0)] interspersed with multiple commensurate lock-in transitions at temperatures above the ferromagnetic ground state. For small applied fields along $\vec{a}$, $\mu_{0}H$~$\approx$~0.05~T, the commensurate lock-in phases are destabilized in favor of the incommensurate SDW, while slightly larger applied fields restore the commensurate lock-in phase with $\delta$~=~1/2 and yield a new commensurate phase with $\delta$~=~2/5. For measurements in an applied field, higher-order scattering is observed that corresponds to the second-harmonic.