Session G15: Focus Session: Quasi-Triangular Frustrated Magnets

Phase transition in Ba$_{2}$Ti$_{13}$O$_{22}$ with Ti$^{3+}$ quasi-triangular lattice

In Ba$_{2}$Ti$_{13}$O$_{22}$, Ti$^{3+}$ ($3d^{1}$) ions form quasi-triangular lattices, and three layers of them (``trilayer'') compose a building block for the crystal structure. We found that this compound exhibits a phase transition at $T_{c} \sim 200$ K, below which the electrical resistivity increases and magnetic susceptibility decreases. We found by electron diffraction measurement that the space group changes at $T_{c}$ from $Cmce$ to $C2/m$, which means that one trilayer and the next trilayer become inequivalent. We also found that a pseudogap appears in the optical conductivity spectra below 0.3 eV at low temperatures. These experimental results suggest that the phase transition is caused by the formation of charge density wave (CDW). However, the almost $T$-linear dependence in the decrease of the magnetic susceptibility below $T_{c}$ is not what is observed in the conventional CDW state, and suggest an exotic nature of the state below $T_{c}$ in the present compound.   

Magnetic-filed and angular dependence of magnetism in the triangular Mott insulator k-(BEDT-TTF)2Cu2(CN)3 investigated by 13C NMR

The organic conductors, $\kappa$-(BEDT-TTF)$_{2}$X, are prototype for investigating Mott physics and spin frustrations. The X=Cu[N(CN)$_{2}$]Cl is a Mott insulator which undergoes an antiferromagnetic phase transition. On the other hand, title compound which has a triangular lattice does not show a long range magnetic ordering. This suppression is believed to deeply relate to strong spin frustrations. While X=Cu$_{2}$(CN)$_{3}$ does not show magnetic ordering, we previously reported anomalous behaviors in $^{13}$C NMR around 6 K, where heat capacity, thermal conductivity and lattice constant show anomalies as well. So, the 6 K anomaly is a key phenomenon for understanding the origin of absence of magnetic order. We have measured external-filed angular dependence of $^{13}$C NMR under magnetic fields up to 15 T for clarifying the origins of the line broadening and the $1/T_{1}$ anomaly around 6K. At room temperature, angular dependence of spectra is well explained by a crystal structure. We will show the detailed experimental results and discuss the low temperature states.   

Frustration dependence of elementary excitation in a quantum spin liquid

A quantum spin liquid state (QSL) with a magnetic gapless excitation has been found in the organic Mott insulator EtMe$_3$Sb[Pd(dmit)$_2$]$_2$ with nearly identical 2D triangular lattices of $S = 1/2$ [1]. To examine the nature of the QSL, it is essential to determine the phase diagram, especially how the gapless QSL evolves when the degree of frustration is changed. Although the gapless QSL is shown to be robust against deuteration of the cation EtMe$_3$Sb [1], the difference of frustration caused by the deuteration is not clear. We study the frustration dependence of the elementary excitation in the mixed-cation materials (Me$_4$Sb)$_{1-x}$(EtMe$_3$Sb)$_x$[Pd(dmit)$_2$]$_2$ in which the degree of frustration is directly reduced by mixing the smaller cation. Magnetic torque measurements showed that spin susceptibilities of the mixed cation ($x$ = 0.32 and 0.35) were temperature independent down to 30~mK and were almost the same with that of $x = 1$, indicating that the QSL exists as a quantum critical phase, rather than a point, when the frustration is varied. We will also present magnetic torque and thermal transport measurements of mixed-cation materials with different $x$.\\[4pt] [1] D. Watanabe \textit{et al.}, Nat. commun. \textbf{3}, 1090 (2012).   

Spin freezing in the quasi-triangular layered magnet, Cu$_2$(OH)$_3$NO$_3$

We have investigated the structural and magnetic properties of the spin S $=$ 1/2 antiferromagnetic quasi-triangular lattice materials: Cu$_{\mathrm{2(1-x)}}$Zn$_{\mathrm{2x}}$(OH)$_{3}$NO$_{3}$ (0 \textless\ x \textless\ 0.65) using a.c. susceptibility, heat capacity [1,2] and neutron scattering. The spin 1/2 Cu planes in these layered compounds form a very slightly ($\sim$ 1{\%}) distorted triangular lattice. We will briefly describe the techniques for synthesizing the hydrogenated, deuterated and intercalated forms of these compounds and also present a brief introduction to the bulk properties of this family of materials. We will discuss recent neutron scattering results from the pure compound. 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 several magnetic Bragg reflections and a Q-independent mode are observed at finite energy. We will contrast these results with those seen in triangular systems with a Kagome motif. \\[4pt] [1] J. Wu, et. al., Europhys Lett, 93, 67001 (2011).\\[0pt] [2] J. Wu, et. al., J. Phys.: Condens. Matter 22, 334211 -- 334222 (2010).   

Microscopic models of Pd(dmit)$_2$-based organic charge transfer salts

Organic charge transfer salts based on the molecule Pd(dmit)$_2$ display strong electronic correlations and geometrical frustration, leading to spin liquid, valence bond solid, and superconducting states, amongst other interesting phases. The low energy electronic degrees of freedom of these materials are often described by a single band model; a triangular lattice with a molecular orbital representing a Pd(dmit)$_2$ dimer on each site. We use \textit{ab initio} electronic structure calculations to construct and parametrize low energy effective model Hamiltonians for a class of Me$_{4-n}$Et$_{n}X$[Pd(dmit)$_2$]$_2$ ($X$=N, As, Sb) salts and investigate how well these systems are described by an anisotropic triangular lattice.   

Spin-liquid versus spiral-order phases in the anisotropic triangular lattice

We study the competition between magnetic and spin-liquid phases in the Hubbard model on the anisotropic triangular lattice, which is described by two hopping parameters $t$ and $t'$ in different spatial directions and is relevant for layered organic charge-transfer salts. By using a variational approach that includes spiral magnetic order, we provide solid evidence that a spin-liquid phase is stabilized in the strongly-correlated regime and close to the isotropic limit $t'/t=1$. Otherwise, a magnetically ordered spiral state is found, connecting the (collinear) N\'eel and the (coplanar) $120^\circ$ phases. The pitch vector of the spiral phase obtained from the unrestricted Hartree-Fock approximation is substantially renormalized in presence of electronic correlations, and the N\'eel phase is stabilized in a wide regime of the phase diagram, i.e., for $t'/t < 0.75$. We discuss these results in the context of organic charge-transfer salts   

Magnetic Soft Modes in the Distorted Triangular Antiferromagnet $\alpha$-CaCr$_{2}$O$_{4}$

We have explored the phase diagram and excitations of a distorted triangular lattice antiferromagnet. The unique two-dimensional distortion considered here is very different from the ``isosceles''-type distortion that has been extensively investigated. We show that suprisingly it is able to stabilize the 120$^{\circ}$ spin structure (typical of the undistorted triangular antiferromagnet) for a large range of exchange interaction values, with new structures found only for extreme distortions. A physical realization of this model is $\alpha$-CaCr$_{2}$O$_{4}$. Despite its highly symmetric 120$^{\circ}$ spin structure, the magnetic excitation spectrum of $\alpha$-CaCr$_{2}$O$_{4}$ is very complex. The unique pattern of nearest-neighbor exchange interactions as well as the substantial next-nearest-neighbor interactions place it close to the phase boundary of the 120$^{\circ}$ structure as is clearly revealed by the presence of low energy modes acting as soft modes of the neighboring structure. Indeed, fitting to linear spin-wave theory favors a set of exchange parameters within the nearby multi-$k$ phase in contradiction to the observed 120$^{\circ}$ order, and quantum fluctuations may be necessary to stabilize $\alpha$-CaCr$_{2}$O$_{4}$ within the 120$^{\circ}$ phase.   

Phase diagram and unusual magnetic excitations in distorted triangular lattice antiferromagnet $\alpha$-$CaCr_20_4$

While it is well known that the ground state of the isotropic Heisenberg model on a triangular lattice is the so called 120$^\circ$ structure, its appearance on the distorted triangular lattice is rather unusual. This case has been recently observed in the distorted triangular lattice antiferromagnet $\alpha$-CaCr$_2$O$_4$ [S. Toth et al, PRB 84, 054452 (2011)] which shows the onset of the 120$^\circ$ long-range magnetic order below $T_N = 42.6 K$. Recent neutron scattering experiments also revealed that this compound has unusual magnetic excitations with a dispersion with roton-like minima at momenta different from those corresponding to its 120$^\circ$-magnetic order [S. Toth et al, PRL 109, 127203 (2012)]. Motivated by these experimental findings, we calculate a magnetic phase diagram and excitation spectrum of anisotropic Heisenberg Hamiltonian on triangular lattice. We showed that at the parameters characterizing $\alpha$-CaCr$_2$O$_4$ compound, the ground state is indeed the 120$^\circ$-structure, however, other possible magnetic orderings are very close in energy. We compute the dispersion of magnetic excitations to order 1/S and compare it with the neutron scattering data.   

Spin dynamics of the triangular lattice antiferromagnet ${\alpha}$-SrCr$_2$O$_4$

We study the spin dynamics of the layered $S=3/2$ triangular lattice antiferromagnet ${\alpha}$-SrCr$_2$O$_4$ by means of inelastic neutron scattering on powder and single-crystal specimen. While the incommensurate long-range order observed below $T_{\rm N}$=43K resembles the usual 120$^\circ$-structure predicted for the perfect triangular lattice antiferromagnet, a spin-wave theory fit to the entire single-crystal dataset reveals strongly distorted exchange interactions. The extreme sensitivity of direct-exchange interactions to the small static Cr$^{3+}$-Cr$^{3+}$ distance variations reported by neutron diffraction, is quantitatively confirmed by {\it ab-initio} calculations that corroborate the spin-wave theory results.   

Low-lying magnetic excitations in the distorted triangular lattice antiferromagnet $\alpha$-CaCr$_2$O$_4$

We will discuss our results on $\alpha$-CaCr$_2$O$_4$ obtained by FIR and Terahertz spectroscopy. This compound orders below $T_{\mathrm{N}}$ = 42.6 K in a proper screw 120$^\circ$ magnetic order, but shows additional low-lying magnetic modes indicative for the vicinity of a more complex magnetic order [1-2]. Our spectra obtained by FTIR and THz-TD spectroscopy show several optical magnons appearing below the magnetic ordering with anomalous temperature dependence. We will discuss their polarization dependence and a possible magnetoelastic coupling of these modes.\\[4pt] [1] S. Toth et al., Phys. Rev. B 84, 054452 (2011)\\[0pt] [2] S. Toth et al., PRL 109, 127203 (2012)   

Raman Evidence for Symmetry Breaking in SrCr$_{2}$O$_{4}$

Raman spectra from 400 cm$^{-1}$ to 650 cm$^{-1}$ were acquired from single crystalline SrCr$_{2}$O$_{4}$ to probe magneto-elastic effects on its frustrated magnetism. The compound contains two-dimensional sheets of CrO$_{2}$, where spin-3/2 Cr atoms with direct exchange interactions form a distorted triangular lattice with helical magnetic order below 43K [1]. Even in the paramagnetic phase, the spectra show mode splitting beyond predictions for space-group P$_{mmn}$ that describes powder x-ray diffraction data. This splitting occurs at the 480 cm$^{-1}$ A$_{g}$ mode and is enhanced below T$_{N}$, which suggests it may be associated with magneto-elastic effects. \\[4pt] [1] S E Dutton, E Climent-Pascual, P W Stephens, J P Hodges, A Huq, C L Broholm, and R J Cava, J. Phys.: Condens. Matter 23 (2011) 246005   

Geometric frustration on a 1/9$^{th}$ site depleted triangular lattice

In the searches both for new spin liquid and spin ice (artificial and macroscopic) candidates, geometrically frustrated two-dimensional spin systems have played a prominent role. Here we present a study of the classical antiferromagnetic Ising (AFI) model on the sorrel net, a 1/9$^{th}$ site depleted and 1/7$^{th}$ bond depleted triangular lattice. The AFI model on this corner-shared triangle net is found to have a large residual entropy per spin $\frac{S}{N} = 0.48185 \pm 0.00008$, indicating the sorrel net is highly geometrically frustrated. Anticipating that it may be difficult to achieve perfect bond depletion, we investigate the physics resulting from turning back on the depleted bonds ($J_2$). We present the phase diagram, analytic expressions for the long range partially ordered ground state spin structure for antiferromagnetic $J_2$ and the short range ordered ground state spin structure for ferromagnetic $J_2$, the magnetic susceptibility and the static structure factor. We briefly comment on the possibility that artificial spin ice on the sorrel lattice could by made, and on a recent report [T. D. Keene $\it{et}$ $\it{al.}$, Dalton Trans. ${\bf{40}}$ 2983 (2011)] of the creation of a 1/9$^{th}$ depleted cobalt hydroxide oxalate.   

Hints of possible spin-liquid state in the spin-1/2 triangular-lattice Heisenberg antiferromagnet

We calculate magnetic susceptibility of the triangular-lattice quantum antiferromagnet in the correlated paramagnet regime and reveal surprising microscopic correspondence between quantum and classical models at all accessible temperatures $T>0.375J$. Namely, we observe a perfect match between the quantum static (zero Matsubara frequency) response $\chi (r)$, where $r$ is the spatial coordinate, and its classical counterpart calculated at temperature $T_{cl}(T)$. The correspondence curve is rather featureless and smoothly extrapolates to a finite value of $T_{cl} = 0.28J$ when $T/J \to 0$. If this extrapolation indeed holds true, then finite value of $T_{cl}(0)$ implies that spins are not ordered in the ground state and form a spin liquid. Existing numerical evidence would $not$ be in contradiction with the spin liquid state because the spin correlation length for the classical Heisenberg model at $T_{cl} \approx 0.28J$ is $>1000$ lattice periods and simulations dealing with small system sizes $L< 10$ would misidentify the ground state as ordered. Our results are based on the high-order skeleton Feynman diagrams within the fermionization framework.