Session Q8: Focus Session: Spin Liquids II

Absence of magnetic order and unusual spin dynamics in the spin liquid candidate Na$_4$Ir$_3$O$_8$

Na$_4$Ir$_3$O$_8$ has spin-1/2 iridium ions on a hyperkagome lattice of corner sharing triangles and is a candidate three-dimensional spin liquid [1], which has led to active theoretical study [2,3,4]. Previously reported measurements have shown no evidence for magnetic ordering down to temperatures around three orders of magnitude below the Curie-Weiss constant of $\sim 650$K [1]. We have carried out muon-spin relaxation measurements which exclude magnetic ordering above 55mK. The field dependence of the muon spin relaxation rate provides further information on the spin dynamics with a temperature-dependent crossover between power laws at intermediate fields suggesting that more than one energy scale is relevant to the fluctuations in this system. $[1]$ Y.\ Okamoto {\em et al.}, Phys.\ Rev.\ Lett. {\bf 99}, 137207 (2007). $[2]$ M.\ J.\ Lawler {\em et al.}, Phys.\ Rev.\ Lett. {\bf 100}, 227201 (2008). $[3]$ Yi Zhou {\em et al.}, Phys.\ Rev.\ Lett. {\bf 101}, 197201 (2008). $[4]$ E.\ J.\ Bergholtz {\em et al.}, Phys.\ Rev.\ Lett. {\bf 105}, 237202 (2010).   

Short range magnetic correlations and the possible role of frustration in the heavy-fermion CeCu$_{4}$Ga

Neutron scattering, longitudinal and transverse resistivity, heat capacity, ac susceptibility, and magnetization measurements on single and polycrystalline samples of the heavy-fermion compound CeCu$_{4}$Ga suggest that its hexagonal lattice and disorder due to Ga substitution may frustrate formation of the long-range magnetic order found in its parent CeCu$_{5}$. The absence of magnetic Bragg peaks in neutron diffraction data, field-dependent specific heat data, a Weiss temperature of $\sim $ 10 K, and diffuse scattering below\textit{ $\sim $}1 K which can be fit to an isotropic model describing spin-spin correlations between third through fifth nearest neighbors support this suggestion. Kondo behavior also plays a role in determining the physical properties of CeCu$_{4}$Ga.   

Mean field phase digram of the layered perovskite $\mathrm{(Li,Na)_2IrO_3}$ in the strong interaction limit: possible realization of spin liquid phases

We study the phase diagram of layered perovskite $\mathrm{(Li,Na)_2IrO_3}$ with an underlying honeycomb lattice structure in the strongly interacting limit. Because of the strong spin-orbit coupling of iridium, the effective spin exchange model is highly anisotropic and frustrated. We use the Schwinger fermion approach to map out the phase diagram of the model. At the mean field level several spin liquid phases are found: a gapless spin liquid , a chiral spin liquid, and a helical spin liquid phase. Moreover, in the strong exchange coupling limit we obtain a dimerized phase. The gapless spin liquid phase is characterized by Dirac nodes. In the chiral phase the Dirac nodes are gapped in the bulk, and the system possess a nonzero Chern number signifying existence of chiral modes along the boundary of the system. The helical phase preserves time reversal symmetry, has a bulk gap, and features helical gapless edge modes along boundary analogous to those in topological insulators with a nontrivial invariant. We further investigate the nature of the spin liquid phase by considering the gauge fluctuations above the mean field solution. The chiral spin liquid phase is stable as it breaks time reversal symmetry and acquires a nonzero Chern-Simon term in the effective low energy theory.   

Magnetic torque measurement in a gapless spin-liquid state of EtMe$_3$Sb[Pd(dmit)$_2$]$_2$

The organic Mott insulator EtMe$_3$Sb[Pd(dmit)$_2$]$_2$ with nearly identical 2D triangular lattice of $S = 1/2$ is the most promising candidate material of a quantum spin liquid, in which gapless spin excitations have been reported by the thermal transport measurements [1]. However, it remains unsettled whether the observed gapless excitations are magnetic ($S \geq 1/2$) or nonmagnetic ($S=0$). The magnetic torque measurement is a powerful tool to probe the magnetic properties down to very low temperature, because it is not affected by isotropic impurities. We report the magnetic torque measurements of EtMe$_3$Sb[Pd(dmit)$_2$]$_2$ down to 30 mK up to 32 T. A finite magnetic susceptibility is observed at the lowest temperature. Magnetization increases linearly with the magnetic field up to 32 T without exhibiting any anomarly. These results indicate that the gapless excitations reported by the thermal conductivity measurements are magnetic and that the present system is in an algebraic spin liquid phase. \\[4pt] [1] M. Yamashita \textit{et al.}, Science \textbf{328}, 1246 (2010).   

Field-Dependent Instability of the Candidate Quantum Spin Liquid in EtMe$_3$Sb[Pd(dmit)$_2$]$_2$ as Revealed by NMR

In recent years, the two-dimensional spin-$1/2$ triangular lattice of the organic salt EtMe$_3$Sb[Pd(dmit)$_2$]$_2$ has emerged as a candidate for the realization of a quantum spin liquid. Furthermore, thermal conductivity and nuclear magnetic resonance (NMR) experiments unveiled the presence of a low-temperature instability in the spin liquid state, the opening of a spin gap. We performed a detailed $^{13}$C NMR study on this material at low temperatures \mbox{($30$mK$\leq T\leq 1.5$ K)} and for a wide range of external magnetic field values \mbox{($B_0=0.6-9$T)}. In finite fields, a clear break in the temperature derivative of the spin-lattice relaxation is observed at a temperature $T_m(B_0)$, with $T_m$ following the empirical form $T_m(B_0)\sim \left| B_0-B_c\right| ^{1/2}$. Moreover, a uniform broadening of the NMR line for finite fields suggests the presence of a small field-induced staggered magnetization. We discuss these results in the context of possible instabilities, and existing thermodynamic data.   

Low temperature magnetodielectric coupling in the spin-liquid candidate $\kappa $-(BEDT-TTF)$_{2}$Cu$_{2}$(CN)$_{3}$

In the context of geometrical frustration of exchange coupling between spins on dimer orbitals, the possibility of a quantum spin-liquid state has been inferred for the quasi-2D organic Mott insulator $\kappa $-(BEDT-TTF)$_{2}$ Cu$_{2}$ (CN)$_{3}$. However, because the geometrical frustration effect is not strong, it has been proposed that the suppression of magnetic order could result from fluctuating quantum electric dipoles. Indeed, non-trivial charge degrees of freedom survive in this dimer Mott insulator as observed in dielectric measurements. Here, we report in-plane microwave dielectric measurements that reveal a coupling of the electric dipoles to the spins at low temperatures. Anomalies in the complex dielectric permittivity are observed at 6 K and around 3-4 K. The one at 6 K is in clear correlation with the thermal expansion measurements for which a second-order phase transition was inferred. The second dielectric anomaly is frequency dependent and cannot be associated to a phase transition; however, it is rapidly modified and ultimately suppressed by a magnetic field which effects are highly anisotropic. These results could be consistent with the scenario of a dipolar-spin liquid phase where spins couple to the dipoles through the interdimer charge fluctuation. Such a phase appears inhomogeneous since the dielectric anomalies are sensitive to thermal cycling and small pressures.   

Coulombic Quantum Liquids in Spin-1/2 Pyrochlores

We develop a non-perturbative ``gauge Mean Field Theory'' (gMFT) method to study a general effective spin-$1/2$ model for magnetism in rare earth pyrochlores. gMFT is based on a novel exact slave-particle formulation, and matches both the perturbative regime near the classical spin ice limit and the semiclassical approximation far from it. We show that the full phase diagram contains two exotic phases: a quantum spin liquid and a coulombic ferromagnet, both of which support deconfined spinon excitations and emergent quantum electrodynamics. Phenomenological properties of these phases are discussed.   

Spin Liquid Ground State of Spin-1/2 Square J$_{1}$-J$_{2}$ Heisenberg Model

We perform highly accurate density matrix renormalization group (DMRG) simulations to investigate the ground state properties of the spin-1/2 antiferromagnetic (AFM) square lattice Heisenberg J$_{1}$-J$_{2}$ model on numerous long cylinders with circumference up to 10 lattice spacings. Besides finding the conventional Neel AFM phase at small J$_{2}$/J$_{1}<$0.41 and the stripe AFM phase at large J$_{2}$/J$_{1}>$0.62, we establish an intriguing gapped quantum spin liquid phase within the parameter space 0.41$<$ J$_{2}$/J$_{1}<$0.62 by showing the absence of various conventional broken symmetries as well as by identifying topological features such as finite topological entanglement entropy and topological ``even-odd'' effect.   

Projected wave function study of Z$_2$ spin liquids on the kagome lattice for the spin-1/2 quantum Heisenberg antiferromagnet

Within the class of Gutzwiller projected fermionic wave functions, by using quantum variational Monte Carlo simulations, we investigated the energetics of all possible $Z_2$ spin liquids that can potentially occur as ground states of the nearest-neighbor S=1/2 quantum Heisenberg model on the Kagome lattice [1]. We conclusively show that all gapped and gapless $Z_2$ spin liquids are higher in energy compared to the U(1) gapless states in whose neighborhoods they lie. In particular, the most promising gapped $Z_2$ spin liquid (the so-called $Z_2[0,\pi]\beta$ state), conjectured to describe the ground state [2], is always higher in energy compared to the U(1) Dirac spin liquid. We also extended the U(1) Dirac state and the uniform RVB spin liquid to include next-nearest-neighbor hopping terms, and studied its local and global stability towards various valence bond crystal patterns. We found that a non-trivial 36-site VBC is stabilized upon addition of a small ferromagnetic exchange coupling [3]. \\[4pt] [1] Y. Iqbal, F. Becca, and D. Poilblanc, Phys. Rev. B 84, 020407(R) (2011)\\[0pt] [2] Y.-M. Lu, Y. Ran, and P.A. Lee. Phys. Rev. B 83, 224413 (2011)\\[0pt] [3] Y. Iqbal, F. Becca, and D. Poilblanc, Phys. Rev. B 83, 100404(R) (2011)   

Edge states of $(2+1)D$ $BF$ theory as the effective field theory of translational invariant ${\bf Z}_{2}$ spin liquids

We classify (2+1)-dimensional $BF$ field theory as the effective field theory for the achiral and translation invariant ${\bf Z}_{2}$ spin liquids on the square lattice. We use the pattern of the crystal momenta of $BF$ field theory to find the corresponding ${\bf Z}_{2}$ spin liquid. Then, we show that some classes of ${\bf Z}_{2}$ spin liquids can support gapless helical edge states depending on the projective symmetry group of the effective $BF$ theory. We supplement this effective theory by studying possible projective symmetry group of ${\bf Z}_{2}$ spin liquids.   

ESR lineshape for a two-dimensional spin liquid state with spinon Fermi surface

We propose that ESR experiment can be an informative probe of a putative spin-liquid ground state with a spinon Fermi surface. Our proposal is based on the assumption that in addition to strong and frustrated Heisenberg exchange interactions, the spins also interact via an asymmetric Dzyaloshinskii-Moriya interaction (DMI). We argue that in a spin-liquid state the DMI plays the role of a spin-orbit interaction well-known in low-dimensional conductors. Assuming further spin-orbit interaction to be of Rashba type, we calculate the ESR absorption spectrum of a two-dimensional fermion gas subject to Zeeman and spin-orbit fields. Finite spin-orbit coupling translates into a finite absorption spectrum width. Remarkably, the ESR signal diverges as an inverse square-root at the edges of the spectrum.   

Loop condensation in quantum dimer models

The formation of topological order is well understood in terms of the mechanism of loop condensation in systems with loop-like degrees of freedom. Various quantum dimer models posses exotic liquid states, including topologically ordered phases. Dimer models and can be mapped to loop models and these dimer liquid phases may be described as loop condensates. We present a numerical study of the geometric properties of the loop condensates in quantum dimer models and related models using classical Monte Carlo as well as ground state quantum Monte Carlo calculations.   

Gapless fractionalized vortex liquids in frustrated quantum antiferromagnets

The standard theoretical approach to gapless spin liquid phases of two-dimensional frustrated quantum antiferromagnets invokes the concept of fermionic slave particles into which the spin fractionalizes. As an alternate we explore new kinds of gapless spin liquid phases in frustrated quantum magnets with $XY$ anisotropy where the vortex of the spin fractionalizes into gapless itinerant fermions. The resulting gapless fractionalized vortex liquid phases are studied within a slave particle framework that is dual to the usual one. We demonstrate the stability of some such phases and describe their properties.   

Properties of the short ranged RVB wavefunction on non-bipartite lattices via Pfaffian Monte Carlo

We introduce a Monte Carlo scheme to investigate the nearest neighbor version of Anderson's resonating-valence-bond (RVB) spin-$1/2$ wave function on the kagome and on the triangular lattice. For the kagome lattice there exists a parent Hamiltonian for this state, but even in the absence of a known Hamiltonian, wave functions of RVB type are interesting as such. The corresponding RVB wave function on the square lattice has recently enjoyed much attention, and it was shown that earlier findings about the criticality of the { \it dimer}-liquid wave function on the square lattice qualitatively carry over to the analogous {\it spin}-liquid wave function on this lattice. On bipartite lattices, the spin-$1/2$ RVB wave functions are amenable to MC methods based on a loop gas picture. For other lattices, this method has a sign problem. We present a method that is free of this sign problem, making use of a Pfaffian presentation of the wave function in the orthogonal Ising basis. Our results for both open and periodic boundary conditions show that spin-spin and ``dimer-dimer'' type correlation function are exponentially decaying. Time and/or results permitting, we also comment on the behavior of the monomer correlations, and mention possible applications of our method to other problems.   

Exact Chiral Spin Liquid on the Ruby Lattice and Mean-Field Perturbations of Gamma Matrix Models

We report recent results on the study of an exactly solvable spin-3/2 model of the Kitaev type [A. Kitaev, Ann. Phys. 321, 2 (2006)] and related mean-field studies. The model is a Yao-Zhang-Kivelson Gamma Matrix (GM) extension [H. Yao, S.C. Zhang, and S.A. Kivelson, Phys. Rev. Lett.102, 217202 (2009)] on the ruby or rhombihexadeltille lattice. We show that the model admits an exact chiral spin liquid ground state solution with emergent free spinon excitations and interesting bandstructure. Specifically, we find gapped phases with chiral edge modes resulting from topologically non-trivial Chern numbers and gapless phases with interesting spinon Fermi surfaces. We have also studied the addition of perturbations to this and other GM Kitaev systems (kagome, square) which leads to weakly interacting spinons. We have applied a mean-field analysis to explore the interplay between these interactions and the gapless spin liquid phases.