Session Z8: Focus Session: Frustrated Magnetism - Magnetic insulators

Nature of the low energy excitations in the spin liquid state of Cs$_2$CuCl$_4$

We have performed detailed measurements as a function of temperature and applied magnetic field of the NMR rate in the spin liquid phase of the spin-1/2 frustrated antiferromagnet Cs$_2$CuCl$_4$. Comparison of the magnetization and relaxation rate to the spin-1/2 antiferromagnetic chain $\alpha$-CuNSal and to variational calculations using Gutzwiller-projected mean-field theory implies that the low energy excitations in Cs$_2$CuCl$_4$ are characterized by gapless fermionic excitations in the spin liquid phase at non-zero temperature and applied field. To investigate the ability of one dimensional versus two dimensional models to reproduce the low energy properties of Cs$_2$CuCl$_4$ \footnote{M.-A. Vachon {\it et al.}, New J. Phys. {\bf 13} 093029 (2011)} we compare the measured T$^{-1}$ NMR rate to a field theoretical description of a Luttinger liquid\footnote{H. K{\"u}hne {\it et al.}, Phys. Rev. B {\bf 83} 100407(R) (2011)}.   

ESR studies of the quasi-2D frustrated Cs2CuBr4

We report low-temperature electron spin resonance (ESR) studies of single-crystalline samples of Cs$_2$CuBr$_4$, a spin-1/2 antiferromagnet with a triangular spin-lattice structure. A remarkable angular dependence of the resonance field, including the splitting of the ESR line for some orientations of the magnetic field, and the presence of a gap in the ESR excitation spectrum at temperatures above $T_N\sim$ 1.3~K have been revealed. Our observations suggest that uniform Dzyaloshinskii-Moriya interaction affects the low-energy excitation spectrum in this frustrated compound. The results are compared with that obtained recently for the isostructural material Cs$_2$CuCl$_4$ [Povarov et al., Phys. Rev. Lett. 107, 037204 (2011)].   

NMR Spectra in 2D Anisotropic Triangle Lattice

The spin 1/2 Heisenberg antiferromagnet on the 2D anisotropic triangle lattice represents an important example of the frustrated quantum magnets. The materials Cs$_2$CuCl$_4$ and Cs$_2$CuBr$_4$ are relevant model systems of such a frustrated magnet. In these compounds magnetic fields can induce numerous exotic quantum states. The states are characterized by their spin texture, which can be measured by NMR. In this talk, we discuss how to model different spin arrangements, and compute the resulting NMR spectra.   

Distinct magnetic regimes through site-selective atom substitution in the frustrated quantum antiferromagnet Cs$_2$CuCl$_{4-x}$Br$_x$

A good realization of a spatially anisotropic spin 1/2 triangular lattice is provided by the isostructural layered compounds Cs2CuCl4 and Cs2CuBr4. In this talk, we shall present electronic structure calculations and magnetic susceptibility measurements[1] for the mixed systems Cs$_2$CuCl$_{4-x}$Br$_x$. We shall discuss the existence of three distinct concentration regimes which are separated by critical concentrations x$_{c1}$ = 1 and x$_{c2}$ = 2. The role of frustration effects at the critical concentrations will be highlighted. \\[4pt] [1] Cong et al. Phys. Rev. B 83, 064425 (2011)   

Two-stage orbital order and dynamical spin frustration in KCuF$_3$

Results from our x-ray and Raman scattering studies on KCuF$_3$, a model orbital order system, strongly link a low-temperature orbital order transition to a previously unidentified structural phase transition at 50 K. Raman scattering shows softening of phonon modes linked to F---ions that ceases upon a splitting of a degenerate E$_g$ mode at 50 K. This, along with the emergence of diffuse scattering around an orbital order Bragg peak at low temperature, suggests an onset of GdFeO$_3$-like octahedral tilting, which serves to stabilize the Neel spin order at 39 K. To explain these effects, we have added to the Kugel-Khomskii model a term for direct orbital exchange driven by electron-electron interactions and ligand distortions. This term creates a near degeneracy, which dynamically frustrates the spin order at high temperature, that is lifted by orbital---lattice interactions at low temperature.   

Evidence for pressure-tuned quantum structural fluctuations in KCuF$_{3}$

Frustrated magnetic systems are currently of great interest because of the possibility that these materials exhibit novel ground states such as orbital and spin liquids. We provide evidence in the orbital-ordering material KCuF$_{3}$ for pressure-tuned quantum melting of a static structural phase to a phase that dynamically fluctuates even near T $\sim $ 0K.[1] Pressure-dependent Raman scattering measurements show that applied pressure above P* $\sim $ 7kbar reverses a low temperature structural distortion in KCuF$_{3}$, resulting in the development of a $\omega \quad \sim $ 0 fluctuational (quasielastic) response near T $\sim $ 0K. This pressure-induced fluctuational response is temperature independent and exhibits a characteristic fluctuation rate that is much larger than the temperature, $\Gamma \quad >>$ K$_{B}$T, consistent with quantum fluctuations of the CuF$_{6}$ octahedra. We show that a previous developed model of pseudospin-phonon coupling qualitatively describes both the temperature- and pressure-dependent evolution of the Raman spectra of KCuF$_{3}$. Work supported by the U.S. Department of Energy under Award No. DE-FG02-07ER46453 and by the National Science Foundation under Grant NSF DMR 08-56321. \\[4pt] [1] S. Yuan et al., arXiv:1107.1433 (2011).   

Bose metal phase in a simple honeycomb lattice model

The existence of quantum spin liquids was first conjectured by Pomeranchuk some 70 years ago, who argued that frustration in simple antiferromagnetic theories could result in a Fermi-liquid-like state for spinon excitations. Here we present evidence that a simple quantum spin model on a honeycomb lattice hosts the long sought for Bose metal with a clearly identifiable Bose surface. The complete phase diagram of the model is determined via exact diagonalization and is shown to include four distinct phases separated by three quantum phase transitions. The stability of the Bose metal phase in the presence of other interactions is also discussed.   

Spin liquid ground state on the honeycomb Heisenberg spin 1/2 model with nearest and next nearest neighbor interaction

We numerically investigate the S=1/2 Heisenberg model on the honeycomb lattice with nearest ($J_1$) and next-nearest neighbor ($J_2$) interactions with the density matrix renormalization group (DMRG). We are able to study open cylinders with widths up to 12 lattice spacings. For $J_2/J_1$ near $0.3$, we find a spin liquid phase, bordered by an antiferromagnetic phase for smaller $J_2$ and a valence bond crystal for larger $J_2$. For the spin liquid phase we find finite spin singlet and triplet gaps and short spin-spin and bond-bond correlation lengths. We also find that the energy splitting between the two different topological sectors decays exponentially with the system width, consistent with a gapped Z2 spin liquid.   

Exact Spin Liquid Ground States of the Quantum Dimer Model on the Square and Honeycomb Lattices

We study a generalized quantum hard-core dimer model on the square and honeycomb lattices, allowing for first and second neighbor dimers. At generalized Rohksar-Kivelson points, the exact ground states can be constructed, and ground-state correlation functions can be equated to those of interacting 1+1 dimensional Grassmann fields. When the concentration of second neighbor dimers is small, the ground state correlations are shown to be short-ranged corresponding to a (gaped) spin liquid phase. On a 2-torus, the ground states exhibit fourfold topological degeneracy. On a finite cylinder we have found a dramatic even-odd effect depending on the circumference, and propose that this can be used as a numerical diagnostic of the phase, more generally.   

Physical solutions of the Kitaev honeycomb model

We have investigated the exact solution of the honeycomb model proposed by Kitaev and derived an explicit formula for the projector onto the physical subspace. The physical states are simply characterized by the parity of the total occupation of the fermionic eigenmodes. We consider a general lattice on a torus and show that the physical fermion parity depends in a nontrivial way on the vortex configuration and the choice of boundary conditions. In the vortex-free case with a constant gauge field we are able to obtain an analytical expression of the parity. For a general configuration of the gauge field the parity can be easily evaluated numerically, which allows the exact diagonalization of large spin models. We consider physically relevant quantities, as in particular the vortex energies, and show that their true value and associated states can be substantially different from the one calculated in the unprojected space, even in the thermodynamic limit.   

An FRG approach to the Heisenberg-Kitaev model

We apply the Functional Renormalization Group (FRG) method to frustrated spin-1/2 systems on two dimensional lattices such as the Heisenberg-Kitaev model. In order to be able to perform diagrammatic approximations, we use the pseudo fermion representation of spin operators. The FRG provides a systematic scheme for infinite order resummations in different interaction channels and hence allows to treat magnetic order and disorder effects on an equal footing. Calculating the magnetic susceptibility we identify different magnetically ordered and paramagnetic phases. In particular, the Heisenberg-Kitaev model exhibits magnetically ordered states well beyond the isotropic Heisenberg limit as well as an extended gapless spin-liquid phase around the highly anisotropic Kitaev limit. From the RG flow of the magnetic susceptibility we extract both, the Curie-Weiss scale and the critical ordering scale (for the magnetically ordered states). The Curie-Weiss scale changes sign, indicating a transition of the dominant exchange from antiferromagnetic to ferromagnetic, deep in the magnetically ordered regime. We discuss our results in light of recent experimental susceptibility measurements for Na$_2$IrO$_3$ and Li$_2$IrO$_3$.   

DMRG study of the spin-1/2 J1-J2 Honeycomb Antiferromagnetic Heisenberg Lattice

A possible quantum spin liquid state has been revealed in the Hubbard model on the honeycomb lattice by Quantum Monte Carlo study, which has stimulated a lot of recent interest in the quantum spin models on honeycomb lattice. In a recent paper (Phys. Rev. B 84, 024406 (2011)), the J1-J2-J3 Heisenberg model has been studied by exact diagonalization (ED), establishing a rich phase diagram with the Neel and plaquette valence-bond crystal (VBC) phases depending on couplings J2 and J3, while it remains unclear if there is a featureless spin liquid phase exists in such a model. Here, by implementing DMRG method with the full rotational SU(2) symmetry, we study the J1-J2 Heisenberg model on honeycomb lattice at larger sizes. We analyze the spin-spin and dimer-dimer correlation functions for different sizes and extrapolate the structure factors to the thermodynamic limit to determine the nature of the quantum state. Our results suggest that the intermediate phase (with J2/J1$\sim $0.2-0.35) may be a spin liquid phase with vanishing spin/dimer correlations at the large distance limit. The nature of such a phase will be explored based on comparison with variational wavefunctions.   

Searching for the Topological Degeneracy in the Hubbard Model on a Honeycomb Lattice

Recent quantum Monte Carlo calculations by Meng, et. al [1] have produced strong numerical evidence for a topological Z2 spin liquid on the Hubbard model on the honeycomb lattice. One feature of these spin liquids is the presence of a ground state degeneracy that depends on the manifold on which the system lives. Using finite temperature QMC calculations, we identify what states can live in the low-lying spectra, constraining the options for topologically degenerate ground states. In this talk we discuss these bounds and the implications for the Z2 spin liquid. [1] Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad, and A. Muramatsu, Quantum spin-liquid emerging in two-dimensional correlated Dirac fermions," Nature 464, 847 (2010).   

Emergent critical phase in a 2D frustrated Heisenberg model

It is well-known that a discrete Ising ($Z_2$) order parameter emerges in the frustrated square lattice $J_1$-$J_2$-Heisenberg model, which may be broken at finite temperature. We ask whether a different discrete symmetry $Z_q$ with $q>2$ may be found in other frustrated Heisenberg models, giving rise to a different finite temperature phase transition. Indeed, we identify an emergent $Z_6$ symmetry at low temperatures in a frustrated Heisenberg model on a 2D lattice that contains both the sites of the triangular and its dual honeycomb lattice. Our analysis combines a spin-wave expansion, susceptible to short-distance physics, with renormalization group arguments of the corresponding long-wavelength non-linear sigma model. Our results are even more appealing since the $Z_6$ clock model has a rich finite temperature phase diagram with two distinct Berezinskii-Kosterlitz-Thouless (BKT) phase transitions separated by a massless critical phase. We also discuss possible realizations of this phenomenon using cold-atoms in optical lattices.