Classical Degeneracy Research Articles

Field control of symmetry-broken and quantum disordered phases in frustrated moiré bilayers with population imbalance

We determine the ground states and excitation spectra of the paradigmatic four-flavor Heisenberg model with nearest- and next-nearest-neighbor exchange couplings on the triangular lattice in a field controlling the population imbalance of flavor pairs. Such a system arises in the strongly correlated limit of moiré bilayers of transition metal dichalcogenides in an electric displacement field or in-plane magnetic field, and can be simulated via ultracold alkaline-earth atoms. We argue that the field tunes between effective SU(4) and SU(2) symmetries in the balanced and fully polarized limits and employ a combination of mean-field calculations, flavor-wave theory, and exact diagonalization to analyze the intermediate, imbalanced regime. We find different symmetry-broken phases with simultaneous spin and excitonic order depending on the field and next-nearest-neighbor coupling. Furthermore, we demonstrate that there is a strongly fluctuating regime without long-range order that connects candidate spin liquids of the SU(2) and SU(4) limit. The strong fluctuations are facilitated by an extensive classical degeneracy of the model, and we argue that they are also responsible for a strong polarizability at 1/3 polarization that survives from the mean-field level to the exact spectrum. Published by the American Physical Society 2024

Classical and quantum phases of the pyrochlore S=12 magnet with Heisenberg and Dzyaloshinskii-Moriya interactions

We investigate the ground state and critical temperature phase diagrams of the classical and quantum $S=1/2$ pyrochlore lattice with nearest-neighbor Heisenberg and Dzyaloshinskii-Moriya interactions (DMI). We consider ferromagnetic and antiferromagnetic Heisenberg exchange as well as direct and indirect DMI. Classically, three ground states are found: all-in/all-out, ferromagnetic and a locally ordered $XY$ phase, known as $\Gamma_5$, which displays an accidental classical U(1) degeneracy. Quantum zero-point energy fluctuations are found to lift the classical ground state degeneracy and select the $\psi_3$ state in most parts of the $\Gamma_5$ regime. Likewise, thermal fluctuations treated classically, select the $\psi_3$ state at $T=0^+$. In contrast, classical Monte Carlo finds that the system orders at $T_c$ in the $\psi_2$ state of $\Gamma_5$ for antiferromagnetic Heisenberg exchange and indirect DMI with a transition from $\psi_2$ to $\psi_3$ at a temperature $T_{\Gamma_5} <T_c$. The same method finds that the system orders via a single transition at $T_c$ directly into the $\psi_3$ state for most of the region with ferromagnetic Heisenberg exchange and indirect DMI. Such ordering behavior at $T_c$ for the $S=1/2$ quantum model is corroborated by high-temperature series expansion. To investigate the $T=0$ quantum ground states, we apply the pseudo-fermion functional renormalization group (PFFRG). The quantum paramagnetic phase of the pure antiferromagnetic $S=1/2$ Heisenberg model is found to persist over a finite region in the phase diagram for both direct or indirect DMI. We find that near the boundary of ferromagnetism and $\Gamma_5$ antiferromagnetism the system may potentially realize a quantum ground state lacking conventional magnetic order. Otherwise, for the largest portion of the phase diagram, PFFRG finds the same ordered phases as in the classical model.

Magnetic excitations, phase diagram, and order-by-disorder in the extended triangular-lattice Hubbard model

The dynamical structure factor is an important observable of quantum magnets but due to numerical and theoretical limitations, it remains a challenge to make predictions for Hubbard-like models beyond one dimension. In this work, we study the magnetic excitations of the triangular lattice Hubbard model including next-nearest-neighbor hopping. Starting from the expected ${120}^{\ensuremath{\circ}}$ and stripe magnetic orders, we compute the magnon spectra within a self-consistent random phase approximation. In the stripe phase, we generically find accidental zero modes related to a classical degeneracy known from the corresponding ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. We extend the order-by-disorder mechanism to Hubbard systems and show how quantum fluctuations stabilize the stripe order. In addition, the frustration-induced condensation of magnon modes allows us to map out the entire phase diagram which is in remarkable agreement with recent numerical works. We discuss connections to experiments on triangular lattice compounds and the relation of our results to the proposed chiral spin liquid phase.

Oscillations of a partially wetting bubble

We study the linear stability of a compressible sessile bubble in an ambient fluid that partially wets a planar solid support, where the gas is assumed to be an ideal gas that obeys the adiabatic law. The frequency spectrum is computed from an integrodifferential boundary value problem and depends upon the wetting conditions through the static contact angle$\alpha$, the dimensionless equilibrium bubble pressure$\varPi$, and the contact-line dynamics that we assume to be either (i) pinned or (ii) freely moving with fixed contact angle. Corresponding mode shapes are defined by the polar-azimuthal mode number pair$[k,\ell ]$with$k+\ell =\mathbb {Z}^{+}_{even}$. We report instabilities to the (i)$[0,0]$breathing mode associated with volume change, and (ii)$[1,1]$mode that is linked to horizontal centre-of-mass motion of the bubble. Stability diagrams and instability growth rates are computed, and the respective instability mechanisms are revealed through an energy analysis. The zonal$\ell =0$modes are associated with volume change, and we show that there is a complex dependence between the classical volume and shape change modes for wetting conditions that differ from neutral wetting$\alpha =90^\circ$. Finally, we show how the classical frequency degeneracy for the Rayleigh–Lamb modes of the free bubble splits for the azimuthal modes$\ell \neq 0,1$.

Ground state of the S=12 triangular lattice Heisenberg-like antiferromagnet Ba3CoSb2O9 in an out-of-plane magnetic field

Spin-$1/2$ triangular lattice Heisenberg antiferromagnet has been accepted as an ideal system for quantum magnetism studies and quantum simulations. This system, for which the classical ground state degeneracy is lifted by quantum fluctuations, exhibits a series of novel spin structures for a field applied in-plane and out-of-plane. It has been found that both anisotropy and interlayer interaction play an important role in the stabilization of the spin configurations in a magnetic field. Conversely, the phase transitions and spin-state evolution in a field along various orientations can provide a deep insight into physics of the triangular lattice Heisenberg antiferromagnet system. While the quantum magnetization process in an in-plane field has been studied extensively, the ground state evolution in the field along the $c$ axis requires further investigation. Here we performed high field magnetization and neutron scattering investigations on a model system of spin-$1/2$ triangular lattice Heisenberg antiferromagnet ${\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}$ with field along $c$ axis and with a small offset angle. For $H\ensuremath{\parallel}c$, the magnetization reveals a narrow plateau prompting a UUD-like phase, which could be suppressed by tilting the field away from the $c$ axis. From the neutron data, a phase transition ${\ensuremath{\mu}}_{0}{H}_{c1}\ensuremath{\sim}12$ T is detected and interpreted as a transition from an umbrella to a coplanar phase. Around about 22.5 T (${\ensuremath{\mu}}_{0}{H}_{c2}$) for $H\ensuremath{\parallel}c$, another transition is observed which might be attributed to a transition between the coplanar $V$ and ${V}^{\ensuremath{'}}$ phases based on a comparison with the calculations and previous results. Theoretical calculations using the large-size cluster mean-field plus scaling method predicts a similar phase evolution as the previous semiclassical analysis, and agree with experiment well. The discrepancies between theory and experiment are also discussed, suggesting the physics of a triangular lattice Heisenberg antiferromagnet in a field along $c$ axis has not been fully unraveled.

Effective field theory approach for the S=32 bilayer honeycomb antiferromagnet

The spin-$\frac{3}{2}$ Heisenberg antiferromagnet on the bilayer honeycomb lattice is a minimal model to describe the magnetic behavior of ${\mathrm{Bi}}_{3}{\mathrm{Mn}}_{4}{\mathrm{O}}_{12}({\mathrm{NO}}_{3})$. We study this model with frustrating interlayer second-neighbor couplings taking into account quantum and thermal fluctuations. We use a path integral formulation in terms of coherent states to describe the low-energy physics of the model. We show that for a particular point in the parameter space, close to the experimental estimated couplings, a continuum classical degeneracy is lifted by both quantum and thermal fluctuations, and a collinear state is then selected by an order by disorder mechanism. Our results provide a global perspective in the understanding of the experimental observations.

Spin liquid in twisted homobilayers of group-VI dichalcogenides

Twisted transition metal dichalcogenide (TMD) homobilayers have recently emerged as a powerful platform for studying correlated insulating states. In the strongly correlated limit, we construct an effective spin Hamiltonian on a honeycomb lattice that includes the Heisenberg interaction and nonsymmetric interactions such as a Dzyaloshinskii-Moriya interaction and a Kane-Mele coupling for the Mott-insulating phase at half-filling. For the twisted TMD homobilayers, the spin-orbit coupling in the Hubbard model, which is expected to induce the antisymmetric exchange couplings in the effective spin Hamiltonian, is a highly tunable and experimentally accessible quantity that can be tuned by an applied electric field. In this study, we investigate classical and quantum phase diagrams of the effective spin Hamiltonian using analytical and numerical methods. We show that the model exhibits a rich classical phase diagram including an antiferromagnetic (AFM) phase, a planar spiral ordered phase with high classical degeneracy, a $z$-AFM phase, a noncoplanar phase, a noncollinear phase, and a 120$^{\circ}$-AFM phase. In the quantum treatment, we calculate low-energy magnon excitation spectrum, ground state energy, and static spin structure factor using linear spin-wave theory and density matrix renormalization group methods to compose the quantum phase diagram of the effective spin Hamiltonian. Beyond the Heisenberg interaction, we find that the existence of these antisymmetric couplings is responsible for the quantum spin liquid, $z$-AFM, noncoplanar, and 120$^{\circ}$ phases. Twisted TMD homobilayers, therefore, offer rich platforms for realizing rich phases of matter such as quantum spin liquid, noncoplanar, and 120$^{\circ}$, resulting from the spin-orbit coupling.

Generic spiral spin liquids

Spiral spin liquids are unique classical spin liquids that occur in many frustrated spin systems, but do not comprise a new phase of matter. Owing to extensive classical ground-state degeneracy, the spins in a spiral spin liquid thermally fluctuate cooperatively from a collection of spiral configurations at low temperatures. These spiral propagation wavevectors form a continuous manifold in reciprocal space, i.e., a spiral contour or a spiral surface, that strongly governs the low-temperature thermal fluctuations and magnetic physics. In this paper, the relevant spin models conveying the spiral spin liquid physics are systematically explored and the geometric origin of the spiral manifold is clarified in the model construction. The spiral spin liquids based on the dimension and the codimension of the spiral manifold are further clarified. For each class, the physical properties are studied both generally and for specific examples. The results are relevant to a wide range of frustrated magnets. A survey of materials is given and future experiments are suggested.

Evolution of magnetic field induced ordering in the layered quantum Heisenberg triangular-lattice antiferromagnet Ba3CoSb2O9

Quantum fluctuations in the effective spin one-half layered structure triangular-lattice quantum Heisenberg antiferromagnet Ba$_3$CoSb$_2$O$_9$ lift the classical degeneracy of the antiferromagnetic ground state in magnetic field, producing a series of novel spin structures for magnetic fields applied within the crystallographic ab plane. Theoretically unresolved, however, are the effects of interlayer antferromagnetic coupling and transverse magnetic fields on the ground states of this system. To address these issues, we have used specific heat, neutron diffraction, thermal conductivity, and magnetic torque measurements to map out the phase diagram as a function of magnetic field intensity and orientation relative to the crystallographic ab plane. For H parallel to the ab plane, we have discovered an additional, previously unreported magnetic-field-induced phase transition at low temperature and an unexpected tetracritical point in the high field phase diagram, which - coupled with the apparent second-order nature of the phase transitions - eliminates several theoretically proposed spin structures for the high field phases. Our calorimetric measurements as a function of magnetic field orientation are in general agreement with theory for field-orientation angles close to plane parallel but diverge at angles near plane perpendicular; a predicted convergence of two phase boundaries at finite angle and a corresponding change in the order of the field induced phase transition is not observed experimentally. Our results emphasize the role of interlayer coupling in selecting and stabilizing field-induced phases, provide new guidance into the nature of the magnetic order in each phase, and reveal the need for new physics to account for the nature of magnetic ordering in this archetypal 2D spin one-half triangular lattice quantum Heisenberg antiferromagnet.

Order by disorder in frustration-free systems: Quantum Monte Carlo study of a two-dimensional PXP model

In this study, an order by disorder mechanism has been proposed in a two-dimensional PXP model, where the extensive degeneracy of the classical ground-state manifold is due to strict occupation constraints instead of geometrical frustrations. By performing an unbias large-scale quantum monte carlos simulation, we find that local quantum fluctuations, which usually work against long-range ordering, lift the macroscopic classical degeneracy and give rise to a compressible ground state with charge-density-wave long-range order. A simple trial wavefunction has been proposed to capture the essence of the ground-state of the two-dimensional PXP model. The finite temperature properties of this model have also been studied, and we find a thermal phase transition with an universality class of two-dimensional Ising model.

Hierarchy of energy scales and field-tunable order by disorder in dipolar-octupolar pyrochlores

Dipolar-octupolar pyrochlore magnets in a strong external magnet field applied in the [110] direction are known to form a `chain' state, with subextensive degeneracy. Magnetic moments are correlated along one-dimensional chains carrying effective Ising degrees of freedom which are noninteracting on the mean-field level. Here, we investigate this phenomenon in detail, including the effects of quantum fluctuations. We identify two distinct types of chain phases, both featuring distinct subextensive, classical ground state degeneracy. Focussing on one of the two kinds, we discuss lifting of the classical degeneracy by quantum fluctuations. We map out the ground-state phase diagram as a function of the exchange couplings, using linear spin wave theory and real-space perturbation theory. We find a hierarchy of energy scales in the ground state selection, with the effective dimensionality of the system varying in an intricate way as the hierarchy is descended. We derive an effective two-dimensional anisotropic triangular lattice Ising model with only three free parameters which accounts for the observed behavior. Connecting our results to experiment, they are consistent with the observation of a disordered chain state in Nd$_2$Zr$_2$O$_7$. We also show that the presence of two distinct types of chain phases has consequences for the field-induced breakdown of the apparent $U(1)$ octupolar quantum liquid phase recently observed in Ce$_2$Sn$_2$O$_7$.

Description of unimolecular reaction rates of Lennard-Jones clusters

The unimolecular rate constant of Lennard-Jones clusters with N = 54 − 56 atoms has been investigated in constant energy molecular dynamics simulations and compared with the detailed balance predictions. Decays were found to be statistical and absolute rate constants agree quantitatively with detailed balance predictions. The level densities required for the comparison of simulated and calculated rate constants were obtained from separate simulations using the integrated microcanonical temperature. The quantitative agreement between simulations and theory requires that the classical degeneracy of the ground state is included into the level densities.

Effective theories for quantum spin clusters: Geometric phases and state selection by singularity

Magnetic systems with frustration often have large classical degeneracy. We show that their low-energy physics can be understood as dynamics within the space of classical ground states. We demonstrate this mapping in a family of quantum spin clusters where every pair of spins is connected by an $XY$ antiferromagnetic bond. The dimer with two spin-$S$ spins provides the simplest example, it maps to a quantum particle on a ring ($S^1$). The trimer is more complex, equivalent to a particle that lives on two disjoint rings ($S^1\otimes \mathbb{Z}_2$). It has an additional subtlety for half-integer $S$ values, wherein both rings must be threaded by $\pi$-fluxes to obtain a satisfactory mapping. This is a consequence of the geometric phase incurred by spins. For both the dimer and the trimer, the effective theory can be seen from a path-integral-based derivation. This approach cannot be extended to the quadrumer which has a non-manifold ground state space, consisting of three tori that touch pairwise along lines. In order to understand the dynamics of a particle in this space, we develop a tight-binding model with this connectivity. Remarkably, this successfully reproduces the low-energy spectrum of the quadrumer. For half-integer spins, a geometric phase emerges which can be mapped to two $\pi$-flux tubes that reside in the space between the tori. The non-manifold character of the space leads to a remarkable effect - the dynamics at low energies is not ergodic as the particle is localized around singular lines of the ground state space. The low-energy spectrum consists of an extensive number of bound states formed around singularities. Physically, this manifests as an order-by-disorder-like preference for collinear ground states. However, unlike order-by-disorder, this `order by singularity' persists even in the classical limit. We discuss consequences for field theoretic studies of magnets.

Tunneling-induced restoration of classical degeneracy in quantum kagome ice

Quantum effect is expected to dictate the behaviour of physical systems at low temperature. For quantum magnets with geometrical frustration, quantum fluctuation usually lifts the macroscopic classical degeneracy, and exotic quantum states emerge. However, how different types of quantum processes entangle wave functions in a constrained Hilbert space is not well understood. Here, we study the topological entanglement entropy (TEE) and the thermal entropy of a quantum ice model on a geometrically frustrated kagome lattice. We find that the system does not show a $Z_2$ topological order down to extremely low temperature, yet continues to behave like a classical kagome ice with finite residual entropy. Our theoretical analysis indicates an intricate competition of off-diagonal and diagonal quantum processes leading to the quasi-degeneracy of states and effectively, the classical degeneracy is restored.

Cubic symmetry and magnetic frustration on the fcc spin lattice in K2IrCl6

Cubic crystal structure and regular octahedral environment of Ir$^{4+}$ render antifluorite-type K$_2$IrCl$_6$ a model fcc antiferromagnet with a combination of Heisenberg and Kitaev exchange interactions. High-resolution synchrotron powder diffraction confirms cubic symmetry down to at least 20 K, with a low-energy rotary mode gradually suppressed upon cooling. Using thermodynamic and transport measurements, we estimate the activation energy of $\Delta\simeq 0.7$ eV for charge transport, the antiferromagnetic Curie-Weiss temperature of $\theta_{\rm CW}\simeq -43$ K, and the extrapolated saturation field of $H_s\simeq 87$ T. All these parameters are well reproduced \textit{ab initio} using $U_{\rm eff}=2.2$ eV as the effective Coulomb repulsion parameter. The antiferromagnetic Kitaev exchange term of $K\simeq 5$ K is about one half of the Heisenberg term $J\simeq 13$ K. While this combination removes a large part of the classical ground-state degeneracy, the selection of the unique magnetic ground state additionally requires a weak second-neighbor exchange coupling $J_2\simeq 0.2$ K. Our results suggest that K$_2$IrCl$_6$ may offer the best possible cubic conditions for Ir$^{4+}$ and demonstrates the interplay of geometrical and exchange frustration in a high-symmetry setting.

Classical mean-field, spin-wave, and perturbative continuous unitary transformation analyses of the squaric acid system H2SQ

We study an organic Hydrogen bonded material $\rm{H_2SQ}$ analytically and map out the phase diagram as well as low energy excitations in the relevant parameter space. At zeroth order the dynamics is governed by plaquette interaction (product of $\sigma_z$ over a plaquette) which defines a $Z_2$ gauge theory and a deconfinement phase satisfying "ice rules". The system is studied under additional interactions such as an external Zeeman field (with strength $K$) in $x$-direction, a inter-molecular interaction (with strength $J_1$) and a dipole-dipole interaction with strength $J_2$ such that $K>J_1>J_2$. The effect of dipole-dipole interaction removes the local $Z_2$ symmetry and gives rise to four global degenerate states with Ferroeletric order. Using meanfield analysis we chart out the phase diagram for classical version of the model and find $K_c$ which defines the transition from disordered phase to ordered phase in $J_1, K$ plane for various values of $J_2$. We find that presence of $J_1$ and $J_2$ tends to stabilize the deconfined phases. Over the classical ground states we perform spin-wave analysis and surprisingly find that quantum fluctuations does not remove the classical degeneracy at all at quadratic level. The spin-wave spectrum is found for the four global degenerate ground states which shows both gapped spectrum and gapless spectrum (near the vicinity of CDT transition) for $J_2=0$. However for $J_2$ finite, the spectrum is always gapped. We perform PCUT analysis to improve the results of spin-wave analysis and calculate ground state energy and one particle dispersion and gap at high symmetry point. Using this we draw the phase boundary between confined and deconfined phase in the $K-J_1$ plane. The effect of $J_2$ is also discussed in the resulting phase boundary.

Short-range order in the quantum XXZ honeycomb lattice material BaCo2(PO4)2

We present observations of highly frustrated quasi two-dimensional (2D) magnetic correlations in the honeycomb lattice layers of the S$_{eff}$ = 1/2 compound $\gamma$-BaCo$_2$(PO$_4$)$_2$ ($\gamma$-BCPO). Specific heat shows a broad peak comprised of two weak kink features at $T_{N1} \sim$ 6 K and $T_{N2} \sim$ 3.5 K, the relative weights of which can be modified by sample annealing. Neutron powder diffraction measurements reveal short range quasi-2D order that is established below $T_{N1}$ and $T_{N2}$, at which two separate, incompatible, short range magnetic orders onset: commensurate antiferromagnetic correlations with correlation length $\xi_c = 60\pm2$ \AA \ ($T_{N1}$) and in quasi-2D helical domains with $\xi_h = 350 \pm 11$ \AA \ ($T_{N2}$). The ac magnetic susceptibility response lacks frequency dependence, ruling out spin freezing. Inelastic neutron scattering data on $\gamma$-BCPO is compared with linear spin wave theory, and two separate parameter regions of the XXZ $J_1$-$J_2$-$J_3$ model with ferromagnetic nearest-neighbor exchange $J_1$ are favored, both near regions of high classical degeneracy. High energy coherent excitations ($\sim 10$ meV) persist up to at least 40 K, suggesting strong in-plane correlations persist above $T_N$. These data show that $\gamma$-BCPO is a rare highly frustrated, quasi-2D S$_{eff}$ = 1/2 honeycomb lattice material which resists long range magnetic order and spin freezing.

String theory duals of Wilson loops from Higgsing

For three-dimensional ABJ(M) theories and mathcal{N}=4 Chern-Simons-matter quiver theories, we construct two sets of 1/2 BPS Wilson loop operators by applying the Higgsing procedure along independent directions of the moduli space, and choosing different massive modes. For theories whose dual M-theory description is known, we also determine the corresponding spectrum of 1/2 BPS M2-brane solutions. We identify the supercharges in M-theory and field theory, as well as the supercharges preserved by M2-/anti-M2-branes and 1/2 BPS Wilson loops. In particular, in mathcal{N}=4 orbifold ABJM theory we find pairs of different 1/2 BPS Wilson loops that preserve exactly the same set of supercharges. In field theory they arise by Higgsing with the choice of either particles or antiparticles, whereas in the dual description they correspond to a pair of M2-/anti-M2-branes localized at different positions in the compact space. This result enlightens the origin of classical Wilson loop degeneracy in these theories, already discussed in arXiv:1506.07614. A discussion on possible scenarios that emerge by comparison with localization results is included.

Staircase of crystal phases of hard-core bosons on the kagome lattice

We study the quantum phase diagram of a system of hard-core bosons on the Kagome lattice with nearest-neighbor repulsive interactions, for arbitrary densities, by means of the hierarchical mean field theory and exact diagonalization techniques. This system is isomorphic to the spin S=1/2 XXZ model in presence of an external magnetic field, a paradigmatic example of frustrated quantum magnetism. In the non-frustrated regime, we find two crystal phases at densities 1/3 and 2/3 that melt into a superfluid phase when increasing the hopping amplitude, in semi-quantitative agreement with quantum Monte Carlo computations. In the frustrated regime and away from half-filling, we find a series of plateaux with densities commensurate with powers of 1/3. The broader density plateaux (at densities 1/3 and 2/3) are remnants of the classical degeneracy in the Ising limit. For densities near half-filling, this staircase of crystal phases melts into a superfluid, which displays finite chiral currents when computed with clusters having an odd number of sites. Both the staircase of crystal phases and the superfluid phase prevail in the non-interacting limit, suggesting that the lowest dispersionless single-particle band may be at the root of this phenomenon.

Generic model for the hyperkagome iridateNa4Ir3O8in the local-moment regime

Hyperkagome iridate, Na$_4$Ir$_3$O$_8$, has been regarded as a promising candidate material for a three-dimensional quantum spin liquid. Here the three-dimensional network of corner-sharing triangles forms the hyperkagome lattice of Ir$^{4+}$ ions. Due to strong spin-orbit coupling, the local moments of Ir$^{4+}$ ions are described by the pseudospin $j_{\rm eff} = 1/2$ Kramers doublet. The Heisenberg model on this lattice is highly frustrated and quantum/classical versions have been studied in earlier literature. In this work, we derive a generic local-moment model beyond the Heisenberg limit for the hyperkagome iridate by considering multi-orbital interactions for all the $t_{2g}$ orbitals and spin-orbit coupling. The lifting of massive classical degeneracy in the Heisenberg model by various spin-anisotropy terms is investigated at the classical level and the resulting phase diagram is presented. We find that different anisotropy terms prefer distinct classes of magnetically ordered phases, often with various discrete degeneracy. The implications of our results for recent $\mu$SR and NMR experiments on this material and possible quantum spin liquid phases are discussed.