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arXiv:0912.4414 · Physical Review B

Possibility of S=1 spin liquids with fermionic spinons on triangular lattices

Zheng-Xin Liu, Yi Zhou, Tai Kai Ng

Published Jun 9, 2010• 34 citations
arXivHTMLPDF

Abstract

In this paper, we generalize the fermionic representation for $S=1/2$ spins to arbitrary spins. Within a mean field theory we obtain several spin liquid states for spin $S=1$ antiferromagnets on triangular lattices, including gapless $f$-wave spin liquid and topologically nontrivial ${p}_{x}+i{p}_{y}$ spin liquid. After considering different competing orders, we construct a phase diagram for the ${J}_{1}\ensuremath{-}{J}_{3}\ensuremath{-}K$ model. The application to recently discovered material ${\text{NiGa}}_{2}{\text{S}}_{4}$ is discussed.

DOI
10.1103/physrevb.81.224417

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In this paper, we generalize the fermionic representation for S=1/2S=1/2S=1/2 spins to arbitrary spins. Within a mean field theory we obtain several spin liquid states for spin S=1S=1S=1 antiferromagnets on triangular lattices, including gapless fff-wave spin liquid and topologically nontrivial px+ipy{p}_{x}+i{p}_{y}px​+ipy​ spin liquid. After considering different competing orders, we construct a phase diagram for the J1\ensuremath−J3\ensuremath−K{J}_{1}\ensuremath{-}{J}_{3}\ensuremath{-}KJ1​\ensuremath−J3​\ensuremath−K model. The application to recently discovered material NiGa2S4{\text{NiGa}}_{2}{\text{S}}_{4}NiGa2​S4​ is discussed.
The fermion representation for S=12S=\frac{1}{2}S=21​ spins is generalized to spins with arbitrary magnitudes. The symmetry properties of the representation is analyzed where we find that the particle-hole symmetry in the spinon Hilbert space of S=12S=\frac{1}{2}S=21​ fermion representation is absent for S>12S>\frac{1}{2}S>21​. As a result, different path-integral representations and mean-field theories can be formulated for spin models. In particular, we construct a Lagrangian with restored particle-hole symmetry, and apply the corresponding mean-field theory to one-dimensional (1D) S=1S=1S=1 and S=3/2S=3/2S=3/2 antiferromagnetic (AFM) Heisenberg models, with results that agree with Haldane's conjecture. For a S=1S=1S=1 open chain, we show that Majorana fermion edge states exist in our mean-field theory. The generalization to spins with arbitrary magnitude SSS is discussed. Our approach can be applied to higher dimensional spin systems. As an example, we study the geometrically frustrated S=1S=1S=1 AFM on triangular lattice. Two spin liquids with different pairing symmetries are discussed: the gapped px+ipy{p}_{x}+i{p}_{y}px​+ipy​-wave spin liquid and the gapless fff-wave spin liquid. We compare our mean-field result with the experiment on NiGa2S4{\text{NiGa}}_{2}{\text{S}}_{4}NiGa2​S4​, which remains disordered at low temperature and was proposed to be in a spin-liquid state. Our fermionic mean-field theory provide a framework to study S>12S>\frac{1}{2}S>21​ spin liquids with fermionic spinon excitations.
Motivated by recent experiments on the material Ba3{}_{3}3​NiSb2{}_{2}2​O9{}_{9}9​, we consider a spin-one quantum antiferromagnet on a triangular lattice with the Heisenberg bilinear and biquadratic exchange interactions and a single-ion anisotropy. Using a fermionic ``triplon'' representation for spins, we study the phase diagram within mean-field theory. In addition to a fully gapped spin-liquid ground state, we find a state where one gapless triplon mode with a Fermi surface coexists with d+idd+idd+id topological pairing of the other triplons. Despite the existence of a Fermi surface, this ground state has fully gapped bulk spin excitations. Such a state has linear in-temperature specific heat and constant in-plane spin susceptibility, with an unusually high Wilson ratio.
The ruby lattice has been the subject of much interest recently due its realization in Rydberg atom arrays, where its rich variety of frustrated interactions gives rise to topologically ordered quantum spin liquids. Similarly, numerical studies of ruby-lattice spin models, with both isotropic and anisotropic interactions, have provided evidence of gapped and gapless spin-liquid ground states with different low-energy gauge structures. Motivated by these findings, we perform a projective symmetry group (PSG) classification of U(1) and Z2\mathbb{Z}_{2}Z2​ fermionic spinon mean-field theories$\unicode{x2014}$respecting space-group and time-reversal symmetries$\unicode{x2014}$for S=1/2S=1/2S=1/2 spins. We obtain a total of 50 U(1) and 64 Z2\mathbb{Z}_{2}Z2​ PSGs, and upon restricting their realization via mean-field Ansa¨tze\textit{Ansätze}Ansa¨tze with up to second-nearest-neighbor singlet amplitudes (relevant to the models studied here), only 8 U(1) and 18 Z2\mathbb{Z}_{2}Z2​ distinct states are obtained. We present the singlet fields for all Ansa¨tze\textit{Ansätze}Ansa¨tze up to third-nearest-neighboring bonds and discuss their spinon dispersions as well as their dynamical spin structure factors. Building on this information, we also obtain the phase diagram of the Heisenberg model in the presence of first (J1J_{1}J1​), second (J1′J_{1}'J1′​), and third (J2J_{2}J2​) neighbor antiferromagnetic couplings within a self-consistent mean-field approximation.
Motivated by recent experiments on Ba3{}_{3}3​NiSb2{}_{2}2​O9{}_{9}9​, we investigate possible quantum spin liquid ground states for spin S=1S=1S=1 Heisenberg models on the triangular lattice. We use variational Monte Carlo techniques to calculate the energies of microscopic spin liquid wave functions where spin is represented by three flavors of fermionic spinon operators. These energies are compared with the energies of various competing three-sublattice ordered states. Our approach shows that the antiferromagnetic Heisenberg model with biquadratic term and single-ion anisotropy does not have a low-temperature spin liquid phase. However, for an SU(3)-invariant model with sufficiently strong ring-exchange terms, we find a paired chiral quantum spin liquid with a Fermi surface of deconfined spinons that is stable against all types of ordering patterns we considered. We discuss the physics of this exotic spin liquid state in relation to the recent experiment and suggest new ways to test this scenario.
Recent work has shown that the triangular lattice spin-$\frac{1}{2} {J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg and XXZ antiferromagnets may exhibit coplanar or supersolid orders proximate to a gapless Dirac spin-liquid phase. We explore a distinct $\text{SU}(2N)\ifmmode\times\else\texttimes\fi{}\text{SU}(M)$ fermionic parton approach, complemented by variational Monte Carlo calculations for the spin-12\frac{1}{2}21​ model, to study the phase diagram of these models. We also calculate their dynamical spin response including parton interactions within a random-phase approximation, and discuss implications for neutron scattering on triangular lattice cobaltates Ba3CoSb2O9,Na2BaCo(PO4)2,K2Co(SeO3)2,Rb2Co(SeO3)2{\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}, {\mathrm{Na}}_{2}\mathrm{BaCo}{({\mathrm{PO}}_{4})}_{2}, {\mathrm{K}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}, {\mathrm{Rb}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}Ba3​CoSb2​O9​,Na2​BaCo(PO4​)2​,K2​Co(SeO3​)2​,Rb2​Co(SeO3​)2​, and Yb-based magnet KYbSe2{\mathrm{KYbSe}}_{2}KYbSe2​.
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