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arXiv:1110.3328 · Physical Review Letters

Spin Liquid Phases for Spin-1 Systems on the Triangular Lattice

Cenke Xu, Fa Wang, Yang Qi, Leon Balents, Matthew P. A. Fisher

Published Feb 23, 2012• 54 citations
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Abstract

Motivated by recent experiments on material ${\mathrm{Ba}}_{3}{\mathrm{NiSb}}_{2}{\mathrm{O}}_{9}$, we propose two novel spin liquid phases ($A$ and $B$) for spin-1 systems on a triangular lattice. At the mean field level, both spin liquid phases have gapless fermionic spinon excitations with quadratic band touching; thus, in both phases the spin susceptibility and $\ensuremath{\gamma}={C}_{v}/T$ saturate to a constant at zero temperature, which are consistent with the experimental results on ${\mathrm{Ba}}_{3}{\mathrm{NiSb}}_{2}{\mathrm{O}}_{9}$. On the lattice scale, these spin liquid phases have $\mathrm{Sp}(4)\ensuremath{\sim}\mathrm{SO}(5)$ gauge fluctuation, while in the long wavelength limit this Sp(4) gauge symmetry is broken down to $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ in the type $A$ spin liquid phase, and broken down to ${Z}_{4}$ in the type $B$ phase. We also demonstrate that the $A$ phase is the parent state of the ferroquadrupole state, nematic state, and the noncollinear spin density wave state.

DOI
10.1103/physrevlett.108.087204

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For certain crystalline systems, most notably the organic compound EtMe3{}_{3}3​Sb[Pd(dmit)2{}_{2}2​]2{}_{2}2​, experimental evidence has accumulated of an insulating state with a high density of gapless neutral excitations that produce Fermi-liquid-like power laws in thermodynamic quantities and thermal transport. This has been taken as evidence of a fractionalized spin-liquid state. In this Rapid Communication, we argue that if the experiments are taken at face value, the most promising spin-liquid candidates are a Z4{Z}_{4}Z4​ spin liquid with a spinon-Fermi surface and no broken symmetries, or a Z2{Z}_{2}Z2​ spin liquid with a spinon-Fermi surface and at least one of the following spontaneously broken: (a) time-reversal and inversion, (b) translation, or (c) certain point-group symmetries. We present a solvable model on the triangular lattice with an (a) type Z2{Z}_{2}Z2​ spin-liquid ground state.
Motivated by recent numerical discovery of a gapped spin liquid phase in spin-1/21/21/2 triangular-lattice ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model, we classify symmetric Z2{Z}_{2}Z2​ spin liquids on triangular lattice in the Abrikosov-fermion representation. We find 20 phases with distinct spinon symmetry quantum numbers, eight of which have their counterparts in the Schwinger-boson representation. Among them we identify two promising candidates (#1 and #20), which can realize a gapped Z2{Z}_{2}Z2​ spin liquid with up to next nearest neighbor mean-field amplitudes. We analyze their neighboring magnetic orders and valence bond solid patterns, and find one state (#20) that is connected to 120-degree Neel order by a continuous quantum phase transition. We also identify gapped nematic Z2{Z}_{2}Z2​ spin liquids in the neighborhood of the symmetric states and find three promising candidates (#1, #6, and #20).
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.
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The U(1)U(1)U(1) Dirac spin liquid might realize an exotic phase of matter whose low-energy properties are described by quantum electrodynamics in 2+12+12+1 dimensions, where gapless modes exist but spinons and gauge fields are strongly coupled. Its existence was proposed in frustrated Heisenberg models in the presence of frustrating superexchange interactions by the (Abrikosov) fermionic representation of the spin operators [Wen, Phys. Rev. B 65, 165113 (2002)], supplemented by the Gutzwiller projection. Here, we construct charge-QQQ monopole excitations in the Heisenberg model on the triangular lattice with nearest-neighbor (J1{J}_{1}J1​) and next-nearest-neighbor (J2{J}_{2}J2​) couplings. In the highly frustrated regime, singlet and triplet monopoles with Q=1Q=1Q=1 become gapless in the thermodynamic limit; in addition, the energies for generic QQQ agree with field-theoretical predictions, obtained for a large number of gapless fermion modes. Finally, we consider localized gauge excitations, in which magnetic $\ensuremath{\pi}$ fluxes are concentrated in the triangular plaquettes (in analogy with Z2{\mathbb{Z}}_{2}Z2​ visons), showing that these kinds of states do not play a relevant role at low energies. All our findings lend support to a stable U(1)U(1)U(1) Dirac spin liquid in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice.
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