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arXiv:1102.3690 · Physical Review B

SU(2)-invariant spin liquids on the triangular lattice with spinful Majorana excitations

Rudro R. Biswas, Liang Fu, Chris R. Laumann, Subir Sachdev A

Published Jun 28, 2011• 54 citations
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Abstract

We describe a new class of spin liquids with global SU(2) spin-rotation symmetry in spin-1/2 systems on the triangular lattice, which have real Majorana fermion excitations carrying spin $S=1$. The simplest translationally invariant mean-field state on the triangular lattice breaks time-reversal symmetry and is stable to fluctuations. It generically possesses gapless excitations along three Fermi lines in the Brillouin zone. These intersect at a single point where the excitations scale with a dynamic exponent $z=3$. An external magnetic field has no orbital coupling to the SU(2) spin-rotation-invariant fermion bilinears that can give rise to a transverse thermal conductivity, thus leading to the absence of a thermal Hall effect. The Zeeman coupling is found to gap out two-thirds of the $z=3$ excitations near the intersection point and this leads to a suppression of the low-temperature specific heat, the spin susceptibility, and the Wilson ratio. We also compute physical properties in the presence of weak disorder and discuss possible connections to recent experiments on organic insulators.

DOI
10.1103/physrevb.83.245131

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We describe a new class of spin liquids with global SU(2) spin-rotation symmetry in spin-1/2 systems on the triangular lattice, which have real Majorana fermion excitations carrying spin S=1S=1S=1. The simplest translationally invariant mean-field state on the triangular lattice breaks time-reversal symmetry and is stable to fluctuations. It generically possesses gapless excitations along three Fermi lines in the Brillouin zone. These intersect at a single point where the excitations scale with a dynamic exponent z=3z=3z=3. An external magnetic field has no orbital coupling to the SU(2) spin-rotation-invariant fermion bilinears that can give rise to a transverse thermal conductivity, thus leading to the absence of a thermal Hall effect. The Zeeman coupling is found to gap out two-thirds of the z=3z=3z=3 excitations near the intersection point and this leads to a suppression of the low-temperature specific heat, the spin susceptibility, and the Wilson ratio. We also compute physical properties in the presence of weak disorder and discuss possible connections to recent experiments on organic insulators.
We revisit the fermionic parton approach to S=1/2S=1/2S=1/2 quantum spin liquids with SU(2)\mathrm{SU}(2)SU(2) spin-rotation symmetry, and the associated projective symmetry group (PSG) classification. We point out that the existing PSG classification is incomplete; upon completing it, we find spin-liquid states with S=1S=1S=1 and S=0S=0S=0 Majorana fermion excitations coupled to a deconfined Z2{Z}_{2}Z2​ gauge field. The crucial observation leading us to this result is that, like space group and time-reversal symmetries, spin rotations can act projectively on the fermionic partons; that is, a spin rotation may be realized by simultaneous SU(2)\mathrm{SU}(2)SU(2) spin and gauge rotations. We show that there are only two realizations of spin rotations acting on fermionic partons: the familiar naive realization where spin rotation is not accompanied by any gauge transformation, and a single type of projective realization. We discuss the PSG classification for states with projective spin rotations. To illustrate these results, we show that there are four such PSGs on the two-dimensional square lattice. We study the properties of the corresponding states, finding that one---with gapless Fermi points---is a stable phase beyond mean-field theory. In this phase, depending on parameters, a small Zeeman magnetic field can open a partial gap for the Majorana fermion excitations. Moreover, there are nearby gapped phases supporting Z2{Z}_{2}Z2​ vortex excitations obeying non-Abelian statistics. We conclude with a discussion of various open issues, including the challenging question of where such S=1S=1S=1 Majorana spin liquids may occur in models and in real systems.
We construct an exactly solvable spin-orbital model on a decorated square lattice that realizes an SU(2)-invariant Majorana spin liquid with parton Fermi surfaces, of the kind discussed recently by Biswas et al. [Phys. Rev. B. 83, 245131 (2011)]. We find power-law spin correlations as well as power-law spin-nematic correlations with the same dominant 1/∣r∣31/{|\mathbf{r}|}^{3}1/∣r∣3 envelope. The model is solvable also in the presence of Zeeman magnetic field. One fermion species carries Sz=0{S}^{z}=0Sz=0 quantum number and its Fermi surface is not altered in the field, while the Fermi surfaces of the other species evolve and can disappear. In particular, we find an interesting half magnetization plateau phase in which spin excitations are gapful while there remain spinless gapless excitations that still produce metal-like thermal properties. In the fully magnetized phase, the model reduces to the one proposed by Baskaran et al. [e-print arXiv:0908.1614 (to be published)] in terms of the orbital degrees of freedom.
The recent experimental realizations of spin-1/2 gapless quantum spin liquids in two-dimensional triangular lattice organic compounds EtMe3Sb[Pd(dmit)2]2 and κ-(ET)2Cu2(CN)3 have stimulated the investigation of the gapless spin liquid theories. The models in dimensions greater than one (D>1) usually involve multispin interactions, such as ring exchange interactions, that are difficult to study, while effective gauge theory descriptions are not well-controlled to give reliable physics information. Driven by the need for a systematic and controlled analysis of such phase, such models on ladders are seriously studied. This thesis first focuses on such ladder models. We propose that the gapless spin liquid phase can be accessed from a two-band interacting electron model by metal-Mott insulator phase transition. We use Bosonization analysis and weak-coupling Renormalization Group to further study the gapless spin liquid state in the presence of Zeeman magnetic fields or orbital magnetic fields. Several new exotic gapless spin liquids with dominant spin nematic correlations are predicted. In such a ladder spin liquid, we also consider the impurity effects. We conclude that the local energy textures and oscillating spin susceptibilities around the impurities are nontrivial and can be observed in the experiments. We then shift our focus to another theoretical candidate, an SU(2)-invariant spin liquid with Majorana excitations, which can also qualitatively explain the experimental phenomenology. We construct an exactly solvable Kitaev-type model realizing the long-wavelength Majorana spin liquid state and study its properties. We find that the state has equal power-law spin and spin-nematic correlations and behaves nontrivially in the presence of Zeeman magnetic fields. Finally, we realize such Majorana spin liquid states on a two-leg ladder and further explore their stability. We conclude the states can be stable against short-range interactions and gauge field fluctuations.
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.
Spin liquids represent exotic types of quantum matter that evade conventional symmetry-breaking order even at zero temperature. Exhaustive classifications of spin liquids have been carried out in several systems, particularly in the presence of full SU(2) spin-rotation symmetry. Real magnetic compounds, however, generically break SU(2) spin symmetry as a result of spin-orbit coupling---which in many materials provides an ``order one'' effect. We generalize previous works by using the projective symmetry group method to classify Z2{\mathbb{Z}}_{2}Z2​ spin liquids on the square lattice when SU(2) spin symmetry is maximally lifted. We find that, counterintuitively, the lifting of spin symmetry actually results in vastly more spin-liquid phases compared to SU(2)-invariant systems. A generic feature of the SU(2)-broken case is that the spinons naturally undergo p+ipp+ipp+ip pairing; consequently, many of these Z2{\mathbb{Z}}_{2}Z2​ spin liquids feature a topologically nontrivial spinon band structure supporting gapless Majorana edge states. We study in detail several spin-liquid phases with varying numbers of gapless edge states and discuss their topological protection. The edge states are often protected by a combination of time reversal and lattice symmetries and hence resemble recently proposed topological crystalline superconductors.
We introduce an exactly solvable SU(2)-invariant spin-1/2 model with exotic spin excitations. With time reversal symmetry (TRS), the ground state is a spin liquid with gapless or gapped spin-1 but fermionic excitations. When TRS is broken, the resulting spin liquid exhibits deconfined vortex excitations which carry spin-1/2 and obey non-Abelian statistics. We show that this SU(2) invariant non-Abelian spin liquid exhibits the spin quantum Hall effect with quantized spin Hall conductivity σ(xy)(s)=ℏ/2π, and that the spin response is effectively described by the SO(3) level-1 Chern-Simons theory at low energy. We further propose that a SU(2) level-2 Chern-Simons theory is the effective field theory describing the topological structure of the non-Abelian SU(2) invariant spin liquid.