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arXiv:1106.0028 · Physical Review B

SU(2)-invariant Majorana spin liquid with stable parton Fermi surfaces in an exactly solvable model

Hsin-Hua Lai, Olexei I. Motrunich

Published Aug 30, 2011• 27 citations
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Abstract

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 $1/{|\mathbf{r}|}^{3}$ envelope. The model is solvable also in the presence of Zeeman magnetic field. One fermion species carries ${S}^{z}=0$ 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.

DOI
10.1103/physrevb.84.085141

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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.
We realize a gapless Majorana orbital liquid (MOL) using orbital degrees of freedom and an SU(2)-invariant Majorana spin liquid (MSL) using both spin and orbital degrees of freedom in Kitaev-type models on a two-leg ladder. The models are exactly solvable by Kitaev's parton approach, and we obtain long-wavelength descriptions for both Majorana liquids. The MOL has one gapless mode and power-law correlations in energy at incommensuate wave vectors, while the SU(2) MSL has three gapless modes and power-law correlations in spin, spin-nematic, and local energy observables. We study the stability of such states to perturbations away from the exactly solvable points. We find that both MOL and MSL can be stable against allowed short-range parton interactions. We also argue that both states persist on allowing Z2{Z}_{2}Z2​ gauge-field fluctuations, in that the number of gapless modes is retained, although with an expanded set of contributions to observables compared to the free parton mean field.
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.
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 the recent synthesis of $\ensuremath{\beta}$-Li2{}_{2}2​IrO3{}_{3}3​---a spin-orbit entangled j=1/2j=1/2j=1/2 Mott insulator with a three-dimensional lattice structure of the Ir4+{}^{4+}4+ ions---we consider generalizations of the Kitaev model believed to capture some of the microscopic interactions between the iridium moments on various trivalent lattice structures in three spatial dimensions. Of particular interest is the so-called hyperoctagon lattice---the premedial lattice of the hyperkagome lattice, for which the ground state is a gapless quantum spin liquid where the gapless Majorana modes form an extended two-dimensional Majorana Fermi surface. We demonstrate that this Majorana Fermi surface is inherently protected by lattice symmetries and discuss possible instabilities. We thus provide the first example of an analytically tractable microscopic model of interacting SU(2) spin-12\frac{1}{2}21​ degrees of freedom in three spatial dimensions that harbors a spin liquid with a two-dimensional spinon Fermi surface.
We present a modification of the exactly solvable spin-(1/)2 Kitaev model on the decorated honeycomb lattice, with a ground state of "spin metal" type. The model is diagonalized in terms of Majorana fermions; the latter form a 2D gapless state with a Fermi circle whose size depends on the ratio of exchange couplings. Low-temperature heat capacity C(T) and dynamic spin susceptibility χ(ω,T) are calculated in the case of small Fermi circle. Whereas, C(T)∼T at low temperatures as it is expected for a Fermi liquid, spin excitations are gapped and χ(ω,T) demonstrates unusual behavior with a power-law peak near the resonance frequency. The corresponding exponent as well as the peak shape are calculated.