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arXiv:cond-mat/0202166 · Physical review. B, Condensed matter

Gapless fermions and quantum order

Xiao-Gang Wen, A. Zee

Published Dec 19, 2002• 42 citations
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Abstract

Using a two-dimensional quantum spin-1/2 model as a concrete example, we studied the relation between gapless fermionic excitations (spinons) and quantum orders in some spin liquid states. Using a winding number, we find that the projective symmetry group characterizing the quantum order can directly determine the pattern of Fermi points in the Brillouin zone. Thus quantum orders provide an origin for gapless fermionic excitations. Gapless fermions can appear even from a pure boson system as long as the boson ground state has a proper quantum order.

DOI
10.1103/physrevb.66.235110

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Using a two-dimensional quantum spin-1/2 model as a concrete example, we studied the relation between gapless fermionic excitations (spinons) and quantum orders in some spin liquid states. Using a winding number, we find that the projective symmetry group characterizing the quantum order can directly determine the pattern of Fermi points in the Brillouin zone. Thus quantum orders provide an origin for gapless fermionic excitations. Gapless fermions can appear even from a pure boson system as long as the boson ground state has a proper quantum order.
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We study fermionic quantum spin liquids (QSLs) on the three-dimensional trillium lattice of corner-sharing triangles. We are motivated by recent experimental and theoretical investigations that have explored various classical and quantum spin liquid states on similar networks of triangular motifs with strong geometric frustration. Using the framework of projective symmetry groups (PSG), we obtain a classification of all symmetric Z 2 and U ( 1 ) QSLs on the trillium lattice. We find two Z 2 spin-liquids, and a single U ( 1 ) spin-liquid that is proximate to one of the Z 2 states. The small number of solutions reflects the constraints imposed by the nonsymmorphic symmetries in the space group of the trillium lattice. Using self-consistency conditions of the mean-field equations, we obtain the spinon band-structure and spin structure factors corresponding to these states. All three of our spin liquids are gapless at their saddle points: one of the two Z 2 QSLs is nodal, while the U ( 1 ) case hosts a spinon Fermi surface. One of our Z 2 spin liquids hosts a stable gapless nodal star that is protected by projective symmetries against additions of further neighbor terms in the mean-field ansatz. We comment on directions for further work.
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We study possible spin liquids on square lattice that respect all lattice symmetries and time-reversal symmetry within the framework of Schwinger boson (mean-field) theory. Such spin liquids have spin gap and emergent Z2{Z}_{2}Z2​ gauge field excitations. We classify them by the projective symmetry group method, and find six spin-liquid states that are potentially relevant to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. The properties of these states are studied under mean-field approximation. Interestingly we find a spin-liquid state that can go through continuous phase transitions to either the N\'eel magnetic order or magnetic orders of the wave vector at the Brillouin zone edge center. We also discuss the connection between our results and the Abrikosov fermion spin liquids.
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.