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arXiv:1112.0586 · Physical Review B

Majorana spin liquids and projective realization of SU(2) spin symmetry

Gang Chen, Andrew M. Essin, Michael Hermele

Published Mar 13, 2012• 32 citations
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Abstract

We revisit the fermionic parton approach to $S=1/2$ quantum spin liquids with $\mathrm{SU}(2)$ 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 $S=1$ and $S=0$ Majorana fermion excitations coupled to a deconfined ${Z}_{2}$ 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 $\mathrm{SU}(2)$ 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 ${Z}_{2}$ vortex excitations obeying non-Abelian statistics. We conclude with a discussion of various open issues, including the challenging question of where such $S=1$ Majorana spin liquids may occur in models and in real systems.

DOI
10.1103/physrevb.85.094418

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Projective symmetry groups beyond spin-1/21/21/2: Classifying integer-spin quantum spin liquids
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We perform a projective symmetry group (PSG) classification of symmetric quantum spin liquids with different gauge groups on the square-octagon lattice. Employing the Abrikosov fermion representation for spin 12\frac{1}{2}21​, we obtain 32SU(2),1808U(1),32\phantom{\rule{4pt}{0ex}}\text{SU}(2), 1808\phantom{\rule{4pt}{0ex}}\text{U}(1),32SU(2),1808U(1), and 384Z2384\phantom{\rule{4pt}{0ex}}{\mathbb{Z}}_{2}384Z2​ algebraic PSGs. Constraining ourselves to mean-field parton ans\"atze with short-range amplitudes, the classification reduces to a limited number, with 4 SU(2)\text{SU}(2)SU(2), 24 U(1),\text{U}(1),U(1), and 36 Z2{\mathbb{Z}}_{2}Z2​, distinct phases. We discuss their ground-state properties and spinon dispersions within a self-consistent treatment of the Heisenberg Hamiltonian with frustrating couplings.
Quantum spin liquids are novel phases of matter whose physical properties cannot be captured within the conventional framework of phase transitions. These quantum states are characterized by excitations, called spinons, carrying a fractional value of the integer spin. A description based on a mean-field approximation leads to effective gauge theory. Due to this gauge freedom symmetry group operations act via projective symmetry group (PSG) representations, which can be used to classify quantum states beyond the conventional method. This introducing discussion is found in Chapters 2-6. A preceding PSG classification on the square lattice, which took spin rotational symmetry breaking terms into account, showed that the resulting mean-field states could exhibit topologically non-trivial spinon bands. In Chapter 7, only those el- ements of this Classification are studied, which lead to short-ranged mean-field models. The mutual effect of spinons and visons is scrutinized by adding static configurations of a background gauge field. Spinons and visons form composite ob- jects, which in the case of topological spinon bands, are described by Majorana zero modes. An exact mapping corroborates the relation to topological supercon- ductors and vortices. The effect of multiple visons is investigated, and a possible experimental signature is exposed. Chapter 8 explains how spinon mean-field theory can be used to derive a phe- nomenological effective model for the compound Ca 10 Cr 7 O 28 . A material that was recently proposed as a quantum spin liquid. This low-energy theory consists of spinon hopping processes. The underlying bilayer kagome structure can be simpli- fied to an effective honeycomb lattice. The dynamical spin structure factor is calcu- lated and compared to neutron scattering data. It is discussed how f -wave pairing of spinons can explain small deviations from a linear temperature dependency observed in heat capacity measurements at the low temperatures. This pairing mechanism breaks the U (1) invariance group of the pure hopping model and renders a coherent picture of Ca 10 Cr 7 O 28 being a Z 2 quantum spin liquid. Chapter 9 carries out a PSG classification for the simple-, body centered-, and face centered cubic lattices. Despite a large number of possible PSG representations, only very few describe short-ranged mean-field states. These mean-field models are constructed up to third neighbor couplings and calculated in a self-consistent manner. It is found that the energy is lower for non-trivial sign patterns induced by non-trivial representations of the symmetry generators. Higher neighbor extensions, including pairing terms, yield lower mean-field ground state energies compared to extensions with only hopping terms. For the fcc lattice, a first neighbor state could be identified, which exhibits symmetry protected zero-energy modes.
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.