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arXiv:1005.5626 · Physical Review B

Fermionic theory for quantum antiferromagnets with spin S>12

Zheng-Xin Liu, Yi Zhou, Tai Kai Ng

Published Oct 13, 2010• 37 citations
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Abstract

The fermion representation for $S=\frac{1}{2}$ spins is generalized to spins with arbitrary magnitudes. The symmetry properties of the representation is analyzed where we find that the particle-hole symmetry in the spinon Hilbert space of $S=\frac{1}{2}$ fermion representation is absent for $S>\frac{1}{2}$. As a result, different path-integral representations and mean-field theories can be formulated for spin models. In particular, we construct a Lagrangian with restored particle-hole symmetry, and apply the corresponding mean-field theory to one-dimensional (1D) $S=1$ and $S=3/2$ antiferromagnetic (AFM) Heisenberg models, with results that agree with Haldane's conjecture. For a $S=1$ open chain, we show that Majorana fermion edge states exist in our mean-field theory. The generalization to spins with arbitrary magnitude $S$ is discussed. Our approach can be applied to higher dimensional spin systems. As an example, we study the geometrically frustrated $S=1$ AFM on triangular lattice. Two spin liquids with different pairing symmetries are discussed: the gapped ${p}_{x}+i{p}_{y}$-wave spin liquid and the gapless $f$-wave spin liquid. We compare our mean-field result with the experiment on ${\text{NiGa}}_{2}{\text{S}}_{4}$, which remains disordered at low temperature and was proposed to be in a spin-liquid state. Our fermionic mean-field theory provide a framework to study $S>\frac{1}{2}$ spin liquids with fermionic spinon excitations.

DOI
10.1103/physrevb.82.144422

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The fermion representation for S=12S=\frac{1}{2}S=21​ spins is generalized to spins with arbitrary magnitudes. The symmetry properties of the representation is analyzed where we find that the particle-hole symmetry in the spinon Hilbert space of S=12S=\frac{1}{2}S=21​ fermion representation is absent for S>12S>\frac{1}{2}S>21​. As a result, different path-integral representations and mean-field theories can be formulated for spin models. In particular, we construct a Lagrangian with restored particle-hole symmetry, and apply the corresponding mean-field theory to one-dimensional (1D) S=1S=1S=1 and S=3/2S=3/2S=3/2 antiferromagnetic (AFM) Heisenberg models, with results that agree with Haldane's conjecture. For a S=1S=1S=1 open chain, we show that Majorana fermion edge states exist in our mean-field theory. The generalization to spins with arbitrary magnitude SSS is discussed. Our approach can be applied to higher dimensional spin systems. As an example, we study the geometrically frustrated S=1S=1S=1 AFM on triangular lattice. Two spin liquids with different pairing symmetries are discussed: the gapped px+ipy{p}_{x}+i{p}_{y}px​+ipy​-wave spin liquid and the gapless fff-wave spin liquid. We compare our mean-field result with the experiment on NiGa2S4{\text{NiGa}}_{2}{\text{S}}_{4}NiGa2​S4​, which remains disordered at low temperature and was proposed to be in a spin-liquid state. Our fermionic mean-field theory provide a framework to study S>12S>\frac{1}{2}S>21​ spin liquids with fermionic spinon excitations.
In this paper, we generalize the fermionic representation for S=1/2S=1/2S=1/2 spins to arbitrary spins. Within a mean field theory we obtain several spin liquid states for spin S=1S=1S=1 antiferromagnets on triangular lattices, including gapless fff-wave spin liquid and topologically nontrivial px+ipy{p}_{x}+i{p}_{y}px​+ipy​ spin liquid. After considering different competing orders, we construct a phase diagram for the ${J}_{1}\ensuremath{-}{J}_{3}\ensuremath{-}K$ model. The application to recently discovered material NiGa2S4{\text{NiGa}}_{2}{\text{S}}_{4}NiGa2​S4​ is discussed.
We study 1-dimension (1-d) and 2-dimensional (2-d) quantum anti-ferromagnets. Following AKLT's theory of valent bond solid (VBS) states, we find a family of topologically nontrivial 1-d VBS states for spin systems and provide a method to construct their parent Hamiltonians. Then we generalize the fermionic representation of S = 1/2 to arbitrary spin, and find an important difference between integer spins and half-odd-integer spins. Depending on different particle number constrains we develop different versions of mean field theory for spin models. We find that the mean field with restored particle-hole symmetry can reproduce Haldane's conjecture for 1-d Heisenberg models. Applying the mean field theory to 2-d anti-ferromagnetic models, we find that S = 1 spin liquids may exist in triangle lattice. Comparing with the experimental results of NiGa2S4, we argue that the antiferro-nematic f-wave spin liquid is a possible ground state. On the other hand, to check whether the material ZnCu3(OH)6Cl2 is really an U(1) Dirac spin liquid, we derive the Raman scattering operators of different symmetry channels for Hubbard model on Kagome lattice. Then we numerically calculate the Raman cross-section according to the mean field spin liquid state. The low energy continuum part of our results qualitatively match the experimental data.
We develop a mean-field theory of the 1D spin-1 antiferromagnetic Heisenberg model, using a two-orbital fermionic representation for the spin-1 operators. Two ansatz with different orbital pairing symmetries are discussed and the projected mean-field ground states with their corresponding parent Hamiltonians have also been obtained. Then we study the U(1) spin liquid state with large spinon Fermi surfaces. The physical conductivity obtained through Ioffe-Larkin composition rule can support surface plasmon modes propagating along the interface between the spin liquid and a linear medium, which can be excited using a Kretschmann-Raether configuration. Next we generalize the linear current response to including nonlinear effects under the perturbation of a time dependent but spatially uniform electric field in the framework of an effective U(1) gauge theory derived through a Fermi-liquid-like approach. The nonlinear susceptibility can support waves with triple frequency inside the spin liquids through the process of third harmonic generation. As a byproduct during the Fermi-liquid approach, we generalize the standard Landau Fermi liquid theory to describe the intrinsic anomalous Hall conductivity, based on some general symmetric considerations. Fitting result of the anomalous Hall conductivity is well consistent with the experimental data, and behaves better in lower frequency range than the simple two-band model.
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 ruby lattice has been the subject of much interest recently due its realization in Rydberg atom arrays, where its rich variety of frustrated interactions gives rise to topologically ordered quantum spin liquids. Similarly, numerical studies of ruby-lattice spin models, with both isotropic and anisotropic interactions, have provided evidence of gapped and gapless spin-liquid ground states with different low-energy gauge structures. Motivated by these findings, we perform a projective symmetry group (PSG) classification of U(1) and Z2\mathbb{Z}_{2}Z2​ fermionic spinon mean-field theories$\unicode{x2014}$respecting space-group and time-reversal symmetries$\unicode{x2014}$for S=1/2S=1/2S=1/2 spins. We obtain a total of 50 U(1) and 64 Z2\mathbb{Z}_{2}Z2​ PSGs, and upon restricting their realization via mean-field Ansa¨tze\textit{Ansätze}Ansa¨tze with up to second-nearest-neighbor singlet amplitudes (relevant to the models studied here), only 8 U(1) and 18 Z2\mathbb{Z}_{2}Z2​ distinct states are obtained. We present the singlet fields for all Ansa¨tze\textit{Ansätze}Ansa¨tze up to third-nearest-neighboring bonds and discuss their spinon dispersions as well as their dynamical spin structure factors. Building on this information, we also obtain the phase diagram of the Heisenberg model in the presence of first (J1J_{1}J1​), second (J1′J_{1}'J1′​), and third (J2J_{2}J2​) neighbor antiferromagnetic couplings within a self-consistent mean-field approximation.
Despite rapid progress in understanding gapped topological states, much less is known about gapless topological phases of matter, especially in strongly correlated electrons. In this work, we discuss a large class of robust gapless quantum spin liquids in frustrated magnets made of half-integer spins, which are described by gapless fermionic spinons coupled to dynamical Z2{\mathbb{Z}}_{2}Z2​ gauge fields. Requiring U(1)\text{U}(1)U(1) spin conservation, time-reversal, and certain space-group symmetries, we show that certain spinon symmetry fractionalization class necessarily leads to a gapless spectrum. These gapless excitations are stable against any perturbations, as long as the required symmetries are preserved. Applying these gapless criteria to spin-12\frac{1}{2}21​ systems on square, triangular, and kagome lattices, we show that all gapped symmetric Z2{\mathbb{Z}}_{2}Z2​ spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.