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arXiv:cond-mat/9603179 · Physical review. B, Condensed matter

Majorana fermion representation for an antiferromagnetic spin- chain

B. Sriram Shastry, Diptiman Sen

Published Feb 1, 1997• 35 citations
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Abstract

We study the one-dimensional Heisenberg antiferromagnet with s= using a Majorana representation of the s= spins. A simple Hartree-Fock approximation of the resulting model gives a bilinear fermionic description of the model. This description is rotationally invariant and gives power-law correlations in the ``ground state'' in a natural fashion. The excitations are a two-parameter family of particles, which are spin-1 objects. These are contrasted to the ``spinon'' spectrum, and the technical aspects of the representation are discussed, including the problem of redundant states.

DOI
10.1103/physrevb.55.2988

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We study the one-dimensional Heisenberg antiferromagnet with s= using a Majorana representation of the s= spins. A simple Hartree-Fock approximation of the resulting model gives a bilinear fermionic description of the model. This description is rotationally invariant and gives power-law correlations in the ground state'' in a natural fashion. The excitations are a two-parameter family of particles, which are spin-1 objects. These are contrasted to the spinon'' spectrum, and the technical aspects of the representation are discussed, including the problem of redundant states.
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A novel approach to S=1/2 antiferromagnets with strong quantum fluctuations based on the representation of spin-1/2 operators as bilinear forms of real (Majorana) fermions is suggested. This representation does not contain unphysical states and thus does not require the imposition of constraints on the fermionic Hilbert space. This property allows one to construct a simple and effective mean field theory of the spin liquid state. As an example illustrating the basic properties of this state I consider a model of Kondo lattice. It is shown that this model has a singlet ground state; elementary excitations have a spectral gap and are S=1 real (Majorana) fermions.
The Majorana representation of spin 1/21/21/2 could be useful in the theory of strongly correlated electrons because it respects isotropy in spin space and has no constraints. To judge its reliability, we apply it to the O(N)O(N)O(N) quantum Heisenberg antiferromagnet for general N.N.N. At $N\ensuremath{\rightarrow}\ensuremath{\infty}$ we find a resonant valence bond-like spin-liquid ground state in precise agreement with previous results from an SU(\ensuremath{\infty}) extrapolation. At N=3,N=3,N=3, however, the representation fails and gives an incorrect gap in the O(3)O(3)O(3) Heisenberg chain. We trace this failure at N=3N=3N=3 to (i) a redundancy of states and (ii) the presence of a local gauge invariance both of which are intrinsic properties of the Majorana representation of spins at any NNN but irrelevant at $N=\ensuremath{\infty}.$ Specifically at N=3,N=3,N=3, there is an unphysical ``longitudinal'' mode. A correct treatment of these difficulties must precede applications of the Majorana representation of spin-1/21/21/2 operators to systems of physical interest.
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.