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arXiv:cond-mat/9707067 · Journal of Physics Condensed Matter

A Majorana fermiont-Jmodel in one dimension

Diptiman Sen, B. Sriram Shastry

Published Sep 22, 1997• 3 citations
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Abstract

We study a rotation invariant Majorana fermion model in one dimension using diagrammatic perturbation theory and numerical diagonalization of small systems. The model is inspired by a Majorana representation of the antiferromagnetic spin-1/2 chain, and it is similar in form to the t-J model of electrons, except that the Majorana fermions carry spin-1 and Z_2 charge. We discuss the implications of our results for the low-energy excitations of the spin-1/2 chain. We also discuss a generalization of our model from 3 species of Majorana fermions to N species; the SO(4) symmetric model is particularly interesting.

DOI
10.1088/0953-8984/9/38/005

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We study a rotation invariant Majorana fermion model in one dimension using diagrammatic perturbation theory and numerical diagonalization of small systems. The model is inspired by a Majorana representation of the antiferromagnetic spin-1/2 chain, and it is similar in form to the t-J model of electrons, except that the Majorana fermions carry spin-1 and Z_2 charge. We discuss the implications of our results for the low-energy excitations of the spin-1/2 chain. We also discuss a generalization of our model from 3 species of Majorana fermions to N species; the SO(4) symmetric model is particularly interesting.
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.
In this work, we study interacting spinless fermions on a two-chain ladder with interchain pair tunneling while single-particle tunneling is suppressed at low energy. The model embodies a Z2{\mathbb{Z}}_{2}Z2​ symmetry associated with the fermion parity on each chain. We find that when the system is driven to the strong-coupling phase by the pair tunneling, Majorana excitations appear on the boundary. Such Majorana edge states correspond to two-fold degeneracy of ground states distinguished by different fermion parity on each chain, thus representing a generalization of one-dimensional topological superconductors. We also characterize the stability of the ground-state degeneracy against local perturbations. Lattice fermion models realizing such an effective field theory are discussed.
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.
A diagrammatic formalism for lattices of 1/2 is developed. It is based on an unconstrained mapping between spin and Majorana operators. This allows the use of standard tools of diagrammatic quantum many-body theory without requiring projections. We derive, in particular, the Feynman rules for the expansion around a color-preserving mean-field theory. We then present the numerical results obtained by computing the corrections up to second order for the Heisenberg model in one and two dimensions, showing that perturbative corrections are not only numerically important, but also qualitatively improve the results of mean-field theory. These results pave the way for the use of Majorana diagrammatic tools in theoretical and numerical studies of quantum spin systems.
The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low energy spectrum of noniteracting fermions is identical to the one of a harmonic chain. This fermion-boson transmutation allows to describe interacting fermions as a system of coupled oscillators. Tomonaga's model of interacting nonrelativistic fermions on a ring is presented and first discussed in low order perturbation theory. After introducing the concept of two independent species of right and left moving fermions the exact solution of the Tomonaga-Luttinger model is discussed in detail. The momentum distribution and spectral functions are calculated using the method of the bosonization of the field operator. The general Luttinger liquid phenomenology is shortly discussed.