On the possibility of fractional statistics in the two-dimensional t-J model at low doping

Abstract

The purpose of this work is the derivation of an effective field theory for the low-energy magnetic modes of the t-J model on a two-dimensional square lattice and for small values of doping, which is of relevance to the physics of high-temperature superconductors. In particular, we address the possibility of low-energy excitations that obey fractional spin and statistics. For temperatures where kBT is considerably smaller than the magnetic exchange coupling J and for low values of doping, the system is close to an antiferromagnetically ordered Néel state and the spin correlation length takes values significantly larger than the lattice constant a, such that a field theoretic analysis is justified. The effective model is obtained by means of a gradient expansion around the antiferromagnetically ordered reference state. Dimensional analysis shows that in (2+1) dimensions only terms up to order O(a2) in the effective action are relevant to the behaviour at large scales, whereas O(a3)-terms are marginal and higher order terms are irrelevant. Even though marginal, the O(a3)-contributions may drastically influence the properties of excitations, as they might feature a term of topological nature which endows field histories with a statistical phase factor. In fact, all field histories can be characterized as mappings from compactified spacetime, the three-sphere S3, to the order parameter space S2 and thus fall into different homotopy classes. The topological invariant characterizing these homotopy classes is given by the Hopf invariant. Consequently, if the Hopf invariant emerges in the effective field theory, the spin and statistics of low-energy excitations fractionalize. The analysis is based on a path integral representation of the t-J model which was obtained recently by means of Dirac quantization. After introducing a staggered quantization axis, the single occupancy constraint which is inherent to the t-J model can be taken into account exactly. We perform a long-wavelength, low-frequency gradient expansion of the effective action and integrate over the fermionic degrees of freedom as well as the fast-fluctuating bosonic modes. Since our derivation is based on a microscopic model, we obtain an effective action where the doping dependence of the coupling constants is made explicit.