Electron fractionalization and unconventional order parameters of the t-J model

In the $t-J$ model, the electron fractionalization is unique due to the non-perturbative phase string effect. We formulated a lattice field theory taking this effect into full account. Basing on this field theory, we introduced a pair of Wilson loops which constitute a complete set of order parameters determining the phase diagram in the underdoped regime. We also established a general composition rule for electric transport expressing the electric conductivity in terms of the spinon and the holon conductivities. The general theory is applied to studies of the quantum phase diagram. We found that the antiferromagnetic and the superconducting phases are dual: in the former, holons are confined while spinons are deconfined, and {\it vice versa} in the latter. These two phases are separated by a novel phase, the so-called Bose-insulating phase, where both holons and spinons are deconfined and the system is electrically insulating.

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