Pairing-induced kinetic energy lowering in doped antiferromagnets

We analyse lowering of the kinetic energy in doped antiferromagnets at the transition to the superconducting state. Measurements of optical conductivity indicate that such unconventional behaviour takes place in underdoped Bi-2212. We argue that the definition of the operator representing the kinetic energy is determined by experimental conditions. The thermodynamic average of that operator is related to the integrated spectral weight of the optical conductivity and thus depends on the cut-off frequency limiting that integral. If the upper limit of the integral lies below the charge transfer gap the spectral weight represents the average of the hopping term in the space restricted to the energy range below the gap. We show that the kinetic energy is indeed lowered at the superconducting transition in the t–J model ( tJ M), which is an effective model defined in the restricted space. That result is in agreement with experimental observations and may be attributed to the formation of spin polarons and the change of roles which are played by the kinetic and the potential energy in the tJ M and in some effective model for spin polarons. The total spectral weight represents the kinetic energy in a model defined in a broader space if the upper limit in the integral of the optical conductivity is set above the gap. We demonstrate that the kinetic energy in the Hubbard model is also lowered in the superconducting state. That result does not agree with experimental observations, indicating that the spectral weight is conserved for all temperatures if the upper limit of the integral is set above the charge transfer gap. This discrepancy suggests that a single band model is not capable of describing in some respects the physics of excitations across the gap.