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Effects of the pseudo-gap on the field-induced kinetic energy density of Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ single crystals

We report on magnetization experiments from which we obtain the field-induced kinetic energy density, $E_k$, in the superconducting phase of several Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ single crystal samples. The kinetic energy magnitude changes according to the characteristic reduction of the single-electron density of states produced by the pseudogap in the underdoped limit. Moreover, a remarkable peak of $E_k$ occurring at the specific holes density $p\sim0.18$ is related to a van Hove singularity due to the pseudogap closure. We also extracted the superfluid density, $ρ_s$. We conclude that $E_k$ and $ρ_s$ are related to the pseudogap energy scale. This result is understood as an evidence of the coexistence between superconductivity and the pseudogap phenomenon in the Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ cuprate compound.