The doping dependence of the superfluid density ${\rho }_{\text{s}}$ exhibits distinct behaviors in the underdoping and overdoping regimes of cuprates, while the superconducting (SC) transition temperature $T_c$ generally scales with ${\rho }_{\text{s}}$. In this paper, we present a unified understanding of the superconducting transition temperature $T_c$ and ${\rho }_{\text{s}}$ across the entire doping range by incorporating the underlying mutual Chern-Simons gauge structure that couples the spin and charge degrees of freedom in a doped Mott insulator. Within this framework, the SC phase fluctuations are deeply intertwined with the spin dynamics, such that thermally excited neutral spins determine $T_c$, while quantum spin excitations effectively reduce the superfluid density at zero temperature. As a result, a Uemura-like scaling of $T_c$ vs ${\rho }_{\text{s}}$ in the underdoped regime naturally emerges, while the suppression of both $T_c$ and ${\rho }_{\text{s}}$ at overdoping is attributed to a drastic reduction of antiferromagnetic spin correlations.

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• Superfluid density
• High-temperature superconductors
• Chern-Simons gauge theory
• t-J model