Emergence of Finite-Temperature Long-Range Order in the One-Dimensional Attractive Hubbard Model with Power-Law Hopping

The one-dimensional attractive Hubbard model with short-range (nearest-neighbor) hopping is known to obey the Mermin-Wagner theorem, precluding true long-range order at any temperature. However, at zero temperature, it exhibits quasi-long-range order characterized by algebraically decaying correlation functions. In this work, we investigate how extending the hopping amplitude to follow a power-law decay, t(r) ∝ 1/rα, modifies this paradigm. Using quantum Monte Carlo simulations and analytical considerations, we explore how the exponent α controls the effective dimensionality of the system and the possible emergence of genuine off-diagonal long-range order (ODLRO) at finite temperature. We present evidence that sufficiently slow-decaying hopping (α ≲ 1) stabilizes a finite-temperature phase with true ODLRO. These results highlight the long-range Hubbard chain as a minimal model to realize finite-temperature superconducting order in one dimension and provide a tunable platform for exploring higher dimensions.