Logarithmic Quasi-Long-Range Order and Novel Continuous Phase Transition in the Two-Dimensional $XY$ Model with $1\mathrm{/}{r}^4$ Interaction

We investigate the phase diagram and critical properties of the two-dimensional classical XY model with interactions decaying as $1\mathrm{/}{r}^{2+\sigma }$. At the marginal case $\sigma =2$, we show that the low-temperature phase is characterized by logarithmic quasi-long-range order (log-QLRO), where spin correlations decay as a power of the logarithm of distance, $C(r)\sim (\ln r{)}^{-{\eta }_{\mathrm{\ell }}}$ with ${\eta }_{\mathrm{\ell }}\propto T$, and that the transition into this phase is a continuous transition beyond both the Ginzburg-Landau-Wilson paradigm and the Berezinskii-Kosterlitz-Thouless mechanism. Our analysis combines a near-exact Gaussian spin-wave theory, an adiabatic renormalization-group analysis, and large-scale Monte Carlo simulations. We find that the $1\mathrm{/}{r}^4$ interaction logarithmically modifies the spin-wave stiffness kernel as $\gamma (k)\simeq \kappa k^2\ln (1\mathrm{/}k)$, which suppresses vortex proliferation and gives rise to a nonuniversal correlation-length exponent $\nu \propto 1\mathrm{/}\sqrt{2\pi \kappa }$. Crucially, our simulations reveal that the short-range algebraic QLRO phase is unstable against a weak long-range perturbation for $\sigma \le 2$. These results establish $\sigma =2$ as the crossover boundary between the long-range and short-range universality classes and provide a critical evaluation of recent theoretical proposals.