BKT-like Correlation Scaling and Twist Responses in a One-Dimensional Fractional $U(1)$ Ginzburg--Landau Model
We study a one-dimensional fractional $U(1)$ Ginzburg--Landau model whose quadratic part has Fourier multiplier $|k|^\sigma$, focusing on the marginal case $\sigma=1$. This dispersion yields logarithmic spin-wave fluctuations, suggesting BKT-like behavior despite the one-dimensional setting. We sample the equilibrium Gibbs measure using stochastic Gross--Pitaevskii dynamics and analyze correlation functions, dimensionless ratios, effective exponents, and twist responses. The correlation function shows a low-temperature algebraic branch with a temperature-dependent exponent, while the high-temperature regime exhibits a nonlocal-kernel-induced tail consistent with $C(r)\sim r^{-2}$. The correlation and Binder ratios are nearly size independent at low temperature and collapse with the BKT-type variable $(T-T_{\rm BKT})(\log L)^2$; finite-size effects set in around $T\simeq0.35\text{--}0.4$, consistent with $T_{\rm BKT}\simeq0.35$. Unlike the two-dimensional XY model, twist responses do not yield a finite helicity modulus: the ordinary linear-response quantity grows with system size, whereas the cusp twist response scales as $L^{-\eta(T)}$, like the squared zero-mode order parameter. Thus, the transition is BKT-like in correlation scaling, but lacks a universal helicity-modulus jump.