RG theory and parameter dependence of the fractional BKT-like transition

Develop a systematic renormalization-group derivation for the compact fractional phase model that clarifies the relationship among phase-slip fugacity, the correlation exponent \(\eta(T)\), and the observed BKT-like finite-size scaling; determine the behavior of the one-dimensional fractional \(U(1)\) Ginzburg--Landau model for values of \(\sigma\) other than \(1\), particularly across the crossover from the marginal case \(\sigma=1\) to \(\sigma<1\), where true long-range order may appear; and establish the relationship between fractional nonlocality, stochastic Gross--Pitaevskii dynamics, and possible realizations in long-range superfluid or condensate systems.

Background

The paper studies a one-dimensional fractional $U(1)$ Ginzburg--Landau model with quadratic Fourier dispersion $|k|^\sigma$, focusing on the marginal case $\sigma=1$. Numerical evidence suggests BKT-like algebraic correlations and logarithmic finite-size scaling, but the theoretical interpretation is based primarily on a heuristic renormalization-group argument involving logarithmically interacting phase slips. A systematic RG treatment is therefore needed to connect phase-slip fugacity and the temperature-dependent exponent $\eta(T)$ quantitatively to the observed scaling behavior.

The authors also identify unresolved extensions to other fractional exponents, especially the transition from $\sigma=1$ to $\sigma<1$, where the spin-wave approximation does not exclude true long-range order. In addition, the broader connection between the fractional nonlocal model, the stochastic Gross--Pitaevskii sampling dynamics, and experimentally or physically realizable long-range superfluid and condensate systems remains to be clarified.