Luttinger liquid parameters in one-dimensional Rydberg arrays
We investigate Berezinskii-Kosterlitz-Thouless (BKT) transitions in one-dimensional Rydberg chains, where commensurate critical regimes associated with the melting of crystalline orders with period larger than five and incommensurate floating phases are both described by Luttinger liquid theory. The central quantity is the Luttinger liquid parameter $K$, which characterizes the universal low-energy theory and controls the relevance of perturbations driving BKT transitions. We extract $K$ using Friedel oscillations and the recently developed crosscap method introduced in Phys. Rev. Lett. 134, 076501 (2025). As benchmarks, we first apply the crosscap method to a $\mathbb{Z}_3$ dual hard-core boson chain and a spin-1 XY chain with single-ion anisotropy, obtaining BKT transition points consistent with previous results after finite-size extrapolation. We then compute $K$ in the Rydberg chain along lines with fixed correlation-oscillation period near BKT transitions. Along the commensurate period-five line, the critical values of $K$ predicted by sine-Gordon theory reveal two BKT transitions separating the disordered phase, the critical phase, and the $\mathbb{Z}_5$ crystalline phase. The results from Friedel oscillations and the crosscap method agree with each other and are further supported by energy-gap scaling and Binder-cumulant analysis. To obtain reliable values of $K$ from Friedel oscillations, we use a multi-harmonic fitting scheme throughout the analysis. Along incommensurate lines, the BKT points obtained from Friedel oscillations agree with those extracted from entanglement entropy. Finally, we show that the values of $K$ obtained from the two methods are mutually consistent inside the incommensurate floating phase.