Fig. 3: Correlation functions and structure factors for different phases.

a The experimental correlation matrix \(\langle {m}_{i}{m}_{{i}^{{\prime} }}\rangle\) employing the order parameter m_i = n_i,1 − n_i,0 for a system size of L = 21. The presented results correspond to (Δ/Ω, R_b/a) = (0, 2.4) in the disordered phase, (3.5, 2.1) in the \({{\mathbb{Z}}}_{3}\times {{\mathbb{Z}}}_{2}\) phase, (3.5, 2.45) in the floating phase, and (3.5, 2.7) in the \({{\mathbb{Z}}}_{4}\) phase. Black boxes indicate examples of minimal repeating patterns for \({{\mathbb{Z}}}_{3}\times {{\mathbb{Z}}}_{2}\) and \({{\mathbb{Z}}}_{4}\) orders, which repeat every three or four sites, respectively. The floating phase lacks a repeating pattern due to an incommensurate wavelength. b The mean correlator C(r) is determined by averaging correlation functions \(\langle {m}_{i}{m}_{{i}^{{\prime} }}\rangle\) over the same relative distance \(r=i-{i}^{{\prime} }\). Both numerical and experimental results are presented, showing nearly identical oscillation periods. c The structure factor S(k) is derived from the Fourier transform of \(\langle {m}_{i}{m}_{{i}^{{\prime} }}\rangle\) with respect to \(i-{i}^{{\prime} }\). While weak signals are observed in the disordered phase, both numerical and experimental results display robust matching signals for \({{\mathbb{Z}}}_{3}\times {{\mathbb{Z}}}_{2}\), floating, and \({{\mathbb{Z}}}_{4}\) phases. Some detailed features of the figures such as beat patterns and peak asymmetries are discussed extensively in Supplementary Note 1.