Fig. 7

The probability distribution of the spin overlaps parameter, described by Eq. (5). The spin overlap parameter is calculated 30 times, each for two independent \(16 \times 16\) lattice, as explained in the results section. We do this at an interaction strength ratio of \(\frac{J}{{J^{\prime } }} = 3.5\), as a representative of the ratio greater than 2, and at an interaction strength ratio of \(\frac{J}{{J^{\prime } }} = 0.5\), as a representative of the ratio less than 2. From the phase diagram (Fig. 5), we expect the glassy state behavior at low temperatures of \(\frac{J}{{J^{\prime } }} > 2\). As it is shown, at high temperature of \(\frac{J}{{J^{\prime } }} = 3.5\) (a), we see two peaks at the values 1 and − 1, indicating the checkerboard order. However, at low temperature (b), we see a broad distribution of the overlap parameter, which is a sign of glassy state. At the high temperature corresponding to (c), we observe a distribution with a peak at zero, which is the expected behavior of a completely disordered (paramagnetic) state. Another important point to note is that, in panel (d), one might not initially expect a broad distribution, as the system is in an ordered phase—the checker-stripy phase—rather than a glassy state. However, the checker-stripy phase is a quasi-ordered phase characterized by near-degenerate configurations. In each simulation run, the system can settle into one of these energetically similar configurations, leading to a spread in the measured values. This explains the observed broadness in the distribution. Nevertheless, the spread is not as extensive as in the glassy phase; instead, it clusters around a few specific values corresponding to different checker-stripy arrangements.