Crossover effects on the phase transitions phenomena translated by arborecences and spectral properties

Abstract

This study investigates how visibility graphs constructed from Monte Carlo Markov Chain time series of spin models capture the critical behavior of the system. More precisely, we show that this approach identifies continuous phase transitions as well as important nuances, such as crossover effects occurring in the transition from a critical line to a first-order line through a tricritical point, as observed, for example, in the Blume--Emery--Griffiths model or, in a simpler setting, in the Blume--Capel model. By applying Kirchhoff's theorem, we show that the number of spanning trees of the resulting graphs serves as a sensitive indicator of these phase transitions. Furthermore, a qualitative analysis of the adjacency matrices based on random matrix theory provides additional evidence for these phenomena. The methodology developed here can potentially be extended to the analysis of criticality in empirical time series from complex systems, such as climate, financial, and epidemiological data, where the Hamiltonian governing the dynamics is not necessarily known.

Figures (7)

• Figure 1: Plot A: A possible time evolution ($j-$th evolution) of the magnetization of the BC model for $N_{\text{steps}}=4$. Plot B: Visibility graph corresponding to the time series. Plot C: All spanning trees of the visiblity graph.
• Figure 2: Structural entropy and its derivative as a function of $T/T_{C}$ for different points along the critical line of the BC model. The tree entropy (structural entropy) exhibits a plateau in the vicinity of the critical temperature. In contrast, the derivative of this quantity shows a pronounced peak at $T=T_{C}$ for points far from the tricritical point. In the vicinity of the tricritical point ($D_{C}=1.9501$) and at the tricritical point itself ($D_{C}=1.9655$), crossover effects affect the estimates, as expected.
• Figure 3: Effects of the initial conditions on the tree entropy. As $m_{0}\to 0$, the peak gradually transforms into a plateau that includes the value $T=T_{C}$.
• Figure 4: (a) Zoom of the plateau region for the case $D=1$. This region is well described by a Boltzmann (sigmoidal) function. (b) Boltzmann fits for the other values of $D$. The tricritical point and the nearby point are not well fitted due to crossover effects, which are characterized by the parameter $x_{0}$.
• Figure 5: Density of eigenvalues of the adjacency matrix of the VGs for different temperatures. Panels (a) and (b) correspond to critical points, while panel (c) corresponds to the tricritical point. Panel (d) shows a comparison between the critical and tricritical cases at low and high temperatures, together with the eigenvalue density of a visibility graph obtained from uncorrelated Gaussian noise.