Persistent homology analysis of phase transitions

Persistent homology analysis, a recently developed computational method in algebraic topology, is applied to the study of the phase transitions undergone by the so-called mean-field $\mathit{XY}$ model and by the ${\displaystyle {\varphi }^4}$ lattice model, respectively. For both models the relationship between phase transitions and the topological properties of certain submanifolds of configuration space are exactly known. It turns out that these a priori known facts are clearly retrieved by persistent homology analysis of dynamically sampled submanifolds of configuration space.

Figure 1

Persistence diagram for the ${\displaystyle \mathrm{MF}XY}$ model. Persistence distributions of ${\displaystyle H_1}$ generators below (${\displaystyle \varepsilon =0.6}$), at (${\displaystyle {\varepsilon }_c=0.75}$), and above (${\displaystyle \varepsilon =0.88}$) the phase transition.

Figure 2

Persistence diagram for the ${\displaystyle {\varphi }^4}$ model. Persistence distributions of ${\displaystyle H_1}$ generators below (${\displaystyle \varepsilon =25}$) and above (${\displaystyle \varepsilon =35}$) the phase transition, occurring at the critical energy density ${\displaystyle {\varepsilon }_c\simeq 31}$.

Figure 3

Homological features of the ${\displaystyle \text{MF}XY}$ model. Raw (inset) and rescaled (main plot) distributions of deaths for the generators of the first homology group ${\displaystyle H_0}$. Note that the width and shape of the distributions change across the transition, becoming more and more narrow as the energy is increased.

Figure 4

Homological features of the ${\displaystyle {\varphi }^4}$ model. Raw (inset) and rescaled (main plot) distributions of deaths for the generators of the first homology group ${\displaystyle H_0}$. At variance with the ${\displaystyle \text{MF}XY}$ model, here there is no appreciable change in the width and shape of the distributions across the transition. The green points refer to ${\displaystyle \varepsilon =25<{\varepsilon }_c}$. The velvet points refer to ${\displaystyle \varepsilon =35>{\varepsilon }_c}$.

Figure 5

Distributions of persistences for the generators of the homology group ${\displaystyle H_1}$ in the case of the ${\displaystyle \text{MF}XY}$ model. In this case the difference in functional forms for the ${\displaystyle H_1}$ persistence distribution below and above the transition is even clearer.

Figure 6

Distributions of persistences for the generators of the homology group ${\displaystyle H_1}$ in the case of the ${\displaystyle {\varphi }^4}$ model. In this case no difference is found in functional form for the ${\displaystyle H_1}$ persistence distributions below and above the transition.

Figure 7

Average persistence landscape of the ${\displaystyle H_1}$ homology for the ${\displaystyle \text{MF}XY}$ model. ${\displaystyle {\mathrm{\Lambda }}_p}$ is the average function (see text) reported as a function of the radius ${\displaystyle \rho }$ of the balls used to construct the Rips-Vietoris simplicial complex. The shadows around solid lines are 95% confidence band.

Figure 8

Average persistence landscape of the ${\displaystyle H_1}$ homology for the ${\displaystyle {\varphi }^4}$ model. ${\displaystyle {\mathrm{\Lambda }}_p}$ is the average function (see text) reported as a function of the radius ${\displaystyle \rho }$ of the balls used to construct the Rips-Vietoris simplicial complex. The shadows around solid lines are 95% confidence band.

Figure 9

A graphic representation of a simplicial complex.

Figure 10

Rips complex filtration. Reproduced with owner's permission from Ref. [6].