Kinked Entropy and Discontinuous Microcanonical Spontaneous Symmetry Breaking

Spontaneous symmetry breaking (SSB) in statistical physics is a macroscopic collective phenomenon. For the paradigmatic $Q$-state Potts model it means a transition from the disordered color-symmetric phase to an ordered phase in which one color dominates. Existing mean field theories imply that SSB in the microcanonical statistical ensemble (with energy being the control parameter) should be a continuous process. Here we study microcanonical SSB on the random-graph Potts model and discover that the entropy is a kinked function of energy. This kink leads to a discontinuous phase transition at certain energy density value, characterized by a jump in the density of the dominant color and a jump in the microcanonical temperature. This discontinuous SSB in random graphs is confirmed by microcanonical Monte Carlo simulations, and it is also observed in bond-diluted finite-size lattice systems.

Figure 1

Schematic drawing of kinked entropy density $s(u)$. As the energy density $u$ of the $Q$-state Potts model decreases to $u_{\text{mic}}$, $s(u)$ changes from concave to convex and its slope drops from ${\beta }_1$ to ${\beta }_2$. The system is color-symmetric at $u_{\text{mic}}+ϵ$ ($ϵ\to 0$) with a lower microcanonical temperature $1/{\beta }_1$, but at $u_{\text{mic}}-ϵ$ it has a highly dominant color and a higher microcanonical temperature $1/{\beta }_2$.

Figure 2

Potts model on regular random graphs, $K=4$ and $Q=6$. (a) Free-energy densities $f(\beta )$ for the disordered symmetric (DS, solid line), the canonical polarized ($CP$, dashed line), and the microcanonical polarized (MP, dotted line) fixed points of the BP equation. The DS solution is stable at inverse temperature $\beta <{\beta }_{\mathrm{DS}}=1.386$, the $CP$ solution exists for $\beta \ge {\beta }_{\mathrm{CP}}=1.147$, and the DS-CP phase transition occurs at ${\beta }_{\mathrm{c}}=1.174$ with the energy density $u$ dropping from $-0.786$ to $-1.523$. (b) Energy density $u(\beta )$, entropy density $s(\beta )$, fixed-point value $q(\beta )$, and density ${\rho }_1(\beta )$ of the dominant color (inset), for the MP fixed point. The maximal achievable MP energy density is $u_{\max }=-0.861$. (c),(d) Entropy density $s$ and dominant color density ${\rho }_1$ vs energy density $u$ for the DS (solid line), MP (dashed line), and $CP$ (dotted line) fixed points. Upper-left and lower-right insets of (c) show an enlarged view of the MP entropy profile and the difference $\mathrm{\Delta }s$ between the MP and DS entropy density values ($\mathrm{\Delta }s=0$ at energy density $u_{\text{mic}}=-0.864$). Symbols in (d) are microcanonical Monte Carlo simulation results obtained on a single RR graph ($N=65536$) and several bond-diluted eight-dimensional periodic hypercubic lattices of side length $L=4$, 5, 6 (D8, $N=L^8$), degree $K=4$ and $Q=6$. The inset of (d) is an enlarged view of the transition region, and the phase transition point $u_{\text{mic}}$ for RR graphs is marked by the vertical dashed line, at which ${\rho }_1$ jumps from $1/6$ to 0.293.