First- and second-order quantum phase transitions of a $q$-state Potts model in fractal lattices

Quantum phase transitions of a $q$-state Potts model in fractal lattices are studied using a continuous-time quantum Monte Carlo simulation technique. For small values of $q$, the transition is found to be second order and critical exponents of the quantum critical point are calculated. The dynamic critical exponent $z$ is found to be greater than one for all fractals studied, which is in contrast to integer-dimensional regular lattices. When $q$ is greater than a certain value $q_c$, the phase transition becomes first order, where $q_c$ depends on the lattice. Further analysis shows that the characteristics of phase transitions are more sensitive to the average number of nearest neighbors than the Hausdorff dimension or the order of ramification.

Figure 1

Fractal lattices studied in this paper: (a) Sierpiński triangle, (b) Sierpiński carpet, (c) Sierpiński tetrahedron, and (d) Sierpiński body-centered cube. In (a)–(c), the result of three recursion steps is shown, whereas the first-step basic structure is drawn in (d).

Figure 2

Data for the Sierpiński triangle, $q=3$. The semilogarithmic plot of the Binder cumulant $U$ is drawn as a function of temperature $T$ for various system sizes $L$. The quantum parameter $\mathrm{\Delta }$ is (a) 0.5812, (b) 0.5817, and (c) 0.5822, respectively. Error bars are smaller than the symbols.

Figure 3

Data for the Sierpiński triangle, $q=3$. (a) Scaling function $\tilde{U}$ drawn against $T{L}^z$ in a semilogarithmic plot. ($\mathrm{\Delta }=0.5817$ and $z=1.23$.) Scaling functions (b) $\tilde{U}$, (c) $\tilde{m}$, and (d) $\tilde{\chi }$ are plotted against $(\mathrm{\Delta }-{\mathrm{\Delta }}_c)L^{1/\nu }$ using the following parameters: ${\mathrm{\Delta }}_c=0.5817$, $T{L}^z=0.8$, $z=1.23$, $\nu =0.53$, $\beta =0.145$, and $\gamma =1.24$. In each graph, it is obvious that all data collapse onto a single scaling curve near the quantum critical point. Error bars are smaller than the symbols.

Figure 4

Data for the Sierpiński triangle, $q=4$. Normalized distribution of (a) quantum energy per spin $e_{\mathrm{\Delta }}$ and (b) total energy per spin $e$, for different system sizes. ($\mathrm{\Delta }=0.42085$ and $T=0.025$ for $L=16$; $\mathrm{\Delta }=0.4205835$ and $T=0.0106$ for $L=32$.)