Quantum critical behavior of the quantum Ising model on fractal lattices

I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.

Figure 1

Sierpiński carpet constructed by the method explained in the text. The recursion process has been repeated five times for this example.

Figure 2

Binder cumulant $U$ as a function of temperature $T$ for (a) $\Delta =2.395$, (b) $\Delta =2.3975$, and (c) $\Delta =2.4$. The system size is $L=3^n$, where $n$ is the number of recursions. The error bars are smaller than the symbols.

Figure 3

Scaling function $\tilde{U}(0,T{L}^z)$ for (a) $z=1.17$, (b) $z=1.22$, and (c) $z=1.27$. The error bars are smaller than the symbols.

Figure 4

Scaling functions (a) $\tilde{U}$, (b) $\tilde{m}$, and (c) $\tilde{\chi }$. I used $z=1.22$ while keeping $T{L}^z=1$ for all system sizes. The critical exponents used here are $\nu =0.70,\beta =0.37$, and $\gamma =1.45$. The error bars are smaller than the symbols.

Figure 5

(a) A Sierpiński gasket can be constructed in a manner similar to the Sierpiński carpet, but using triangles instead of squares. (b) A Sierpiński tetrahedron is constructed out of tetrahedrons and is a three-dimensional analog of a Sierpiński gasket.

Figure 6

Scaling plots of the Binder cumulant for (a) the Sierpiński gasket ($z=1.18$ and $\Delta =1.865$), (b) the Sierpiński tetrahedron ($z=1.30$ and $\Delta =2.707$), and (c) the two-dimensional square lattice ($z=1$ and $\Delta =3.045$).