Quantum critical behavior of the quantum Ising model on fractal lattices

TL;DR

This paper investigates the quantum critical behavior of the transverse-field quantum Ising model on fractal lattices, revealing unique critical exponents and universality classes influenced by fractal geometry.

Contribution

It provides the first detailed analysis of quantum critical points on fractal lattices, showing that critical exponents depend on fractal structure beyond just dimensionality.

Findings

Dynamic critical exponent greater than one for all fractal lattices

Critical exponents differ from classical values and satisfy quantum scaling

Sierpiński tetrahedron belongs to a different universality class

Abstract

I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpi\'nski carpet, Sierpi\'nski gasket, and Sierpi\'nski tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and the 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 Sierpi\'nski tetrahedron, of which the dimension is exactly two, 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.