Quantum phase transition in transverse-field Ising model on Sierpi\'nski gasket lattice

Abstract

We investigate the quantum phase transition in the transverse-field Ising model on the Sierpi\'nski gasket using finite-size scaling (FSS) and numerical renormalization group (NRG). Since next generations of the fractal lattice contain exponentially more spins, which in turn increase exponentially the Hilbert space dimension, we challenge and prove usefulness of small systems in FSS. We identified a quantum critical point at $\lambda_c \approx 2.63 - 2.93$, with critical exponents $ \nu \approx 0.64 - 0.71, \beta \approx 0.30, \gamma \approx 1.67$ and $z \approx 1.33$. The numerical renormalization group method produced results consistent with finite-size scaling approach ($\lambda_c = 2.766$$, \beta = 0.316$), supporting our findings. Compared to the values reported so far in the literature, critical field is in a strong disagreement, while exponents are generally similar excluding $\beta$ and $\gamma$. However, it should be noted, that the lattice investigated in previous works is different from ours, while the latter is in our opinion the standard Sierpi\'nski gasket.