Quantum phase transition in transverse-field Ising model on Sierpiński gasket lattice

TL;DR

This work analyzes quantum phase transitions of the transverse-field Ising model on the fractal Sierpiński gasket using finite-size scaling and numerical renormalization group methods. Benchmarking on the exactly solvable 1D chain validates the methodologies, then applying them to the gasket reveals a quantum critical point at $\lambda_c$ in the range $2.72$–$2.93$ with exponents $z \approx 0.84$, $\nu \approx 1.1$, $\beta \approx 0.30$, and $\gamma \approx 2.54$, corresponding to an effective dimension $d_{\text{eff}} \approx d_H + z \approx 2.43$. The dynamical exponent is significantly reduced by fractal geometry, indicating slower quantum fluctuations and non-mean-field universality distinct from both 1D and mean-field predictions. Numerical renormalization group analyses corroborate the FSS findings, demonstrating reliable extraction of critical properties for fractal lattices with limited system sizes.

Abstract

We study quantum phase transition in the transverse-field Ising model on the Sierpiński gasket. By applying finite-size scaling and numerical renormalization group methods, we determine the critical coupling and the exponents that describe this transition. We first checked our finite-size scaling and the renormalization methods on the exactly solvable one-dimensional chain, where we recovered proper values of critical couplings and exponents. Then, we applied the method to the Sierpiński gasket with 11 and 15 spins. We found a quantum critical point at $λ_c \approx 2.72$ to $2.93$, with critical exponents $z\approx0.84$, $ν\approx 1.12 $, $β\approx 0.30$, and $γ\approx 2.54$. The lower dynamical exponent $z$ indicates that quantum fluctuations slow down due to fractal geometry, yielding an effective critical dimension of about 2.43. The numerical renormalization group method yielded similar results $λ_c = 2.765$, $β= 0.306$, supporting our findings. These exponents differ from those in both the one-dimensional and mean-field cases.

Figures (11)

• Figure 1: The Binder cumulant before (left) and after rescaling (right). As expected from the finite size scaling ansatz, Eq. (\ref{['Binder_scal']}), the data for system sizes $N$=14, 16, 18 fall onto a single curve for $\lambda_c=0.9835, \nu = 1.0044$ (right).
• Figure 2: The magnetization before (left) and after rescaling (right) according to the Eq. \ref{['m_scal']}. The values of $m$ calculated for system sizes $N=14, 16, 18$ fall onto a single curve for $\lambda_c=1.0051$, $\beta=0.1253$, and $\nu=0.9157$ (right). This collapse was obtained with fitted parameters differing by 0.3% from the exact value of $\beta$ and 8% from the exact value of $\nu$ - it showes the robustness of the FSS procedure even for very small systems.
• Figure 3: The gap between ground state and first excited state before (left) and after rescaling (right). The values of the gap calculated for system sizes $N=14, 16, 18$ fall onto a single curve for $\lambda_c=0.9998$ and $\nu=1.0123$ (right). To perform this scaling, $z=1$ was assumed. Taking the value of $\nu$ from the scaling, for example of $m$ and fixing it, yields a slightly lower value of $z$; see the text.
• Figure 4: Sierpiński gasket lattices with periodic boundary conditions.
• Figure 5: The Binder cumulant for the Sierpiński gasket before (left) and after rescaling (right). The data for system sizes $N = 11, 15$ collapse onto a single curve for $\lambda_c = 2.7239$ and $\nu = 1.1296$.