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

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.

💡 Research Summary

The authors investigate the quantum phase transition of the transverse‑field Ising model (TFIM) on a fractal lattice, the Sierpiński gasket, whose Hausdorff dimension is d_H = ln 3/ln 2 ≈ 1.585. The study combines two complementary numerical approaches—finite‑size scaling (FSS) and numerical renormalization group (NRG)—and first validates both methods on the exactly solvable one‑dimensional TFIM. Using exact diagonalization with Lanczos for chains of N = 6, 10, 14, 16, 18 spins, they recover the known critical point λ_c = 1 and critical exponents z = 1, ν = 1, β = 1/8, γ ≈ 1.75, confirming that reliable exponent estimates can be obtained even from very small systems when appropriate scaling windows and fitting procedures are used.

Having established the methodology, the authors turn to the Sierpiński gasket. Because of the fractal geometry, only modest system sizes (N = 11 and 15 spins) are tractable. They exploit the lattice’s rotational (120°) and global spin‑flip symmetries to block‑diagonalize the Hamiltonian, reducing the Hilbert space dimension dramatically. For each size they compute four observables: Binder cumulant, magnetization, energy gap, and susceptibility. Global nonlinear least‑squares fits to the standard scaling forms (U_N = B(ε N^{1/ν}), Δ N^{z}=G(ε N^{1/ν}), m N^{β/ν}=M(ε N^{1/ν}), χ N^{-γ/ν}=X(ε N^{1/ν})) yield consistent estimates across observables. The Binder analysis gives λ_c ≈ 2.724 and ν ≈ 1.13; magnetization scaling gives λ_c ≈ 2.93, β ≈ 0.30 and ν ≈ 1.10; gap scaling provides a dynamical exponent z ≈ 0.84; susceptibility scaling yields γ ≈ 2.54 and ν ≈ 1.55. The spread in λ_c values (2.70–2.93) reflects finite‑size effects, but the correlation‑length exponent ν remains robust around 1.1.

The second approach, NRG, follows the scheme of Jullien et al. for 1D TFIM and is adapted to the gasket by defining blocks B_k consisting of two T_k triangles sharing a corner. Blocks of 3 and 9 spins are diagonalized exactly; the low‑energy subspace is projected onto an effective spin‑½ degree of freedom, yielding renormalized transverse field h_{n+1}=ΔE_n/2 and effective coupling J_{n+1}=ξ_a(ξ_b+ξ_c)J_n, where ξ’s are matrix elements of σ^x between the two lowest eigenstates. Iterating these recursions reproduces the 1D critical point and, when applied to the gasket, gives λ_c = 2.765 and β ≈ 0.306, in excellent agreement with the FSS results.

The combined evidence points to a distinct universality class for the TFIM on the Sierpiński gasket. The dynamical exponent z < 1 indicates that quantum fluctuations are slowed by the fractal connectivity, effectively raising the critical dimension to d_eff ≈ 2.43 (using the quantum‑to‑classical mapping d_eff = d_H + z). The order‑parameter exponent β is substantially larger than the 1D value (0.125) and the susceptibility exponent γ is far above the mean‑field value, signalling strong critical fluctuations.

Limitations of the work stem from the small system sizes accessible to exact diagonalization; while symmetry reduction mitigates the Hilbert‑space explosion, finite‑size corrections remain significant, especially for the gap and susceptibility analyses. Future work could employ tensor‑network methods, quantum Monte‑Carlo on larger fractal clusters, or experimental quantum simulators (e.g., Rydberg atom arrays) to test the scaling predictions and to explore whether the observed exponents persist in the thermodynamic limit. The paper thus establishes a solid computational framework for studying quantum criticality on fractal lattices and opens avenues for probing new universality classes beyond integer dimensions.