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arXiv:2602.02904 · Zeitschrift für Naturforschung A

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

Tymoteusz Braciszewski, Oliwier Urbański, Piotr Tomczak

cond-mat.stat-mechquant-phSubmitted Feb 2, 2026• Announced Feb 4, 2026• 0 citations• Revised Apr 15, 2026 (v2)
v1 · Feb 2, 2026v2 · Apr 15, 2026
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Abstract

We investigate the quantum phase transition in the transverse-field Ising model on the Sierpiński 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 $λ_c \approx 2.63 - 2.93$, with critical exponents $ ν\approx 0.64 - 0.71, β\approx 0.30, γ\approx 1.67$ and $z \approx 1.33$. The numerical renormalization group method produced results consistent with finite-size scaling approach ($λ_c = 2.766$$, β= 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 $β$ and $γ$. 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 Sierpiński gasket.

Journal reference
Z. Naturforsch. A 81 (2026) 505-512
DOI
10.1515/zna-2025-0405

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We investigate the quantum phase transition in the transverse-field Ising model on the Sierpiński 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 λc≈2.63−2.93λ_c \approx 2.63 - 2.93λc​≈2.63−2.93, with critical exponents ν≈0.64−0.71,β≈0.30,γ≈1.67 ν\approx 0.64 - 0.71, β\approx 0.30, γ\approx 1.67ν≈0.64−0.71,β≈0.30,γ≈1.67 and z≈1.33z \approx 1.33z≈1.33. The numerical renormalization group method produced results consistent with finite-size scaling approach (λ_c = 2.766$$, β= 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 βββ and γγγ. 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 Sierpiński gasket.
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.
We identify a quantum critical point with fractal symmetry whose effective theory eludes the renormalization group framework. We consider the Newman-Moore model with three-body interaction subjected to an external transverse field, which exhibits a Kramers-Wannier type self-duality and a fractal Z2Z_2Z2​ symmetry with Ising charge conserved on a fractal subset of sites, i.e., on Sierpinski gaskets. Using large-scale quantum Monte Carlo simulations, we identify a continuous quantum phase transition between a phase with spontaneous fractal symmetry breaking and a paramagnetic phase. This phase transition is characterized by the emergence of a fractal scaling dimension d=ln⁡(3)/ln⁡(2)d=\ln(3)/\ln(2)d=ln(3)/ln(2) at the quantum critical point, where the power-law exponent of the correlation function is related to the fractal dimension of the Sierpinski triangle. We develop a field theory to elucidate such quantum criticality and denote the fractal scaling as a subsequence of UV-IR mixing, where the low energy modes at the critical point are manipulated by short-wavelength physics due to the fractal symmetry.
The transverse-field Ising model on the Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski fractal, which is characterized by the fractal dimension ${log}_{2}^{\phantom{\rule{3.33333pt}{0ex}}}3\ensuremath{\approx}1.585$, is studied by a tensor-network method known as the higher-order tensor renormalization group. We analyze the ground-state energy and the spontaneous magnetization in the thermodynamic limit. The system exhibits the second-order phase transition at the critical transverse field hc=1.865{h}_{\mathrm{c}}^{\phantom{\rule{3.33333pt}{0ex}}}=1.865hc​=1.865. The critical exponents $\ensuremath{\beta}\ensuremath{\approx}0.198$ and $\ensuremath{\delta}\ensuremath{\approx}8.7$ are obtained. Complementary to the tensor-network method, we make use of the real-space renormalization group and improved mean-field approximations for comparison.
We investigate the quantum phase transitions of the transverse-field quantum Ising model on the triangular lattice and Sierpiński fractal lattices by employing the multipartite entanglement and quantum coherence along with the quantum renormalization group method. It is shown that the quantum criticalities of these high-dimensional models closely relate to the behaviors of the multipartite entanglement and quantum coherence. As the thermodynamic limit is approached, the first derivatives of the multipartite entanglement and quantum coherence exhibit singular behaviors, and the consistent finite-size scaling behaviors for each lattice are also obtained from the first derivatives. The multipartite entanglement and quantum coherence are demonstrated to be good indicators for detecting the quantum phase transitions in the triangular lattice and Sierpiński fractal lattices. Furthermore, the dimensions determine the relations between the critical exponents and the correlation length exponents for these lattices.
Adopting concurrence as entanglement measure, we study entanglement and quantum phase transition of the Ising spin systems on the triangular lattice and Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski fractal lattices by using the quantum renormalization-group method. It is found that the ground-state entanglement between two spins (or spin blocks) depends on the following factors: the size of system, the magnetic field, the exchange coupling, and the structure of lattice. As the size of the system becomes large, (a) the range of the magnetic field, in which the entanglement exists, contracts gradually and focuses on the critical point; and (b) the first derivative of entanglement shows singular behavior, and its maximum or minimum is approaching to the critical point gradually. The scaling behaviors of entanglement on the different lattices are similar but the scaling relations are diverse. For the triangular lattice, the space dimensionality determines the scaling relationship between the critical exponent of the entanglement and the critical exponent correlation length. However, for fractal lattices, it is the fractal dimensionality but not the space one to determine this relationship.
We study the critical behavior of the transverse-field quantum Ising model on a fractal structure, namely the Sierpinski carpet. When a magnetic field Δ is applied perpendicular to the Ising spin direction, quantum fluctuations affect the transition between the ferromagnetic and the paramagnetic phases. Employing the continuous-time quantum Monte Carlo simulation method and the finite-size scaling analysis, we investigate the interplay between the quantum fluctuations and the exotic dimensionality of the fractal structure and its effect on the critical behavior. As the transverse magnetic field increases, the critical temperature monotonically decreases until it apparently vanishes at a critical field Δ(c), beyond which the system becomes paramagnetic at all temperatures. However, the critical exponents are independent of Δ and remain the same as in the purely classical(Δ=0) case.