Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1408.5595 · Physical Review E

Quantum critical behavior of the quantum Ising model on fractal lattices

Hangmo Yi

Published Jan 9, 2015• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.91.012118

Related papers

Critical behavior of the quantum Ising model on a fractal structure

Hangmo Yi

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.

Physical Review E•Jul 29, 2013•2 citations

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

Tymoteusz Braciszewski, Oliwier Urbański, Piotr Tomczak

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.

cond-mat.stat-mechCross-listedarXiv:2602.02904•Zeitschrift für Naturforschung A•Feb 4, 2026•0 citations

Quantum criticality from Ising model on fractal lattices

Beni Yoshida, Aleksander Kubica

We study the quantum Ising model on the Sierpiński triangle, whose Hausdorff dimension is $\log 3/ \log 2 \approx 1.585$, and demonstrate that it undergoes second-order phase transition with scaling relations satisfied precisely. We also study the quantum $3$-state Potts model on the Sierpiński triangle and find first-order phase transition, which is consistent with a prediction from $ε$-expansion that the transition becomes first-order for $D > 1.3$. We then compute critical exponents of the Ising model on higher-dimensional Sierpiński pyramids with various Hausdorff dimension via Monte-Carlo simulations and real-space RG analysis for $D\in[1,3]$. We find that only the correlation length exponent $ν$ interpolates the values of integer-dimensional models. This implies that, contrary to a generally held belief, the universality class of quantum phase transition may not be uniquely determined by symmetry and spatial dimension of the system. This work initiates studies on quantum critical phenomena on graphs and networks which may be of significant importance in the context of quantum networks and communication.

arXiv:1404.6311•Apr 25, 2014•2 citations

Fractal Quantum Phase Transitions: Critical Phenomena Beyond Renormalization

Zheng Zhou, Xuefeng Zhang, Frank Pollmann +1 more

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 $Z_2$ 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 $d=\ln(3)/\ln(2)$ 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.

arXiv:2105.05851•May 12, 2021•15 citations

First- and second-order quantum phase transitions of a q -state Potts model in fractal lattices

Hangmo Yi

Quantum phase transitions of a q-state Potts model in fractal lattices are studied using a continuous-time quantum Monte Carlo simulation technique. For small values of q, the transition is found to be second order and critical exponents of the quantum critical point are calculated. The dynamic critical exponent z is found to be greater than one for all fractals studied, which is in contrast to integer-dimensional regular lattices. When q is greater than a certain value q_{c}, the phase transition becomes first order, where q_{c} depends on the lattice. Further analysis shows that the characteristics of phase transitions are more sensitive to the average number of nearest neighbors than the Hausdorff dimension or the order of ramification.

Physical review. E•Dec 5, 2017•0 citations

Multipartite entanglement, quantum coherence, and quantum criticality in triangular and Sierpiński fractal lattices

Jun-Qing Cheng, Jing‐Bo Xu

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.

arXiv:1802.03177•Physical review. E•Jun 19, 2018•11 citations
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 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.
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\approx 0.64 - 0.71, β\approx 0.30, γ\approx 1.67$ and z≈1.33z \approx 1.33z≈1.33. The numerical renormalization group method produced results consistent with finite-size scaling approach (λc=2.766λ_c = 2.766λc​=2.766,β=0.316, β= 0.316,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.
We study the quantum Ising model on the Sierpiński triangle, whose Hausdorff dimension is log⁡3/log⁡2≈1.585\log 3/ \log 2 \approx 1.585log3/log2≈1.585, and demonstrate that it undergoes second-order phase transition with scaling relations satisfied precisely. We also study the quantum 333-state Potts model on the Sierpiński triangle and find first-order phase transition, which is consistent with a prediction from εεε-expansion that the transition becomes first-order for D>1.3D > 1.3D>1.3. We then compute critical exponents of the Ising model on higher-dimensional Sierpiński pyramids with various Hausdorff dimension via Monte-Carlo simulations and real-space RG analysis for D∈[1,3]D\in[1,3]D∈[1,3]. We find that only the correlation length exponent ννν interpolates the values of integer-dimensional models. This implies that, contrary to a generally held belief, the universality class of quantum phase transition may not be uniquely determined by symmetry and spatial dimension of the system. This work initiates studies on quantum critical phenomena on graphs and networks which may be of significant importance in the context of quantum networks and communication.
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.
Quantum phase transitions of a q-state Potts model in fractal lattices are studied using a continuous-time quantum Monte Carlo simulation technique. For small values of q, the transition is found to be second order and critical exponents of the quantum critical point are calculated. The dynamic critical exponent z is found to be greater than one for all fractals studied, which is in contrast to integer-dimensional regular lattices. When q is greater than a certain value q_{c}, the phase transition becomes first order, where q_{c} depends on the lattice. Further analysis shows that the characteristics of phase transitions are more sensitive to the average number of nearest neighbors than the Hausdorff dimension or the order of ramification.
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.