arXiv:2003.09604

Quantum Potts Models on the Sierpiński Pyramid

Roman Krčmár, Mária Zelenayová, Genzor, Jozef, Libor Caha, Peter Rapčan, Tomotoshi Nishino, Andrej Gendiar

Published Mar 21, 2020• 0 citations

Abstract

Phase transition of the two- and three-state quantum Potts models on the Sierpiński pyramid are studied by means of a tensor network framework, the higher-order tensor renormalization group method. Critical values of the transverse magnetic field and the magnetic exponent $β$ are evaluated. Despite the fact that the Hausdorff dimension of the Sierpiński pyramid is exactly two $( = \log_2^{~} 4)$, the obtained critical properties show that the effective dimension is lower than two.

DOI: 10.48550/arxiv.2003.09604

Related papers

Quantum criticality from Ising model on fractal lattices

Beni Yoshida, Aleksander Kubica

arXiv:1404.6311Apr 25, 20142 citations

Tensor-network study of a quantum phase transition on the Sierpiński fractal

Roman Krčmár, Jozef Genzor, Yoju Lee +3 more

arXiv:1711.11401Physical review. EDec 11, 201814 citations

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

Tymoteusz Braciszewski, Oliwier Urbański, Piotr Tomczak

cond-mat.stat-mechCross-listedarXiv:2602.02904Zeitschrift für Naturforschung AFeb 4, 20260 citations

Investigation of the Potts Model on Triangular Lattices by the Second Renormalization of Tensor Network States

Wang Meng-Xiong, Cai Jian-Wei, Z. Y. Xie +3 more

Chinese Physics LettersJul 1, 20108 citations

Quantum critical behavior of the quantum Ising model on fractal lattices

Hangmo Yi

arXiv:1408.5595Physical Review EJan 9, 20159 citations

Quantum Potts chain in alternating field

Péter Lajkó, Wedad Alaaeldin Yehia, Ferenc Iglói

arXiv:2306.09127Physical review. B./Physical review. BOct 2, 20230 citations

arXiv:2003.09604

Quantum Potts Models on the Sierpiński Pyramid

Roman Krčmár, Mária Zelenayová, Genzor, Jozef, Libor Caha, Peter Rapčan, Tomotoshi Nishino, Andrej Gendiar

Published Mar 21, 2020• 0 citations

arXiv HTML PDF

Abstract

β

are evaluated. Despite the fact that the Hausdorff dimension of the Sierpiński pyramid is exactly two

( = \log_2^{~} 4)

, the obtained critical properties show that the effective dimension is lower than two.

DOI: 10.48550/arxiv.2003.09604

Related papers

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.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:1404.6311Apr 25, 20142 citations

Tensor-network study of a quantum phase transition on the Sierpiński fractal

Roman Krčmár, Jozef Genzor, Yoju Lee +3 more

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 ${h}_{\mathrm{c}}^{\phantom{\rule{3.33333pt}{0ex}}}=1.865$ . 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.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 ${h}_{\mathrm{c}}^{\phantom{\rule{3.33333pt}{0ex}}}=1.865$. 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.

arXiv:1711.11401Physical review. EDec 11, 201814 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.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:2602.02904Zeitschrift für Naturforschung AFeb 4, 20260 citations

Investigation of the Potts Model on Triangular Lattices by the Second Renormalization of Tensor Network States

Wang Meng-Xiong, Cai Jian-Wei, Z. Y. Xie +3 more

We employ the second renormalization group method of tensor-network states to investigate thermodynamic properties of the ferromagnetic and antiferromagnetic Potts model on triangular lattices. From the temperature dependence of the internal energy and the specific heat, both the critical temperatures and critical exponents are evaluated. For the q = 3 antiferromagnetic Potts model, the critical temperature is found to be T c = 0.627163±0.000003, which is at least one order of magnitude more accurate than that obtained by other methods.We employ the second renormalization group method of tensor-network states to investigate thermodynamic properties of the ferromagnetic and antiferromagnetic Potts model on triangular lattices. From the temperature dependence of the internal energy and the specific heat, both the critical temperatures and critical exponents are evaluated. For the q = 3 antiferromagnetic Potts model, the critical temperature is found to be T c = 0.627163±0.000003, which is at least one order of magnitude more accurate than that obtained by other methods.

Chinese Physics LettersJul 1, 20108 citations

Quantum critical behavior of the quantum Ising model on fractal lattices

Hangmo Yi

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.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.

arXiv:1408.5595Physical Review EJan 9, 20159 citations

Quantum Potts chain in alternating field

Péter Lajkó, Wedad Alaaeldin Yehia, Ferenc Iglói

The $q$ -state Potts chain with ferromagnetic couplings, $J=1$ , in the presence of a transverse field $\mathrm{\ensuremath{\Gamma}}$ has a quantum phase transition at $\mathrm{\ensuremath{\Gamma}}/q=1$, which is continuous for $q\ensuremath{\le}4$ and of first order for $q>4$ . Here we introduce a $q$ -periodic alternating longitudinal field of strength $h$ and study the phase diagram and the critical properties of the model. For $h<q/(q\ensuremath{-}1)$, there is a ferromagnetic ordered phase, for $\mathrm{\ensuremath{\Gamma}}<{\mathrm{\ensuremath{\Gamma}}}_{c}(h)$, and at $h=q/(q\ensuremath{-}1)$, there is a classical endpoint at $\mathrm{\ensuremath{\Gamma}}=0$, with finite entropy at $T=0$ . We considered the $q=3$ model and using density matrix renormalization group techniques, we calculated the low-lying spectrum of the Hamiltonian, the transverse magnetization, and the spin-spin correlation function, all of which signal a diverging correlation length at the transition point with the exponent of the three-state Potts model. In the vicinity of the classical endpoint, the model is mapped to a quantum hard rod model, which also belongs to the universality class of the ferromagnetic three-state Potts model. Also, the spectrum of the critical Hamiltonian is found in agreement with conformal invariance. At the same time, the correlation function shows a jump at the transition point, and thus the transition is of mixed order for $h<q/(q\ensuremath{-}1)$.The $q$-state Potts chain with ferromagnetic couplings, $J=1$, in the presence of a transverse field $\mathrm{\ensuremath{\Gamma}}$ has a quantum phase transition at $\mathrm{\ensuremath{\Gamma}}/q=1$, which is continuous for $q\ensuremath{\le}4$ and of first order for $q>4$. Here we introduce a $q$-periodic alternating longitudinal field of strength $h$ and study the phase diagram and the critical properties of the model. For $h<q/(q\ensuremath{-}1)$, there is a ferromagnetic ordered phase, for $\mathrm{\ensuremath{\Gamma}}<{\mathrm{\ensuremath{\Gamma}}}_{c}(h)$, and at $h=q/(q\ensuremath{-}1)$, there is a classical endpoint at $\mathrm{\ensuremath{\Gamma}}=0$, with finite entropy at $T=0$. We considered the $q=3$ model and using density matrix renormalization group techniques, we calculated the low-lying spectrum of the Hamiltonian, the transverse magnetization, and the spin-spin correlation function, all of which signal a diverging correlation length at the transition point with the exponent of the three-state Potts model. In the vicinity of the classical endpoint, the model is mapped to a quantum hard rod model, which also belongs to the universality class of the ferromagnetic three-state Potts model. Also, the spectrum of the critical Hamiltonian is found in agreement with conformal invariance. At the same time, the correlation function shows a jump at the transition point, and thus the transition is of mixed order for $h<q/(q\ensuremath{-}1)$.

arXiv:2306.09127Physical review. B./Physical review. BOct 2, 20230 citations

β

are evaluated. Despite the fact that the Hausdorff dimension of the Sierpiński pyramid is exactly two

( = \log_2^{~} 4)

, the obtained critical properties show that the effective dimension is lower than two.

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.

{h}_{\mathrm{c}}^{\phantom{\rule{3.33333pt}{0ex}}}=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.

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.

The

q

-state Potts chain with ferromagnetic couplings,

J=1

, in the presence of a transverse field $\mathrm{\ensuremath{\Gamma}}$ has a quantum phase transition at $\mathrm{\ensuremath{\Gamma}}/q=1$, which is continuous for $q\ensuremath{\le}4$ and of first order for

q>4

. Here we introduce a

q

-periodic alternating longitudinal field of strength

h

and study the phase diagram and the critical properties of the model. For $h<q/(q\ensuremath{-}1)$, there is a ferromagnetic ordered phase, for $\mathrm{\ensuremath{\Gamma}}<{\mathrm{\ensuremath{\Gamma}}}_{c}(h)$, and at $h=q/(q\ensuremath{-}1)$, there is a classical endpoint at $\mathrm{\ensuremath{\Gamma}}=0$, with finite entropy at

T=0

. We considered the

q=3

model and using density matrix renormalization group techniques, we calculated the low-lying spectrum of the Hamiltonian, the transverse magnetization, and the spin-spin correlation function, all of which signal a diverging correlation length at the transition point with the exponent of the three-state Potts model. In the vicinity of the classical endpoint, the model is mapped to a quantum hard rod model, which also belongs to the universality class of the ferromagnetic three-state Potts model. Also, the spectrum of the critical Hamiltonian is found in agreement with conformal invariance. At the same time, the correlation function shows a jump at the transition point, and thus the transition is of mixed order for $h<q/(q\ensuremath{-}1)$.