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arXiv:1711.11401 · Physical review. E

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

Roman Krčmár, Jozef Genzor, Yoju Lee, Hana Čenčariková, Tomotoshi Nishino, Andrej Gendiar

Published Dec 11, 2018• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.98.062114

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The transverse-field Ising model on the Sierpi\ifmmode \acute{n}\else '{n}\fi{}ski fractal, which is characterized by the fractal dimension log23\ensuremath≈1.585{log}_{2}^{\phantom{\rule{3.33333pt}{0ex}}}3\ensuremath{\approx}1.585log2​3\ensuremath≈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 \ensuremathβ\ensuremath≈0.198\ensuremath{\beta}\ensuremath{\approx}0.198\ensuremathβ\ensuremath≈0.198 and \ensuremathδ\ensuremath≈8.7\ensuremath{\delta}\ensuremath{\approx}8.7\ensuremathδ\ensuremath≈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 phase transition of the classical Ising model on the Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpet, which has the fractal dimension ${log}_{3}^{\phantom{\rule{0.16em}{0ex}}}8\ensuremath{\approx}1.8927$, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature ${T}_{\mathrm{c}}^{\phantom{\rule{0.16em}{0ex}}}\ensuremath{\approx}1.478$. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent $\ensuremath{\beta}$ associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas Tc{T}_{\mathrm{c}}^{\phantom{\rule{0.16em}{0ex}}}Tc​ is not affected. Furthermore, we employ automatic differentiation to accurately and efficiently compute the average spontaneous magnetization per site as a first derivative of free energy with respect to the external field, yielding the global critical exponent of $\ensuremath{\beta}\ensuremath{\approx}0.135$.
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.
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 βββ are evaluated. Despite the fact that the Hausdorff dimension of the Sierpiński pyramid is exactly two (=log⁡2 4)( = \log_2^{~} 4)(=log2 ​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≈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.
The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension log_{4}12≈1.792, is investigated using a modified higher-order tensor renormalization group algorithm supplemented with automatic differentiation to compute relevant derivatives efficiently and accurately. The complete set of critical exponents characteristic of a second-order phase transition was obtained. Correlations near the critical temperature were analyzed through two impurity tensors inserted into the system, which allowed us to obtain the correlation lengths and calculate the critical exponent ν. The critical exponent α was found to be negative, consistent with the observation that the specific heat does not diverge at the critical temperature. The extracted exponents satisfy the known relations given by various scaling assumptions within reasonable accuracy. Perhaps most interestingly, the hyperscaling relation, which contains the spatial dimension, is satisfied very well, assuming the Hausdorff dimension takes the place of the spatial dimension. Moreover, using automatic differentiation, we have extracted four critical exponents (α, β, γ, and δ) globally by differentiating the free energy. Surprisingly, the global exponents differ from those obtained locally by the technique of the impurity tensors; however, the scaling relations remain satisfied even in the case of the global exponents.
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.