[Paper Review] Quantum criticality from Ising model on fractal lattices

Beni Yoshida, Aleksander Kubica|arXiv (Cornell University)|Apr 25, 2014

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper demonstrates quantum criticality in the transverse-field Ising model on the Sierpiński triangle, a fractal lattice with Hausdorff dimension ≈1.585, via Monte Carlo simulations and real-space RG analysis. It shows second-order phase transition with precise scaling relations, challenging the conventional belief that universality classes are solely determined by symmetry and spatial dimension, and further confirms first-order transitions in the 3-state Potts model on the same lattice.

ABSTRACT

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.

Motivation & Objective

• To investigate whether quantum criticality can emerge in strongly correlated many-body systems on fractal lattices with non-integer Hausdorff dimension.
• To test the validity of using fractal lattices as probes for higher-dimensional quantum critical phenomena, particularly when standard methods fail due to exponential Hilbert space growth.
• To examine whether universality classes of quantum phase transitions are uniquely determined by symmetry and spatial dimension, especially in systems with finite ramification number.
• To compare critical exponents of the quantum Ising model on Sierpiński pyramids of varying Hausdorff dimension with those of integer-dimensional models.
• To validate the use of fractal lattices as a computational and conceptual tool for studying quantum networks and quantum criticality beyond standard lattices.

Proposed method

• Employed Monte Carlo simulations via the Trotter-Suzuki decomposition to map the quantum Ising and Potts models on the Sierpiński triangle to classical statistical models on a stack of fractal layers.
• Used the Sierpiński triangle and higher-dimensional Sierpiński pyramids as lattices with finite ramification number, enabling efficient real-space renormalization group (RG) analysis.
• Applied real-space RG with a coarse-graining scheme based on self-similarity of the fractal structure, allowing estimation of critical exponents such as the correlation length exponent ν.
• Computed critical exponents for the quantum Ising model on Sierpiński pyramids with Hausdorff dimensions D ∈ [1,3] using both Monte Carlo simulations and real-space RG.
• Leveraged self-duality of the Hamiltonian to identify the critical point at ε = 1/2 for the Ising model on the Sierpiński pyramid.
• Compared critical exponents from fractal models with those of standard integer-dimensional systems to assess interpolation behavior and universality.

Experimental results

Research questions

• RQ1 Can quantum criticality with second-order phase transitions be realized on fractal lattices with non-integer Hausdorff dimension?
• RQ2 Do critical exponents on fractal lattices satisfy precise scaling relations, indicating true quantum criticality?
• RQ3 Does the universality class of a quantum phase transition depend solely on symmetry and spatial dimension, or can other factors such as lattice topology play a role?
• RQ4 How do critical exponents of the Ising model on Sierpiński pyramids with varying Hausdorff dimension interpolate between those of integer-dimensional systems?
• RQ5 Is the transition in the 3-state Potts model on the Sierpiński triangle first-order, consistent with theoretical predictions for D > 1.3?

Key findings

• The quantum Ising model on the Sierpiński triangle exhibits a second-order phase transition with critical exponents satisfying precise scaling relations, confirming the presence of quantum criticality.
• The correlation length exponent ν for the Ising model on the Sierpiński triangle (D ≈ 1.585) was found to be ν ≈ 0.72 via real-space RG, and ν ≈ 0.62 for D = 2 (Sierpiński pyramid), both in good agreement with numerical estimates.
• The critical exponents of the Ising model on higher-dimensional Sierpiński pyramids interpolate smoothly between those of integer-dimensional models, indicating ν is primarily governed by spatial dimension and symmetry.
• The 3-state Potts model on the Sierpiński triangle exhibits a first-order phase transition, consistent with the theoretical prediction that transitions become first-order for D > 1.3.
• The study demonstrates that fractal lattices with finite ramification number can support genuine quantum criticality, challenging previous doubts about their utility due to universality violations.
• The results suggest that universality classes in quantum phase transitions may not be uniquely determined by symmetry and spatial dimension, especially in systems with non-trivial topology or fractal geometry.