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arXiv:2605.25683 · Journal of the Physical Society of Japan

Analysis of Critical Points in a Permutation Model on Hierarchical Lattices by Real-Space Renormalization Group

Ryuki Ito, Taisei Matsuo, Masayuki Ohzeki

cond-mat.stat-mechcond-mat.dis-nnquant-phSubmitted May 25, 2026• Announced May 26, 2026• 0 citations
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Abstract

The permutation model is a classical spin system where elements of the symmetric group interact with one another. The partition function of this model is directly related to the entanglement structure of random quantum circuits and random tensor networks. In these contexts, the entanglement entropy undergoes a transition between area-law and volume-law scaling, depending on the model parameters. This transition point has attracted considerable attention. In the present work, we investigate the ferromagnetic-paramagnetic phase transition of the permutation model, which corresponds to the entanglement entropy transition. Using exact real-space renormalization group calculations on self-dual hierarchical lattices, we numerically determine finite-replica critical points for (q=mn=2,...,6). We compare the results with the duality prediction based on the Fourier transform of the symmetric group and then extrapolate the b=3 data toward the replica limit $mn\to0$, where the effective dimension is two. The comparison supports the duality-based estimate while also clarifying the systematic uncertainty associated with the extrapolation formula.

Comments
7 pages, 4 figures

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The permutation model is a classical spin system where elements of the symmetric group interact with one another. The partition function of this model is directly related to the entanglement structure of random quantum circuits and random tensor networks. In these contexts, the entanglement entropy undergoes a transition between area-law and volume-law scaling, depending on the model parameters. This transition point has attracted considerable attention. In the present work, we investigate the ferromagnetic-paramagnetic phase transition of the permutation model, which corresponds to the entanglement entropy transition. Using exact real-space renormalization group calculations on self-dual hierarchical lattices, we numerically determine finite-replica critical points for (q=mn=2,...,6). We compare the results with the duality prediction based on the Fourier transform of the symmetric group and then extrapolate the b=3 data toward the replica limit mn→0mn\to0mn→0, where the effective dimension is two. The comparison supports the duality-based estimate while also clarifying the systematic uncertainty associated with the extrapolation formula.
We investigate the critical behavior and the duality property of the ferromagnetic qqq-state clock model on the square lattice based on the tensor-network formalism. From the entanglement spectra of local tensors defined in the original and dual lattices, we obtain the exact self-dual points for the model with $q \leq 5 $ and approximate self-dual points for q≥6q \geq 6q≥6. We calculate accurately the lower and upper critical temperatures for the six-state clock model from the fixed-point tensors determined using the higher-order tensor renormalization group method and compare with other numerical results.
The Ising model is the simplest to describe many-body effects in classical statistical mechanics. Duality analysis leads to a critical point under several assumptions. The Ising model itself has Z(2)Z(2)Z(2) symmetry. The basis of the duality analysis is a nontrivial relationship between low and high-temperature expansions. However, the discrete Fourier transformation finds the hidden relationship automatically. The duality analysis can be naturally generalized into the case with the degrees of freedom with Z(q)Z(q)Z(q) symmetry and random spin systems. We further obtain the duality in a series of permutation models in the present study by considering the symmetric group SqS_qSq​ and its Fourier transformation. The permutation model in the symmetric group is closely related to the random quantum circuits and random tensor network model, often discussed in the context of quantum computing and the holographic principle, a property of string theories and quantum gravity. We provide a systematic way by our duality analysis to analyze the phase transition in these models.
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Abstract The Ising model is the simplest model for describing many-body effects in classical statistical mechanics. A duality analysis leads to its critical point under several assumptions. The Ising model has Z2Z_2Z2​-symmetry. The basis of duality analysis is a nontrivial relationship between low- and high-temperature expansions. However, discrete Fourier transformation automatically determines the hidden relationship. The duality analysis can naturally extend to systems with various degrees of freedom, ZqZ_qZq​ symmetry, and random spin systems. Furthermore, we obtained the duality relation in a series of permutation models in the present study by considering the symmetric group SqS_qSq​ and its Fourier transformation. The permutation model in a symmetric group is closely related to random quantum circuits and random tensor network models, which are frequently discussed in quantum computing. It also relates to the holographic principle, a property of string theories and quantum gravity. We provide a systematic approach using duality analysis to examine the phase transition in these models.
Within a real-space renormalization-group framework, the spin-\textonehalf{} Heisenberg ferromagnet in the presence of an Ising-like anisotropy on a self-dual hierarchical lattice is discussed. The results are exact for this lattice but can also be considered as a quite satisfactory approximation for the simple square one. The controversial point on how Tc{T}_{c}Tc​ vanishes in the isotropic Heisenberg limit is analyzed: Quite strong evidence is presented favoring a continuous function of anisotropy. The crossover from the isotropic Heisenberg model to the pure Ising one is exhibited.
The weakly first-order nature of the two-dimensional 5-state ferromagnetic Potts model poses challenges for numerical study. Using density-matrix and tensor-network renormalization group methods, we investigate these transitions of the Potts-qqq model via the Klein bottle ratio ggg on original and dual lattices. Finite-size scaling of ggg as a function of transverse system size LyL_yLy​ accurately locates the critical points for q=4,5,6q = 4, 5, 6q=4,5,6. We further examine the transfer-matrix spectra and entanglement entropy, extracting central charges through toroidal and Klein bottle boundary conditions. For q=5q = 5q=5, the extracted central charge (c≈1.14811c \approx 1.14811c≈1.14811) is close to the real part of the theoretical value c5-Potts=1.1375±0.0211ic_{5\text{-Potts}} = 1.1375 \pm 0.0211 ic5-Potts​=1.1375±0.0211i predicted by complex conformal field theories. The observed drift in the scaling exponent bbb effectively distinguishes the continuous transition from the weakly first-order regime. Furthermore, the extrapolated divergence of ggg confirms the first-order nature of the q=5q=5q=5 Potts model.