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Progress of Theoretical and Experimental Physics

Duality Analysis in a Symmetric Group and its Application to Random Tensor Network Models

Masayuki Ohzeki

Published Nov 14, 2024• 4 citations
Publisher

Abstract

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 $Z_2$-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, $Z_q$ 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 $S_q$ 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.

DOI
10.1093/ptep/ptae171

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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.
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.
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 study a duality analysis in conjunction with the star-triangle transformation for symmetric-group permutation models on the triangular and honeycomb lattices. The calculation is motivated by the permutation-model description of random tensor networks and by earlier duality analyses of replicated spin glasses. The essential point is that the finite-basis unit is not a bare bond but a star-triangle block. Our analysis estimates the critical bond dimension for the honeycomb lattice to be 2.634929344884, and the associated single-bond duality relation yields the triangular-lattice estimate 1.475661534848.
Abstract Exact solutions of model systems are the most reliable source of information. However, there are not many model systems that can be exactly solved and, consequently, many approximate schemes such as series expansions have been developed. It is sometimes possible, though, to extract exact information without directly solving those model systems. Arguments using duality transformations make it possible to derive the exact location of the phase transition point and the exact value of the energy at the transition point when the model is self dual. Duality not only determines the exact location of the transition point of the two-dimensional Ising model and related models but also is useful to rewrite the XYXYXY model into a different form, the Villain model and its dual, which reveals new physical aspects of the system. Classical and quantum dualities can be related by certain mappings and Fourier transformation is at the root of such deep results.
Abstract In a recent series of papers, a duality between orthogonal and symplectic random tensor models has been proven, first for quartic models and then for models with interactions of arbitrary order. However, the tensor models considered so far in the literature had no symmetry under permutation of the indices. In this paper, we generalize these results for tensors models with interactions of arbitrary order which further have non-trivial symmetry under the permutation of the indices. Totally symmetric and anti-symmetric tensors are thus treated as a particular case of our result.
Tensor networks provide a powerful tool for studying many-body quantum systems, particularly making quantum simulations more efficient. In this article, we construct a tensor network representation of the spin network states, which correspond to S U ( 2 ) gauge-invariant discrete field theories. Importantly, the spin network states play a central role in the loop quantum gravity (LQG) approach to the Planck scale physics. Our focus is on the Ising-type spin networks, which provide a basic model of quantum space in LQG. It is shown that the tensor network approach improves the previously introduced methods of constructing quantum circuits for the Ising spin networks by reducing the number of qubits involved. It is also shown that the tensor network approach is convenient for implementing holographic states involving the bulk-boundary conjecture, which contributes to establishing a link between LQG and holographic duality.