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Tensor renormalization group approach to the $O(2)$ models via symmetry-twisted partition functions
Authors:
Shinichiro Akiyama,
Raghav G. Jha,
Jun Maeda,
Yuya Tanizaki,
Judah Unmuth-Yockey
Abstract:
We investigate critical phenomena in the $O(2)$ models using symmetry-twisted partition functions that can be efficiently computed within the tensor renormalization group framework. We first demonstrate, taking the three-dimensional model as an example, that symmetry-twisted partition functions detect the spontaneous breaking of global continuous symmetry. We then consider the same model in two di…
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We investigate critical phenomena in the $O(2)$ models using symmetry-twisted partition functions that can be efficiently computed within the tensor renormalization group framework. We first demonstrate, taking the three-dimensional model as an example, that symmetry-twisted partition functions detect the spontaneous breaking of global continuous symmetry. We then consider the same model in two dimensions, where the Berezinskii--Kosterlitz--Thouless (BKT) transition occurs. Since symmetry-twisted partition functions directly provide the helicity modulus at a finite twist angle, we determine the BKT transition point. These results are presented based on Ref.~\cite{Akiyama:2026dzg}. Finally, in addition to the original paper~\cite{Akiyama:2026dzg}, we apply this approach to the two-dimensional generalized $O(2)$ model and confirm that it successfully identifies the phase transitions between the ferromagnetic and nematic phases, as well as between the nematic and paramagnetic phases.
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Submitted 3 March, 2026;
originally announced March 2026.
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Tensor renormalization group approach to critical phenomena via symmetry-twisted partition functions
Authors:
Shinichiro Akiyama,
Raghav G. Jha,
Jun Maeda,
Yuya Tanizaki,
Judah Unmuth-Yockey
Abstract:
The locality of field theories strongly constrains the possible behaviors of symmetry-twisted partition functions, and thus they serve as order parameters to detect low-energy realizations of global symmetries, such as spontaneous symmetry breaking (SSB). We demonstrate that the tensor renormalization group (TRG) offers an efficient framework to compute the symmetry-twisted partition functions, wh…
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The locality of field theories strongly constrains the possible behaviors of symmetry-twisted partition functions, and thus they serve as order parameters to detect low-energy realizations of global symmetries, such as spontaneous symmetry breaking (SSB). We demonstrate that the tensor renormalization group (TRG) offers an efficient framework to compute the symmetry-twisted partition functions, which enables us to detect the symmetry-breaking transition and also to study associated critical phenomena. As concrete examples of SSB, we investigate the two-dimensional (2D) classical Ising model and the three-dimensional (3D) classical $O(2)$ nonlinear sigma model, and we identify their critical points solely from the twisted partition function. By employing the finite-size scaling argument, we find the critical temperature $T_c=2.2017(2)$ with the critical exponent $ν= 0.663(33)$ for the 3D $O(2)$ model. In addition, we also study the Berezinskii-Kosterlitz-Thouless (BKT) criticality of the 2D classical $O(2)$ model by extracting the helicity modulus from the twisted partition functions, and we obtain the BKT transition temperature, $T_{\mathrm{BKT}}=0.8928(2)$.
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Submitted 3 April, 2026; v1 submitted 5 January, 2026;
originally announced January 2026.
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Twisted Partition Functions as Order Parameters
Authors:
Jun Maeda,
Yuya Tanizaki
Abstract:
For quantum field theories with global symmetry, we can study the behavior of the partition function with the background gauge field to diagnose different quantum phases. For the case of discrete symmetries, we find that the symmetry-twisted partition function works as an order parameter that discriminates spontaneous symmetry breaking (SSB), symmetry-protected topological (SPT) states, and symmet…
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For quantum field theories with global symmetry, we can study the behavior of the partition function with the background gauge field to diagnose different quantum phases. For the case of discrete symmetries, we find that the symmetry-twisted partition function works as an order parameter that discriminates spontaneous symmetry breaking (SSB), symmetry-protected topological (SPT) states, and symmetry-enriched topological (SET) states. We then consider its application to the case of 4d Yang-Mills theory with adjoint matters to understand the relation between the twisted partition function and the Wilson-'t Hooft classification. We also study its behavior for the spontaneously broken U(1) symmetry and interpret the result from the viewpoint of the mixed anomaly with the emergent solitonic symmetry.
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Submitted 1 June, 2025; v1 submitted 22 May, 2025;
originally announced May 2025.
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$N$-ality symmetry and SPT phases in (1+1)d
Authors:
Jun Maeda,
Tsubasa Oishi
Abstract:
Duality symmetries have been extensively investigated in various contexts, playing a crucial role in understanding quantum field theory and condensed matter theory. In this paper, we extend this framework by studying $N$-ality symmetries, which are a generalization of duality symmetries and are mathematically described by $\mathbb{Z}_N$-graded fusion categories. In particular, we focus on an $N$-a…
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Duality symmetries have been extensively investigated in various contexts, playing a crucial role in understanding quantum field theory and condensed matter theory. In this paper, we extend this framework by studying $N$-ality symmetries, which are a generalization of duality symmetries and are mathematically described by $\mathbb{Z}_N$-graded fusion categories. In particular, we focus on an $N$-ality symmetry obtained by gauging a non-anomalous subgroup of $\mathbb{Z}_N\times\mathbb{Z}_N\times\mathbb{Z}_N$ symmetry with a type III anomaly. We determine the corresponding fusion rules via two complementary approaches: a direct calculation and a representation-theoretic method. As an application, we study the symmetry-protected topological (SPT) phases associated with the $N$-ality symmetry. We classify all such SPT phases using the SymTFT framework and explicitly construct lattice Hamiltonians for some of them.
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Submitted 8 December, 2025; v1 submitted 28 April, 2025;
originally announced April 2025.
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Neural network representation of quantum systems
Authors:
Koji Hashimoto,
Yuji Hirono,
Jun Maeda,
Jojiro Totsuka-Yoshinaka
Abstract:
It has been proposed that random wide neural networks near Gaussian process are quantum field theories around Gaussian fixed points. In this paper, we provide a novel map with which a wide class of quantum mechanical systems can be cast into the form of a neural network with a statistical summation over network parameters. Our simple idea is to use the universal approximation theorem of neural net…
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It has been proposed that random wide neural networks near Gaussian process are quantum field theories around Gaussian fixed points. In this paper, we provide a novel map with which a wide class of quantum mechanical systems can be cast into the form of a neural network with a statistical summation over network parameters. Our simple idea is to use the universal approximation theorem of neural networks to generate arbitrary paths in the Feynman's path integral. The map can be applied to interacting quantum systems / field theories, even away from the Gaussian limit. Our findings bring machine learning closer to the quantum world.
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Submitted 17 March, 2024;
originally announced March 2024.
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Synthetic dimension band structures on a Si CMOS photonic platform
Authors:
Armandas Balčytis,
Tomoki Ozawa,
Yasutomo Ota,
Satoshi Iwamoto,
Jun Maeda,
Toshihiko Baba
Abstract:
Synthetic dimensions, which simulate spatial coordinates using non-spatial degrees of freedom, are drawing interest in topological science and other fields for modelling higher-dimensional phenomena on simple structures. We present the first realization of a synthetic frequency dimension on a silicon ring resonator photonic device fabricated using a CMOS process. We confirm that its coupled modes…
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Synthetic dimensions, which simulate spatial coordinates using non-spatial degrees of freedom, are drawing interest in topological science and other fields for modelling higher-dimensional phenomena on simple structures. We present the first realization of a synthetic frequency dimension on a silicon ring resonator photonic device fabricated using a CMOS process. We confirm that its coupled modes correspond to a 1D tight-binding model through acquisition of up to 280 GHz bandwidth optical frequency comb-like spectra, and by measuring the first synthetic band structures on an integrated device. Furthermore, we realized two types of gauge potentials along the frequency dimension, and probed their effects through the associated band structures. An electric field analogue was produced via modulation detuning, whereas effective magnetic fields were induced using synchronized nearest- and second-nearest-neighbor coupling. Creation of coupled mode lattices and two effective forces on a monolithic Si CMOS device represents a key step towards wider adoption of topological principles.
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Submitted 28 May, 2021;
originally announced May 2021.
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Universal temperature dependence of electron number in one-dimensional Hubbard model
Authors:
Jun Maeda,
Sei-ichiro Suga
Abstract:
We investigate the temperature region in which a Tomonaga-Luttinger liquid (TLL) description of the charge sector of the one-dimensional Hubbard model is valid. By using the thermodynamic Bethe ansatz method, electron number is calculated at finite temperatures and fixed chemical potential. We observe maximum electron number as a function of temperature close to the chemical potential of the upp…
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We investigate the temperature region in which a Tomonaga-Luttinger liquid (TLL) description of the charge sector of the one-dimensional Hubbard model is valid. By using the thermodynamic Bethe ansatz method, electron number is calculated at finite temperatures and fixed chemical potential. We observe maximum electron number as a function of temperature close to the chemical potential of the upper critical value that corresponds to half filling. As the chemical potential approaches the upper critical value from below, the temperature $(T_{\rm M})$ at which the electron number shows its maximum asymptotically approaches a universal relation. We show that, below the energy corresponding to $T_{\rm M}$, the charge excitation spectrum nearly obeys a linear dispersion relation. The results demonstrate that $T_{\rm M}$ marks the important temperature below which TLL is realized.
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Submitted 13 October, 2009; v1 submitted 26 August, 2009;
originally announced August 2009.