Publication

Papers
  1. Shinichiro Akiyama, Yannick Meurice, Ryo Sakai
    "Tensor Renormalization Group for fermions"
    [arXiv:2401.08542 [hep-lat]]
    We have reviewed the Grassmann tensor renormalization group (GTRG) approach. We have explained how to reformulate the path integral including fermions as a Grassmann tensor network in two ways and made a comparison between the two formulations. The algorithms of the Levin-Nave TRG and HOTRG and several improved methods including the TNR, bond-weighting method, and multilayered constructions for Nf-flavor fermions have been reviewed. We have also made an overview of the application status of GTRG in the HEP community.

  2. Shinichiro Akiyama, Yoshinobu Kuramashi
    "Critical endpoint of (3+1)-dimensional finite density Z3 gauge-Higgs model with tensor renormalization group"
    JHEP 10 (2023) 077 [arXiv:2304.07934 [hep-lat]]
    This work is an extension of our previous study (JHEP05(2022)102). Using the ATRG method, we have determined the CEP of the (3+1)-dimensional Z3 gauge-Higgs model at finite density. We have made a consistency check, where the plaquette value obtained by the impurity tensor method is compared with that by the dual lattice simulation. We have also made a comparison between the endpoints of Z2 and Z3 models. Please see the last figure in the paper.

  3. Shinichiro Akiyama
    "Matrix product decomposition for two- and three-flavor Wilson fermion: benchmark results in the lattice Gross-Neveu model at finite density"
    Phys. Rev. D108 (2023) 034514 [arXiv:2304.01473 [hep-lat]]
    We find that two- and three-flavor Gross-Neveu-Wilson models can be represented as two- and three-layer tensor networks whose bond dimension is equal to four. Using this description, we developed a coarse-graining algorithm, which allows us to compute the path integral. The calculation is done at finite density and the Silver Blaze feature is clearly captured by the coarse-graining. It must be interesting to consider an exact matrix product decomposition for the initial tensor network because we have a chance to avoid directly dealing with the large tensor whose size scales exponentially with respect to the number of flavors.

  4. Shinichiro Akiyama
    "Bond-weighting method for the Grassmann tensor renormalization group"
    JHEP 11 (2022) 030 [arXiv:2208.03227 [hep-lat]] Sample code
    Recently, the tensor network description with bond weights on its edges has been proposed as a novel improvement for the tensor renormalization group algorithm. The bond weight is controlled by a single hyperparameter, whose optimal value is estimated in the original work via the numerical computation of the two-dimensional critical Ising model. We have developed this bond-weighted tensor renormalization group (BTRG) algorithm to make it applicable to the fermionic system, benchmarking with the two-dimensional massless Wilson fermion. We have confirmed that all the advantages in the BTRG are taken over to the Grassmann BTRG.

  5. Shinichiro Akiyama, Yoshinobu Kuramashi
    "Tensor renormalization group study of (3+1)-dimensional Z2 gauge-Higgs model at finite density"
    JHEP 05 (2022) 102 [arXiv:2202.10051 [hep-lat]]
    We have investigated the critical endpoint of Z2 gauge-Higgs model in three and four dimensions with the TRG method. In three dimensions, the resulting endpoint is consistent with a recent study, and the first-order transition points are located on the self-dual line with high precision. In four dimensions, on the other hand, the endpoint located by the TRG method is not in agreement with the previous result obtained by the MC simulation. We have also studied the four-dimensional model at finite density. The current TRG computation shows that though the critical inverse gauge coupling has little chemical-potential dependence, the critical spin coupling is diminished as the chemical potential increases.

  6. Shinichiro Akiyama, Yoshinobu Kuramashi, Takumi Yamashita
    "Metal-insulator transition in (2+1)-dimensional Hubbard model with tensor renormalization group"
    PTEP 2022 023I01 (2022) [arXiv:2109.14149 [cond-mat.str-el]]
    In this work, we have investigated the metal-insulator transition in (2+1)-dimensional Hubbard model evaluating the number density as a function of finite chemical potential. The Grassmann ATRG algorithm allows us to survey the model almost in the thermodynamic and zero-temperature limits, even away from the half-filling. The numerical results well capture the characteristic feature of the metal-insulator transition and they imply that the model may exhibit the metal-insulator transition in the vast regime of the finite coupling.

  7. Shinichiro Akiyama, Yoshinobu Kuramashi
    "Tensor renormalization group approach to (1+1)-dimensional Hubbard model"
    Phys. Rev. D104, 014504 (2021) [arXiv:2105.00372 [hep-lat]]
    (1+1)-dimensional Hubbard model has been studied within the path-integral formalism. We have derived the Grassmann tensor network representation for the path integral of (d+1)-dimensional Hubbard model. When d=1, the model is exactly solvable based on the Bethe ansatz. We have applied the Grassmann higher-order tensor renormalization group to evaluate the electron density as a function of chemical potential. Performing the scaling analysis with respect to the bond dimension, the numerical results of the critical chemical potential and the critical exponent ν show the consistency with the exact solution.

  8. Shinichiro Akiyama, Yoshinobu Kuramashi, Yusuke Yoshimura
    "Phase transition of four-dimensional lattice φ4 theory with tensor renormalization group"
    Phys. Rev. D104, 034507 (2021) [arXiv:2101.06953 [hep-lat]]
    Using the parallelized ATRG algorithm, we have investigated the lattice φ4 theory with a finite coupling. The location of κc with λ=5 is comparable with the values obtained by various methods, including the MC simulation. We have also investigated κc at λ=40 and λ=100 to observe how κc would approach the transition point in the Ising model, as a function of 1/λ. Moreover, the bond energy and the magnetization at λ=40 on the lattice volumes, up to 10244, show that this model possibly exhibits the weak first-order phase transition. These results are consistent with our previous study of the four-dimensional Ising model with the HOTRG algorithm.

  9. Shinichiro Akiyama, Yoshinobu Kuramashi, Takumi Yamashita, Yusuke Yoshimura
    "Restoration of chiral symmetry in cold and dense NambuーJona-Lasinio model with tensor renormalization group"
    JHEP 01 (2021) 121 [arXiv:2009.11583 [hep-lat]]
    We have investigated the NambuーJona-Lasinio (NJL) model, defined on the lattice with the staggered fermions, at finite density and vanishing temperature in the thermodynamic limit. We have formulated the Grassmann anisotropic tensor renormalization group, which can evaluate the path integral over the Grassmann numbers and whose computational complexity is the same as the anisotropic tensor renormalization group. Measuring the chiral condensate as an order parameter, the first-order chiral phase transition has been observed in the cold and dense region, where the Monte Carlo simulation should be hindered by the severe sign problem. We have also calculated the pressure and number density, which are ingredients in the equation of state.

  10. Shinichiro Akiyama, Daisuke Kadoh
    "More about the Grassmann tensor renormalization group"
    JHEP 10 (2021) 188 [arXiv:2005.07570 [hep-lat]]
    Introducing auxiliary fermion fields, we have discussed the tensor network formulation for lattice fermions. We have provided a formula to derive the tensor network representation for the lattice fermions with nearest-neighbor interaction. We have also reformulated the Grassmann higher-order tensor renormalization group and numerically tested the current formulation.

  11. Shinichiro Akiyama, Daisuke Kadoh, Yoshinobu Kuramashi, Takumi Yamashita, Yusuke Yoshimura
    "Tensor renormalization group approach to four-dimensional complex φ4 theory at finite density"
    JHEP 09 (2020) 177 [arXiv:2005.04645 [hep-lat]]
    This study is the first application of TRG to 4d QFT. Complex scalar field theory at finite density is a typical system with the sign problem and a naive MC simulation on a large lattice is hindered. We have applied the anisotropic tensor renormalization group with parallel computation. TRG approach is essentially free from the sign problem and its computational cost scales logarithmically with respect to system size. We have confirmed the Silver Blaze phenomenon, which is associated with the sign problem in the large volume limit at low temperature. This numerical investigation encourages us to apply TRG to other 4d QFT with or without the sign problem.

  12. Shinichiro Akiyama, Yoshinobu Kuramashi, Takumi Yamashita, Yusuke Yoshimura
    "Phase transition of four-dimensional Ising model with higher-order tensor renormalization group"
    Phys. Rev. D100, 054510 (2019) [arXiv:1906.06060 [hep-lat]]
    We have applied the higher-order tensor renormalization group with parallel computation to 4d Ising model. This is the first application of tensor renormalization group method to 4d system. The transition point is investigated through the degeneracy of the local tensor. Employing the impure tensor method, we have calculated the internal energy and the magnetization. Surprisingly, the current results imply that the order of phase transition is not second but weakly first.

Conference Proceedings
  1. Shinichiro Akiyama, Raghav G. Jha, Judah Unmuth-Yockey
    "Tensor renormalization group study of 3D principal chiral model"
    Proceedings of Science (LATTICE2023) 355 [arXiv:2312.11649 [hep-lat]]
    We have investigated the SU(2) principal chiral model, which is formulated as a tensor network with character expansion. Two TRG algorithms, ATRG and triad RG, have been employed to locate the critical point. The further study will be announced in a different paper. This work was initiated by some discussions during and after the INT workshop on "Tensor Networks in Many Body and Quantum Field Theory". We thank the INT for its kind hospitality and stimulating research environment!

  2. Shinichiro Akiyama
    "Implementation of bond-weighting method for the Grassmann tensor renormalization group"
    Proceedings of Science (LATTICE2023) 370 [arXiv:2311.17691 [hep-lat]]
    Documentation for the Grassmann-BTRG. The sample code is available here.

  3. Shinichiro Akiyama, Yoshinobu Kuramashi, Yusuke Yoshimura
    "Quantum Field Theories with Tensor Renormalization Group"
    Proceedings of Science (LATTICE2021) 530 [arXiv:2111.04240 [hep-lat]]
    We have reported recent progress on the numerical application of the tensor renormalization group approach by our collaboration.

  4. Shinichiro Akiyama, Yoshinobu Kuramashi, Takumi Yamashita, Yusuke Yoshimura
    "Phase transition of four-dimensional Ising model with tensor network scheme"
    Proceedings of Science (LATTICE2019) 138 [arXiv:1911.12954 [hep-lat]]
    We have applied the anisotropic tensor renormalization group (ATRG) with parallel computation to 4d Ising model. Using the ATRG, we have investigated the transition point and the phase transition order. The results seem consistent with our previous study of the Ising model with the higher-order tensor renormalization group. We expect that the ATRG algorithm is very useful to study higher-dimensional systems.

Doctoral Dissertation
    "Tensor renormalization group approach to higher-dimensional lattice field theories" [pdf]
Master Thesis
    「高次テンソル繰り込み群による4次元Ising模型の比熱の解析」
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