Three-dimensional finite temperature Z2 gauge theory with tensor network scheme

  • Yoshinobu Kuramashi
  • Yusuke YoshimuraEmail author
Open Access
Regular Article - Theoretical Physics


We apply a tensor network scheme to finite temperature Z2 gauge theory in 2+1 dimensions. Finite size scaling analysis with the spatial extension up to Nσ = 4096 at the temporal extension of Nτ = 2, 3, 5 allows us to determine the transition temperature and the critical exponent ν at high level of precision, which shows the consistency with the Svetitsky-Yaffe conjecture.


Duality in Gauge Field Theories Lattice QCD 


Open Access

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  1. [1]
    T. Nishino and K. Okunishi, Corner transfer matrix renormalization group method, J. Phys. Soc. Jpn.65 (1996) 891.ADSCrossRefGoogle Scholar
  2. [2]
    N. Maeshima, Y. Hieida, Y. Akutsu, T. Nishino and K. Okunishi, Vertical density matrix algorithm: a higher dimensional numerical renormalization scheme based on the tensor product state ansatz, Phys. Rev.E 64 (2001) 016705 [cond-mat/0101360] [INSPIRE].ADSGoogle Scholar
  3. [3]
    M. Levin and C.P. Nave, Tensor renormalization group approach to 2D classical lattice models, Phys. Rev. Lett.99 (2007) 120601 [cond-mat/0611687] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    Z.-C. Gu and X.-G. Wen, Tensor-entanglement-filtering renormalization approach and symmetry protected topological order, Phys. Rev.B 80 (2009) 155131 [arXiv:0903.1069] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    Y. Shimizu, Tensor renormalization group approach to a lattice boson model, Mod. Phys. Lett.A 27 (2012) 1250035 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    Y. Shimizu and Y. Kuramashi, Grassmann tensor renormalization group approach to one-flavor lattice Schwinger model, Phys. Rev.D 90 (2014) 014508 [arXiv:1403.0642] [INSPIRE].ADSGoogle Scholar
  7. [7]
    Z.-C. Gu, F. Verstraete and X.-G. Wen, Grassmann tensor network states and its renormalization for strongly correlated fermionic and bosonic states, arXiv:1004.2563 [INSPIRE].
  8. [8]
    Z.-C. Gu, Efficient simulation of Grassmann tensor product states, Phys. Rev.B 88 (2013) 115139 [arXiv:1109.4470] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    Y. Shimizu and Y. Kuramashi, Critical behavior of the lattice Schwinger model with a topological term at θ = π using the Grassmann tensor renormalization group, Phys. Rev.D 90 (2014) 074503 [arXiv:1408.0897] [INSPIRE].ADSGoogle Scholar
  10. [10]
    Y. Shimizu and Y. Kuramashi, Berezinskii-Kosterlitz-Thouless transition in lattice Schwinger model with one flavor of Wilson fermion, Phys. Rev.D 97 (2018) 034502 [arXiv:1712.07808] [INSPIRE].ADSGoogle Scholar
  11. [11]
    S. Takeda and Y. Yoshimura, Grassmann tensor renormalization group for the one-flavor lattice Gross-Neveu model with finite chemical potential, Prog. Theor. Exp. Phys.2015 (2015) 043B01.CrossRefGoogle Scholar
  12. [12]
    D. Kadoh, Y. Kuramashi, Y. Nakamura, R. Sakai, S. Takeda and Y. Yoshimura, Tensor network formulation for two-dimensional lattice N = 1 Wess-Zumino model, JHEP03 (2018) 141 [arXiv:1801.04183] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  13. [13]
    R. Sakai, S. Takeda and Y. Yoshimura, Higher order tensor renormalization group for relativistic fermion systems, Prog. Theor. Exp. Phys.2017 (2017) 063B07 [arXiv:1705.07764] [INSPIRE].Google Scholar
  14. [14]
    Y. Yoshimura, Y. Kuramashi, Y. Nakamura, S. Takeda and R. Sakai, Calculation of fermionic Green functions with Grassmann higher-order tensor renormalization group, Phys. Rev.D 97 (2018) 054511 [arXiv:1711.08121] [INSPIRE].ADSMathSciNetGoogle Scholar
  15. [15]
    Y. Liu et al., Exact blocking formulas for spin and gauge models, Phys. Rev.D 88 (2013) 056005 [arXiv:1307.6543] [INSPIRE].ADSGoogle Scholar
  16. [16]
    B. Dittrich, F.C. Eckert and M. Martin-Benito, Coarse graining methods for spin net and spin foam models, New J. Phys.14 (2012) 035008 [arXiv:1109.4927] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    B. Dittrich and F.C. Eckert, Towards computational insights into the large-scale structure of spin foams, J. Phys. Conf. Ser.360 (2012) 012004 [arXiv:1111.0967] [INSPIRE].CrossRefGoogle Scholar
  18. [18]
    B. Dittrich, S. Mizera and S. Steinhaus, Decorated tensor network renormalization for lattice gauge theories and spin foam models, New J. Phys.18 (2016) 053009 [arXiv:1409.2407] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    M. Caselle and M. Hasenbusch, Deconfinement transition and dimensional crossover in the 3D gauge Ising model, Nucl. Phys.B 470 (1996) 435 [hep-lat/9511015] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    B. Svetitsky and L.G. Yaffe, Critical behavior at finite temperature confinement transitions, Nucl. Phys.B 210 (1982) 423 [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    Z.Y. Xie, J. Chen, M.P. Qin, J.W. Zhu, L.P. Yang and T. Xiang, Coarse-graining renormalization by higher-order singular value decomposition, Phys. Rev.B 86 (2012) 045139.ADSCrossRefGoogle Scholar

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Center for Computational SciencesUniversity of TsukubaTsukubaJapan

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