- Open Access
Phase structure of -dimensional dense two-color QCD at in the strong coupling limit with the tensor renormalization group
Phys. Rev. D 113, 034503 – Published 12 February, 2026
DOI: https://doi.org/10.1103/jd1r-cqrc
Abstract
We investigate the phase structure of the ()-dimensional strong coupling two-color QCD at zero temperature () with finite chemical potential using the tensor renormalization group method. The chiral and diquark condensates and the quark number density are evaluated as a function of the chemical potential. We further determine the critical exponents associated with the diquark condensate, which suggest consistency with the predictions of mean-field theory.
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References (53)
- E. Dagotto, F. Karsch, and A. Moreo, The strong coupling limit of SU(2) QCD at finite baryon density, Phys. Lett. 169B, 421 (1986).
- E. Dagotto, A. Moreo, and U. Wolff, Lattice QCD at finite temperature and density in the strong coupling limit, Phys. Lett. B 186, 395 (1987).
- J. U. Klatke and K. H. Mutter, Strong coupling QCD with SU(2) gauge fields at finite baryon number density, Nucl. Phys. 342B, 764 (1990).
- Y. Nishida, K. Fukushima, and T. Hatsuda, Thermodynamics of strong coupling two color QCD with chiral and diquark condensates, Phys. Rep. 398, 281 (2004).
- Y. Nishida, Phase structures of strong coupling lattice QCD with finite baryon and isospin density, Phys. Rev. D 69, 094501 (2004).
- A. Nakamura, Quarks and gluons at finite temperature and density, Phys. Lett. 149B, 391 (1984).
- T. Boz, P. Giudice, S. Hands, and J.-I. Skullerud, Dense two-color QCD towards continuum and chiral limits, Phys. Rev. D 101, 074506 (2020).
- A. Begun, V. G. Bornyakov, V. A. Goy, A. Nakamura, and R. N. Rogalyov, Study of two color QCD on large lattices, Phys. Rev. D 105, 114505 (2022).
- K. Iida, E. Itou, K. Murakami, and D. Suenaga, Lattice study on finite density towards zero temperature, J. High Energy Phys. 10 (2024) 022.
In this paper, the “TRG method” or the “TRG approach” refers to not only the original numerical algorithm proposed by Levin and Nave [11] but also its extensions [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].
- M. Levin and C. P. Nave, Tensor renormalization group approach to two-dimensional classical lattice models, Phys. Rev. Lett. 99, 120601 (2007).
- Y. Shimizu and Y. Kuramashi, Grassmann tensor renormalization group approach to one-flavor lattice Schwinger model, Phys. Rev. D 90, 014508 (2014).
- S. Akiyama, Y. Kuramashi, T. Yamashita, and Y. Yoshimura, Restoration of chiral symmetry in cold and dense Nambu–Jona-Lasinio model with tensor renormalization group, J. High Energy Phys. 01 (2021) 121.
- D. Adachi, T. Okubo, and S. Todo, Bond-weighted tensor renormalization group, Phys. Rev. B 105, L060402 (2022).
- S. Akiyama, Bond-weighting method for the Grassmann tensor renormalization group, J. High Energy Phys. 11 (2022) 030.
- D. Adachi, T. Okubo, and S. Todo, Anisotropic tensor renormalization group, Phys. Rev. B 102, 054432 (2020).
- 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, 045139 (2012).
- G. Evenbly and G. Vidal, Tensor network renormalization, Phys. Rev. Lett. 115, 180405 (2015).
- R. Sakai, S. Takeda, and Y. Yoshimura, Higher order tensor renormalization group for relativistic fermion systems, Prog. Theor. Exp. Phys. 2017, 063B07 (2017).
- S. Yang, Z.-C. Gu, and X.-G. Wen, Loop optimization for tensor network renormalization, Phys. Rev. Lett. 118, 110504 (2017).
- M. Hauru, C. Delcamp, and S. Mizera, Renormalization of tensor networks using graph independent local truncations, Phys. Rev. B 97, 045111 (2018).
- D. Kadoh and K. Nakayama, Renormalization group on a triad network, arXiv:1912.02414.
- Y. Shimizu and Y. Kuramashi, Critical behavior of the lattice Schwinger model with a topological term at using the Grassmann tensor renormalization group, Phys. Rev. D 90, 074503 (2014).
- H. Kawauchi and S. Takeda, Tensor renormalization group analysis of model, Phys. Rev. D 93, 114503 (2016).
- H. Kawauchi and S. Takeda, Loop-TNR analysis of CP(1) model with theta term, EPJ Web Conf. 175, 11015 (2018).
- L.-P. Yang, Y. Liu, H. Zou, Z. Xie, and Y. Meurice, Fine structure of the entanglement entropy in the O(2) model, Phys. Rev. E 93, 012138 (2016).
- Y. Shimizu and Y. Kuramashi, Berezinskii-Kosterlitz-Thouless transition in lattice Schwinger model with one flavor of Wilson fermion, Phys. Rev. D 97, 034502 (2018).
- 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, 043B01 (2015).
- D. Kadoh, Y. Kuramashi, Y. Nakamura, R. Sakai, S. Takeda, and Y. Yoshimura, Tensor network formulation for two-dimensional lattice Wess-Zumino model, J. High Energy Phys. 03 (2018) 141.
- D. Kadoh, Y. Kuramashi, Y. Nakamura, R. Sakai, S. Takeda, and Y. Yoshimura, Investigation of complex theory at finite density in two dimensions using TRG, J. High Energy Phys. 02 (2020) 161.
- S. Takeda, Tensor network approach to real-time path integral, Proc. Sci., LATTICE2019 (2019) 033 [arXiv:1908.00126].
- Y. Kuramashi and Y. Yoshimura, Tensor renormalization group study of two-dimensional U(1) lattice gauge theory with a term, J. High Energy Phys. 04 (2020) 089.
- S. Akiyama, D. Kadoh, Y. Kuramashi, T. Yamashita, and Y. Yoshimura, Tensor renormalization group approach to four-dimensional complex theory at finite density, J. High Energy Phys. 09 (2020) 177.
- K. Nakayama, L. Funcke, K. Jansen, Y.-J. Kao, and S. Kühn, Phase structure of the CP(1) model in the presence of a topological -term, Phys. Rev. D 105, 054507 (2022).
- J. Bloch, R. G. Jha, R. Lohmayer, and M. Meister, Tensor renormalization group study of the three-dimensional O(2) model, Phys. Rev. D 104, 094517 (2021).
- J. Bloch and R. Lohmayer, Grassmann higher-order tensor renormalization group approach for two-dimensional strong-coupling QCD, Nucl. Phys. B986, 116032 (2023).
- S. Akiyama and Y. Kuramashi, Critical endpoint of ()-dimensional finite density gauge-Higgs model with tensor renormalization group, J. High Energy Phys. 10 (2023) 077.
- S. Akiyama and Y. Kuramashi, Tensor renormalization group study of ()-dimensional U(1) gauge-Higgs model at with Lüscher’s admissibility condition, J. High Energy Phys. 09 (2024) 086.
- X. Luo and Y. Kuramashi, Tensor renormalization group approach to ()-dimensional SU(2) principal chiral model at finite density, Phys. Rev. D 107, 094509 (2023).
- M. Hite and Y. Meurice, Quantum real-time evolution using tensor renormalization group methods, Phys. Rev. D 111, 034502 (2025).
- X. Luo and Y. Kuramashi, Quantum phase transition of ()-dimensional O(3) nonlinear sigma model at finite density with tensor renormalization group, J. High Energy Phys. 11 (2024) 144.
- X. Luo and Y. Kuramashi, Critical endpoints of three-dimensional finite density SU(3) spin model with tensor renormalization group, J. High Energy Phys. 07 (2025) 036.
- M. Asaduzzaman, S. Catterall, Y. Meurice, R. Sakai, and G. C. Toga, Tensor network representation of non-Abelian gauge theory coupled to reduced staggered fermions, J. High Energy Phys. 05 (2024) 195.
- K. H. Pai, S. Akiyama, and S. Todo, Grassmann tensor renormalization group approach to ()-dimensional two-color lattice QCD at finite density, J. High Energy Phys. 03 (2025) 027.
- H. Oba, Cost reduction of swapping bonds part in anisotropic tensor renormalization group, Prog. Theor. Exp. Phys. 2020, 013B02 (2020).
- S. Hands, J. B. Kogut, M.-P. Lombardo, and S. E. Morrison, Symmetries and spectrum of SU(2) lattice gauge theory at finite chemical potential, Nucl. Phys. B558, 327 (1999).
- W. Pauli, On the conservation of the Lepton charge, Nuovo Cimento 6, 204 (1957).
- F. Gürsey, Relation of charge independence and baryon conservation to Pauli’s transformation, Nuovo Cimento 7, 411 (1958).
- J. B. Kogut, D. K. Sinclair, S. J. Hands, and S. E. Morrison, Two color QCD at nonzero quark number density, Phys. Rev. D 64, 094505 (2001).
- S. Akiyama and D. Kadoh, More about the Grassmann tensor renormalization group, J. High Energy Phys. 10 (2021) 188.
- D. Weingarten, Asymptotic behavior of group integrals in the limit of infinite rank, J. Math. Phys. (N.Y.) 19, 999 (1978).
- Y. Sugimoto and S. Sasaki, Triad representation for the anisotropic tensor renormalization group in four dimensions, Phys. Rev. D 112, 094514 (2025).
- R. B. Lehoucq, D. C. Sorensen, and C. Yang, Software, Environments, and Tools ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (Society for Industrial and Applied Mathematics, 1998), 10.1137/1.9780898719628.