Grassmann tensor-network coarse-graining for four-dimensional lattice QCD: full gauge-fermion treatment with controlled truncation accuracy

The Grassmann higher-order tensor renormalization group (GHOTRG) of Sakai, Takeda and Yoshimura (PTEP 2017) made deterministic tensor-network coarse-graining available for relativistic lattice fermions, benchmarking on 2D and 3D free Wilson fermions and the 2D finite-density Thirring model, and left open a research agenda with a single objective: bringing such methods to interacting gauge-fermion systems, ultimately four-dimensional lattice QCD. Since 2017 the original cost barrier (contraction scaling $D_{\rm cut}^{4d-1}$) has largely fallen through anisotropic and randomized coarse-graining; four-dimensional fermion systems without gauge fields (the 3+1-dimensional Nambu-Jona-Lasinio model) and strong-coupling limits of QCD are now computed with Grassmann tensor networks; and first tensor-network formulations of non-Abelian gauge fields coupled to fermions exist in two dimensions. What remains open is the composite core: a Grassmann tensor-network coarse-graining computation of a (3+1)-dimensional non-Abelian lattice gauge theory with dynamical fermions at finite gauge coupling — lattice QCD as the flagship — delivering thermodynamic observables at zero and finite chemical potential with quantitatively controlled truncation error.

Background

Monte Carlo importance sampling is the workhorse of lattice field theory, but at nonzero baryon chemical potential the fermion determinant becomes complex, so stochastic methods fail (the sign problem). Tensor-network coarse-graining is the main deterministic alternative. In the tensor renormalization group (TRG) of Levin and Nave, the partition function of a $d$-dimensional Euclidean lattice model is rewritten as the contraction of a network of local tensors; repeated singular value decompositions (SVD) coarse-grain the network, and accuracy is controlled by the bond dimension $D_{\rm cut}$, the number of retained singular values per link. The higher-order TRG (HOTRG) of Xie et al. uses a higher-order SVD that preserves the hypercubic lattice topology and works in $d\ge 3$, at the price of a tensor-contraction cost growing as $D_{\rm cut}^{4d-1}$.

Fermions require Grassmann calculus: the lattice path integral over anticommuting (Grassmann) variables must be performed exactly, not sampled. Grassmann tensor networks encode the fermionic weight, sign structure included, into local tensors carrying auxiliary Grassmann variables on the bonds, making the network amenable to TRG-style coarse-graining. After the Grassmann TRG (GTRG) was benchmarked on the one-flavor Gross-Neveu model (a four-fermion lattice field theory) at finite chemical potential by Takeda and Yoshimura (2015), Sakai, Takeda and Yoshimura introduced the Grassmann higher-order TRG (GHOTRG), the Grassmannization of HOTRG, and validated it on 2D and 3D free Wilson fermions and on the 2D lattice Thirring model (a four-fermion model whose fermions interact through a current-current coupling) at finite chemical potential $\mu$ (Wilson and staggered fermions are the two standard lattice discretizations of continuum quarks). They also supplied a boundary-tensor implementation of the anti-periodic temporal boundary conditions required at finite temperature.

The paper closed with two observations that define the open agenda. First, in 2D systems at finite chemical potential the accuracy degrades around $\mu\approx 1.0$ because the eigenvalue spectra of the intermediate matrices $M^{\pm}$ (formed by contracting tensor pairs during blocking; their decay controls what the truncation can safely discard) have a very poor hierarchy, an effect inherited from GTRG and not cured by increasing $D_{\rm cut}$. Second, although the algorithm extends to any dimension on paper, the $D_{\rm cut}^{4d-1}$ cost made $d=4$ infeasible; the authors suggested randomizing the contraction or hybridizing with Ferris's tensor-network Monte Carlo, and expressed the hope that GHOTRG be a starting point towards four-dimensional lattice QCD. Lattice QCD — SU(3) matrix-valued gauge fields on the links of a 4D Euclidean hypercubic lattice coupled to quark fields on the sites — is precisely the system whose finite-density regime is inaccessible to Monte Carlo, which is why a controlled tensor-network route to it is a stated community goal (the Snowmass 2021 white paper, the US high-energy-physics community's decadal roadmap).

Post-2017 progress redefined the frontier and must be known to judge the open core. The anisotropic TRG (ATRG) drastically reduced the higher-dimensional contraction cost; its Grassmann version computed the (3+1)-dimensional Nambu-Jona-Lasinio model (a four-fermion theory without gauge fields) on $1024^4$ lattices; randomized HOTRG variants lower the scaling further, down to $O(D_{\rm cut}^{d+3})$; strong-coupling QCD — the limit of vanishing gauge coupling, in which the gauge links can be integrated out analytically — has been treated with Grassmann tensor networks in 2D and in (3+1) dimensions; and the first tensor-network representation of a non-Abelian gauge theory coupled to fermions (2D SU(2), reduced staggered fermions, with Gaussian-quadrature and tensor-decomposition truncations of the gauge sector) has appeared. No computation exists at finite gauge coupling with dynamical non-Abelian links and dynamical quarks in four dimensions.

Problem Statement

The problem is to extend Grassmann tensor-network coarse-graining — the GHOTRG program — to interacting lattice gauge-fermion systems, with full four-dimensional lattice QCD as the flagship target, with quantitatively controlled errors. All components below serve one objective: a deterministic, sign-problem-free computation of the thermodynamics of a (3+1)-dimensional non-Abelian lattice gauge theory with dynamical fermions at finite gauge coupling, at zero and finite chemical potential. No component alone completes the objective.

(1) Gauge-fermion tensor construction and coarse-graining: construct local initial tensors encoding dynamical non-Abelian gauge links coupled to dynamical quarks at finite gauge coupling, together with a blocking scheme for coarse-graining such tensors. This goes beyond the present partial results: the strong-coupling limit (links integrated out exactly or to fixed order in the inverse coupling), pure-gauge character-expansion (expansion of gauge-link variables in irreducible representations of the gauge group) and 'armillary sphere' reductions (which strip matrix-index degeneracies from the gauge tensors), and 2D SU(2) formulations that truncate the gauge sector by quadrature.

(2) Four-dimensional scalability: Grassmannize and validate contraction algorithms that beat the $D_{\rm cut}^{4d-1}$ cost — anisotropic, randomized, or tensor-network-Monte-Carlo hybrid schemes — at the bond dimensions needed for $d=4$ thermodynamic-limit computations, and demonstrate their accuracy for fermionic, not only bosonic, networks.

(3) Truncation-accuracy control in hard regimes: determine and validate the algorithmic devices (alternative truncation criteria, bond weighting, entanglement filtering, adaptive schemes, or others) that control the error where the $M^{\pm}$ spectra decay slowly — the finite-density $\mu\approx 1.0$-type pathology of the original benchmarks — including at first-order transitions and at Silver Blaze behavior (observables remaining at their vacuum values until $\mu$ crosses the lightest excitation threshold).

An answer must deliver, for a specified four-dimensional non-Abelian gauge-fermion lattice discretization (lattice QCD, or a staged sequence that demonstrably culminates in it), thermodynamic observables — free energy, chiral condensate, quark number density and equation of state, at zero and finite chemical potential — in the thermodynamic limit, with truncation errors extrapolable in $D_{\rm cut}$. Intermediate systems — four-dimensional pure-gauge or gauge-Higgs models at finite coupling, 2D non-Abelian gauge-fermion systems without quadrature truncation, strong-coupling or strong-coupling-expansion QCD — constitute partial progress, not completion.

References

1. Ryo Sakai, Shinji Takeda, Yusuke Yoshimura, 'Higher-order tensor renormalization group for relativistic fermion systems', Prog. Theor. Exp. Phys. 2017, 063B07 (2017). DOI: 10.1093/ptep/ptx080; arXiv:1705.07764; https://arxiv.org/abs/1705.07764
2. Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, T. Xiang, 'Coarse-graining renormalization by higher-order singular value decomposition', Phys. Rev. B 86, 045139 (2012). DOI: 10.1103/PhysRevB.86.045139; arXiv:1201.1144; https://arxiv.org/abs/1201.1144
3. M. Levin, C. P. Nave, 'Tensor renormalization group approach to two-dimensional classical lattice models', Phys. Rev. Lett. 99, 120601 (2007). DOI: 10.1103/PhysRevLett.99.120601; arXiv:cond-mat/0611687; https://arxiv.org/abs/cond-mat/0611687
4. S. Takeda, 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). DOI: 10.1093/ptep/ptv022; arXiv:1412.7855; https://arxiv.org/abs/1412.7855
5. D. Adachi, T. Okubo, S. Todo, 'Anisotropic tensor renormalization group', Phys. Rev. B 102, 054432 (2020). DOI: 10.1103/PhysRevB.102.054432; arXiv:1906.02007; https://arxiv.org/abs/1906.02007
6. Shinichiro Akiyama, Yoshinobu Kuramashi, Takumi Yamashita, Yusuke Yoshimura, 'Restoration of chiral symmetry in cold and dense Nambu-Jona-Lasinio model with tensor renormalization group', J. High Energy Phys. 2021(1), 121 (2021). DOI: 10.1007/JHEP01(2021)121; arXiv:2009.11583; https://arxiv.org/abs/2009.11583
7. Katsumasa Nakayama, 'Randomized higher-order tensor renormalization group', arXiv:2307.14191 (2023). https://arxiv.org/abs/2307.14191
8. Yuto Sugimoto, Shoichi Sasaki, 'Triad representation for the anisotropic tensor renormalization group in four dimensions', Phys. Rev. D 112, 094514 (2025). DOI: 10.1103/c3qc-tn48; arXiv:2507.21909; https://arxiv.org/abs/2507.21909
9. Pascal Milde, Jacques Bloch, Robert Lohmayer, 'Tensor-network simulation of the strong-coupling U(N) model', PoS LATTICE2021, 462 (2022). DOI: 10.22323/1.396.0462; arXiv:2112.01906; https://arxiv.org/abs/2112.01906
10. Jacques Bloch, Robert Lohmayer, 'Grassmann higher-order tensor renormalization group approach for two-dimensional strong-coupling QCD', Nucl. Phys. B 986, 116032 (2023). DOI: 10.1016/j.nuclphysb.2022.116032; arXiv:2206.00545; https://arxiv.org/abs/2206.00545
11. Thomas Samberger, Jacques Bloch, Robert Lohmayer, Tilo Wettig, 'Tensor-network formulation of QCD in the strong-coupling expansion', Nucl. Phys. B 1024, 117267 (2026). DOI: 10.1016/j.nuclphysb.2025.117267; arXiv:2508.09891; https://arxiv.org/abs/2508.09891
12. Thomas Samberger, Jacques Bloch, Robert Lohmayer, Tilo Wettig, 'Order-separated tensor-network method for QCD in the strong-coupling expansion', arXiv:2603.24487 (2026). https://arxiv.org/abs/2603.24487
13. Yuto Sugimoto, Shinichiro Akiyama, Yoshinobu Kuramashi, 'Phase structure of (3+1)-dimensional dense two-color QCD at $T=0$ in the strong coupling limit with the tensor renormalization group', Phys. Rev. D 113, 034503 (2026). DOI: 10.1103/jd1r-cqrc; arXiv:2509.23637; https://arxiv.org/abs/2509.23637
14. Yuto Sugimoto, Shinichiro Akiyama, Yoshinobu Kuramashi, 'Tensor renormalization group study of cold and dense QCD in the strong coupling limit', Phys. Rev. D 113, 114509 (2026). DOI: 10.1103/v5v2-7mq5; arXiv:2601.20690; https://arxiv.org/abs/2601.20690
15. Muhammad Asaduzzaman, Simon Catterall, Yannick Meurice, Ryo Sakai, Goksu Can Toga, 'Tensor network representation of non-abelian gauge theory coupled to reduced staggered fermions', J. High Energy Phys. 2024(5), 195 (2024). DOI: 10.1007/JHEP05(2024)195; arXiv:2312.16167; https://arxiv.org/abs/2312.16167
16. Atis Yosprakob, 'Reduced tensor network formulation for non-Abelian gauge theories in arbitrary dimensions', Prog. Theor. Exp. Phys. 2024, 073B05 (2024). DOI: 10.1093/ptep/ptae091; arXiv:2311.02541; https://arxiv.org/abs/2311.02541
17. Atis Yosprakob, Kouichi Okunishi, 'Tensor renormalization group study of the 3D SU(2) and SU(3) gauge theories with the reduced tensor network formulation', Prog. Theor. Exp. Phys. 2025, 033B06 (2025). DOI: 10.1093/ptep/ptaf028; arXiv:2406.16763; https://arxiv.org/abs/2406.16763
18. Atis Yosprakob, Jun Nishimura, Kouichi Okunishi, 'A new technique to incorporate multiple fermion flavors in tensor renormalization group method for lattice gauge theories', J. High Energy Phys. 2023(11), 187 (2023). DOI: 10.1007/JHEP11(2023)187; arXiv:2309.01422; https://arxiv.org/abs/2309.01422
19. Shinichiro Akiyama, 'Bond-weighting method for the Grassmann tensor renormalization group', J. High Energy Phys. 2022(11), 30 (2022). DOI: 10.1007/JHEP11(2022)030; arXiv:2208.03227; https://arxiv.org/abs/2208.03227
20. Daisuke Kadoh, Yoshinobu Kuramashi, Ryoichiro Ueno, 'Irregular parameter dependence of numerical results in tensor renormalization group analysis', Prog. Theor. Exp. Phys. 2019, 061B01 (2019). DOI: 10.1093/ptep/ptz056; arXiv:1812.10517; https://arxiv.org/abs/1812.10517
21. Yusuke Yoshimura, Y. Kuramashi, Y. Nakamura, S. Takeda, Ryo Sakai, 'Calculation of fermionic Green functions with Grassmann higher-order tensor renormalization group', Phys. Rev. D 97, 054511 (2018). DOI: 10.1103/PhysRevD.97.054511; arXiv:1711.08121; https://arxiv.org/abs/1711.08121
22. Shinichiro Akiyama, Yannick Meurice, Ryo Sakai, 'Tensor renormalization group for fermions', J. Phys.: Condens. Matter 36, 343002 (2024). DOI: 10.1088/1361-648X/ad4760; arXiv:2401.08542; https://arxiv.org/abs/2401.08542
23. Shinichiro Akiyama, Daisuke Kadoh, 'More about the Grassmann tensor renormalization group', J. High Energy Phys. 2021(10), 188 (2021). DOI: 10.1007/JHEP10(2021)188; arXiv:2005.07570; https://arxiv.org/abs/2005.07570
24. Jian-Gang Kong, Jia-Ji Zhu, Z. Y. Xie, 'Grassmann tensor networks', Ann. Phys. 491, 170511 (2026). DOI: 10.1016/j.aop.2026.170511; arXiv:2605.12907; https://arxiv.org/abs/2605.12907
25. Yannick Meurice, James C. Osborn, Ryo Sakai, Judah Unmuth-Yockey, Simon Catterall, Rolando D. Somma, 'Tensor networks for High Energy Physics: contribution to Snowmass 2021', arXiv:2203.04902 (2022). https://arxiv.org/abs/2203.04902
26. Andrew J. Ferris, 'Unbiased Monte Carlo for the age of tensor networks', arXiv:1507.00767 (2015). https://arxiv.org/abs/1507.00767