Large-$N$ volume independence in the tensor renormalization group beyond two-dimensional pure gauge: self-similar singular-value spectra in $d\ge 3$ and $O(1)$ topological volume independence with matter fields

In exactly solvable two-dimensional pure $U(N)$ and $SU(N)$ lattice gauge theories, Hirasawa, Matsumoto, Nishimura and Yosprakob (JHEP 12 (2021) 011) showed that the tensor renormalization group (TRG) makes large-$N$ volume independence manifest through the singular-value spectrum of the fundamental tensor: the leading large-$N$ part of every singular value carries a common factor $e^{C^{(0)}N^2}$, so the free-energy density becomes volume independent at $N=\infty$ — the Eguchi–Kawai (EK) reduction, reinterpreted as an approximate self-similarity of the spectrum under coarse graining. They also found a new type of volume independence: in the strong-coupling phase with a $\theta$ term, the properly normalized topological charge density becomes independent of the volume at $N=\infty$, a statement about the $O(1)$ term of the free energy that goes beyond the EK reduction, which concerns only the $O(N^2)$ term. Whether these phenomena occur in large-$N$ gauge theories other than the 2D pure ones — in $d\ge 3$, where spontaneous breaking of the center symmetry restricts volume independence to volumes above a critical value $V_{\mathrm{cr}}$, and in theories with matter fields — was left open. Later work removed one enabling obstacle (the Clebsch–Gordan-induced degeneracy of the singular-value spectrum in $d\ge 3$ pure gauge, eliminated by the armillary-sphere reduced tensor network) and demonstrated finite-$N$ TRG simulations of 3D $SU(2)$/$SU(3)$ pure gauge, but no large-$N$ singular-value analysis, no study of the self-similarity mechanism, and no matter-coupled extension at large $N$ exists. The problem is to determine, positively or negatively with a mechanism, whether and how large-$N$ volume independence operates in the TRG framework beyond 2D pure gauge theories.

Background

The tensor renormalization group (TRG) of Levin and Nave is a deterministic, sampling-free coarse-graining method for lattice models: the partition function is rewritten exactly as a contraction of local tensors, and at each coarse-graining step the network is truncated by retaining only the largest singular values of the local tensors, up to a bond-dimension cutoff $D_{\mathrm{cut}}$. The higher-order TRG (HOTRG) of Xie et al. extends the scheme to three and four dimensions. Because it requires no importance sampling, TRG is free from the sign problem that obstructs Monte Carlo methods at finite density and nonzero $\theta$, which is the main motivation for pursuing gauge theories in this framework. For a lattice gauge theory the tensor representation is built from the character expansion: the Boltzmann weight of each plaquette is expanded in irreducible representations of the gauge group, after which the group integrals over the link variables are performed analytically. The resulting fundamental tensor carries representation indices on its legs. Before the source paper, tensor-network studies of gauge theories were essentially limited to $U(1)$ (e.g. the 2D $U(1)$ theory with a $\theta$ term by Kuramashi and Yoshimura) and $SU(2)$ (e.g. the 2D non-Abelian Higgs model with matter by Bazavov et al.), because for higher rank the number of representations needed in the character expansion grows rapidly and must be restricted in a controlled way. The source paper (Hirasawa, Matsumoto, Nishimura, Yosprakob, JHEP 12 (2021) 011) proposed a practical restriction strategy — for each fixed $U(1)$ charge keep only the representations of minimal dimensionality, with a cutoff on the representation dimension — and demonstrated it in 2D $U(N)$ and $SU(N)$ pure gauge theory, which is exactly solvable at large $N$ (Gross–Witten–Wadia). Two large-$N$ results of that paper frame the present problem. First, the free-energy density becomes volume independent at $N=\infty$ because the leading large-$N$ part of every singular value of the fundamental tensor is common to all representation sequences; this is the Eguchi–Kawai (EK) reduction — the 1982 statement that large-$N$ gauge theory on an infinite lattice equals a single-site (reduced) theory — here reinterpreted as a self-similarity of the singular-value spectrum under the TRG coarse graining. In $d\ge 3$ the EK equivalence is expected to hold only above a critical volume $V_{\mathrm{cr}}$: at smaller volumes the $U(1)^d$ center symmetry breaks spontaneously (Bhanot, Heller and Neuberger), which invalidates the original proof of the reduction; large-$N$ reduction in three dimensions was studied nonperturbatively by Narayanan and Neuberger. Second, a new type of volume independence: adding a $\theta$ term (a topological term $i\theta Q$ weighting each gauge configuration by its topological charge $Q$, the winding number of the $U(1)$ part of the gauge field in 2D $U(N)$), in the strong-coupling phase ($\lambda>2$, with $\lambda=2Ng^2a^2$ the dimensionless 't Hooft coupling) the normalized topological charge density $\mathrm{Im}\langle Q\rangle/(V\chi_N)$ — with $\chi_N$ the large-$N$ topological susceptibility, i.e. the variance per volume of $Q$ at $\theta=0$ — tends at $N=\infty$ to a $\theta$-dependent function independent of the volume $V$. Because the topological term contributes to the free energy only at $O(1)$ beneath the $O(N^2)$ vacuum energy, this statement goes beyond the EK reduction; it fails in the weak-coupling phase $\lambda<2$, where representations with nonzero charge survive and introduce volume-dependent factors. The technical obstacle the authors flagged for going beyond 2D pure gauge is that in $d\ge 3$ and with matter the fundamental tensor acquires internal index structure from the Clebsch–Gordan (CG) coefficients — the tensors that couple irreducible representations in group-product decompositions — which enters the singular values, produces degeneracies, and complicates representation selection and truncation. Subsequent work removed this specific obstacle for pure gauge in any dimension (the armillary-sphere reduced tensor network of Yosprakob, in which the gauge-link matrix indices are traced out along a closed loop of vertex tensors encircling each lattice site, removing the degeneracy they would otherwise induce) and carried out finite-$N$ TRG simulations of 3D $SU(2)$/$SU(3)$ pure gauge, but nothing is known at large $N$ in $d\ge 3$, and matter fields have not been combined with the reduced formulation at all.

Problem Statement

Within the tensor renormalization group framework, determine whether and how large-$N$ volume independence operates in large-$N$ gauge theories beyond the exactly solvable two-dimensional pure $U(N)$/$SU(N)$ models, along the directions the source leaves open. (A) Higher dimensions: for $d\ge 3$ pure $SU(N)$/$U(N)$ lattice gauge theory at $N\to\infty$ — where volume independence is expected only above the center-symmetry-breaking volume $V_{\mathrm{cr}}$ — establish the large-$N$ profile of the singular-value spectrum of the fundamental tensor and of the successively coarse-grained tensors (including controlled representation truncation at large $N$ in $d\ge 3$), and decide whether the free-energy density becomes volume independent for $V>V_{\mathrm{cr}}$ through an approximate self-similarity that emerges after finitely many TRG coarse-graining steps, as speculated in the source. (B) Topological and matter-coupled theories: determine whether the new type of volume independence found in 2D $U(N)$ strong coupling — the normalized topological charge density $\mathrm{Im}\langle Q\rangle/(V\chi_N)$ becoming $V$-independent at $N=\infty$, an $O(1)$ effect beneath the $O(N^2)$ free energy that goes beyond the Eguchi–Kawai reduction — occurs in large-$N$ gauge theories other than 2D pure $U(N)$/$SU(N)$: in $d\ge 3$ with a $\theta$ term (which requires a tensor-network representation of the higher-dimensional topological charge, e.g. the four-dimensional clover construction, the standard nearest-neighbor lattice discretization of the field-strength definition of $Q$) or in large-$N$ theories with matter fields (which first requires extending the reduced armillary-sphere tensor network to matter, explicitly still open). The common objective of all components is a characterization, within the TRG singular-value framework, of whether and by what mechanism large-$N$ volume independence generalizes beyond 2D pure gauge; an acceptable answer determines both directions, (A) and (B), positively or negatively, with a mechanism — via controlled TRG computations (with controlled representation cutoff, bond dimension, and $N$- and volume-extrapolations) and/or analytical derivation of the singular-value behavior — and not merely by an unconstrained numerical observation. Within (B), the $\theta$-term setting and the matter-field setting are alternative routes to the same question, so a determination in either one answers that direction, and a determination in both is likewise acceptable. A negative answer for a stated class, with evidence that the responsible structure is absent, answers the question as faithfully as a positive one.

References

1. Mitsuaki Hirasawa, Akira Matsumoto, Jun Nishimura, Atis Yosprakob, 'Tensor renormalization group and the volume independence in 2D U(N) and SU(N) gauge theories', J. High Energy Phys. 12 (2021) 011, doi:10.1007/JHEP12(2021)011, arXiv:2110.05800, https://arxiv.org/abs/2110.05800
2. Michael Levin, Cody P. Nave, 'Tensor renormalization group approach to two-dimensional classical lattice models', Phys. Rev. Lett. 99 (2007) 120601, doi:10.1103/PhysRevLett.99.120601, https://doi.org/10.1103/PhysRevLett.99.120601
3. 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 (2012) 045139, doi:10.1103/PhysRevB.86.045139, https://doi.org/10.1103/PhysRevB.86.045139
4. Yoshinobu Kuramashi, Yusuke Yoshimura, 'Tensor renormalization group study of two-dimensional U(1) lattice gauge theory with a θ term', J. High Energy Phys. 04 (2020) 089, doi:10.1007/JHEP04(2020)089, arXiv:1911.06480, https://arxiv.org/abs/1911.06480
5. Alexei Bazavov, Simon Catterall, Raghav G. Jha, Judah Unmuth-Yockey, 'Tensor renormalization group study of the non-Abelian Higgs model in two dimensions', Phys. Rev. D 99 (2019) 114507, doi:10.1103/PhysRevD.99.114507, https://doi.org/10.1103/PhysRevD.99.114507
6. David J. Gross, Edward Witten, 'Possible third-order phase transition in the large-N lattice gauge theory', Phys. Rev. D 21 (1980) 446, doi:10.1103/PhysRevD.21.446, https://doi.org/10.1103/PhysRevD.21.446
7. Spenta R. Wadia, 'N=∞ phase transition in a class of exactly soluble model lattice gauge theories', Phys. Lett. B 93 (1980) 403, doi:10.1016/0370-2693(80)90353-6, https://doi.org/10.1016/0370-2693(80)90353-6
8. Tohru Eguchi, Hikaru Kawai, 'Reduction of dynamical degrees of freedom in the large-N gauge theory', Phys. Rev. Lett. 48 (1982) 1063, doi:10.1103/PhysRevLett.48.1063, https://doi.org/10.1103/PhysRevLett.48.1063
9. Gyan Bhanot, Urs M. Heller, Herbert Neuberger, 'The quenched Eguchi-Kawai model', Phys. Lett. B 113 (1982) 47, doi:10.1016/0370-2693(82)90106-X, https://doi.org/10.1016/0370-2693(82)90106-X
10. R. Narayanan, H. Neuberger, 'Large-N reduction in the continuum three-dimensional Yang-Mills theory', Phys. Rev. Lett. 91 (2003) 081601, doi:10.1103/PhysRevLett.91.081601, https://doi.org/10.1103/PhysRevLett.91.081601
11. Atis Yosprakob, 'Reduced tensor network formulation for non-Abelian gauge theories in arbitrary dimensions', Prog. Theor. Exp. Phys. 2024 (2024) ptae091, doi:10.1093/ptep/ptae091, arXiv:2311.02541, https://arxiv.org/abs/2311.02541
12. 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 (2025) ptaf028, doi:10.1093/ptep/ptaf028, arXiv:2406.16763, https://arxiv.org/abs/2406.16763
13. Takaaki Kuwahara, Asato Tsuchiya, 'Toward tensor renormalization group study of three-dimensional non-Abelian gauge theory', Prog. Theor. Exp. Phys. 2022 (2022) ptac103, doi:10.1093/ptep/ptac103, arXiv:2205.08883, https://arxiv.org/abs/2205.08883
14. 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. 11 (2023) 187, doi:10.1007/JHEP11(2023)187, arXiv:2309.01422, https://arxiv.org/abs/2309.01422
15. Shinichiro Akiyama, Raghav G. Jha, Judah Unmuth-Yockey, 'SU(2) principal chiral model with tensor renormalization group on a cubic lattice', Phys. Rev. D 110 (2024) 034519, doi:10.1103/PhysRevD.110.034519, arXiv:2406.10081, https://arxiv.org/abs/2406.10081
16. 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. 05 (2024) 195, doi:10.1007/JHEP05(2024)195, arXiv:2312.16167, https://arxiv.org/abs/2312.16167
17. Philip Rouenhoff, Stephan Durr, 'Topology in 2D non-Abelian lattice gauge theories', PoS LATTICE2024 (2025) 0461, doi:10.22323/1.466.0461, arXiv:2411.11593, https://arxiv.org/abs/2411.11593
18. Atis Yosprakob, 'Toward tensor renormalization group study of lattice QCD', PoS LATTICE2024 (2025) 029, doi:10.22323/1.466.0029, arXiv:2501.14293, https://arxiv.org/abs/2501.14293