Matrix product decomposition for two- and three-flavor Wilson fermions: Benchmark results in the lattice Gross-Neveu model at finite density

We formulate the path integral of two- and three-flavor Wilson fermion in two dimensions as a multilayer Grassmann tensor network by the matrix product decomposition. Thanks to this new description, the memory cost scaling is reduced from $\mathrm{O}(\mathrm{e}^{N_{f}})$ for the conventional construction to $\mathrm{O}(N_{f})$. Based on this representation, we develop a coarse-graining algorithm where spatially or temporally adjacent Grassmann tensors are converted into a canonical form along a virtual direction before we carry out the spacetime coarse-graining. Benchmarking with the lattice Gross-Neveu model at finite density, we see that the Silver Blaze phenomenon in the pressure and number density is captured with relatively small bond dimensions.Comment: 16 pages, 13 figure

Implementation of bond weighting method for the Grassmann tensor renormalization group

We demonstrate the efficiency of the bond weighting method for the Grassmann tensor renormalization group (TRG). Benchmarking with the two-dimensional Gross-Neveu model with the Wilson fermion at finite density, we show that the bond weighting method improves the accuracy of the original Grassmann TRG. We also provide a sample code of the bond-weighted TRG that can be applied to the two-dimensional models including fermions on a square lattice.Comment: 7 pages, 4 figures, Proceedings of the 40th International Symposium on Lattice Field Theory (Lattice 2023), 31 July - 4 August 2023, Fermilab, Batavia, Illinois, US

Bond-weighting method for the Grassmann tensor renormalization group

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 develop this bond-weighted tensor renormalization group algorithm to make it applicable to the fermionic system, benchmarking with the two-dimensional massless Wilson fermion. We show that the accuracy with the fixed bond dimension is improved also in the fermionic system and provide numerical evidence that the optimal choice of the hyperparameter is not affected by whether the system is bosonic or fermionic. In addition, by monitoring the singular value spectrum, we find that the scale-invariant structure of the renormalized Grassmann tensor is successfully kept by the bond-weighting technique.Comment: 12 pages, 6 figure

Critical endpoint of (3+1)-dimensional finite density Z3Z_3 gauge-Higgs model with tensor renormalization group

The critical endpoint of the (3+1)-dimensional $Z_3$ gauge-Higgs model at finite density is determined by the tensor renormalization group method. This work is an extension of the previous one on the $Z_2$ model. The vital difference between them is that the $Z_3$ model suffers from the sign problem, while the $Z_2$ model does not. We show that the tensor renormalization group method allows us to locate the critical endpoint for the $Z_3$ gauge-Higgs model at finite density, regardless of the sign problem.Comment: 18 pages, 10 figures. arXiv admin note: text overlap with arXiv:2202.1005

Quantum Field Theories with Tensor Renormalization Group

We report recent progress on the application of the tensor renormalization group (TRG) to quantum field theories pursued by the Tsukuba group. We explain how to treat the scalar, fermion, and gauge theories with the TRG method presenting the results for the phase transitions in the (3+1)-dimensional ((3+1)$d$) complex $\phi^4$ theory at finite density, (1+1)$d$ pure U(1) lattice gauge theory with a $\theta$ term, (3+1)$d$ Nambu--Jona-Lasinio model at finite density and (1+1)$d$ and (2+1)$d$ Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.Comment: 25 pages, 20 figures, Proceedings of the 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July 2021, Zoom/Gather@Massachusetts Institute of Technolog