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

Tensor renormalization group for fermions

“…In practice, however, the size of the resulting Grassmann tensor scales exponentially for N f and a O(e N f ) computational memory is required in the numerical computations. This issue has been started to be addressed recently by Akiyama [103] and also by Yosprakob et al [127].…”

“…The symbol 'gTr' in equation (18) means the Grassmann tensor trace that is analogous to the tensor trace symbol 'tTr' common in the tensor network formulation in the spin systems. This formulation has been applied recently for the free Wilson and staggered fermions [125], Hubbard models [115,116], N f = 1, 2, 3 Gross-Neveu model [103,126], Z n and U(1) gauge theories with N f = 1, 2, 4 Wilson fermions [127], and several public codes for the Grassmann TRG methods [128,129].…”

“…Akiyama [103] has employed the matrix product decomposition (MPD) to introduce a virtual direction so that each flavor degree of freedom is assigned to the different layers orthogonal to the virtual direction. MPD is a common idea in the tensor network methods such as the MPS and matrix product operator [156,157].…”

“…MPD is a common idea in the tensor network methods such as the MPS and matrix product operator [156,157]. Akiyama [103] has particularly utilized a canonical form of the MPD proposed in [158]. Thanks to the MPD, the memory cost for each local Grassmann tensor 9 https://github.com/akiyama-es/Grassmann-BTRG.…”

“…Applications for relativistic models are presented in section 5 which covers the Gross-Neveu model [102,103], QCD at infinite coupling [104], the Nambu-Jona-Lasinio (NJL) model [105], the N = 1 Wess-Zumino model [106] and non-abelian gauge theories with fermion [107]. Finally, the Fermi-Hubbard model which has been extensively studied in condensed matter community [108][109][110][111][112][113][114] is considered in section 6 following the approach of [115] in 1+1 dimensions and [116] in 2+1 dimensions.…”