Real Space Functional Renormalization Group

Studies of symmetry broken phases in material systems and the experimental Moire materials require moving beyond current functional renormalization group (fRG) implementations designed for translationally invariant crystalline materials. The limiting factor in real space fRG implementation is the scaling of the vertex, which requires tracking the flow of N⁴ couplings. We significantly reduce the computational cost and memory footprint of the fRG by utilizing a tensor train decomposition (TTD) of the vertex in the three channels. The TTD decomposition can be constructed from a partial evaluation of the vertex and appears to scale favorably as we vary the dimensionality of the system. We find it effective in treating charge and spin response functions of 3D Hubbard Hamiltonians. We further analyze the computational complexity of potential application of the TTD-fRG to the Hofstadter Hamiltonians in 2D,3D and the twisted Hubbard Hamiltonians.