Tensor Networks at the Fermi Edge

Tensor networks are powerful for one-dimensional quantum systems but quickly become inefficient for quantum transport in real space, where entanglement across spatial cuts grows linearly in time. Ordering the reservoir degrees of freedom by frequency in a mixed representation removes the dominant linear-in-time contribution by aligning the (asymptotically) frequency-resolved scattering states. Empirically, the entanglement then exhibits logarithmic growth, S=k log t. Here, we analytically demonstrate this growth is due to an infrared divergence. Finite time imposes a frequency resolution |dw| < 1/t, so that weak off-diagonal mixing in frequency space is cut off in the infrared; combined with sharp occupation steps, this produces a universal log t growth for both the von Neumann and Renyi entropies. The non-universal prefactor, k, is set by the single-particle scattering matrix and the magnitude of the occupation jumps at the relevant Fermi edges. This helps explain the observed exponential-to-polynomial reduction in computational cost within mixed-basis matrix product state transport simulations.