Matrix Product States and First Quantization

Common wisdom says that the entanglement of fermionic systems can be low in the second quantization formalism but is extremely large in the first quantization. Hence matrix product state (MPS) methods based on moderate entanglement have been overwhelmingly formulated in the second quantization. Here we introduce a first-quantized MPS approach to simulate quantum many-body systems. We show that, by reformulating the way the fermionic antisymmetry is handled, we arrive at MPS with a level of entanglement comparable to the usual one found in the second quantization. We demonstrate our scheme on the one-dimensional t-V model (spinless fermions with density-density interaction) for both ground state and time evolution. For the ground state, we find that the first quantization approach naturally extends to long-range interaction in the Wigner crystal regime. For time evolution, we find that the entanglement entropy in first quantization is significantly smaller than in its second quantization counterpart.