Matrix product states in the first quantization

Jheng-Wei Li^1*, Yuriel Núñez Fernández², Xavier Waintal¹

¹Quantum Photonics, Electronics and Engineering laboratory, Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Grenoble, France
²Institut Néel, Centre national de la recherche scientifique (CNRS), Grenoble, France

* Presenter:Jheng-Wei Li, email:jheng-wei.li@cea.fr

Matrix product states (MPS) have become the cornerstone method for solving quantum many-body problems.
The method is always formulated in the second quantization language, and its success lies in the fact that certain wave functions, such as ground states of short-range Hamiltonians, can have low entanglement in such representation.
An MPS formulation in the first-quantized language, on the other hand, has never been considered.

In this talk, I will introduce a first-quantized MPS approach to explore such a possibility.
The difficulty of this problem manifests itself when considering non-distinguishable particles, such as fermions or bosons.
Due to the (anti)-symmetrization condition with respect to particle label exchange, the first-quantized wavefunction is necessarily entangled and does not admit a low-rank MPS approximation.
We propose a novel strategy to solve this.
Then, we apply our approach to study the one-dimensional $t$-$V$ model of interacting spinless fermions.
I will discuss our results for ground states and time evolutions, and the prospect of this approach to general problems.

Keywords: Matrix product states, Correlated electrons, Tensor network