# Spin-projected matrix product states (SP-MPS): a versatile tool for strongly correlated systems

## TL;DR

The paper introduces spin-projected matrix product states (SP-MPS), a flexible wavefunction ansatz that simplifies achieving spin-adaptation and enables efficient exploration of complex magnetic systems in strongly correlated electronic structures.

## Contribution

It develops a new SP-MPS ansatz combining spin projection with MPS, offering a simpler alternative to non-Abelian symmetry incorporation and facilitating studies of magnetic and open-shell systems.

## Key findings

- Provides a simpler route to spin-adapted wavefunctions.
- Enables straightforward generation of broken symmetry states.
- Supports parallel computation of expectation values.

## Abstract

We present a new wavefunction ansatz that combines the strengths of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as $|\Psi^{(N,S,M)}_{SP-MPS}\rangle=\mathcal{P}_S|\Psi_{MPS}^{(N,M)}\rangle$, where $\mathcal{P}_S$ is the spin projector for total spin $S$ and $|\Psi_{MPS}^{(N,M)}\rangle$ is an MPS wavefunction with a given particle number $N$ and spin projection $M$. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin-adaptation (i.e. to create eigenfunctions of $\hat{S}^2$) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for non-singlet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) Due to the use of $|\Psi_{MPS}^{(N,M)}\rangle$ as its underlying state, the SP-MPS can be closely connected to broken symmetry mean-field states. This allows to straightforwardly generate the large number of broken symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector $\mathcal{P}_S$ naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin projected theories.