Non-local order parameters for the 1D Hubbard model

We characterize the Mott insulator and Luther-Emery phases of the 1D Hubbard model through correlators that measure the parity of spin and charge strings along the chain. These non-local quantities order in the corresponding gapped phases and vanish at the critical point $U_{c}=0$, thus configuring as hidden order parameters. The Mott insulator consists of bound doublon-holon pairs, which in the Luther-Emery phase turn into electron pairs with opposite spins, both unbinding at $U_{c}$. The behavior of the parity correlators is captured by an effective free spinless fermion model.