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We report on ground state phases of a doped one-dimensional Hubbard model,which for large onsite interactions is governed by the $t$-$J$ Hamiltonian,where the extant entanglement is immutable under perturbative or sudden changesof system parameters, a phenomenon termed as adiabatic freezing. We observethat in the metallic Luttinger liquid phase of the model bipartite entanglementdecays polynomially and is adiabatically frozen, in contrast to the variable,exponential decay in the phase-separation and superconducting spin-gap phases.Significantly, at low fixed electron densities, the spin-gap phase showsremarkable affinity to doped resonating valence bond gas, with multipartiteentanglement frozen across all parameter space. We note that entanglement, ingeneral, is sensitive to external perturbation, as observed in several systems,and hitherto, no such invariance or freezing behavior has been reported.

Adiabatic freezing of entanglement with insertion of defects in a one-dimensional Hubbard model