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Microscopically understanding competing orders in strongly correlated systemsis a key challenge in modern quantum many-body physics. For example, the studyof magnetic polarons and their relation to pairing in the Fermi-Hubbard modelin different geometries remains one of the central questions, and may help tounderstand the mechanism underlying unconventional superconductivity incuprates or transition metal dichalcogenides. With recent advances in analogquantum simulation of the Fermi-Hubbard model on non-bipartite lattices,frustrated physics can now be explored experimentally in systems lackingparticle-hole symmetry. Here, we study the singly doped $t-J$ model on thetriangular lattice, focusing on the competition between kinetic and magneticfrustration as a function of temperature. In doublon doped systems, we uncovera crossover between Nagaoka-type ferromagnetic (FM) correlations at hightemperature and exchange mediated antiferromagnetic (AFM) order around thedoublon at low temperature. For hole doped systems, kineticHaerter-Shastry-type AFM at high temperature as well as exchange interactionsat low temperature favor $120^{\circ}$ order, strengthening magneticcorrelations compared to the undoped system. In the ground state, the presenceof AFM correlations throughout a wide range of interactions indicatesconfinement of both types of dopants. In this regime we firmly establish thepresence of linear confining potentials via energy scaling arguments,supporting the picture of geometric strings in the frustrated triangular $t-J$model.

Kinetic-to-magnetic frustration crossover and linear confinement in the doped triangular $t-J$ model