$Q$-Symmetry in Condensed Matter Physics: Effective Generalized Hubbard Model with Quantum Symmetry

We transform the Hubbard model with phonons, which has recently been shown to exhibit under particular circumstances a quantum symmetry $\operatorname{su}(2)\oplus[\operatorname{su}(2)]_q$, into an effective electron hamiltonian, by means of a generalized Lang–Firsov transformation. The effective hamiltonian is explicitly proved to mantain the quantum symmetry, which survives even when relaxing one of the constraints on the hamiltonian parameters which must hold in the model with phonons. We investigate under which circumstances some superconducting eigenstates of the effective hamiltonian, which can be built thanks to the quantum symmetry, could possibly become metastable.