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Double symmetry breaking and 2D quantum phase diagram in spin-boson systems

The quantum ground state properties of two independent chains of spins (two-levels systems) interacting with the same bosonic field are theoretically investigated. Each chain is coupled to a different quadrature of the field, leading to two independent symmetry breakings for increasing values of the two spin-boson interaction constants $Ω_C$ and $Ω_I$. A phase diagram is provided in the plane ($Ω_C$,$Ω_I$) with 4 different phases that can be characterized by the complex bosonic coherence of the ground states and can be manipulated via non-abelian Berry effects. In particular, when $Ω_C$ and $Ω_I$ are both larger than two critical values, the fundamental subspace has a four-fold degeneracy. Possible implementations in superconducting or atomic systems are discussed.