We theoretically investigate the emerging phase of the strongly correlated spin 1/2 Fermi gas of Rydberg dressing atoms in a one-dimensional optical lattice. For weak coupling, we use the bosonization description to demonstrate the ability to alternately undergo quantum phase transitions between different Luttinger liquids. At strong coupling, the ground state develops a non-trivial phase separation exhibiting Rattinger liquid ”puddles” separated by domain walls due to length-scale interactions of the incommensurate packing and Rydberg core. These phases can be detected in the ultracold gas of Rydberg atoms made from $^6$Li.

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