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Homogeneous and domain wall topological Haldane conductors with dressed Rydberg atoms

The interplay between antiferromagnetic interaction and hole motion is capable of inducing intriguing conducting topological Haldane phases described by a finite non-local string order parameter. Here we show that these states of matter are captured by the one dimensional $t-J_z$ model which can be experimentally realized with dressed Rydberg atoms trapped onto a one dimensional optical lattice. In the sector with vanishing total magnetization exact Bethe ansatz calculations associated to bosonization technique allow to predict that both metallic and superconducting topological Haldane states can be achieved. With the addition of an appropriate magnetic field the system enters in a domain wall structure with finite total magnetization. In this regime conducting topological Haldane states are confined in domains separated by regions where fully polarized Luttinger liquid occurs. A procedure to dynamically stabilize such Haldane topological phases starting from a confined Ising state is also described