Fig. 2: Experimental obtained band structures of tight-binding lattices and the Hall ladder.

The band structures exhibit tight-binding lattice characters when two resonators are either not coupled (J^V = 0, a) or fully coupled into a large resonator whose length is doubled and FSR is halved (b). \({J}_{B}^{H}\) is set to zero in the left panels of (b), while set to be equal to \({J}_{A}^{H}\) in the right panels. The right panel in (a) and the lower panels in (b) are the corresponding numerical calculations. c Illustration of the Hall ladder where the crossing coupling is absent. d–g The heat maps are the experimentally obtained band structures of the Hall ladder with coupling strength ratio \({r}_{VH}={J}_{V}/2{J}_{H}=0.75\), J^H = 0.06Ω and an effective magnetic flux ϕ₁ being  ± π/4 or  ± 3π/4. h, i are the numerical calculations of (f and g). The population information on the two spins (resonators) are encoded in the normalized intensities. Choosing the opposite flux (ϕ₁) is equivalent to selectively exciting the opposite spin. For one spin (d and e), the negative (positive) k states in the first Brillouin zone predominate the upper (lower) bands. Meanwhile, complementary patterns appear in the other spin (f and g), signifying the spin-momentum locking.