Symmetry-protected topological phases in spin ladders with two-body interactions

Spin-1/2 two-legged ladders respecting interleg exchange symmetry sigma and spin rotation symmetry D-2 have new symmetry-protected topological (SPT) phases which are different from the Haldane phase. Three of the new SPT phases are t(x),t(y),t(z), which all have symmetry-protected twofold degenerate edge states on each end of an open chain. However, the edge states in different phases have different responses to magnetic field. For example, the edge states in the tz phase will be split by the magnetic field along the z direction, but not by the fields in the x and y directions. We give the Hamiltonian that realizes each SPT phase and demonstrate a proof-of-principle quantum simulation scheme for Hamiltonians of the t(0) and t(z) phases based on the coupled-QED-cavity ladder.