Numerical evidence for continuous transition between Ising ferromagnet and valence bond solid in one dimension

We perform a numerical study of a spin-1/2 model with Z2 x Z2 symmetry in one dimension which demonstrates an interesting similarity to the physics of two-dimensional deconfined quantum critical points (DQCP). Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working directly in the thermodynamic limit using uniform matrix product states, we find evidence for a direct continuous phase transition which lies outside of the Landau-Ginzburg paradigm. In our model, the continuous transition is found everywhere on the phase boundary. We find that the magnetic and VBS correlations show identical power law exponent, which is expected from the self-duality of the parton description of this DQCP. Critical exponents vary continuously along the phase boundary and are in agreement with predictions of the field theory for this transition. We also find a regime where the phase boundary splits with the appearance of an intermediate phase of coexisting ferromagnetic and VBS order parameters, as suggested by the theory.