Symmetry-protected topological phases and competing orders in a spin-1/2 XXZ ladder with a four-spin interaction

We study a spin-1/2 XXZ model with a four-spin interaction on a two-leg ladder. By means of effective field theory and matrix product state calculations, we obtain rich ground-state phase diagrams that consist of eight distinct gapped phases. Four of them exhibit spontaneous symmetry breaking with either a magnetic or valence-bond-solid (VBS) long-range order. The other four are featureless, i.e., the bulk ground state is unique and does not break any symmetry. The featureless phases include the rung singlet (RS) and Haldane phases as well as their variants, the RS* and Haldane* phases, in which twisted singlet pairs (vertical bar up down arrow > + vertical bar up down arrow >)/root 2 are formed between the two legs. We argue and demonstrate that Gaussian transitions with the central charge c = 1 occur between the featureless phases and between the ordered phases while Ising transitions with c = 1/2 occur between the featureless and ordered phases. The two types of transition lines cross at the SU(2)-symmetric point, where the criticality is described by the SU(2)(2) Wess-Zumino-Witten theory with c = 3/2. The RS-Haldane* and RS*-Haldane transitions give examples of topological phase transitions. Interestingly, the RS* and Haldane* phases, which have highly anisotropic nature, appear even in the vicinity of the isotropic case. We demonstrate that all the four featureless phases are distinguished by topological indices in the presence of certain symmetries.

Automated summary

In this study, a spin-1/2 XXZ model on a two-leg ladder with a four-spin interaction is investigated, revealing eight distinct gapped phases consisting of symmetry-breaking and featureless phases. The presence of Gaussian and Ising transitions between the different phases, as well as the identification of topological phase transitions, highlights the complexity of the system's behavior under certain symmetries.