Staircase to higher-order topological phase transitions

We find a series of topological phase transitions of increasing order, beyond the more standard second-order phase transition in a one-dimensional topological superconductor. The jumps in the order of the transitions depend on the range of the pairing interaction, which is parametrized by an algebraic decay with exponent $\alpha$. Remarkably, in the limit $\alpha =1$ the order of the topological transition becomes infinite. We compute the critical exponents for the series of higher-order transitions in exact form and find that they fulfill the hyperscaling relation. We also study the critical behavior at the boundary of the system and discuss potential experimental platforms of magnetic atoms in superconductors.

Figure 1

(a) Second derivative of the bulk thermodynamic potential, obtained from Eq. (5) for the LRKC on the $\mu /t\text{−}\alpha$ plane. The dashed line indicates the value of $\alpha$ below which our numerical results are not accurate anymore. (b) Staircase of higher-order topological phase transition at $\mu /t=-2$, indicating how the order of the transition increases as one approaches $\alpha =1$.

Figure 2

First derivative of the boundary thermodynamic potential as a function of $\mu /t$, given by Eq. (6). The dashed line indicates the value of $\alpha$ below which our numerical results are not accurate anymore. Although the topological phase transition at $\mu /t=2$ remains of first order in the entire regime of $\alpha$, in the thermodynamic limit the phase transition at $\mu /t=-2$ terminates at a quantum critical point located at $\alpha =1$. However, numerical limitations prevent us from going below $\alpha =3/2$.