Thermodynamic signature of topological quantum phase transition in a two-leg Kitaev ladder

We study the topological quantum phase transitions (QPTs) and quantum criticality of a two-leg Kitaev ladder. In the absence of superconducting phase difference, the thermal Drude weight portrays well the gapped topological quantum phases (Dth=0) and gapless quantum criticality (Dth>0), which is further manifested by the critical energy spectrum with different gapless linear relation modes. The self-dual quantum criticality is demonstrated by the Grüneisen ratio (GR) Γ= constant irrelevant of temperature. Tuning the superconducting phase difference Φ=π, the system undergoes a gapped topological nontrivial phase transition into a gapless trivial one, manifested by the quadratic contact point at the center of Brillouin zone. Meanwhile, it is identified by the temperature dependent power-law divergence: Γ∼±T−1 at low temperature, which is similar to the general QPT. In particular, with the chemical potential vanishing, one can obtain full flat bands, giving rise to the discontinuous topological QPT. Furthermore, it is worth noting that the scaling of GR provides a new thermodynamic means to check the order of topological QPTs, which falls on a universal curve or not, indicating the continuous or discontinuous transition.