Generalized Fidelity Susceptibilities as Applied to the $J_1-J_2$ Heisenberg Chain

In this talk slightly generalized quantum fidelity susceptibilities for the antiferromagnetic Heisenberg $J_1-J_2$ chain will be introduced. The differential change in these fidelities differ from the typical fidelity in that they are measured with respect to a term other than the one used for driving the system towards a quantum phase transition. We study three fidelity susceptibilities; $\chi_{\rho}$, $\chi_D$ and $\chi_{AF}$, which are related to the spin stiffness, the dimer order and antiferromagnetic order, respectively. I will discuss how these quantities can accurately identify the quantum critical point at $J_2$=0.241167$J_1$ in this model. This phase transition, being in the Berezinskii-Kosterlitz-Thouless universality class, is controlled by a marginal operator and is therefore particularly difficult to observe. In addition more recent work on the anisotropic Heisenberg triangular model will be discussed.