Oscillating fidelity susceptibility near a quantum multicritical point

We study scaling behavior of the geometric tensor ${\chi }_{\alpha ,\beta }({\lambda }_1,{\lambda }_2)$ and the fidelity susceptibility $({\chi }_{\mathrm{F}})$ in the vicinity of a quantum multicritical point (MCP) using the example of a transverse XY model. We show that the behavior of the geometric tensor (and thus of ${\chi }_{\mathrm{F}}$) is drastically different from that seen near a critical point. In particular, we find that is highly non-monotonic function of $\lambda$ along the generic direction ${\lambda }_1\sim {\lambda }_2=\lambda$ when the system size $L$ is bounded between the shorter and longer correlation lengths characterizing the MCP: $1\mathrm{/}\mathrm{\mid }\lambda {\mathrm{\mid }}^{{\nu }_1}\ll L\ll 1\mathrm{/}\mathrm{\mid }\lambda {\mathrm{\mid }}^{{\nu }_2}$, where ${\nu }_1<{\nu }_2$ are the two correlation length exponents characterizing the system. We find that the scaling of the maxima of the components of ${\chi }_{\alpha \beta }$ is associated with emergence of quasi-critical points at $\lambda \sim 1\mathrm{/}{L}^{1\mathrm{/}{\nu }_1}$, related to the proximity to the critical line of finite momentum anisotropic transition. This scaling is different from that in the thermodynamic limit $L\gg 1/|\lambda|^{\nu_2}$, which is determined by the conventional critical exponents. We use our results to calculate the defect density following a rapid quench starting from the MCP and show that it exerts a step-like behavior for small quench amplitudes. Study of heat density and diagonal entropy density also show signatures of quasi-critical points.