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Fidelity susceptibility and general quench near an anisotropic quantum critical point

We study the scaling behavior of fidelity susceptibility density $(χ_{\rm f})$ at or close to an anisotropic quantum critical point characterized by two different correlation length exponents $ν_{||}$ and $ν_{\bot}$ along parallel and perpendicular spatial directions, respectively. Our studies show that the response of the system due to a small change in the Hamiltonian near an anisotropic quantum critical point is different from that seen near an isotropic quantum critical point. In particular, for a finite system with linear dimension $L_{||}$ ($L_{\bot}$) in the parallel (perpendicular) directions, the maximum value of $χ_{\rm f}$ is found to increases in a power-law fashion with $L_{||}$ for small $L_{||}$, with an exponent depending on both $ν_{||}$ and $ν_{\bot}$ and eventually crosses over to a scaling with $L_{\bot}$ for $L_{||}^{1/ν_{||}} \gtrsim L_{\bot}^{1/ν_{\bot}}$. We also propose scaling relations of heat density and defect density generated following a quench starting from an anisotropic quantum critical point and connect them to a generalized fidelity susceptibility. These predictions are verified exactly both analytically and numerically taking the example of a Hamiltonian showing a semi-Dirac band-crossing point.