Oscillating fidelity susceptibility near a quantum multicritical point

We study scaling behavior of the geometric tensor $χ_{α,β}(λ_1,λ_2)$ and the fidelity susceptibility $(χ_{\rm F})$ 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 $χ_{\rm F}$) is drastically different from that seen near a critical point. In particular, we find that is highly non-monotonic function of $λ$ along the generic direction $λ_1\simλ_2 = λ$ when the system size $L$ is bounded between the shorter and longer correlation lengths characterizing the MCP: $1/|λ|^{ν_1}\ll L\ll 1/|λ|^{ν_2}$, where $ν_1<ν_2$ are the two correlation length exponents characterizing the system. We find that the scaling of the maxima of the components of $χ_{αβ}$ is associated with emergence of quasi-critical points at $λ\sim 1/L^{1/ν_1}$, related to the proximity to the critical line of finite momentum anisotropic transition. This scaling is different from that in the thermodynamic limit $L\gg 1/|λ|^{ν_2}$, 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.

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