Oscillating fidelity susceptibility near a quantum multicritical point

Abstract

We study scaling behavior of the geometric tensor ${\ensuremath{\chi}}_{\ensuremath{\alpha},\ensuremath{\beta}}({\ensuremath{\lambda}}_{1},{\ensuremath{\lambda}}_{2})$ and the fidelity susceptibility ${\ensuremath{\chi}}_{F}$ in the vicinity of a quantum multicritical point (MCP) using the example of a transverse $\mathit{XY}$ model. We show that the behavior of the geometric tensor (and thus of ${\ensuremath{\chi}}_{F}$) is drastically different from that seen near a critical point. In particular, we find that it is a highly nonmonotonic function of $\ensuremath{\lambda}$ along the generic direction ${\ensuremath{\lambda}}_{1}~{\ensuremath{\lambda}}_{2}=\ensuremath{\lambda}$ when the system size $L$ is bounded by the shorter and longer correlation lengths characterizing the MCP: $1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{1}}\ensuremath{\ll}L\ensuremath{\ll}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, where ${\ensuremath{\nu}}_{1}<{\ensuremath{\nu}}_{2}$ are the two correlation-length exponents characterizing the system. We find that the scaling of the maxima of the components of ${\ensuremath{\chi}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ is associated with the emergence of quasicritical points at $\ensuremath{\lambda}~1/{L}^{1/{\ensuremath{\nu}}_{1}}$, related to the proximity to the critical line of the finite-momentum anisotropic transition. This scaling is different from that in the thermodynamic limit $L\ensuremath{\gg}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, 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 steplike behavior for small quench amplitudes. A study of heat density and diagonal entropy density also shows signatures of quasicritical points.