The information-geometric origin of fidelity susceptibility and its utility as a universal probe of quantum criticality in many-body settings have been widely discussed. Here, we explore the metric response of quantum relative entropy (QRE), by tracing out all but $n$ adjacent sites from the ground state of spin chains of finite length $N$ as a parameter of the corresponding Hamiltonian is varied. The diagonal component of this metric defines a susceptibility of the QRE that diverges at quantum critical points (QCPs) in the thermodynamic limit, even for small $n$. We study two spin-$1/2$ models as examples, namely, the integrable transverse field Ising model (TFIM) and a nonintegrable Ising chain with three-spin interactions. We demonstrate distinct scaling behaviors for the peak of the QRE susceptibility as a function of $N$ using $n\le 3$: namely, a square logarithmic divergence in TFIM and a power-law divergence in the nonintegrable chain. We further show that this susceptibility diverges even at finite $N$ if the subsystem size, $n$, exceeds a certain value when the Hamiltonian is tuned to its classical limits due to the rank of the corresponding reduced density matrices being finite; unlike the divergence associated with the QCPs, which requires $N\to \infty$.

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• Quantum phase transitions
• Density matrix methods
• Quantum spin chains