Metric response of relative entropy: A universal indicator of quantum criticality

Abstract

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. We study two spin-$1/2$ models as examples, namely the integrable transverse field Ising model (TFIM) and a non-integrable Ising chain with three-spin interactions. We demonstrate distinct scaling behaviors for the peak of the QRE susceptibility as a function of $N$: namely a square logarithmic divergence in TFIM and a power-law divergence in the non-integrable chain. This susceptibility encodes uncertainty of entanglement Hamiltonian gradients and is also directly connected to other information measures such as Petz-Rényi entropies. 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 RDMs being finite; unlike the divergence associated with the QCPs which require $N\to \infty$.

v2; 30 pages, 8 figures; this version is close to the published one and supersedes an earlier submission - arXiv:2412.02236

References in corpus (9)

• Quantum critical scaling of the geometric tensors
• QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems part I: spin chains
• Measuring entanglement entropy of a generic many-body system with a quantum switch
• Single-Site Entanglement of Fermions at a Quantum Phase Transition
• Quenching a Quantum Critical State by the Order Parameter: Dynamical Quantum Phase Transitions and Quantum Speed Limits
• Local Measures of Entanglement and Critical Exponents at Quantum Phase Transitions
• Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula
• Weak universality, quantum many-body scars and anomalous infinite-temperature autocorrelations in a one-dimensional spin model with duality
• Integral formula for quantum relative entropy implies data processing inequality