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Quantum Criticality as Geometric Singularity: Universal Fidelity Scaling, Temporal Metrics in Discrete Time Crystals, and Experimental Tests of Information-Driven Geometry

Francis Procaccia

Published Feb 14, 2026• 0 citations
Publisher

Abstract

We demonstrate that quantum information geometry—specifically the fidelity susceptibility metric on parameter manifolds—exhibits finite-size scaling and curvature singularities at quantum critical points that are structurally identical to geometric response equations, and we propose that Floquet discrete time crystals (DTCs) extend this framework to include temporal metric components whose divergence at DTC melting transitions can be experimentally tested as a probe of emergent spacetime structure in driven quantum matter.

DOI
10.5281/zenodo.18646241

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We demonstrate that quantum information geometry—specifically the fidelity susceptibility metric on parameter manifolds—exhibits finite-size scaling and curvature singularities at quantum critical points that are structurally identical to geometric response equations, and we propose that Floquet discrete time crystals (DTCs) extend this framework to include temporal metric components whose divergence at DTC melting transitions can be experimentally tested as a probe of emergent spacetime structure in driven quantum matter.
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 nnn adjacent sites from the ground state of spin chains of finite length NNN, 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-1/21/21/2 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 NNN: 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 NNN if the subsystem size, nnn, 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→∞N \rightarrow \inftyN→∞.
From the Aharonov-Bohm effect to general relativity, geometry plays a central role in modern physics. In quantum mechanics, many physical processes depend on the Berry curvature. However, recent advances in quantum information theory have highlighted the role of its symmetric counterpart, the quantum metric tensor. In this paper, we perform a detailed analysis of the ground state Riemannian geometry induced by the metric tensor, using the quantum XYXYXY chain in a transverse field as our primary example. We focus on a particular geometric invariant, the Gaussian curvature, and show how both integrals of the curvature within a given phase and singularities of the curvature near phase transitions are protected by critical scaling theory. For cases where the curvature is integrable, we show that the integrated curvature provides a new geometric invariant, which like the Chern number characterizes individual phases of matter. For cases where the curvature is singular, we classify three types (integrable, conical, and curvature singularities) and detail situations where each type of singularity should arise. Finally, to connect this abstract geometry to experiment, we discuss three different methods for measuring the metric tensor: via integrating a properly weighted noise spectral function or by using leading-order responses of the work distribution to ramps and quenches in quantum many-body systems.
This paper introduces a novel framework for analog gravity in which an effective (2+1)-dimensional Lorentzian spacetime emerges dynamically on the information geometry of a driven open quantum system. Unlike traditional fluid-based analogs, the spatial fabric is provided by the Quantum Fisher Information Metric (QFIM) over a control-parameter manifold, while laboratory time serves as the temporal axis. Using Keldysh nonequilibrium field theory, the study demonstrates that the Lorentzian signature and thus the existence of causal light cones-originates fundamentally from the thermodynamic passivity of the quantum state. This profound connection makes the causal structure a tunable, falsifiable property: driving the system into population inversion triggers a signature-breakdown transition that destroys the hyperbolicity of the emergent spacetime. Supported by exact derivations for a two-qubit model and concrete experimental protocols involving continuous homodyne detection and FPGA-based feedback, the research offers a fully measurable laboratory platform to explore the intersection of quantum information geometry, nonequilibrium thermodynamics, and emergent causal structure. By Gökay Yıldız gokay@krekpot.com
Dirac exceptional points (EPs) represent a novel class of non-Hermitian singularities that, unlike conventional EPs, reside entirely within the parity-time unbroken phase and exhibit linear energy dispersion. Here, we theoretically investigate the quantum geometry of Dirac EPs realized in nitrogen-vacancy centers in diamond, utilizing fidelity susceptibility as a probe. We demonstrate that despite the absence of a symmetry-breaking phase transition, the Dirac EP induces a pronounced geometric singularity, confirming the validity of the fidelity in characterizing non-Hermitian EPs. Specifically, the real part of the fidelity susceptibility diverges to negative infinity, which serves as a signature of non-Hermitian criticality. Crucially, however, we reveal that this divergence exhibits a distinct anisotropy, diverging along the non-reciprocal coupling direction while remaining finite along the detuning axis. Furthermore, we establish that this anisotropy, characterized by at least one exact dark direction coexisting with divergent directions, is a generic consequence of the Dirac EP structure whenever the parameter derivatives collectively span the off-diagonal operator space at the Dirac EP. This behavior stands in stark contrast to the omnidirectional divergence observed in conventional EPs. Our findings provide a comprehensive picture of the fidelity probe near the Dirac EP, highlighting the critical role of parameter directionality in exploiting Dirac EPs for quantum control and sensing applications.