This paper examines the averaged quantum metric for multiparametric random-matrix Hamiltonians in the vicinity of an integrable point, revealing a local conical geometry that arises from specific spectral conditions. The central object is the eigenstate-averaged quantum geometric tensor, which defines a Riemannian metric on the parameter space of the Hamiltonian family. For the two-parameter model studied by Sharipov and colleagues, the averaged metric components admit the asymptotic expansion Grr∼a/rGrr∼a/r and Gϕϕ∼brGϕϕ∼br as the integrable point is approached, with aa and bb positive. By embedding this surface of revolution into Euclidean three-space, the author demonstrates that the metric is locally isometric to a right circular cone whose opening angle satisfies tanα=b/atanα=b/a. Thus, the ratio of the two leading coefficients determines the cone's aperture, linking geometry directly to the relative weight of radial and angular perturbation matrix elements at low frequencies. The paper then turns to the spectral origin of this conical singularity. Using the spectral representation of the metric, the radial component GrrGrr is expressed as an integral over the spectral density of squared matrix elements divided by ω2ω2. The divergence 1/r1/r requires that this density retains finite weight at frequencies on the order of the mean level spacing, a condition characteristic of Poissonian spectra where level repulsion is absent. However, if a hard spectral gap ΔΔ is introduced in the matrix-element spectrum, the integral acquires a lower cutoff, yielding a bound Grr≤C/Δ2Grr≤C/Δ2 that removes the singularity entirely. This regularization mirrors classical mechanisms in which spectral gaps suppress low-order resonances and confine instability. Between the conical near-integrable regime and the hemispherical geometry of the fully ergodic phase, an intermediate scaling window is identified with the non-ergodic extended phase, where multifractal estimates suggest a smooth interpolation. The discussion draws structural parallels to classical theories of thin versus thick chaos, where low-frequency spectral weight determines whether local instability remains confined or becomes transport-effective. However, the author emphasizes that the quantum calculation only establishes kinematic consistency, not a quantitative dictionary with classical measures of confinement. Two propositions are rigorously derived: the local conical embedding follows directly from the metric expansion, and a hard spectral gap bounds the radial component. All conclusions are strictly limited to the specific ensembles studied, and the paper makes no claim of universality beyond those models.
This paper examines the averaged quantum metric for multiparametric random-matrix Hamiltonians in the vicinity of an integrable point, revealing a local conical geometry that arises from specific spectral conditions. The central object is the eigenstate-averaged quantum geometric tensor, which defines a Riemannian metric on the parameter space of the Hamiltonian family. For the two-parameter model studied by Sharipov and colleagues, the averaged metric components admit the asymptotic expansion Grr∼a/rGrr∼a/r and Gϕϕ∼brGϕϕ∼br as the integrable point is approached, with aa and bb positive. By embedding this surface of revolution into Euclidean three-space, the author demonstrates that the metric is locally isometric to a right circular cone whose opening angle satisfies tanα=b/atanα=b/a. Thus, the ratio of the two leading coefficients determines the cone's aperture, linking geometry directly to the relative weight of radial and angular perturbation matrix elements at low frequencies. The paper then turns to the spectral origin of this conical singularity. Using the spectral representation of the metric, the radial component GrrGrr is expressed as an integral over the spectral density of squared matrix elements divided by ω2ω2. The divergence 1/r1/r requires that this density retains finite weight at frequencies on the order of the mean level spacing, a condition characteristic of Poissonian spectra where level repulsion is absent. However, if a hard spectral gap ΔΔ is introduced in the matrix-element spectrum, the integral acquires a lower cutoff, yielding a bound Grr≤C/Δ2Grr≤C/Δ2 that removes the singularity entirely. This regularization mirrors classical mechanisms in which spectral gaps suppress low-order resonances and confine instability. Between the conical near-integrable regime and the hemispherical geometry of the fully ergodic phase, an intermediate scaling window is identified with the non-ergodic extended phase, where multifractal estimates suggest a smooth interpolation. The discussion draws structural parallels to classical theories of thin versus thick chaos, where low-frequency spectral weight determines whether local instability remains confined or becomes transport-effective. However, the author emphasizes that the quantum calculation only establishes kinematic consistency, not a quantitative dictionary with classical measures of confinement. Two propositions are rigorously derived: the local conical embedding follows directly from the metric expansion, and a hard spectral gap bounds the radial component. All conclusions are strictly limited to the specific ensembles studied, and the paper makes no claim of universality beyond those models.
The quantum geometric tensor (QGT) characterizes the Hilbert space geometry of the eigenstates of a parameter-dependent Hamiltonian. In recent years, the QGT and related quantities have found extensive theoretical and experimental utility, in particular for quantifying quantum phase transitions both at and out of equilibrium. Here, we consider the symmetric part (quantum Riemannian metric) of the QGT for various multiparametric random matrix Hamiltonians and discuss the possible indication of ergodic or integrable behavior. We found for a two-dimensional parameter space that, while the ergodic phase corresponds to the smooth manifold, the integrable limit marks itself as a singular geometry with a conical defect. Our study thus provides more support for the idea that the landscape of the parameter space yields information on the ergodic-nonergodic transition in complex quantum systems, including the intermediate phase.
We consider the geometry of quantum states associated with random matrix Hamiltonians belonging to ensembles that exhibit an integrable-to-chaotic transition in terms of the nearest-neighbour energy level spacing distribution, focusing on the β-Gaussian ensembles with generic Dyson index. For the tridiagonal Gaussian β-ensemble, which shows chaos-to-integrability transition with varying Dyson index, we first calculate an analytical expression for the ensemble-averaged fidelity susceptibility for a two-by-two matrix representation of the Hamiltonian for generic values of the Dyson index, and show that it diverges as the total Hamiltonian, which is the sum of a diagonal matrix with independent elements and a tridiagonal β-matrix, goes over to the integrable phase. Next, for large-dimensional matrices, we numerically compute the fidelity susceptibility and the quantum metric tensor, respectively, for a one-parameter and a two-parameter class of Hamiltonians that we construct from the β-ensemble, and find scaling relations of these quantities for chaotic and integrable phases. We also consider variations of the β-ensemble, such as the one that preserves the rotational invariance of the ensemble, and compute the relevant ensemble-averaged fidelity susceptibility to show that it has similar features as the tridiagonal ensemble in the integrable as well as the chaotic phase, thereby establishing the universality of these properties. Finally, the presence of non-vanishing non-diagonal elements, which arises due to the rotational non-invariance of the β-ensembles, is a specific feature of the corresponding metric tensor, and we use this to illuminate the difference between the quantum state space geometry of these ensembles and the rotationally invariant ones, such as the classical Gaussian ensembles.
A classical particle under spatial constraints is strictly confined to live on a specific space manifold or path, but this assumption is incompatible with the zero-point fluctuations of a quantum particle. One way to describe quantum mechanics under constraints is the confinement potential approach (CPA). For a non-relativistic particle, the CPA maps the problem onto the solution of a Schrödinger-type equation in an isometrically embedded Riemannian submanifold of Euclidean space while the motion along orthogonal directions are decoupled and spatially confined. This approach respects quantum uncertainty, and one of its key results is the appearance of geometry- and metric-induced potentials that affect the stationary states and the dynamics of the particle. For particles constrained to different spaces, such as structures hosting sharp bents, vertices, wedges, conical apices, tips, or self-intersections, a formalism beyond the CPA is needed. Here, a step towards a CPA extension for irregular spaces is presented. After classifying the possible geometric irregularities concerning the CPA formalism, the presentation is focused on a sharply bent quantum wire modeled as an embedded curve with singular (but absolute integrable) curvature. For a subclass fulfilling the additional requirement that the geometric potential is a distribution of first order, a solution scheme for the confined Schrödinger equation is presented based on singular Sturm-Liouville theory and operator theoretic methods. The analytical considerations and numerical simulations evidence the existence of curvature-induced bound states with non-differentiable wave functions localized around the singular point, with an extension well beyond the singularity. Furthermore, a multitude of scattering states appear that may affect the transport and optical properties of the system.
We elucidate the Bures metric in quantum state space near a rank-changing point of the density matrix and show contrasting behavior for two-level (N=2) systems versus higher-level systems. Due to the smooth pure-state boundary for N=2, we prove the apparent metric divergences to be merely coordinate artifacts and present three Lindblad processes exhibiting qualitatively different evolution near rank-changing points, showing geodesic approach, power-law scaling, and pure-state escape law. For higher-dimensional (N≥3) systems, the geometry near a rank-changing point differs fundamentally. Under suitable restrictions of the density matrix and its approach towards a pure state, the Bures metric reduces to a conical metric with the pure state at the cone tip. Such a conic geometry leads to genuine curvature singularities: A two-dimensional cone exhibits a Dirac delta-function curvature near the tip while a higher-dimensional cone shows a power-law divergence of the curvature towards the cone tip. A construction of Lindblad evolution for N=3 systems with conic singularities is presented, along with possible implications for future experimental and theoretical research.
We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence ∼∣εμ∣−2, while the Berry curvature shows a weaker, subleading divergence ∼∣εμ∣−3/2, with εμ denoting the parameter displacement from the EP along an individual parameter axis μ. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.
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 XY 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.