The quantum geometric tensor is a fundamental property of quantum states, with broad applications in condensed matter physics, topological phases, and quantum phase transitions. The eigenvalues characterize the scale, anisotropy, and effective rank of random quantum geometry, going beyond scalar quantities such as the trace. Here we study the eigenvalue statistics of the quantum geometric tensor in finite-dimensional parameter-dependent random Hamiltonians. We obtain exact analytical results for the first two nontrivial cases,
N=2 and
N=3, with
N=3 already showing genuine shape fluctuations. We further propose a finite-
N, arbitrary-
D description of QGT eigenvalue statistics and verify it by numerical simulations. Our results provide exact benchmarks and a practical framework for random quantum geometry in finite-dimensional disordered and chaotic systems.
This manuscript explores the Riemannian geometry of quantum state manifolds and its physical implications. We establish that quantum state manifolds, as submanifolds of Hilbert space, possess intrinsic Riemannian geometry characterized by a metric tensor derived from the Hilbert space inner product. This geometric framework reveals fundamental connections: the distance between nearby states scales with quantum fluctuations, curvature indicates state robustness, and geometric phase transitions occur in many-body systems. We review historical developments from early quantum geometry to modern quantum information geometry, and present methodologies for computing geometric quantities and their physical interpretations. Results include distance-fluctuation relations for coherent states, curvature calculations for SU(2) manifolds, and applications to quantum optimization and metrology via the quantum Fisher information metric. The discussion interprets geometric quantities in terms of uncertainty principles, state robustness, and quantum criticality, while outlining future research directions in quantum chaos-geometry correlations, non-associative geometry, and geometric quantum algorithm design. We conclude that Riemannian geometry provides a powerful framework for understanding and exploiting the structure of quantum state space, with applications across quantum information processing, metrology, and foundational physics.
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 geometry of multiparameter families of quantum states is important in numerous contexts, including adiabatic or nonadiabatic quantum dynamics, quantum quenches, and the characterization of quantum critical points. Here, we discuss the Hilbert space geometry of eigenstates of parameter-dependent random matrix ensembles, deriving the full probability distribution of the quantum geometric tensor for the Gaussian unitary ensemble. Our analytical results give the exact joint distribution function of the Fubini-Study metric and the Berry curvature. We discuss relations to Levy stable distributions and compare our results to numerical simulations of random matrix ensembles as well as electrons in a random magnetic field.
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.
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.