We introduce the Hessian of the negative ground-state energy as a Riemannian metric on the parametric family of the ground states of a parameterized Hamiltonian of a quantum system and study the critical behavior of the scalar curvature of this new metric. Taking the anisotropic XY chain in a transverse field as an example, we study the critical behaviors of the scalar curvature associated with the quantum phase transitions both numerically and analytically. The behaviors are found to be different from those of the Fubini–Study metric but in agreement with the scalar curvature for the Bogoliubov–Kubo–Mori metric in thermodynamic geometry from the perspective of the universality class. We also briefly discuss the Legendre structure concerning this Hessian metric.
We introduce the Hessian of the negative ground-state energy as a Riemannian metric on the parametric family of the ground states of a parameterized Hamiltonian of a quantum system and study the critical behavior of the scalar curvature of this new metric. Taking the anisotropic XY chain in a transverse field as an example, we study the critical behaviors of the scalar curvature associated with the quantum phase transitions both numerically and analytically. The behaviors are found to be different from those of the Fubini–Study metric but in agreement with the scalar curvature for the Bogoliubov–Kubo–Mori metric in thermodynamic geometry from the perspective of the universality class. We also briefly discuss the Legendre structure concerning this Hessian metric.
Unlike for classical many-body systems, the scalar curvature of the exponential family for quantum many-body systems has been not so investigated, and its physical meaning remains unclear. In this paper, we analytically study the scalar curvature of the space of Gibbs thermal states, belonging to the quantum exponential family, equipped with the Bogoliubov-Kubo-Mori metric for the zero- and one-dimensional transverse-field Ising model at low and high temperatures. We find that these scalar curvatures converge to zero in the high-temperature limit whereas they exponentially diverge approaching zero temperature. This divergence is a consequence of quantumness. Furthermore, if we can reconsider the criticality of the scalar curvatures at zero temperature, they both can be considered to show a critical behavior with an exponent of 1, and this critical exponent is consistent with the quantum-classical correspondence of the Ising model.
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 study the classical analog of the quantum metric tensor and its scalar curvature for two well-known quantum physics models. First, we analyze the geometry of the parameter space for the Dicke model with the aid of the classical and quantum metrics and find that, in the thermodynamic limit, they have the same divergent behavior near the quantum phase transition, as opposed to their corresponding scalar curvatures which are not divergent there. On the contrary, under resonance conditions, both scalar curvatures exhibit a divergence at the critical point. Second, we present the classical and quantum metrics for the Lipkin-Meshkov-Glick model in the thermodynamic limit and find a perfect agreement between them. We also show that the scalar curvature is only defined on one of the system's phases and that it approaches a negative constant value. Finally, we carry out a numerical analysis for the system's finite sizes, which clearly shows the precursors of the quantum phase transition in the metric and its scalar curvature and allows their characterization as functions of the parameters and of the system's size.
This paper introduces the Negative-Curvature Energy Phase (NCEP), a novel theoretical framework that reformulates quantum many-body phase transitions and state classification through the intrinsic geometry of the energy density landscape. By defining a Riemannian metric tensor on the parameter space via the second derivatives of the ground state energy, we construct the associated scalar curvature R_E. We establish that when the scalar curvature satisfies the condition R_E < 0, the quantum system enters a fundamentally distinct energy organization phase. This negative-curvature regime signals an inherent geometric instability, leading to striking physical phenomena such as anomalous energy aggregation, pronounced non-Gaussian fluctuations, and novel critical behaviors. The proposed framework transcends traditional symmetry-breaking paradigms, offering a powerful, model-independent geometric taxonomy for complex quantum states of matter.
We investigate the ground-state Riemannian metric and the cyclic quantum distance of an inhomogeneous quantum spin-1/2 chain in a transverse field. This model can be diagonalized by using a general canonical transformation to the fermionic Hamiltonian mapped from the spin system. The ground-state Riemannian metric is derived exactly on a parameter manifold ring S1, which is introduced by performing a gauge transformation to the spin Hamiltonian through a twist operator. The cyclic ground-state quantum distance and the second derivative of the ground-state energy are studied in different exchange coupling parameter regions. Particularly, we show that, in the case of exchange coupling parameter Ja = Jb, the quantum ferromagnetic phase can be characterized by an invariant quantum distance and this distance will decay to zero rapidly in the paramagnetic phase.
A Riemannian metrical structure of the parameter space has been introduced and investigated for magnetic systems described in the framework of quantum statistics. The introduced metric is based on the conception of the relative information. Two contrasting models have been investigated in detail: the one-dimensional Ising model, with short-range interactions, and the mean-field model of Kac, with long-range interactions. In the second case the metric tensor degenerates. The degeneration has been removed by adding the lattice energy to the original magnetic Hamiltonian. It turns out that in both cases the scalar curvature of parameter space tends toward plus infinity while approaching the critical points. The inverse of the scalar curvature, given by the second and third moments of stochastic variables, has been interpreted as a measure of the stability of the considered magnetic systems. The scalar curvature represents a joint part of fluctuations caused by the interactions of spins.