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.
The complete quantum metric of a parametrized quantum system has a real part (usually known as the Provost-Vallee metric) and a symplectic imaginary part (known as the Berry curvature). In this paper, we first investigate the relation between the Riemann curvature tensor of the space described by the metric, and the Berry curvature, by explicit parallel transport of a vector in Hilbert space. Subsequently, we write a generating function from which the complex metric, as well as higher-order geometric tensors (affine connection, Riemann curvature tensor), can be obtained in terms of gauge-invariant cumulants. The generating function explicitly relates the quantities which characterize the geometry of the parameter space to quantum fluctuations. We also show that for a mixed quantum-classical system both real and imaginary parts of the quantum metric contribute to the dynamics, if the mass tensor is Hermitian. A many-operator generalization of the uncertainty principle results from taking the determinant of the complex quantum metric. We also calculate the quantum metric for a number of Lie group coherent states, including several representations of the
SU(1,1) group. In our examples nontrivial complex geometry results for generalized coherent states. A pair of oscillator states corresponding to the
SU(1,1) group gives a double series for its spectrum. The two minimal uncertainty coherent states show trivial geometry, but, again, for generalized coherent states nontrivial geometry results.
The quantum geometric tensor sets the local relation between variations of external parameters and distances between states in Hilbert space. Its real and imaginary parts are, respectively, the quantum Fisher information matrix, which determines the ultimate precision bound in parameter estimation, and the Berry curvature. We study the geometry of SU(1,1) Hamiltonian eigenstates and propose two dynamical methods for extracting the components of the SU(1,1) quantum geometric tensor. A time-separated quench method extracts both the real and imaginary parts of the quantum geometric tensor from transition probabilities following two sudden parameter changes separated by free evolution, while a weak resonant modulation method relates quasiparticle production to the same geometric tensor. These results provide an operational method for measuring the local geometry of single-mode squeezed states and, in particular, for extracting the quantum Fisher information matrix directly from experimental observables.
The geometric properties of quantum states are fully encoded by the quantum geometric tensor. The real and imaginary parts of the quantum geometric tensor are the quantum metric and Berry curvature, which characterize the distance and phase difference between two nearby quantum states in Hilbert space, respectively. For conventional Hermitian quantum systems, the quantum metric corresponds to the fidelity susceptibility and has already been used to specify quantum phase transitions from the geometric perspective. In this paper, we extend this wisdom to the non-Hermitian systems for revealing non-Hermitian critical points. To be concrete, by employing numerical exact diagonalization and analytical methods, we calculate the quantum metric and corresponding order parameters in various non-Hermitian models, which include two non-Hermitian generalized Aubry-Andr\'e models and non-Hermitian cluster and mixed-field Ising models. We demonstrate that the quantum metric of eigenstates in these non-Hermitian models exactly identifies the localization transitions, mobility edges, and many-body quantum phase transitions with gap closings, respectively. We further show that this strategy is robust against the finite-size effect and different boundary conditions.
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.