The riemannian structure on the states of quantum-like systems

Jean-Pierre Provost, Gérard Vallée

Pablo Páez-Velasco

Tensor networks provide a powerful framework for efficiently representing high-dimensional data and many-body quantum states. Endowing tensor networks with a Riemannian manifold structure provides a natural setting for numerical optimization and analysis. A central feature of tensor networks is their gauge freedom, whose characterisation (captured by so-called fundamental theorems) underlies both their intrinsic structure and the design of numerical algorithms. In this work, we study the interaction between the Riemannian manifold structure and the gauge freedom for several families of tensor networks. Using group actions and Riemannian submersions, we establish a Riemannian fundamental theorem for the tensor network families studied.

N. S. Tonchev

The proposed in J. Math. Phys. v.57,071903 (2016) analytical expansion of monotone (contractive) Riemannian metrics (called also quantum Fisher information(s)) in terms of moments of the dynamical structure factor (DSF) relative to an original intensive observable is reconsidered and extended. The new approach through the DSF which characterizes fully the set of monotone Riemannian metrics on the space of Gibbs thermal states is utilized to obtain an extension of the spectral presentation obtained for the Bogoliubov-Kubo-Mori metric (the generalized isothermal susceptibility) on the entire class of monotone Riemannian metrics. The obtained spectral presentation is the main point of our consideration. The last allows to present the one to one correspondence between monotone Riemannian metrics and operator monotone functions (which is a statement of the Petz theorem in the quantum information theory) in terms of the linear response theory. We show that monotone Riemannian metrics can be determined from the analysis of the infinite chain of equations of motion of the retarded Green's functions. Inequalities between the different metrics have been obtained as well. It is a demonstration that the analysis of information-theoretic problems has benefited from concepts of statistical mechanics and might cross-fertilize or extend both directions, and vice versa. We illustrate the presented approach on the calculation of the entire class of monotone (contractive) Riemannian metrics on the examples of some simple but instructive systems employed in various physical problems.

Эрнест Ахмеджанович Тагиров

Bruno Mera

The localization tensor is a measure of distinguishability between insulators and metals. This tensor is related to the quantum metric tensor associated with the occupied bands in momentum space. In two dimensions and in the thermodynamic limit, it defines a flat Riemannian metric over the twist-angle space, topologically a torus, which endows this space with a complex structure, described by a complex parameter $\ensuremath{\tau}$. It is shown that the latter is a physical observable related to the anisotropy of the system. The quantity $\ensuremath{\tau}$ and the Riemannian volume of the twist-angle space provide an invariant way to parametrize the flat quantum metric obtained in the thermodynamic limit. Moreover, if, by changing the couplings of the theory, the system undergoes quantum phase transitions in which the gap closes, the complex structure $\ensuremath{\tau}$ is still well defined, although the metric diverges (metallic state), and it is fixed by the form of the Hamiltonian near the gap closing points. The Riemannian volume is responsible for the divergence of the metric at the phase transition.

Julian Schwinger