# Theoretical physicists unveil one of the most

Even though we study them in school, oxidation numbers have so far eluded any rigorous quantum mechanical definition. A new SISSA study, published in *Natural Physics*, reverses this state of affairs by proposing such a definition, based on the theory of topological quantum numbers, which was rewarded with the 2016 Nobel Prize in Physics awarded to Thouless, Haldane and Kosterlitz. This result, combined with recent advances in transport theory made at SISSA, paves the way for accurate, yet tractable, numerical simulation of a broad class of materials important in energy-related technologies and planetary sciences.

Every undergraduate natural science student learns to associate an integer oxidation number with a chemical species participating in a reaction. Unfortunately, the very concept of oxidation state has so far escaped rigorous quantum mechanical definition, so that no method was known until now to calculate oxidation numbers from fundamental laws. of nature, let alone demonstrate that their use in the simulation of load transport does not spoil the quality of numerical simulations. At the same time, the evaluation of electric currents in ionic conductors, necessary to model their transport properties, currently relies on a heavy quantum mechanical approach which strongly limits the feasibility of large-scale computer simulations. Scientists have recently noticed that a simplified model where each atom carries a charge equal to its oxidation number can yield results in surprising good agreement with rigorous but much more expensive approaches. By combining the new topological definition of the oxidation number with the so-called “gauge invariance” of the transport coefficients, recently discovered at SISSA, Federico Grasselli and Stefano Baroni have proven that what was considered a simple coincidence is in fact based on solid theoretical grounds, and that the simple integer charge model captures the electrical transport properties of ionic conductors without any approximation.

In addition to solving a fundamental puzzle in condensed matter physics, this result, obtained as part of the MAX European Center of Excellence for supercomputing applications, also represents a breakthrough for applications, enabling computationally feasible quantum simulations charge transport in ionic systems of paramount importance in energy-related technologies, in the automotive and telecommunications sectors, as well as in planetary sciences. These applications range from ionic mixtures adopted in electrolytic cells and heat exchangers of power plants, to solid electrolyte batteries for electric cars and electronic devices, and even to the exotic conductive phases of water occurring indoors. ice giants, which are believed to be related to the origin of the magnetic fields in these planets.

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