# Topological quantization and gauge invariance of charge transport in liquid insulators

## Abstract

According to the Green–Kubo theory of linear response, the conductivity of an electronically gapped liquid can be expressed in terms of the time correlations of the adiabatic charge flux, which is determined by the atomic velocities and Born effective charges. We show that topological quantization of adiabatic charge transport and gauge invariance of transport coefficients allow one to rigorously express the electrical conductivity of an insulating fluid in terms of integer-valued, scalar, and time-independent atomic oxidation numbers, instead of real-valued, tensor and time-dependent Born charges.

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The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We are grateful to R. Resta for insightful discussions and to R. Bertossa for technical assistance. This work was partially funded by the EU through the max Centre of Excellence for supercomputing applications (project nos. 676598 and 824143).

## Author information

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### Contributions

Both authors contributed to all aspects of this work.

### Corresponding author

Correspondence to Stefano Baroni.

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Grasselli, F., Baroni, S. Topological quantization and gauge invariance of charge transport in liquid insulators. Nat. Phys. 15, 967–972 (2019). https://doi.org/10.1038/s41567-019-0562-0

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