is it possible to modify the abbrvnat.bst style from the natbib package so it underlines automatically the doi number in the generated bibliography?
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The abbrvnat bibliography style uses \doi for formatting the contents of doi fields. It is possible to write a custom definition for \doi. The example below also loads ulem which defines \uline for underlining text. The command works in some situations where \underline fails.

\documentclass{article}
\usepackage{natbib}
\usepackage[normalem]{ulem}% \uline
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@article{Rutledge19082014,
author = {Rutledge, Robb B. and Skandali, Nikolina and Dayan, Peter and
Dolan, Raymond J.},
title = {A computational and neural model of momentary subjective well-being},
volume = {111},
number = {33},
pages = {12252--12257},
year = {2014},
doi = {10.1073/pnas.1407535111},
journal = {Proceedings of the National Academy of Sciences},
}
\end{filecontents}
\newcommand{\doi}[1]{doi: \uline{#1}}
\begin{document}
\bibliographystyle{abbrvnat}
\nocite{*}
\bibliography{\jobname}
\end{document}