We discuss pycnonuclear burning of highly exotic atomic nuclei in deep crusts of neutron stars, at densities up to 1e13 g/cc. As an application, we consider pycnonuclear burning of matter accreted on a neutron star in a soft X-ray transient (SXT, a compact binary containing a neutron star and a low-mass companion). The energy released in this burning, while the matter sinks into the stellar crust under the weight of newly accreted material, is sufficient to warm up the star and initiate neutrino emission in its core. The surface thermal radiation of the star in quiescent states becomes dependent of poorly known equation of state (EOS) of supranuclear matter in the stellar core, which gives a method to explore this EOS. Four qualitatively different model EOSs are tested against observations of SXTs. They imply different levels of the enhancement of neutrino emission in massive neutron stars by (1) the direct Urca process in nucleon/hyperon matter; (2) pion condensates; (3) kaon condensates; (4) Cooper pairing of neutrons in nucleon matter with the forbidden direct Urca process. A low level of the thermal quiescent emission of two SXTs, SAX J1808.4-3658 and Cen X-4, contradicts model (4). Observations of SXTs test the same physics of dense matter as observations of thermal radiation from cooling isolated neutron stars, but the data on SXTs are currently more conclusive.
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