  The existence of a phase transition between two distinct liquid phases in
single-component network-forming liquids (e.g., water, silica, silicon) has
elicited considerable scientific interest. The challenge, both for experiments
and simulations, is that the liquid-liquid phase transition occurs under deeply
supercooled conditions, where crystallization occurs very rapidly. Thus, early
evidence from numerical equation of state studies was challenged, with the
argument that slow spontaneous crystallization had been misinterpreted as
evidence of a second liquid state. Rigorous free energy calculations have
subsequently confirmed the existence of a liquid-liquid phase transition in
some models of water, and exciting new experimental evidence has since
supported these computational results. Similar results have so far not been
found for silicon. Here, we present results from free energy calculations
performed for silicon modelled with the classical, empirical Stillinger-Weber
potential. Through a careful study employing state-of-the-art constrained
simulation protocols and numerous checks for thermodynamic consistency, we find
that there are two distinct metastable liquid states and a phase transition.
Our results resolve a long-standing debate concerning the existence of a
liquid-liquid transition in supercooled liquid silicon and address key
questions regarding the nature of the phase transition and the associated
critical point.
