  The slope of the coexistence line of the liquid-liquid phase transition
(LLPT) can be positive, negative, or zero. All three possibilities have been
found in Monte-Carlo simulations of a modified spherically symmetric two-scale
Jagla model. Since the liquid-liquid critical point (LLCP) frequently lies in a
region of the phase diagram that is difficult to access experimentally, it is
of great interest to study critical phenomena in the supercritical region. We
therefore study the properties of the Widom line, which is defined in the
one-phase region above the critical point as the locus of maximum correlation
length as function of the ordering field at constant thermal field.
Asymptotically close to the critical point, the Widom line coincides with the
loci of the response function extrema, because all response functions can be
asymptotically expressed as functions of the diverging correlation length. We
find that the method of identifying the Widom line as the loci of heat capacity
maxima becomes unfruitful when the slope of the coexistence line approaches
zero in the $T$-$P$ plane. In this case the specific heat displays no maximum
in the one-phase region because for a horizontal phase coexistence line,
according to the Clapeyron equation, the enthalpy difference between the
coexisting phases is zero, and thus there can be no contribution to enthalpy
fluctuations from the critical fluctuations. The extension of the coexistence
line beyond the critical point into the one-phase region must in this case be
performed using density fluctuations; the line of compressibility maxima
remains well defined, though it bifurcates into a symmetrical pair of lines.
These findings agree well with the linear scaling theory of the LLCP by
Anisimov and collaborators.
