  We propose a phenomenological spin fluctuation theory for antiferromagnetic
quantum tricritical point (QTCP), where the first-order phase transition
changes into the continuous one at zero temperature. Under magnetic fields,
ferromagnetic quantum critical fluctuations develop around the
antiferromagnetic QTCP in addition to antiferromagnetic ones, which is in sharp
contrast with the conventional antiferromagnetic quantum critical point. For
itinerant electron systems,} we show that the temperature dependence of
critical magnetic fluctuations around the QTCP are given as chiQ \propto
T^{-3/2} (chi0\propto T^{-3/4}) at the antiferromagnetic ordering
(ferromagnetic) wave number q=Q (q=0). The convex temperature dependence of
chi0^{-1} is the characteristic feature of the QTCP, which is never seen in the
conventional spin fluctuation theory. We propose that the general theory of
quantum tricriticality that has nothing to do with the specific Kondo physics
itself, solves puzzles of quantum criticalities widely observed in
heavy-fermion systems such as YbRh2Si2, CeRu2Si2, and beta-YbAlB4. For
YbRh2Si2, our theory successfully reproduces quantitative behaviors of the
experimental ferromagnetic susceptibility and the magnetization curve by
choosing the phenomenological parameters properly. The quantum tricriticality
is also consistent with singularities of other physical properties such as
specific heat, nuclear magnetic relaxation time 1/T_1T, and Hall coefficient.
For CeRu2Si2 and beta-YbAlB4, we point out that the quantum tricriticality is a
possible origin of the anomalous diverging enhancement of the uniform
susceptibility observed in these materials.
