  The discovery of high incidence of hot Jupiters in dense clusters challenges
the field-based hot Jupiter formation theory. In dense clusters, interactions
between planetary systems and flyby stars are relatively common. This has a
significant impact on planetary systems, dominating hot Jupiter formation. In
this paper, we perform high precision, few-body simulations of stellar flybys
and subsequent planet migration in clusters. A large parameter space
exploration demonstrates that close flybys that change the architecture of the
planetary system can activate high eccentricity migration mechanisms:
Lidov-Kozai and planet-planet scattering, leading to high hot Jupiter formation
rate in dense clusters. Our simulations predict that many of the hot Jupiters
are accompanied by "ultra-cold Saturns", expelled to apastra of thousands of
AU. This increase is particularly remarkable for planetary systems originally
hosting two giant planets with semi-major axis ratios $\sim$ 4 and the flyby
star approaching nearly perpendicular to the planetary orbital plane. The
estimated lower limit to the hot Jupiter formation rate of a virialized cluster
is $\sim 1.6\times10^{-4}({\sigma}/{\rm 1kms^{-1}})^5({a_{\rm p}}/{\rm 20
AU})({M_{\rm c}}/{\rm 1000M_\odot})^{-2}$Gyr$^{-1}$ per star, where $\sigma$ is
the cluster velocity dispersion, $a_{\rm p}$ is the size of the planetary
system and $M_{\rm c}$ is the mass of the cluster. Our simulations yield a hot
Jupiter abundance which is $\sim$ 50 times smaller than that observed in the
old open cluster M67. We expect that interactions involving binary stars, as
well as a third or more giant planets, will close the discrepancy.
