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 $sim 1.6times10^{-4}({sigma}/{rm 1kms^{-1}})^5({a_{rm p}}/{rm 20 AU})({M_{rm c}}/{rm 1000M_odot})^{-2}$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.
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