The paper by Takashi Oka and Hideo Aoki explores the photovoltaic Hall effect in graphene under the influence of intense circularly polarized light and a weak DC bias. The authors use the Floquet method to formulate the response of electronic systems to AC electric fields and DC source-drain fields. They show that the non-linear effect of circularly polarized light can open a gap in the Dirac cone, leading to a photo-induced dc Hall current. This phenomenon is confirmed numerically for a graphene ribbon attached to electrodes using the Keldysh Green's function formalism. The study combines the geometric phase argument with the physics of graphene, focusing on the chiral states associated with the two Dirac cones. The geometric phase, which has become an important concept in modern electric transport theory, is extended to non-adiabatic situations through the Aharonov-Anandan (AA) phase. The authors derive the Kubo formula for electric transport in strong AC fields, showing that the Hall conductivity can be expressed in a TKNN-like form, where the Berry curvature is replaced by Floquet states and depends on the AA phase. For a Dirac band, the authors demonstrate that the AC field opens gaps in the quasi-energy band structure, reflecting an analog of the AC-Wannier-Stark ladder. In graphene, the circularly polarized light induces a new gap at the Dirac point, leading to a non-trivial Berry curvature. The Keldysh Green's function analysis of transport properties in graphene irradiated by circularly polarized light and attached to electrodes reveals a photo-induced DC Hall current, which grows linearly with the bias voltage but saturates and decreases for larger voltages. The Hall current is inverted when the circularly polarized light is changed from right to left polarization or when the bias voltage is inverted. The results suggest that the photovoltaic Hall effect in graphene can be observed under realistic experimental conditions, with the typical laser intensity corresponding to $F \sim 0.001w$. The study highlights the potential for practical applications in optoelectronics and quantum transport.The paper by Takashi Oka and Hideo Aoki explores the photovoltaic Hall effect in graphene under the influence of intense circularly polarized light and a weak DC bias. The authors use the Floquet method to formulate the response of electronic systems to AC electric fields and DC source-drain fields. They show that the non-linear effect of circularly polarized light can open a gap in the Dirac cone, leading to a photo-induced dc Hall current. This phenomenon is confirmed numerically for a graphene ribbon attached to electrodes using the Keldysh Green's function formalism. The study combines the geometric phase argument with the physics of graphene, focusing on the chiral states associated with the two Dirac cones. The geometric phase, which has become an important concept in modern electric transport theory, is extended to non-adiabatic situations through the Aharonov-Anandan (AA) phase. The authors derive the Kubo formula for electric transport in strong AC fields, showing that the Hall conductivity can be expressed in a TKNN-like form, where the Berry curvature is replaced by Floquet states and depends on the AA phase. For a Dirac band, the authors demonstrate that the AC field opens gaps in the quasi-energy band structure, reflecting an analog of the AC-Wannier-Stark ladder. In graphene, the circularly polarized light induces a new gap at the Dirac point, leading to a non-trivial Berry curvature. The Keldysh Green's function analysis of transport properties in graphene irradiated by circularly polarized light and attached to electrodes reveals a photo-induced DC Hall current, which grows linearly with the bias voltage but saturates and decreases for larger voltages. The Hall current is inverted when the circularly polarized light is changed from right to left polarization or when the bias voltage is inverted. The results suggest that the photovoltaic Hall effect in graphene can be observed under realistic experimental conditions, with the typical laser intensity corresponding to $F \sim 0.001w$. The study highlights the potential for practical applications in optoelectronics and quantum transport.