This paper presents a study on the photovoltaic Hall effect in graphene under the influence of circularly polarized light. The authors use the Floquet method to analyze the response of electronic systems to intense AC fields and apply this formalism to graphene. They show that a non-linear effect of circularly polarized light can open a gap in the Dirac cone, leading to a photo-induced DC Hall current. This is numerically confirmed for a graphene ribbon attached to electrodes using the Keldysh Green's function. The paper begins with an introduction to non-linear phenomena in electronic systems and their potential to lead to distinct transport properties. It discusses the geometric phase argument extended to electron transport in intense AC fields and the physics of graphene involving chiral states associated with two Dirac cones. The authors then derive the Kubo formula for electric transport in strong AC fields, which is accomplished using the Floquet-matrix formalism. They show that a TKNN-like formula for the Hall conductivity is obtained, where the Berry curvature is expressed in terms of Floquet states which depend on the AA phase. The paper then applies this formula to a single Dirac band and then to graphene. It concludes that a photo-induced Hall current should appear in graphene irradiated by circularly polarized light and attached to two electrodes, where the Hall current can exceed the longitudinal current in magnitude. The paper also discusses the Kubo formula in the presence of strong light fields, deriving the formula for DC transport in an intense AC background field. It shows that the Hall conductivity can be simplified to a TKNN-like formula, where the Berry curvature is expressed in terms of Floquet states. The authors then apply this formula to a Dirac band and show that gaps open at specific frequencies in the quasi-energy band structure, reflecting a Dirac-band analog of the AC-Wannier-Stark ladder. The paper further discusses the Keldysh approach to photovoltaic transport in graphene, analyzing the transport properties of a graphene ribbon irradiated by circularly polarized light and attached to two electrodes. The authors show that the polarized light induces locally circulating currents in the absence of a DC bias, and that a photo-induced DC Hall current appears when a bias voltage is applied. The Hall current is naturally inverted when the polarization of the light is changed or the bias voltage is inverted. The I-V characteristics of the longitudinal and Hall currents are shown, with the Hall current growing linearly with the bias voltage but saturating and then decreasing when the voltage becomes large. The paper concludes that a combined effect of an intense AC field of circularly polarized light and a weak DC bias can produce a photo-voltaic DC Hall current in graphene, despite the absence of a uniform magnetic field. The typical intensity of laser conceived here is within experimental feasibility. The authors also mention that including dissipation etc. will be an interesting future problem.This paper presents a study on the photovoltaic Hall effect in graphene under the influence of circularly polarized light. The authors use the Floquet method to analyze the response of electronic systems to intense AC fields and apply this formalism to graphene. They show that a non-linear effect of circularly polarized light can open a gap in the Dirac cone, leading to a photo-induced DC Hall current. This is numerically confirmed for a graphene ribbon attached to electrodes using the Keldysh Green's function. The paper begins with an introduction to non-linear phenomena in electronic systems and their potential to lead to distinct transport properties. It discusses the geometric phase argument extended to electron transport in intense AC fields and the physics of graphene involving chiral states associated with two Dirac cones. The authors then derive the Kubo formula for electric transport in strong AC fields, which is accomplished using the Floquet-matrix formalism. They show that a TKNN-like formula for the Hall conductivity is obtained, where the Berry curvature is expressed in terms of Floquet states which depend on the AA phase. The paper then applies this formula to a single Dirac band and then to graphene. It concludes that a photo-induced Hall current should appear in graphene irradiated by circularly polarized light and attached to two electrodes, where the Hall current can exceed the longitudinal current in magnitude. The paper also discusses the Kubo formula in the presence of strong light fields, deriving the formula for DC transport in an intense AC background field. It shows that the Hall conductivity can be simplified to a TKNN-like formula, where the Berry curvature is expressed in terms of Floquet states. The authors then apply this formula to a Dirac band and show that gaps open at specific frequencies in the quasi-energy band structure, reflecting a Dirac-band analog of the AC-Wannier-Stark ladder. The paper further discusses the Keldysh approach to photovoltaic transport in graphene, analyzing the transport properties of a graphene ribbon irradiated by circularly polarized light and attached to two electrodes. The authors show that the polarized light induces locally circulating currents in the absence of a DC bias, and that a photo-induced DC Hall current appears when a bias voltage is applied. The Hall current is naturally inverted when the polarization of the light is changed or the bias voltage is inverted. The I-V characteristics of the longitudinal and Hall currents are shown, with the Hall current growing linearly with the bias voltage but saturating and then decreasing when the voltage becomes large. The paper concludes that a combined effect of an intense AC field of circularly polarized light and a weak DC bias can produce a photo-voltaic DC Hall current in graphene, despite the absence of a uniform magnetic field. The typical intensity of laser conceived here is within experimental feasibility. The authors also mention that including dissipation etc. will be an interesting future problem.