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The effect of an external electric field on the quark matter is an importantquestion due to the presence of strong electric fields in heavy ion collisions.In the lattice QCD approach, the case of a real electric field suffers from the`sign problem', and a classical electric field is often used similar as thecase of chemical potential. Interestingly, in axial gauge a uniform classicalelectric field actually can correspond to an inhomogeneous imaginary chemicalpotential that varies with coordinate. On the other hand, with imaginarychemical potential, Roberge-Weiss~(R-W) phase transition occurs. In this work,the case of a uniform classical electric field is studied by using lattice QCDapproach, with the emphasis on the properties of the R-W phase. Novel phenomenashow up at high temperatures. It is found that, the chiral condensationoscillates with $z$ at high temperatures, and so is the absolute value of thePolyakov loop. It is verified that the charge density also oscillates with $z$at high temperatures. The Polyakov loop can be described by an ansatz $A_p+\sum_{q=u,d} C_q\exp\left(L_{\tau} Q_q iazeE_z\right)$, where $A_p$ is a complexnumber and $C_d>0,C_u\geq 0$ are real numbers that are fitted for differenttemperatures and electric field strengths. As a consequence, the behavior ofthe phase of Polyakov loop is different depending on whether the Polyakov loopencloses the origin, which implies a possible phase transition.

Study of the Roberge-Weiss phase caused by external uniform classical electric field using lattice QCD approach