Black holes in 4D Einstein-Maxwell-Gauss-Bonnet gravity coupled with scalar fields

Einstein-Maxwell-Gauss-Bonnet-axion theory in $4$-dimensional spacetime is investigated in this paper through a "Kaluza-Klein-like" process. Dual to systems at finite temperature with background magnetic field on three dimensions, the four-dimensional dyonic black hole solution coupled with higher derivative terms is obtained. After the tensor-type perturbation is added, the shear viscosity to entropy density ratio is calculated at high temperature and low temperature separately. The behaviour of shear viscosity to entropy density ratio of uncharged black holes is found to be similar with that in $5$-dimensional spacetime, violating the Kovtun-Starinets-Son bound as well when temperature becomes lower. In addition, the main feature of this ratio remains almost unchanged in $4$ dimensions, which is characterised by $(T/\Delta)^2$ at low temperature $T$, with $\Delta$ proportional to the coefficient $\beta$ from scalar fields. The difficulty in causal analysis is also discussed, which is mainly caused by the vanishing momentum term in equations of motion.