Holographic Description of Chiral Symmetry Breaking in a Magnetic Field in 2+1 Dimensions with an Improved Dilaton

We consider a holographic description of the chiral symmetry breaking in an external magnetic field in $ (2+1) $-dimensional gauge theories from the softwall model using an improved dilaton field profile given by $\Phi(z) = - kz^2 + (k+k_1)z^2\tanh (k_{2}z^2)$. We find inverse magnetic catalysis for $B<B_c$ and magnetic catalysis for $B>B_c$, where $B_c$ is the pseudocritical magnetic field. The transition between these two regimes is a crossover and occurs at $B=B_c$, which depends on the fermion mass and temperature. We also find spontaneous chiral symmetry breaking (the chiral condensate $\sigma \not=0$) at $T=0$ in the chiral limit ($m_q\to 0$) and chiral symmetry restoration for finite temperatures. We observe that changing the $k$ parameter of the dilaton profile only affects the overall scales of the system such as $B_c$ and $\sigma$. For instance, by increasing $k$ one sees an increase of $B_c$ and $\sigma$. This suggests that increasing the parameters $k_1$ and $k_2$ will decrease the values of $B_c$ and $\sigma$.