Thermodynamic topology of 4D Euler-Heisenberg-AdS black hole in different ensembles

We study the thermodynamic topology of 4D Euler-Heisenberg-AdS (EHAdS) black hole and higher-order QED corrected Euler-Heisenberg-AdS black hole in different ensembles using generalized off-shell free energy. In this approach, black holes are viewed as defects in the thermodynamic space. We work in two ensembles: canonical ensemble in which the charge is kept fixed and grand canonical ensemble in which the conjugate potential $\phi_e$ is kept fixed. In each case, the local and global topology of the thermodynamic space is investigated via the computation of winding numbers at the defects. For 4D Euler-Heisenberg-AdS black hole in canonical ensemble, the topological class is found to be different depending on the Euler-Heisenberg (EH) parameter $a$. The topological numbers for $a<0$ and $a>0$ cases are found to be $W=+1$ and $W=0$ respectively. The topological number is found to be independent of the variation in pressure $P$ and charge $Q$ of the black hole. With the introduction of higher order QED correction, the difference in the topological class of 4D EHAdS black hole with the sign of $a$ is observed to go away.The topological number in this case is found to be $W=+1$ irrespective of the values of $a$, $P$ and $Q$. In the grand canonical ensemble, the topological number for both EHAdS and higher order QED corrected EHAdS black hole is found to be $W=0$, independent of the values of $P$, $\phi_e$ and $a$. Therefore, we infer that the topological class of both 4D EHAdS black hole and higher order QED corrected EHAdS black hole is ensemble dependent. Moreover, in the canonical ensemble, higher order QED correction alters the topological class of the black hole for positive values of EH parameter $a$. In the grand canonical ensemble, the higher order corrections do not change the thermodynamic topology of the black hole.