Euler-Heisenberg Black Holes in Hernquist Dark Matter Environment: Thermodynamics and Shadow

In this work, we construct a static, spherically symmetric Euler-Heisenberg (EH) black hole immersed in a Hernquist dark matter (HDM) halo. We show that the event horizon radius increases monotonically with the characteristic parameters of the HDM distribution. We then analyze the thermodynamic properties of the system by computing the Hawking temperature, entropy, specific heat, and Gibbs free energy. Using the generalized Gibbs free energy framework, we investigate how the HDM halo and nonlinear electrodynamics modify the thermodynamic topology. Moreover, we include thermal fluctuations and derive entropy corrections up to second order only. We further explore observational aspects by studying the photon sphere and black hole shadow, emphasizing the influence of the HDM halo and geometric parameters. A topological analysis of the photon sphere is also performed to quantify these effects. Finally, we examine neutral particle dynamics and determine the innermost stable circular orbits (ISCOs), highlighting their dependence on the underlying spacetime parameters.