Barrow Holographic Dark Energy in $f(Q,L_{m})$ gravity: A dynamical system perspective

In this work, we investigate the cosmological implications of the Barrow Holographic Dark Energy (BADE) model within the framework of $f(Q,L_{m})$ gravity, specifically considering the model $f(Q,L_{m})=\alpha Q+\beta L_{m}$. Using a dynamical system approach for both non-interacting and interacting scenarios, we identify critical points corresponding to different phases of the Universe's evolution, including matter domination, radiation domination and dark energy-driven accelerated expansion. Our analysis reveals two stable critical points in the non-interacting case and three stable critical points in the interacting case, each indicating a transition to a stable phase dominated by BADE. The phase plots clearly demonstrate the evolution of the Universe's dynamics toward these stable points. At these stable points, the deceleration parameter is negative, consistent with accelerated expansion and the equation of state parameter suggests that BADE behaves as a dark energy component. These findings highlight the BADE model's strength as a viable explanation for the Universe's late-time acceleration inside $f(Q, L_{m})$ gravity, and they provide novel perspectives on the cosmic development of dark energy-matter interactions.