Holographic Dark Energy as a Source for Wormholes in Modified Gravity

Traversable wormhole solutions are explored in $f(\mathcal{R},\mathbb{T})$ gravity, a curvature--matter extension in which $\mathcal{R}$ is the Ricci scalar and $\mathbb{T}$ denotes the trace of the energy--momentum tensor. To generate explicit wormhole models, we prescribe holographic dark-energy densities based on entropy formalism proposed by R\'enyi, Moradpour, and Bekenstein--Hawking, namely $$\rho_{\textit{R}} = \frac{\alpha}{4\alpha_1 r^4 c^2 \kappa}\ln\!\left(1+\pi \alpha_1 r^2\right), \qquad \rho_{\textit{M}} = \frac{\alpha}{4\pi r^2 c^2 \kappa \left(\pi \alpha_1 r^2 + 1\right)}, \qquad \rho_{\textit{BH}} = \frac{\alpha}{4 c^2 \kappa r^2},$$ with $\alpha$ and $\beta$ carrying dimensions of $L^{-2}$. The corresponding shape functions obtained from the field equations satisfy the standard throat and flare-out requirements for traversability. We then study how varying $\alpha$ and $\beta$ affects (i) the balance of forces associated with equilibrium and (ii) the status of the energy conditions. In particular, the null energy condition is found to be violated, indicating that exotic matter (or an effective exotic sector) is required to support the wormhole geometry. The spatial structure of the solutions is further visualized through embedding surfaces.