Models for chain homotopy category of relative acyclic complexes

Let $(\mathcal{X},\mathcal{Y})$ be a balanced pair in an abelian category $\mathcal{A}$. Denote by ${\mathbf{K}}_{\mathcal{E}\text{-}\mathrm{a}\mathrm{c}}(\mathcal{X})$ the chain homotopy category of right $\mathcal{X}$-acyclic complexes with all items in $\mathcal{X}$, and dually by ${\mathbf{K}}_{\mathcal{E}\text{-}\mathrm{a}\mathrm{c}}(\mathcal{Y})$ the chain homotopy category of left $\mathcal{Y}$-acyclic complexes with all items in $\mathcal{Y}$. We establish realizations of ${\mathbf{K}}_{\mathcal{E}\text{-}\mathrm{a}\mathrm{c}}(\mathcal{X})$ and ${\mathbf{K}}_{\mathcal{E}\text{-}\mathrm{a}\mathrm{c}}(\mathcal{Y})$ as homotopy categories of model categories under mild conditions. Consequently, we obtain relative versions of recollements of Krause and Neeman-Murfet. We further give applications to Gorenstein projective and Gorenstein injective modules.