We investigate the role of electron correlation and disorder on the electronic structure of layered nickelate ${({\mathrm{La}}_{0.5}{\mathrm{Sr}}_{0.5})}_{2}{\mathrm{NiO}}_{4}$ using core level and valence band photoemission spectroscopy in conjunction with density functional theory (DFT) and dynamical mean-field theory (DMFT) calculations. Sr $3d$ and La $4d$ core level spectra exhibit multiple features associated with photoemission final-state effects. An increase of unscreened features in the Sr $3d$ and La $4d$ core level spectra with lowering temperature suggests the reduction in density of states (DOS) at the Fermi level, ${E}_{F}$. Valence band spectra collected using different photon energies reveal finite intensity at ${E}_{F}$ and overall spectra are well captured by $\mathrm{DFT}+\mathrm{DMFT}$. Strong renormalization of partially filled ${e}_{g}$ bands in $\mathrm{DFT}+\mathrm{DMFT}$ result indicates strong correlation in this system. Mass enhancement factor, ${m}^{*}/{m}_{\text{DFT}}\ensuremath{\sim}$ 3, agrees well with values obtained from specific heat measurements. High-resolution spectra in the vicinity of ${E}_{F}$ show monotonically decreasing spectral intensity with lowering temperature, which evolves to exhibit a Fermi cutoff at low temperatures indicating metallic character in contrast to insulating transport, suggesting Anderson insulating state. $|E\ensuremath{-}{E}_{F}{|}^{1/2}$ dependence of the spectral DOS and square-root temperature dependence of spectral DOS at ${E}_{F}$ evidences the role of disorder in the electronic structure of ${({\mathrm{La}}_{0.5}{\mathrm{Sr}}_{0.5})}_{2}{\mathrm{NiO}}_{4}$.
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