In this paper, we study the second-order Hamiltonian systems \t\t\tu¨−L(t)u+∇W(t,u)=0,t∈R,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\ddot{u}-L(t)u+\\nabla W(t,u)=0,\\quad t\\in \\mathbb{R}, $$\\end{document} where Lin C(mathbb{R},mathbb{R}^{Ntimes N}) is a T-periodic and positive definite matrix for all tin mathbb{R} and W is superquadratic but does not satisfy the usual Ambrosetti–Rabinowitz condition at infinity. One ground homoclinic solution is obtained by applying the monotonicity trick of Jeanjean and the concentration–compactness principle. The main result improves the recent result of Liu–Guo–Zhang (Nonlinear Anal., Real World Appl. 36:116–138, 2017).