In quantum spin systems, a phase transition is studied from the perspective of magnetization curve and a magnetic susceptibility. We propose a new method for studying the anomaly of magnetic susceptibility χ that indicates a phase transition. In addition, we introduce the fourth derivative A of the lowest-energy eigenvalue per site with respect to magnetization, i.e., the second derivative of χ-1. To verify the validity of this method, we apply it to an S =1 /2 X X Z antiferromagnetic chain. The lowest energy of the chain is calculated by numerical diagonalization. As a result, the anomalies of χ and A exist at zero magnetization. The anomaly of A is easier to observe than that of χ , indicating that the observation of A is a more efficient method of evaluating an anomaly than that of χ . The observation of A reveals an anomaly that is different from the Kosterlitz-Thouless (KT) transition. Our method is useful in analyzing critical phenomena.