Digital twin, which enables emulation, evaluation, and optimization of physical entities through synchronized digital replicas, has gained increasing attention as a promising technology for intricate wireless networks. For 6G, numerous innovative wireless technologies and network architectures have posed new challenges in establishing wireless network digital twins. To tackle these challenges, artificial intelligence (AI), particularly the flourishing generative AI, emerges as a potential solution. In this article, we discuss emerging prerequisites for wireless network digital twins considering the complicated network architecture, tremendous network scale, extensive coverage, and diversified application scenarios in the 6G era. We further explore the applications of generative AI, such as Transformer and diffusion model, to empower the 6G digital twin from multiple perspectives including physical-digital modeling, synchronization, and slicing capability. Subsequently, we propose a hierarchical generative AI-enabled wireless network digital twin at both the message-level and policy-level, and provide a typical use case with numerical results to validate the effectiveness and efficiency. Finally, open research issues for wireless network digital twins in the 6G era are discussed.