Capacity-achieving polar codes have gained significant attention in recent years. In general, polar codes can be decoded by either successive cancellation (SC) or the belief propagation (BP) algorithm. However, unlike SC decoders, performance optimizations for BP decoders have not been fully explored yet. In this paper, we explore novel early stopping criteria for polar BP decoding to significantly reduce energy dissipation and decoding latency. First, we propose two detection-type novel early stopping criteria for detecting valid outputs. For polar (1024, 512) codes, these two stopping criteria can reduce the number of iterations by up to 42.5% at 3.5 dB. Then, we propose a novel channel condition estimation approach, which can help select different stopping criteria in different SNR regions. Furthermore, the hardware architectures of polar BP decoders with the proposed stopping criteria are presented and developed. Synthesis results show that with the use of the proposed stopping criteria, the energy dissipation, and average latency of polar (1024, 512) BP decoder can be reduced by 10% ~ 30% with 2% ~ 5% hardware overhead, and average throughput can be increased by 20% ~ 55%.