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In Natural language processing, what are the major applications of Hidden Markov Chain (HMM), and what are the alternatives that usually can outperform HMM, is RNN and LSTM always the choice right now?

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Transformer based architectures are some of the most popular in NLP right now. You can check this blog post for more information:

https://ai.googleblog.com/2017/08/transformer-novel-neural-network.html

Other than performance, one major advantage of transformers is that operations can be parallelized, making it much faster than RNNs/LSTMs.

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