Will my new Xiaomi RedMi Note 2 prime work for LTE network?
No. Keep the phone, it's all right. You will get 4G if your APN settings are configured correctly and you have a working 4G plan.

But for your information, what T-Mobile calls "4G" is actually a kind of enhanced 3G (HSPA+) whereas "real" 4G is LTE.

This phone can access 4G in India, on a hardware basis it should be capable of accessing 4G in the US, unless you got a carrier-locked handset.