next minute可以表示从现在或者过去某一时刻起“下一分钟”，the next minute只能表示从过去某一时刻起“下一分钟”。所以The flying carpet travelled so fast that next minute they were in Zimbabwe.= The flying carpet travelled so fast that the next minute they were in Zimbabwe. ( 这个句子使用了 “so ... that ...”这个句型)

next day, next week, next year, next month, next term等有时带定冠词，有时不带定冠词。
▲当以现在为时间为背景来谈论下周、下月、第二天、明年是，应该使用不带定冠词的next。
Is he coming this weekend or next weekend? 他是在这周末来，还是下周末来。
I shall go there next year. 我将于明年去那里。
By the end of next year, they will have finished work on the new stadium. 到明年年底他们将完成新体育场的工程。
▲当以过去某一时间点或将来某个时间点为背景时，则next常常带定冠词the。
He came again the next morning.
He will spend the first week of his holiday in Egypt and the next week in Italy.
▲但是，请注意，next前不加定冠词the也可以以过去时间为时间背景。例如：
Next morning she was down early. 第二天早晨她下楼很早。
They predicted a crisis next month. 他们语言了下个月的危机。
——《英语常见问题解答大词典》