The negative of must is don't have to.
When must means that something is obligatory, the negative is don’t have to because don’t have to means that there is no obligation. You can do something or not, as you wish. Mustn’t, on the other hand, means that you must NOT do something. In other words, the thing is forbidden.
You must stay home today. You have to take care of your little sister. (It is obligatory.)
You don’t have to stay home today. I'll be home to take care of your sister. (You have the option to go out.)
You mustn’t stay home today. They’re going to spray strong insecticide in the whole building. (It is obligatory that you NOT stay home. You must go out.)
Then again, when must is used for deduction, British English makes the negative with can’t. This means that something is logically impossible or that there is strong evidence to believe that it is not the case. This is the only form taught in the Hello! series.
Bernie must be home. All the lights are on.
Bernie can’t be home. All the lights are off.