"Hang your head" is an idiomatic expression used to describe someone who is feeling down or ashamed. There are numerous synonyms that can be used instead of "hang your head" that convey similar emotions and feelings. For instance, you could use "bow your head", "lower your gaze", or "lower your head" to describe someone who is feeling embarrassed or defeated. "Hang your chin", "bow your chin", or "droop your chin" could be used in place of "hang your head" to show someone who is feeling down or disheartened. Other phrases like "slump your shoulders" or "shrink away" could also be used instead of "hang your head" to depict someone who is feeling defeated or dejected.