You haven't told it what files you want to put in the archive.
- Told it where to create the file
- Told it what directory to change to
But haven't told it what to put in there.
The simplest solution might be
tar cJvf /mnt/1234/a.tar.xz -C ~/ .
Which will then put all the files into the archive with names starting with
Be careful when extracting an archive like this, though, because it will extract to the current directory and may overwrite existing files, including changing permissions on the current directory.