What is the source of Earth's magnetic field?

Circulating neutral particles will not by themselves create a magnetic field. However, if the neutral particles are moving through an existing magnetic field, and the neutral medium is conducting, then the magnetic field will induce a current via the Lorentz force. That induced current will in turn create it's own magnetic field, which may enhance the existing magnetic field. If things work out right you have a self-reinforcing dynamo where motion thru the magnetic fields drives currents and those currents in turn support the magnetic field. However, there had to be some sort of "seed" field to get the thing started in the beginning.