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A magnetic dipole moment has the same magnetic field as if there were a current circulating in a loop in the same area. When doing problems involving magnetized materials, we often use this picture of "bound currents" to solve Maxwell's equations.

Now, imagine we have a ferromagnetic metal. When we apply a voltage bias to this sample, do the bound currents affect in any way the flow of the transport current? If this metal had magnetic domains in it (which would each have a bound current circulating along its edge), would the spatial distribution of the transport current somehow reflect these circulating bound currents?

I have also heard the term "equilibrium currents" used in the context of quantum Hall physics. Are these bound currents also "equilibrium" currents? What similarities/differences are there between quantum Hall edge states and these magnetization bound currents? I am particularly interested in how one would tease out the differences between these bound currents and quantum Hall edges states in a quantum anomalous Hall system (magnetic).

I hope that someone can give me some insight into any one of these many questions! Thanks a lot!

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The "bound current" are not real currents but are "volume current density" $J_m$ equivalent to the magnetization vector field $M$ they represent $J_m=\nabla\times M$.

So to answer you questions: \ "When we apply a voltage bias to this sample, do the bound currents affect in any way the flow of the transport current?" : they affect the current through the magnetic field generated by the magnetisation vector field.

"If this metal had magnetic domains in it (which would each have a bound current circulating along its edge), would the spatial distribution of the transport current somehow reflect these circulating bound currents?" : no

"Are these bound currents also "equilibrium" currents? What similarities/differences are there between quantum Hall edge states and these magnetization bound currents? " : The quantum Hall edge states are real current contrary to the bond current which are just a way to represent the magnetization vector field.

I recommand this reading https://link.springer.com/referenceworkentry/10.1007%2F978-0-387-23062-7_2#Sec1

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