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It is always true that $\boldsymbol{\nabla}\cdot \textbf{B}=0$ (implying that there are no magnetic monopoles). However, $\boldsymbol{\nabla}\cdot \textbf{H}\neq 0$ when $\boldsymbol{\nabla}\cdot \textbf{M} \neq 0$. Does it mean that, in those cases $\textbf{H}$-field has poles although $\textbf{B}$-field does not?

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The short answer is yes. $\textbf{H}$-field can have sources and sinks - these are what the "poles" of a bar magnet are.

In an LIH medium $\textbf{B}=\mu_0\mu_r\textbf{H}$. Even though $\boldsymbol{\nabla}\cdot\textbf{B} = 0$ (always), this does not mean that $\boldsymbol{\nabla}\cdot \textbf{H} =0$ because there is an instantaneous change in $\mu_r$ at the boundaries between media. Lines of $\textbf{H}$-field begin and end on these interfaces, whereas lines of $\textbf{B}$-field are continuous.

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The expressions above deal with macroscopic electrodynamics, the theory of electromagnetism in some matter. It stands as convenient extention of Maxwell-Lorentz vacuum elecrodynamics, for studying fields in both conducting and insulating media, but it bases on same picture of charges in a free space. It would be strange magnetic monopoles to occur.

It is not straightforward, however, that $\vec B$(induction), not $\vec H$, plays the role of the strength of magnetic field in the theory of macroscopic electrodynamics. Vector $\vec H$ is analog of $\vec D$("electric induction") for magnetic field. In some sense, it plays supporting role in theory. So, there indeed no divergence of magnetic field. But we can calculate divergence for vector $\vec H$.

There were no magnetic monopoles detected anywhere anytime by the moment.

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