Anomalous spontaneous induction of magnetic and electric fields in dense quark matter

In this paper, we will demonstrate that a dense quark-matter system in the dual chiral density wave (DCDW) phase behaves as a ferromagnet in the sense that its magnetic-field dependent magnetization remains different from zero even at <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>B</a:mi> <a:mo stretchy="false">→</a:mo> <a:mn>0</a:mn> </a:math> . The corresponding permanent magnetization is a function of the baryonic chemical potential <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi>μ</d:mi> </d:math> , decreasing up to zero as <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>μ</f:mi> </f:math> increases in the range of intermediate densities ( <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mrow> <h:mn>312</h:mn> <h:mtext> </h:mtext> <h:mtext> </h:mtext> <h:mi>MeV</h:mi> <h:mo>⩽</h:mo> <h:mi>μ</h:mi> <h:mo>⩽</h:mo> <h:mn>342</h:mn> <h:mtext> </h:mtext> <h:mtext> </h:mtext> <h:mi>MeV</h:mi> </h:mrow> </h:math> ) and then increasing from zero in the higher density interval <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mrow> <j:mn>490</j:mn> <j:mtext> </j:mtext> <j:mtext> </j:mtext> <j:mi>MeV</j:mi> <j:mo>⩽</j:mo> <j:mi>μ</j:mi> <j:mo>⩽</j:mo> <j:mn>550</j:mn> <j:mtext> </j:mtext> <j:mtext> </j:mtext> <j:mi>MeV</j:mi> </j:mrow> </j:math> . We will show that this system’s ability to generate permanent magnetization, together with the existence of the axial anomaly, open up the possibility of spontaneously generating a magnetic field coupled to a collinear electric field. The generated magnetic field can reach values up to <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:msup> <l:mn>10</l:mn> <l:mn>16</l:mn> </l:msup> <l:mtext> </l:mtext> <l:mtext> </l:mtext> <l:mi mathvariant="normal">G</l:mi> </l:math> , depending on <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>μ</o:mi> </o:math> , and the electric field will be 3 orders smaller. The fact that the DCDW phase is able to induce a magnetic field can be seen as its spontaneous tendency to remove the so called Landau-Peierls instability that is present in this single-modulated phase in the absence of a magnetic field. The spontaneous induction of a strong magnetic field at intermediate to high densities can be of interest for the astrophysics of compact stellar objects exhibiting strong magnetic fields as magnetars.