Almost half-quantized planar Hall effects in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> -wave magnets with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=

The planar Hall effect is a phenomenon that the Hall conductivity emerges perpendicular to the electric field in the presence of an in-plane magnetic field. We investigate the planar Hall effect in two-dimensional metal coupled with higher symmetric <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>X</a:mi> </a:math> -wave magnets with <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>X</b:mi> <b:mo>=</b:mo> <b:mi>p</b:mi> <b:mo>,</b:mo> <b:mi>d</b:mi> <b:mo>,</b:mo> <b:mi>f</b:mi> <b:mo>,</b:mo> <b:mi>g</b:mi> <b:mo>,</b:mo> <b:mi>i</b:mi> </b:mrow> </b:math> , where those with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>X</c:mi> <c:mo>=</c:mo> <c:mi>d</c:mi> <c:mo>,</c:mo> <c:mi>g</c:mi> <c:mo>,</c:mo> <c:mi>i</c:mi> </c:mrow> </c:math> are altermagnets. The <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>X</d:mi> </d:math> -wave magnet is characterized by the number <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mi>N</e:mi> <e:mi>X</e:mi> </e:msub> </e:math> of the nodes in the band structure, where <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:msub> <f:mi>N</f:mi> <f:mi>X</f:mi> </f:msub> <f:mo>=</f:mo> <f:mn>1</f:mn> <f:mo>,</f:mo> <f:mn>2</f:mn> <f:mo>,</f:mo> <f:mn>3</f:mn> <f:mo>,</f:mo> <f:mn>4</f:mn> <f:mo>,</f:mo> <f:mn>6</f:mn> </f:mrow> </f:math> corresponds to <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>X</g:mi> <g:mo>=</g:mo> <g:mi>p</g:mi> <g:mo>,</g:mo> <g:mi>d</g:mi> <g:mo>,</g:mo> <g:mi>f</g:mi> <g:mo>,</g:mo> <g:mi>g</g:mi> <g:mo>,</g:mo> <g:mi>i</g:mi> </g:mrow> </g:math> . Although the system is metallic, provided the Dirac gap is tiny, we demonstrate that the Hall conductivities are almost half quantized and well approximated by the formula <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:msub> <h:mi>σ</h:mi> <h:mrow> <h:mi>x</h:mi> <h:mi>y</h:mi> </h:mrow> </h:msub> <h:mo>=</h:mo> <h:mo>±</h:mo> <h:mrow> <h:mo>(</h:mo> <h:msup> <h:mi>e</h:mi> <h:mn>2</h:mn> </h:msup> <h:mo>/</h:mo> <h:mn>2</h:mn> <h:mi>h</h:mi> <h:mo>)</h:mo> </h:mrow> <h:mspace width="0.16em"/> <h:mi>sgn</h:mi> <h:mspace width="0.16em"/> <h:mrow> <h:mo>(</h:mo> <h:mi>J</h:mi> <h:mo form="prefix">sin</h:mo> <h:msub> <h:mi>N</h:mi> <h:mi>X</h:mi> </h:msub> <h:mi mathvariant="normal">Φ</h:mi> <h:mo>)</h:mo> </h:mrow> </h:mrow> </h:math> , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>J</m:mi> </m:math> is the coefficient of the coupling between the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mi>X</n:mi> </n:math> -wave magnet and the electrons and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mi mathvariant="normal">Φ</o:mi> </o:math> is the direction of the applied magnetic field. Hence, the Hall conductivity is periodic in <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"> <q:mi mathvariant="normal">Φ</q:mi> </q:math> , and the periodicity is equal to the number <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"> <s:msub> <s:mi>N</s:mi> <s:mi>X</s:mi> </s:msub> </s:math> of the nodes. This property may be used to confirm that the target material is indeed an <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:mi>X</t:mi> </t:math> -wave magnet. Furthermore, the sign of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML"> <u:mi>J</u:mi> </u:math> may be used as a bit for antiferromagnetic spintronics.