Higher-order bulk photovoltaic effects, quantum geometry, and application to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -wave magnets

The injection and shift currents are generalized to the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>ℓ</a:mi> <a:mi>th</a:mi> </a:mrow> </a:math> -order injection and shift currents for the longitudinal conductivities in the two-band model, where <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>ℓ</b:mi> </b:math> is the power of the applied electric field. In addition, the formulas for the higher-order injection current are expressed in terms of the quantum metric, and those for the higher-order shift current are expressed in terms of the higher-order quantum connection. Then, they are applied to <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>p</c:mi> </c:math> -wave magnets, which are well described by the two-band Hamiltonian as far as low-energy physics is concerned. It is shown that the injection and shift currents are zero. On the other hand, the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>ℓ</d:mi> <d:mi>th</d:mi> </d:mrow> </d:math> -order injection and shift currents with odd <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>ℓ</e:mi> </e:math> are nonzero when the direction of the spin-splitting vector of the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>p</f:mi> </f:math> -wave magnet points to an in-plane direction.