Electric, thermal, and thermoelectric magnetoconductivity for Weyl/multi-Weyl semimetals in planar Hall set-ups induced by the combined effects of topology and strain

We continue our investigation of the response tensors in planar Hall (or planar thermal Hall) configurations where a three-dimensional Weyl/multi-Weyl semimetal is subjected to the combined influence of an electric field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi></mml:math> (and/or temperature gradient <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>∇</mml:mi> <mml:mi>r</mml:mi></mml:msub> <mml:mi>T</mml:mi></mml:mrow> </mml:math> ) and an effective magnetic field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>B</mml:mi> <mml:mi>χ</mml:mi></mml:msub> </mml:math> , generalizing the considerations of Phys. Rev. B 108 (2023) 155132 and Physica E 159 (2024) 115914. The electromagnetic fields are oriented at a generic angle with respect to each other, thus leading to the possibility of having collinear components, which do not arise in a Hall set-up. The net effective magnetic field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>B</mml:mi> <mml:mi>χ</mml:mi></mml:msub> </mml:math> consists of two parts-(a) an actual/physical magnetic field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math> applied externally; and (b) an emergent magnetic field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>B</mml:mi> <mml:mn>5</mml:mn></mml:msub> </mml:math> which quantifies the elastic deformations of the sample. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>B</mml:mi> <mml:mn>5</mml:mn></mml:msub> </mml:math> is an axial pseudomagnetic field because it couples to conjugate nodal points with opposite chiralities with opposite signs. Using a semiclassical Boltzmann formalism, we derive the generic expressions for the response tensors, including the effects of the Berry curvature (BC) and the orbital magnetic moment (OMM), which arise due to a nontrivial topology of the bandstructures. We elucidate the interplay of the BC-only and the OMM-dependent parts in the longitudinal and transverse (or Hall) components of the electric, thermal, and thermoelectric response tensors. Especially, for the co-planar transverse components of the response tensors, the OMM part acts exclusively in opposition (sync) with the BC-only part for the Weyl (multi-Weyl) semimetals.