我们计算了三维节点环半金属（漩涡节点环和 PT 对称节点环）和三带霍普夫半金属的纵向磁电导率（σ zz）和非线性电化学响应（ECR），应用半经典玻尔兹曼形式法。虽然前者的节点曲线被取在k z = 0平面，后者的节点点在其 Berry 曳度（BC）分布中具有偶极子，其偶极子轴沿k z轴对齐。所有这些系统都是拓扑的，并且在它们的带具有消失的陈数方面统一。线性响应σ zz是从当系统受到沿各向异性轴（即z轴）的平行电场和磁场作用时的精确解获得的。非线性部分涉及三阶张量，代表二阶响应系数，将电流与化学势梯度和外部电场的联合效应联系起来。我们分析了漩涡节点环和霍普夫半金属响应的相似性，这可以追溯到它们的 BC 场中偶极子源。
英文摘要：
We investigate the longitudinal magnetoelectric conductivity and nonlinear electrochemical response (ECR) of three-dimensional nodal-ring semimetals, including vortex nodal-rings (VNRs) and $\mathcal{PT}$-symmetric nodal-rings (PTNRs), together with three-band Hopf semimetals (HSMs), using the semiclassical Boltzmann formalism. The nodal-rings of the VNRs and PTNRs lie in the $k_z=0$ plane, whereas the nodal points of HSMs host isolated Berry-curvature (BC) dipoles with axes along the $k_z$ direction. In VNRs, such dipoles may be viewed as forming a continuous distribution along the nodal-ring, connecting the two classes of systems. For collinear electric and magnetic ($\mathcal{B}$) fields along the anisotropy axis $\boldsymbol {\hat{z}}$, the linear response is computed exactly and displays odd powers of $B_z$ for a single ring or node. For a two-node HSM with oppositely-oriented BC-dipole axes, these odd terms cancel in the sum. We analyse the nonlinear ECR, described by third-rank response tensors that relate the current to the combined action of an electric field and a chemical-potential gradient. The surviving tensor components are determined by the tilt direction of the bandstructure, and we obtain explicit results for simultaneous in-plane and axial tilts. A comparison of the three systems reveals a close correspondence between the responses of VNRs and HSMs, while PTNRs behave qualitatively differently. We attribute these similarities and differences to the topology of the BC field, namely dipole-like sources in VNRs and HSMs versus the ring-confined BC structure of PTNRs.
