Nonlinear pole topology of slow currents in holographic probe-brane metals

Abstract

Low-frequency Drude response does not by itself determine which nearly conserved sector carries an electric current. We classify the leading cubic retarded response of an isolated, inversion-symmetric slow vector by its pole topology. Constitutive nonlinearity produces only incoming-frequency poles, whereas nonlinear relaxation adds a propagator at the emitted frequency; coupling to a second scalar slow mode generates pair-frequency poles. We then compute the full complex third-harmonic response of finite-density Dirac--Born--Infeld probe branes in Lifshitz black-brane backgrounds. In the marginal $z=2$ theory, the dense infrared response approaches the relaxation topology, with a signed infrared weight $p_{\rm IR}=0.994\pm0.002$ and a normalization-independent zero at $\omega_\times/\Gamma_J=0.2918$. Nonmarginal $z=1$ and $z=3/2$ backgrounds instead realize mixed constitutive--relaxation weights, demonstrating that the topology is selected dynamically rather than imposed by the DBI square root. The exact nonlinear dc solution further yields a parameter-free dense-limit scaling function and a crossover field $E_{\rm nl}\propto T^{3/2}$ at $z=2$. These results connect emergent higher-form slow currents, nonlinear response theory, and holographic transport, while the two-mode extension identifies the additional pole structures expected when energy or deformation modes remain slow.