Drude weight, historically associated with the longitudinal Drude conductivity, can be generalized to describe the transverse or Hall component of the extrinsic conductivity tensor. In particular, transverse Drude weights, such as band geometric quantities Berry curvature dipole and spin vorticity, manifest themselves through the extrinsic second-order nonlinear Hall effect and extrinsic linear spin Hall effect (SHE) in diffusive transport, respectively. In this work, we uncover a new class of intrinsic Hall effects in quantum transport regime, termed the quantum intrinsic Hall effect (QIHE), which is the manifestation of system symmetry through intrinsic transport phenomena. For a given Hamiltonian, its transport characteristics can be revealed either intrinsically through QIHE in quantum ballistic transport or extrinsically via the transverse Drude weight in diffusive transport, where both intrinsic and extrinsic effects share the same salient transport features governed by symmetry of the Hamiltonian. The physical origin of QIHE is attributed to quantum boundary scattering of the measurement setup that respects the system symmetry, as exemplified by the contact resistance of a two-terminal ballistic conductor. We demonstrate our findings by studying the $T$-even ($T$, time-reversal) nonlinear charge Hall effect induced by Berry curvature dipole and the $T$-odd spin Hall effect in altermagnets, thereby revealing the quantum intrinsic nature of these effects. Our work explicitly resolves the intrinsic-extrinsic classification paradox when describing quantum transport properties using the language of bulk transport theory, and paves a way towards quantum transport manifestation of band geometric characteristics.

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• Hall effect
• Transport phenomena
• Altermagnets