Electrons moving in a Bloch band are known to acquire an anomalous Hall velocity proportional to the Berry curvature of the band [1] which is responsible for the intrinsic linear Hall effect in materials with broken time-reversal symmetry [2]. Here, we demonstrate that there is also an anomalous correction to the electron acceleration which is proportional to the Berry curvature dipole [3] and is responsible for the Non-linear Hall effect recently discovered in materials with broken inversion symmetry [4,5]. This allows us to uncover a deeper meaning of the Berry curvature dipole as a nonlinear version of the Drude weight that serves as a measurable order parameter for broken inversion symmetry in metals. We also derive a "Quantum Rectification Sum Rule" in time reversal invariant materials by showing that the integral over frequency of the rectification conductivity depends solely on the Berry connection and not on the band energies. The intraband spectral weight of this sum rule is exhausted by the Berry curvature dipole Drude-like peak, and the interband weight is also entirely controlled by the Berry connection. This sum rule opens a door to search for alternative photovoltaic technologies based on the Berry geometry of bands. We also describe the rectification properties of Weyl semimetals which are a promising platform to investigate these effects.

We reveal that the p-wave superfluid can be realized in a mixture of fermionic and F=1 bosonic gases. We derive a general set of the gap equations for gaps in the s-and p-channels. It is found that the spin-spin bose-fermi interactions favor the p-wave pairing and naturally suppress the pairing in the s-channel. The gap equations for the polar phase of p-wave superfluid fermions are numerically solved. It is shown that a pure p-wave superfluid can be observed in a well-controlled environment of atomic physics.

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We introduce an operational definition of the Berry Phase Rectification Tensor as the second order change of polarization of a material in response to an ideal short pulse of electric field. Under time reversal symmetry this tensor depends exclusively on the Berry phases of the Bloch bands and not on their energy dispersions, making it an intrinsic property to each material which contains contributions from both the inter-band shift currents and the intra-band Berry Curvature Dipole. We also introduce the Solar Rectification Vector as a technologically relevant figure of merit for bulk photo-current generation which counts the number of electrons contributing to the rectified current per incoming photon under ideal black-body radiation in analogy with the classic solar cell model of Shockley and Queisser. We perform first principle calculations of the Berry Phase Rectification Tensor and the Solar Rectification Vector for the Weyl semi-metal TaAs and the insulator LiAsSe2 which features large shift currents close to the peak of solar radiation intensity. We also generalize the formula for the Glass coefficient to include the spectral distribution of the incoming radiation, the directionality dependence of the conductivity of the material and the reflectivity at its surface.

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