We rederive the semiconductor Bloch equations emphasizing the close link to the Berry connection. Our rigorous derivation reveals the existence of a third contribution to the (longitudinal) current in addition to the traditional intraband and polarization-related interband terms. The novel term becomes sizable in situations where the dipole-matrix elements are strongly wave-number dependent. We apply the formalism to high-harmonic generation for a Dirac metal. The novel term adds to the frequency-dependent emission intensity (high-harmonic spectrum) significantly at certain frequencies reaching up to 90% of the total signal.

We identify a (pseudo) relativistic spin-dependent analogue of the celebrated quantum phase transition driven by the formation of a bright soliton in attractive one-dimensional bosonic gases. In this new scenario, due to the simultaneous existence of the linear dispersion and the bosonic nature of the system, special care must be taken with the choice of energy region where the transition takes place. Still, due to a crucial adiabatic separation of scales, and identified through extensive numerical diagonalization, a suitable effective model describing the transition is found. The corresponding mean-field analysis based on this effective model provides accurate predictions for the location of the quantum phase transition when compared against extensive numerical simulations. Furthermore, we numerically investigate the dynamical exponents characterizing the approach from its finite-size precursors to the sharp quantum phase transition in the thermodynamic limit.

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