We derive the field correction to the Berry curvature of Bloch electrons, which can be traced back to a positional shift due to the interband mixing induced by external electromagnetic fields. The resulting semiclassical dynamics is accurate to second order in the fields, in the same form as before, provided that the wave packet energy is derived up to the same order. As applications, we discuss the orbital magnetoelectric polarizability and predict nonlinear anomalous Hall effects. PACS numbers: 72.15. 72.10.Bg, The response of Bloch electrons to external fields has been a central topic in solid state physics. Due to the Berry curvature of Bloch states, the semiclassical dynamics acquires a non-canonical structure [1][2][3]. This is manifested as an anomalous velocity and a modification of the phase-space density of states, with important consequences on the thermodynamic and transport properties [2,[4][5][6]. Together with a first order correction to the band energy due to the orbital magnetic moment, the Berry curvature provides the essential ingredient for a full theory of the electron response to first order in external fields.However, response functions such as electric polarizability, magnetic susceptibility, and magnetoelectric polarizability would require a theory that is accurate up to second order in external fields. The difficulty in establishing this type of theory originates from the unboundedness of the perturbative Hamiltonian. Blount pioneered the work of systematically extending semiclassical theory up to second order by using phase space quantum mechanics [7]. However, his method uses variables which are not fully gauge invariant with respect to the phase choice in the basis Bloch states, rendering it difficult to understand the physical meaning of his results, especially so because of some unresolved gauge issues. [8].In this letter, we present a second order semiclassical theory for Bloch electrons under uniform electromagnetic fields in terms of physical position and crystal momentum which are fully gauge invariant. A central concept is a gauge-invariant positional shift due to field induced interband mixing. It leads to a field correction to the Berry curvature, and modifies the relationship between the physical position and crystal momentum with the canonical ones. However, to our surprise and delight, the resulting equations of motion up to second order still retain the same form as in the first order theory, provided that the band energy is also corrected to second order in the fields.The field induced positional shift of Bloch electrons has profound implications. It is solely responsible for the cross-gap part of the orbital magnetoelectric polarizability [9][10][11][12][13][14]. Moreover, its resulting field correction to the Berry curvature also leads to nonlinear anomalous Hall effects, with a Hall conductivity proportional to external electric or magnetic field. The electric nonlinear anomalous Hall conductivity is intimately related to the orbital magnetoelectric polarizability, ...

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We report on a study of the process e+ e- → π± (DD*)∓ at sqrt[s] = 4.26  GeV using a 525 pb(-1) data sample collected with the BESIII detector at the BEPCII storage ring. A distinct charged structure is observed in the (DD*)∓ invariant mass distribution. When fitted to a mass-dependent-width Breit-Wigner line shape, the pole mass and width are determined to be Mpole = (3883.9±1.5(stat)±4.2(syst))  MeV/c2 and Γpole = (24.8±3.3(stat)±11.0(syst))  MeV. The mass and width of the structure, which we refer to as Zc(3885), are 2σ and 1σ, respectively, below those of the Zc(3900) → π± J/ψ peak observed by BESIII and Belle in π+ π- J/ψ final states produced at the same center-of-mass energy. The angular distribution of the πZc(3885) system favors a JP = 1+ quantum number assignment for the structure and disfavors 1- or 0-. The Born cross section times the DD* branching fraction of the Zc(3885) is measured to be σ(e+ e- → π± Zc(3885)∓)×B(Zc(3885)∓ → (DD*)∓) = (83.5±6.6(stat)±22.0(syst))   pb. Assuming the Zc(3885) → DD* signal reported here and the Zc(3900) → πJ/ψ signal are from the same source, the partial width ratio (Γ(Zc(3885) → DD*)/Γ(Zc(3900) → πJ/ψ)) = 6.2±1.1(stat)±2.7(syst) is determined.

Within the wave-packet semiclassical approach, the Bloch electron energy is derived to second order in the magnetic field and classified into gauge-invariant terms with clear physical meaning, yielding a fresh understanding of the complex behavior of orbital magnetic susceptibility. The Berry curvature and quantum metric of the Bloch states give rise to a geometrical magnetic susceptibility, which can be dominant when bands are filled up to a small energy gap. There is also an energy polarization term, which can compete with the Peierls-Landau and Pauli magnetism on a Fermi surface. All these, and an additional Langevin susceptibility, can be calculated from each single band, leaving the Van Vleck susceptibility as the only term truly from interband coupling.

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