Computing observables without eigenstates: Applications to Bloch Hamiltonians

Abstract

Calculating the observables of a Hamiltonian requires taking matrix elements of operators in the eigenstate basis. Since eigenstates are only defined up to arbitrary phases that depend on Hamiltonian parameters, analytical expressions for observables are often difficult to simplify. In this paper, we show how for small Hilbert space dimension $N$ all observables can be expressed in terms of the Hamiltonian and its eigenvalues using the properties of the $\mathrm{SU}(N)$ algebra, and we derive explicit expressions for $N=2,3,4$. Then we present multiple applications specializing the case of Bloch electrons in crystals, including the computation of Berry curvature, quantum metric, and orbital moment, as well as a more complex observable in nonlinear response, the linear photogalvanic effect (LPGE). As a physical example, we consider multiband Hamiltonians with nodal degeneracies to show first how constraints between these observables are relaxed when going from two- to three-band models, and second how quadratic dispersion can lead to constant LPGE at small frequencies.