Computing observables without eigenstates: applications to Bloch Hamiltonians

TL;DR

This paper introduces a method to compute quantum observables directly from Hamiltonians and eigenvalues without explicitly using eigenstates, simplifying calculations for small systems and applying to Bloch electrons.

Contribution

It derives explicit formulas for observables in terms of Hamiltonian and eigenvalues for systems with up to four levels, and applies these to Bloch electrons and nonlinear response phenomena.

Findings

Explicit formulas for N=2,3,4 Hamiltonians.

Application to Berry curvature, quantum metric, and orbital moment.

Analysis of linear photogalvanic effect in multiband systems.

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 work, 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 SU(N) algebra, and we derive explicit expressions for N=2,3,4. Then we present multiple applications specializing to 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 non-linear 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.