2D Invariants Research Articles

Invariant-based inverse engineering of time-dependent, coupled harmonic oscillators

Two-dimensional systems with time-dependent controls admit a quadratic Hamiltonian modelling near potential minima. Independent, dynamical normal modes facilitate inverse Hamiltonian engineering to control the system dynamics, but some systems are not separable into independent modes by a point transformation. For these "coupled systems" 2D invariants may still guide the Hamiltonian design. The theory to perform the inversion and two application examples are provided: (i) We control the deflection of wave packets in transversally harmonic waveguides; and (ii) we design the state transfer from one coupled oscillator to another.

Homotopy characterization of non-Hermitian Hamiltonians

We revisit the problem of classifying topological band structures in non-Hermitian systems. Recently, a solution has been proposed, which is based on redefining the notion of energy band gap in two different ways, leading to the so-called "point-gap" and "line-gap" schemes. However, simple Hamiltonians without band degeneracies can be constructed which correspond to neither of the two schemes. Here, we resolve this shortcoming of the existing classifications by developing the most general topological characterization of non-Hermitian bands for systems without a symmetry. Our approach, which is based on homotopy theory, makes no particular assumptions on the band gap, and predicts several amendments to the previous classification frameworks. In particular, we show that the 1D invariant is the noncommutative braid group (rather than $\mathbb{Z}$ winding number), and that depending on the braid group invariants, the 2D invariants can be cyclic groups $\mathbb{Z}_n$ (rather than $\mathbb{Z}$ Chern number). We interpret these novel results in terms of a correspondence with gapless systems, and we illustrate them in terms of analogies with other problems in band topology, namely the fragile topological invariants in Hermitian systems and the topological defects and textures of nematic liquids.

Relation between 3D invariants and 2D invariants

Polynomial relations are established between the invariants of certain mixed sets of points and lines and the invariants of their projected images. The relations are obtained using the properties of a rational curve which is a covariant of the given set of points and lines. Numerical experiments indicate the level of performance which can be expected from a recognition system based on one of the relations.