290S/290K Quantum Materials Seminar: Aleksander Avdoshkin (MIT); Wednesday, February 14 at 2:00 PM Pacific Time in 402 Physics South
Time/Venue: Wednesday, February 14 at 2:00 PM Pacific Time in 402 Physics South and via Zoom:
https://berkeley.zoom.us/j/99523499113pwd=REovb3pyam03WXQwbEhrU3dqNHZvdz09
Meeting ID: 995 2349 9113 Passcode: 600704
Host: Ehud Altman / Joel Moore
Title: Geometry of degenerate states
Abstract: It is known that a collection of non-degenerate quantum states can be fully characterized using the Fubini-Study (quantum) metric and symplectic (Berry curvature) form. This geometric representation is commonly used to capture the effects of Bloch state shapes in solid state physics. However, one is often also interested in degenerate states, e.g. in PT-symmetric band structures and in quantum state preparation. To provide the tools for addressing such problems, in this talk, I will show how to reduce the geometry of degenerate states to the non-Abelian (Wilczek-Zee) curvature $F$ and a previously unexplored, matrix-valued metric tensor $G$. Choosing to represent the subspaces via orthogonal projectors, I begin with describing all invariants associated with a collection of $m$-dimensional subspaces of a Hilbert space $\mathbb{C}^n$ from the perspective of linear algebra. For two subspaces, the configuration is described by a set of $m$ principal angles that generalize the notion of quantum distance. For more subspaces, there are $3 m^2 - 3 m + 1$ additional invariants associated with triples of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces. After that, I present a procedure for calculating global invariants as integrals of $F$ and $G$ over $Gr_{m,n}$ submanifolds constructed from geodesics. This work introduces a new mathematical structure, the matrix-valued metric tensor, and provides a comprehensive geometric framework for dealing with degenerate states.
See the Physics Department calendar for future seminars.