Deformation of Matrix Geometry via Landau Level Evolution

TL;DR

This work develops a unitarity-preserving, non-perturbative scheme to deform matrix geometries by following spectral flow between Landau-model realizations on symmetric and non-symmetric manifolds. Matrix coordinates are constructed explicitly from Landau-level eigenstates, with the deformation parameter guiding a continuous transition from a fuzzy sphere to ellipsoidal geometries, including both non-relativistic and relativistic (Dirac) Landau models. Key results include explicit ellipsoidal matrix geometries per Landau level, preservation of the zeroth-mode index in line with the Atiyah-Singer theorem, and a clear quantitative distinction between ellipsoidal matrix geometries and fuzzy ellipsoids. The framework provides practical tools for analyzing non-symmetric non-commutative spaces and has potential extensions to higher dimensions and topology-aware deformations.

Abstract

We propose a scheme for the construction of deformed matrix geometries using Landau models. The Landau models are practically useful tools to extract matrix geometries. The level projection method however cannot be applied straightforwardly to the Landau models on deformed manifolds, as they do not generally exhibit degenerate energy levels. We overcome this problem by exploiting the idea of spectral flow. Taking a symmetric matrix geometry as a reference point of the spectral flow, we evolve the matrix geometry by deforming the Landau model. In this process, unitarity is automatically preserved. The explicit matrix realization of the coordinates is derived mechanically even for a non-perturbative deformation. We clarify basic properties of the deformed matrix geometries through a concrete analysis of the non-relativistic and relativistic Landau models on expanding two-sphere and elongating ellipsoid. The obtained ellipsoidal matrix geometries show behaviors quantitatively different in each Landau level, but qualitatively similar to their classical counterpart. We also numerically investigate the differences between the ellipsoidal matrix geometry and the fuzzy ellipsoid.

Figures (19)

• Figure 1: Evolution of the Landau levels. Upper: the degeneracy of the Landau level on a symmetric manifold (left) is lifted by a deformation (right). Lower: there are an infinite number of paths in the parameter space that connect the initial value of the deformation parameter and the final value. (In the figure, only two paths are depicted.)
• Figure 2: Left: the matrix geometry of the fuzzy sphere (the dashed circle is added as a guide for the eyes). The eigenvalues of ${X}_3/\alpha$ are plotted on the $m$-axis. The lengths of the latitudes correspond the eigenvalues of $2\sqrt{{X}^2+{Y}^2}$. Right: the distributions of the absolute values of the monopole harmonics for $m=\pm 2$ (green), $\pm 1$ (purple), and $0$ (blue).
• Figure 3: The spectral flow of $E_{N}^{(\mu)}$ (\ref{['muenesta']}) for $I/2=2$ and $N=0,1,2$ (green, orange, blue). The degeneracy in each Landau level is not lifted.
• Figure 4: Deformation from sphere to ellipsoids. The $\mu_{\text{ini}}=1$ is the reference point at which the two-sphere is realized. The $\mu<1$ represents an oblate spheroid (the limit $\mu \rightarrow 0$ corresponds to the squashed sphere Andronache-Steinacker-2015), while the $\mu>1$ denotes a prolate spheroid.
• Figure 5: Left: the distribution of the gauge field $\boldsymbol{A}$ on the ellipsoid (\ref{['gaugefiele']}). Middle: the distribution of the magnetic field $\boldsymbol{B}$ (\ref{['magcar']}). Right: the distribution of the magnitude $B$ (\ref{['magmags']}) with non-trivial dependence on $z$. The two peaks move to the directions of the blue arrows as $\mu$ increases.