Signatures of Green's function zeros and their topology using impurity spectroscopy

Abstract

Topology without quasiparticles has emerged as a key framework for understanding Mott insulators, where Green's-function zeros encode nontrivial topological structure. Yet, experimental detection of these zeros represents a challenge. Using exact diagonalization of the one-dimensional Hubbard model with an impurity and Zeeman field, supported by exact analytic results, we show that Green's-function zeros manifest as an in-gap spectral weight in the unitary scattering regime. In this limit, we map the impurity problem onto a doped Mott insulator and identify the resulting in-gap state as a "zeron" excitation which is a localized doublon (holon) for an attractive (repulsive) potential. The zeron spectral weight and its associated zero vanish above a critical Zeeman field. Our results imply that Green's function zeros have in fact already been observed in experiments, and establish impurity and magnetic-field tuning as practical tools for controlling their topology.

Figures (5)

• Figure 1: Schematic figure illustrating the key results: (a) variation of the impurity-induced bound state energy for a Dirac delta impurity potential, in metals and in band insulators showing its divergence in the unitary ($V \rightarrow \infty$) limit. (b) In contrast for a Mott insulator with upper and lower Hubbard bands (UHB, LHB), and Green's function zeros (GFZs), the impurity band asymptotes for $|V|>U$ due to the GFZs forming a zeron excitation. The disks and circles denote the content of initial and final eigenvectors leading to various observed spectral weights -- the filled circles are electron and empty are holes. The arrows indicate single particle addition $cc^\dagger$ and removal $c^\dagger c$ and the pink color indicates the added electron/hole; (c) If a Zeeman field is applied, the zerons and GFZs disappear for a critical field; the arrows indicate spin-down electrons and $k$ is the wave-vector, inset figure showing the spectral function across the Brillouin zone. (d) For $|V|<U$, the critical field can be made arbitrarily small in the unitary limit ($t/U\rightarrow 0$), making it experimentally accessible; (e) Spectral function of a single-hole–doped Mott insulator with $N$ sites, showing two in-gap zeron states above the LHB; in the unitary limit the impurity problem maps onto this doped system, identifying the zeron excitations as a manifestation of GFZs.
• Figure 2: Plot of $A_\sigma(k,\omega)$ and $\Sigma_\sigma(k,\omega)$ vs $V$ for $\sigma=\downarrow\text{ or }\uparrow, k=3/10\times 2\pi$ (no Zeeman field), calculated for a 10-site lattice, highlighting the UHB, LHB, impurity band and GFZ, and also the saturation of the impurity band above the LHB. The bands are colored based on whether they originate from the particle addition or the removal term.
• Figure 3: Plot of $A_\sigma(k,\omega)$ and $\Sigma_\sigma(k,\omega)$ vs $H_b$ for $\sigma=\downarrow, k=5/10\times 2\pi$, and $V/U=-10/15$. The UHB, LHB, and the impurity band are highlighted, showing the disappearance of impurity band for $H_b>H_b^*$. The inset plots are across the BZ for $H_b=0$ and $H_b=0.6>H_b^*$.
• Figure 4: $H_b^*/U$ vs (a)$t/U$ and (b)$V/U$ in the unitarity limit ($U/t\rightarrow\infty$). $H_b^*$ is parabolic on $t$ for $V<U$ and linear for $V>U$. In the unitarity limit,$H_b^*$ is nearly constant close to zero for $V<U$ and linear for $V>U$.
• Figure 5: Probing boundary GFZs: (Left) Projection of bulk-boundary states with impurity. As the bulk is an ordinary insulator without GFZs while the boundary has GFZs associated with interaction-gapped edge states, zerons occur only when the impurity is localized at the boundary. (Right) Proposed STM/STS experiment in Mott-insulating materials with engineered dopants added on Sn/Si(111) Weitering2017Weitering2023, where the tip is used to tune the impurity potential.