Kapteyn Astronomical Institute, University of Groningen, NL

# Star & planet formation: Upcoming opportunities in the space-based infrared

Floris van der Tak<EMAIL_ADDRESS>SRON Netherlands Institute for Space
Research, The Netherlands;

###### Abstract

While ALMA and JWST are revolutionizing our view of star and planet formation
with their unprecedented sensitivity and resolution at submillimeter and near-
IR wavelengths, many outstanding questions can only be answered with
observations in the thermal (mid- and far-) infrared domain. Many of these
questions require space-based observations, to achieve the necessary
sensitivity and/or wavelength coverage. In particular, how do interstellar
clouds develop filamentary structures and dense cores? What are the masses and
luminosities of objects at the earliest stages of star formation? What are the
gas masses of planet-forming disks, and how do these disks disperse during
planet formation? How is refractory and volatile material distributed within
the disks, and how does this evolve with time? This article reviews how
upcoming and planned balloon-borne and space-based telescopes for the mid- and
far-infrared will address these questions, and outlines which further missions
will be needed beyond 2030, when the ELTs will be in full operation.

##  1 Introduction: Why the space-based infrared

Observations in the thermal (mid- and far-) infrared, defined here as the
3–300 $\mu$m wavelength range, are uniquely able to probe the cold obscured
Universe. One area of astrophysics requiring such observations is the study of
the evolution of galaxies from the formation of the first stars (‘cosmic
dawn’, $z$=6–20) via the peak epoch of cosmic star formation (‘cosmic noon’,
$z$=2–3) into the present-day galaxy population (Aravena, this volume). Key
questions include the universality and physical drivers of galaxy scaling
relations such as the ’star formation main sequence’ (e.g., Pearson et al.,
2018), and the origin of dust and heavy elements in galaxies (Henning, this
volume). Atomic fine structure lines in the mid- and far- infrared are key
diagnostics of ISM conditions at redshifts out to $\approx$6 (Ramos Padilla et
al., 2021, 2022).

The second area of astrophysics where thermal infrared observations play a
large role is the physics of galaxies, in particular the so-called baryon
cycle (Van der Tak et al., 2018). Key questions in this area are how the star
formation process depends on galaxy properties, and the role of feedback from
stars and AGN (Saintonge & Catinella, 2022). Spectroscopic maps in the mid-
and far-infrared are crucial to address these questions, both of atomic ([C
ii], [N ii], [O i], [O iii], …) and of molecular (HD, OH, H2O, high-$J$ CO, …)
tracers, as outlined by Madden (this volume) and Rubio (this volume).

The third area is the formation of stars and planets, which is the topic of
this conference and the focus of this article. In the area of star formation,
ALMA has allowed great progress, in particular for the structure of accretion
disks (e.g., Cesaroni et al., 2017; Maud et al., 2019), protostellar
multiplicity (e.g., Tobin et al., 2022), and the chemical structure of pre-
stellar cores (e.g., Caselli et al., 2022). Similarly, breakthroughs are
expected from JWST, if its first results are any guide (e.g., Yang et al.,
2022). However, most ALMA results so far are on individual sources or small
samples, which has the risk of creating a biased view. Larger surveys to build
up statistics and remove selection biases are planned, in particular ATOMS
(Liu et al., 2020), ALMA-IMF (Stutz, this volume) and ALMA-GAL (Fuller, this
volume). Furthermore, several outstanding issues in star formation require
far-infrared observations to be resolved, in particular the very early stages.
Witnessing the formation of molecular clouds out of atomic gas requires wide-
field velocity-resolved maps of the fine structure lines of [C ii], [N ii],
and [O i], as pioneered with SOFIA (Pabst et al., 2019; Kavak et al., 2022)
attempted with the STO-2 balloon mission (Seo et al., 2019) and planned with
its successor GUSTO (Walker et al., 2022). Herschel surveys (Molinari et al.,
2016; Elia et al., 2021) have shown that these clouds develop a filamentary
structure with hubs and branches, but unraveling the mechanism of their
formation also requires wide-field maps of the same fine structure lines at
$\sim$km s-1 spectral resolution (Hacar et al., 2022). Magnetic fields may
also play a role in creating these structures, and polarimetric maps of
arcmin-sized fields are needed to find out. SOFIA has pioneered this field
(Stephens et al., 2022), ground-based single dishes (e.g., CCAT) and balloon-
borne missions (e.g., BlastPol) may take initial steps (Pattle et al., 2022),
but space-based observations are needed to fully address this question (André
et al., 2019). Finally, studying the collapse of clumps inside filaments and
the fragmentation of clumps into cores, which is crucial to understand the
origin of the observed low star formation efficiency and Initial Mass Function
shape, needs sub-arcsecond imaging around 100 $\mu$m, which requires an
interferometer or a large dish in space.

The formation of planets is another area where ALMA has allowed great
progress, both in terms of disk physical structure (Andrews et al., 2018; Van
der Marel et al., 2021) and in terms of disk chemistry (Öberg et al., 2021;
Brunken et al., 2022). Many JWST programs focus on disks, and progress is
expected especially for inner disks (${{}_{<}\atop{{}^{\sim}}}$10 au), where
planets are thought to form. The two facilities complement each other: ALMA
probes the outer and JWST the inner parts of disks. However, while both
telescopes excel in revealing the detailed structure of individual objects,
they struggle to carry out surveys to build up statistics.

Even with JWST and ALMA in operation, far-infrared observations are essential
to address key questions about protoplanetary disks (Kamp et al., 2021). First
among these is the gas mass, which is best probed by the HD molecule, through
its $J$=1–0 and 2–1 transitions at 112 and 56 $\mu$m (Trapman et al., 2017).
Spatially and/or spectrally resolved HD observations of disks are needed to
understand how their gas masses evolve during the process of planet formation.
A full view of the mid- and far-IR spectra of disks is needed to understand
the evolution of the H2O gas and ice abundances, and the distribution of
minerals and ices within the disk; and to link their dust composition to that
of asteroids. See Miotello et al. (2022) for a review of bulk disk properties
and the physical environment in which planets form.

##  2 Outstanding questions and instrumentation needs

While the need for mid- and far-IR observations is clear, specific instrument
requirements strongly depend on the science case at hand. In particular, if
high angular and/or spectral resolution is needed, ground-based telescopes are
the natural choice, as such performance scales with physical size. Large
mirrors are able to push diffraction limits, and large gratings offer high
resolving powers. A case in point is the METIS instrument for the e-ELT, which
just passed FDR and is scheduled for first light in 2027 (Brandl et al.,
2022).

Space missions offer two advantages: superior sensitivity (due to the low
thermal background), and broad wavelength coverage (by lack of atmospheric
absorption). The current state of the art are the Spitzer and Herschel
missions, which respectively offered a small cold mirror (85 cm, 5.5 K) and a
large warm mirror (3.5 m, 80 K). The natural next step is a large
(${{}_{>}\atop{{}^{\sim}}}$2 m) cryogenic (${{}_{<}\atop{{}^{\sim}}}$10 K)
mirror. In the past decade, two such concepts were developed: SPICA in
Europe/Japan (Roelfsema et al., 2018) and OST in the US (Meixner et al.,
2019), neither of which however made it to agency adoption.

All three world-leading space agencies have plans for infrared space missions,
but their timescales and levels of maturity differ greatly. In Japan, JAXA is
considering the Grex-Plus mission, which would offer a 1.2 m mirror cooled to
50 K, a 1,400 arcmin2 camera for the 2–10 $\mu$m range, and an $R$=30,000
spectrometer for the 12–18 $\mu$m band (Inoue et al., 2022). The concept
derives from the SPICA/SMI instrument, and launch is foreseen for the early
2030s.

In Europe, ESA is starting its Voyage 2050
program111https://www.cosmos.esa.int/web/voyage-2050. No mid- or far-IR
concepts have survived the ongoing selection of the M7 mission, with launch in
the late 2030s222https://www.cosmos.esa.int/web/call-for-missions-2021/update-
on-the-f2-and-m7-mission-opportunity. The L5 mission is very likely to be in
the infrared: either a near-IR successor to the Gaia astrometry mission, or a
mid-IR mission for exoplanet characterization. However, neither concept covers
the far-IR nor addresses the above science topics; and in any case, the launch
of L5 is not foreseen until the 2040s.

Third, following the Decadal report, NASA has announced an opportunity for
Probe-class missions, with a cost cap of 1 B$ excluding launch and GO
program333https://explorers.larc.nasa.gov/2023APPROBE/pdf_files/NNH22ZDA008L.pdf.
Up to 30% of the cost may be externally paid (similar to e.g. HST and JWST),
but given its current financial troubles, ESA is unlikely to participate.
Launch of the Probe should be in the early 2030s, which requires a high
technology readiness level at the proposal stage (2023-2025). The announcement
specifies that the Probe should be either an X-ray or a far-IR mission.

The instrumentation needs of a far-IR space telescope strongly depend on the
science case. Galaxy evolution studies generally need maximum sensitivity
($\sim$10-19 W m-2) and survey speed (ideally of contiguous fields with
complementary data), while modest spectral resolution ($\sim$200) is enough to
measure line intensities. In contrast, most studies of the ISM of local
galaxies need a reasonably high mapping speed ($\sim$arcmin/hr) and spectral
resolution ($\sim$3000), while moderate angular resolution ($\sim$10′′) tends
to be sufficient. Infrared studies of Galactic star formation typically
require wide ($\sim$degree) fields and $\sim$km s-1 spectral resolution, while
today’s sensitivity levels are adequate. Planet formation studies require high
($\sim$10-19 W m-2) sensitivity and $\sim$km s-1 spectral resolution, while
survey speed is irrelevant as known sources are observed.

Clearly, the disparity in requirements between the four main areas of mid/far-
infrared astrophysics is too large to bridge by compromise. Adding other areas
where space-based infrared observations are important, in particular Solar
system science and exoplanet characterization, only exacerbates the situation.
Given finite budgets, future infrared missions therefore are unlikely to be
general-purpose observatories like Spitzer and Herschel, but will instead be
optimized to address selected scientific areas.

##  3 NASA opportunities

This section describes the four infrared concepts that are currently being
developed for NASA’s Probe-class opportunity. Note that the detailed
specifications of the telescopes and instruments are likely to evolve as the
concepts mature. For a review of the underlying detector physics, see Staguhn
(this volume).

The PRobe Infrared Mission for Astrophysics (PRIMA; Bradford et al. 2022) is a
cryogenic 2 m telescope which will be confusion limited at wavelengths
${{}_{>}\atop{{}^{\sim}}}$70 $\mu$m. To optimally benefit from the cold
mirror, the team plans to use Kinetic Inductance Detectors (KIDs) with an NEP
of $\sim$10-19 W Hz-1. Such detectors have been demonstrated in the lab
(Baselmans et al., 2022), and space qualification is underway. Two main
instruments are foreseen for PRIMA: a 25–230 $\mu$m imager with polarimetric
capabilities at the long-wavelength end; and a grating spectrometer giving
$R$=60–250 over the 25–330 $\mu$m range. In addition a high-resolution
(Fourier transform or Fabry-Pérot) spectrometer is planned, giving
$R$=3000–5000 at a reduced sensitivity, mapping speed, and spectral coverage.

The science case for PRIMA is broad, but the main focus is on galaxy
evolution. The high sensitivity, spectral capability, and high blind spectral
survey speed are powerful tools to study the coevolution of star formation and
black hole accretion in galaxies (Bisigello et al., 2021), as well as the
buildup of heavy elements and dust at high redshift. The imaging capability
allows surveys of cosmological deep fields, with matching photometry at
optical/UV and radio wavelengths. For local galaxies, PRIMA is well suited for
studies of ISM conditions and dust content. For Galactic star formation, PRIMA
is especially powerful to study protostellar accretion variability (Johnstone
et al., 2022) and the origin of magnetized ISM filaments. For planet
formation, PRIMA’s grating spectrometer is useful to study disk mineralogy,
while gas masses can be measured with the high-resolution mode.

The FIRSST concept (Far-Infrared Spectroscopic Survey Telescope) is also a
cryogenic $\sim$2 m telescope, with a raw sensitivity similar to PRIMA, i.e.,
$\sim$100$\times$ better than Herschel/PACS. Its main instrument is a direct-
detection imaging spectrometer, which in its broad-band mode covers the 30–270
$\mu$m range at $R$=200, suitable to measure fine structure line intensities
in targeted surveys of high-redshift galaxies. The medium-resolution mode
offers $R\sim$2000 which is suitable to study feeding and feedback processes
in local galaxies, as pioneered in OH lines by Sturm et al. (2011). The high-
resolution mode uses VIPAs (virtual phased arrays) to provide $R$=105 over a
10% bandwidth, centered on lines of interest such as HD, [C ii], [O i], H2O,
and HDO. This mode is especally useful to measure the gas masses of
protoplanetary disks (in HD), and to study their dispersal (in [O i]). The
VIPAs are the most unique part of FIRSST, but they still require demonstration
in the lab and space qualification. In addition, FIRSST is planned to have a
heterodyne spectrometer, offering $R$=3$\times$107, with H2O and HDO chemistry
as its main science goals. The added value of this instrument seems somewhat
limited, especially compared to Herschel/HIFI, which had higher angular
resolution. The sensitivity of heterodyne systems is quantum-limited, so that
they do not fully benefit from the capabilities of a cryogenic mirror.

The Single Aperture Large Telescope for Universe Studies (SALTUS; Kim et al.
2022) concept uses an inflatable 20-m non-cryogenic (45 K) telescope with a
deployable boom/torus structure. This design builds on heritage from the 14-m
Inflatable Aperture Experiment (IAE) in 1996 and the successful deployment of
the 6.5-m JWST in early 2022 (see Quach et al. 2021). The sensitivity is
similar to PRIMA and FIRSST, i.e. 10$\times$ better than SPICA, but the
angular resolution is 1.4′′, i.e. 10$\times$ better than PRIMA and FIRSST, and
close to the optical and radio views of galaxies, rather than being an order
of magnitude worse typically. Three instruments are planned: (i) a 4-band KID-
based grating spectrometer covering the 30–300 $\mu$m range at $R$=300, which
is suitable to study the origin of dust and heavy elements in galaxies, as
well as interstellar ices and minerals, and dust in planet-forming disks; (ii)
a set of tunable 8-pixel Hot Electron Bolometer arrays, covering selected
bands in the THz window at $R$=105–106, suitable to study H2O chemistry in
regions of star and planet formation; (iii) a set of tunable SIS arrays
covering the 520–650 $\mu$m band for ISM spectroscopy and the 870–1300 $\mu$m
band to provide long baselines to the Event Horizon Telescope (EHT
Collaboration et al., 2019). The improved angular resolution would increase
the EHT target list from 2 to $\approx$100 sources. Optimizing the orbit of
SALTUS will be a challenge: while the EHT connection requires proximity to
Earth, the other science goals are better served further away, where the
thermal environment is more benign. While SALTUS is not suitable for blind
surveys, its strength will be to observe large samples of individual (point)
sources, e.g. to build up statistics for galaxy evolution studies. With
careful sample construction, the impact of selection bias can be limited.

The SPICE concept (SPace Interferometer for Cosmic
Evolution)444https://asd.gsfc.nasa.gov/spice/ is an interferometer with two
connected elements on a maximum baseline of 36 m, providing an angular
resolution of 0.3′′ at 100 $\mu$m. The 1 m dishes are cooled to 4 K, giving a
line sensitivity of $\sim$4$\times$10-19 W m-2. Spectral coverage is 25–400
$\mu$m at $R$=3000. Such a telescope is especially powerful for planet
formation, as it can spatially resolve gas masses (in HD) as well as H2O gas
and ice abundances for $\sim$100 disks. The concept is also of interest for
galaxy evolution studies, as it measures atomic fine structure line emission
as well as far-IR dust continuum at a resolution matched to optical and radio
surveys. The drawback is the poor instantaneous uv coverage, so that a fully
sampled image of the 1′ field takes $\approx$24 hours to make. Deep
extragalactic fields will require several passes, which limits the statistics
that can be built up during the 3-year lifetime. Observations of brighter
sources such as QSOs, lensed galaxies, and local AGN will be much more
efficient, as are spectra of T Tauri stars and protoplanetary disks. In
continuum, the SPICE team is planning four-band multi-wavelength synthesis,
enhancing its efficiency for e.g. signs of exoplanets in debris disks.

##  4 Conclusions

While the decommissioning of SOFIA is a setback, JWST is a success, and in
$\sim$6 years, METIS will provide superb angular and spectral resolution in
the ground-based 2.9–14 $\mu$m windows. For planet formation, significant
steps are expected for the physical structure and chemical evolution of
protoplanetary disks, including signatures of protoplanets. The METIS
instrument will also be powerful for high-mass star formation, especially the
nature of accretion flows and disks, and the formation of stellar clusters.

Beyond 2030, the future for mid- and far-infrared studies of star and planet
formation is less clear, although several ideas exist. If adopted, JAXA’s
Grex-Plus mission will sharpen our view on planet formation, in particular the
structure, kinematics, and composition of inner disks. Later in the 2030s,
NASA’s Probe mission will be powerful for star and planet formation, provided
an infrared concept is selected. For this field, SALTUS is the preferred
option, as it provides high angular and spectral resolution. The deep high-
resolution spectra from FIRSST will be useful to measure disk masses and water
abundances. SPICE can provide high-resolution dust maps of protostellar
clusters and ice maps of protoplanetary disks. With PRIMA, the dust and ice
mineralogy of disks can be studied, as well as magnetized interstellar
filaments and protostellar accretion variability.

The Probe concepts differ in their telescope optics: PRIMA and FIRSST are made
for mapping, while SALTUS and SPICE aim at point sources. They also have
different technological challenges: SALTUS the deployment, and the others the
cooling of one large or several small mirrors. In terms of detectors, KIDs and
VIPAs need further development, while heterodyne systems are ready for space
(e.g., Gan et al., 2021).

Parallel to these developments, suborbital platforms will be valuable to carry
out specific single-instrument science cases. Stratospheric balloon missions
such as ASTHROS (and, if adopted, POEMM) will also be useful for prototyping
and space qualification of new instrument and detector technology. Significant
work is needed to turn these concepts into reality. We conclude that the near-
and mid-term future for the near- and mid-IR is bright, but that the far
future is more uncertain, especially for the far infrared.

Acknowledgements: The author thanks Inga Kamp (Kapteyn), Peter Roelfsema and
Lingyu Wang (SRON), and Lee Mundy (Maryland) for helpful comments.

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#  A protected spin-orbit induced absorption divergence in distorted Landau
levels

Dominik Sidler<EMAIL_ADDRESS>Max Planck Institute for the Structure and
Dynamics of Matter and Center for Free-Electron Laser Science, Luruper
Chaussee 149, 22761 Hamburg, Germany  The Hamburg Center for Ultrafast
Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany  Michael Ruggenthaler
<EMAIL_ADDRESS>Max Planck Institute for the Structure and
Dynamics of Matter and Center for Free-Electron Laser Science, Luruper
Chaussee 149, 22761 Hamburg, Germany  The Hamburg Center for Ultrafast
Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany  Angel Rubio
<EMAIL_ADDRESS>Max Planck Institute for the Structure and Dynamics of
Matter and Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761
Hamburg, Germany  The Hamburg Center for Ultrafast Imaging, Luruper Chaussee
149, 22761 Hamburg, Germany  Center for Computational Quantum Physics,
Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA  Nano-Bio
Spectroscopy Group, University of the Basque Country (UPV/EHU), 20018 San
Sebastián, Spain

###### Abstract

The effect of spin-orbit (and Darwin) interaction on a 2D electron gas subject
to a radial symmetric, inhomogeneous $1/r$-magnetic field is discussed
analytically in a perturbative and non-perturbative manner. For this purpose,
we investigate the radial Hall conductivity that emerges from an additional
homogeneous electric field perturbation perpendicular to the 2D electron gas,
which solely interacts via spin-orbit coupling. Numerical calculations of the
absorptive spin-orbit spectra show for an ideal InSb electron gas a behaviour
that is dominated by the localized (atomic) part of the distorted Landau
levels. In contrast, however, we also find analytically that a (non-local)
divergent static response emerges for Fermi energies close to the ionization
energy in the thermodynamic limit. The divergent linear response implies that
the external electric field is entirely absorbed outside the 2D electron gas
by induced radial spin-orbit currents, as it would be the case inside a
perfect conductor. This spin-orbit induced polarization mechanism depends on
the effective $g^{*}$-factor of the material for which it shows a critical
behaviour at $g^{*}_{c}=2$, where it abruptly switches direction. The
diverging absorption relies on the presence of degenerate energies with
allowed selection rules that are imposed by the radial symmetry of our
inhomogeneous setup. We show analytically the presence of a discrete Rydberg-
like band structure that obeys these symmetry properties. While in our case
this structure turns out to be of minor relevance, it is a promising property,
which may facilitate the experimental realization in the future. In a last
step, we investigate the robustness of the spectra by solving analytically the
Dirac equation expanded up to order $1/(mc)^{2}$. We find that the distorted
Landau-levels, and thus the divergent spin-orbit polarization, remain
protected with respect to slow changes of the applied $1/r$-magnetic field.

††preprint: AIP/123-QED

##  I Introduction

The Hall effect has been a cornerstone of solid state physics for almost 150
years since its first discovery.[1] The basic mechanism, that a charge
currents get deflected in a perpendicular magnetic field, does not only hold
on a macroscopic scale, but numerous seminal experiments have been designed,
where delicate quantum features of the materials become apparent in all sorts
of different Hall setups. For example, the integer[2] and fractional[3, 4]
quantum Hall effect are potentially the most famous among those experiments.
With growing experimental capabilities, more delicate experimental conditions
have been realized over time, where even the quantization of the electro-
magnetic fields can become decisive, as for example under ultra-strong
coupling conditions in optical cavities. [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]
All of which experiments have in common that they fundamentally rely on the
emergence of quantized Landau levels for a two dimensional electron gas, [15,
16] which provides a paradigmatic analytic model for materials in an (at least
locally) homogeneous magnetic field.

Only very recently, a simple analytic solution was provided for a non-
interacting 2D electron gas in a truly inhomogeneous magnetic field, by
assuming a rotationally symmetric and $1/r$-decaying magnetic field with
respect to the distance $r$ from the origin.[17] It has been shown that this
$B$-field ”impurity” introduces strongly distorted Landau levels that can
fundamentally alter physical observables, compared with Landau-level physics
or the homogeneous electron gas.[17] When creating distorted Landau levels
with a strong spin-dependency, it is now tempting to ask what spin-orbit
effects could emerge in such a setting. In the subsequent manuscript we will
uncover those basic mechanisms analytically and numerically for the spin-orbit
Hall conductivity. Notice that our chosen setup is fundamentally different to
the anomalous Hall effect in ferromagnetic materials[18] or the related spin
Hall effect,[19] which also emerge from spin-orbit interaction and are of
particular interest for the flourishing field of spintronics.[20] In those
cases, three dominant mechanisms are known: The intrinsic scattering
contributions[21], which emerge from the topological Fermi liquid
property,[22] and two extrinsic contributions (skew [23, 24] and side jump
scattering[25]) that are related to the scattering at a scalar potential well
(e.g. charged impurity). All of these effects can jointly be described by
Kubo’s linear response theory,[26] which we will also rely on subsequently.
However, in our case, the significant spin-orbit effects will not emerge from
a scalar potential, instead they arise from the subtle interplay of a magnetic
field-impurity (affecting the Zeeman interaction) with its corresponding
vector-potential that couples to the momentum operator.

The manuscript is structured as follows: First, we shortly recapitulate the
recently introduced analytical solution for distorted Landau levels in an
inhomogeneous magnetic field. Based on this, we determine the radial Hall
conductivity that emerges from spin-orbit interaction when applying a small
electric field perturbation perpendicular to the 2D electron gas. In the next
section, we focus on the static (DC) absorptive features of the radial Hall
conductivity, which shows local as well as non-local (diverging) features
depending on the Fermi energy. Furthermore, we discuss the influence of the
effective $g^{*}$-factor on the results, which gives rise to different phases.
Afterwards, we focus on the robustness of the previously derived results with
respect to slow variations of the externally applied magnetic fields, by
solving analytically the corresponding quasi-static eigenvalue problem.
Finally, we summarize our results and assess them with respect to their
potential experimental verification.

##  II Radial Hall conductivity from spin-orbit interaction

The starting point of our investigation of transversal spin-orbit Hall
conductivity effects in a non-interacting 2D electron gas subject to a
inhomogeneous magnetic field $\bm{B}(\bm{r})$, we use the Pauli Hamiltonian
[27, 17]

$\displaystyle\hat{H}_{0}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\frac{\hat{\bm{\Pi}}_{j}^{2}}{2m^{*}}-\frac{g^{*}q\hbar}{4m^{*}}\hat{\bm{\sigma}}_{j}\cdot\bm{B}(\bm{r}_{j}).$
(1)

The recently introduced bound state solution is given by,[17]

$\displaystyle
E_{n,l,s}=\frac{q^{2}A_{\phi}^{2}}{2m^{*}}\bigg{(}1-\bigg{[}\frac{2l+g^{*}s}{2n+1}\bigg{]}^{2}\bigg{)},\
n\geq l,\ l+g^{*}s/2>0.$ (2)

with Hydrogen-like eigenstates,

$\displaystyle\Psi_{n,l,s}(t)=\frac{1}{\sqrt{N_{n,l,s}}}e^{il\phi}e^{-\frac{x(r)}{2}}x^{l}(r)L_{n-l}^{2l}(x(r))\chi(s),$
(3)

which are written in terms of generalized Laguerre polynomials $L_{w}^{\nu}$.
We have introduce an energy dependent radial scaling factor,

$\displaystyle x(r):=\frac{2qA_{\phi}}{\hbar}\frac{2l+g^{*}s}{2n+1}r,$ (4)

and the normalization constant,

$\displaystyle N_{n,l,s}$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\int_{0}^{\infty}\Psi_{n,l,s}^{*}\Psi_{n,l,s}rdrd\phi$
(5) $\displaystyle=$ $\displaystyle
2\pi\frac{(n+l)!}{(n-l)!}(2n+1)\bigg{(}\frac{2qA_{\phi}}{\hbar}\frac{2l+g^{*}s}{2n+1}\bigg{)}^{-2}.$

Notice that the inequality conditions for the allowed quantum numbers
distinguishes the bound states from the unknown continuum solution, for which
$l+g^{*}s/2\leq 0$ holds.[17] The canonical momentum operator is defined as
$\hat{\bm{\Pi}}_{j}:=\hat{\bm{p}}_{j}-q\bm{A}(\bm{r}_{j})$ and the effective
electron mass is indicated by $m^{*}$ with negative unit charge $q=-e$. The
integer quantum numbers are labeled as $n,l$, where $l$ corresponds to the
angular momentum quantum number. Furthermore, the electron spin-half quantum
numbers are indicated by $s$. We denote the usual position operator of
particle $j$ as $\bm{r}_{j}$ and the corresponding momentum operator as
$\hat{\bm{p}}_{j}$. The anisotropic external vector potential is denoted by
$\bm{A}(\bm{r})$ and an effective g-factor is introduced as $g^{*}$, which in
our definition explicitly excludes modifications due to the effective mass of
the electrons, i.e. it only accounts for relativistic corrections or other
higher order host material effects (e.g. influence of Coulomb interaction or
applied magnetic fields). Those mechanisms will not be discussed further. The
Pauli vector for electron $j$ is labeled by $\bm{\sigma}_{j}$. In cylindrical
coordinates the vector potential and corresponding magnetic field are given
as,[17]

$\displaystyle\bm{A}(\bm{r})$ $\displaystyle:=$ $\displaystyle
A_{\phi}\bm{e}_{\phi}$ (6) $\displaystyle\bm{B}(\bm{r})$ $\displaystyle=$
$\displaystyle\bm{\nabla}\wedge\bm{A}=\frac{1}{r}\frac{\partial(rA_{\phi})}{\partial
r}\bm{e}_{z}=\frac{A_{\phi}}{r}\bm{e}_{z}.$ (7)

such that $A_{\phi}$ is constant, with
$\bm{e}_{\phi}=\frac{1}{r}(-y\bm{e}_{x}+x\bm{e}_{y})$ indicating the unit
vector along $\phi$-direction. Notice, our inhomogeneous magnetic field
$\bm{B}(\bm{r})$ corresponds to a radial external current density of the form
$J_{\mathrm{ext}}=\frac{\bm{\nabla}\wedge\bm{B}}{\mu_{0}}=\frac{A_{\phi}}{\mu_{0}}\frac{1}{r^{2}}\bm{e}_{\phi}$
if considered in free space, i.e., it does not correspond to a magnetic
monopole. Eq. (2) gives rise to a distorted Landau level structure (see Fig. 1
for different $g^{*}$-factors), which introduces a plethora of features such
as infinite degeneracies, charge or current oscillations as well as Hall
conductivity phase transitions for in-plane static electric field
perturbations which have been previously been discussed in Ref. 17. Notice
also that likewise, hydrogen related spectra emerge for two dimensional
magnetic quantum dots.[28, 29]

(a) $g^{*}=1$

(b) $g^{*}=2$

(c) $g^{*}=3$

Figure 1: Radially resolved bound state eigenvalue spectrum for an InSb 2D
electron gas assuming different effective $g^{*}$ factors and
$A_{\phi}=-0.0006$ [a.u.]. A distorted Landau level structure becomes apparent
caused by the applied $1/r$-decaying magnetic field (orange).[17] The up spin
(red) and down spin (blue) dependencies are visualized as well as the
ionization energy at $E_{A^{2}}=q^{2}A_{\phi}^{2}/(2m^{*})$ (dashed black
line). An increase in the externally applied vector potential $A_{\phi}$ will
result in an increase of the ionization threshold and simultaneously it will
squeeze the states spatially more densely, i.e., it reduces the expected
radial distance $r_{n,l,s}$ with respect to the origin.

In contrast to established properties within the context of Hall conductivity
for in plane perturbations, we will investigate the impact of a time-
dependent, homogeneous electric field $\mathbb{E}_{z}(t)=E_{z}e^{-{i\mkern
1.0mu}\omega t}\bm{e}_{z}$ perpendicular (!) to the 2D electron gas. For this
purpose, we apply Kubo’s linear response formalism to our setup. The leading
order in-plane effects from $\mathbb{E}_{z}(t)$ are induced by the spin-orbit
(and Darwin) interaction (according to the Foldy-Wouthuysen expansion of the
Dirac equation up to order $1/(m^{*}c)^{2}$). The corresponding time-dependent
perturbing Hamiltonian can be written as, [27]

$\displaystyle\hat{H}_{1}(t)$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{N}\frac{q\hbar}{8m^{*2}c^{2}}\Big{[}(\hat{\bm{p}}_{j}-q\bm{A}(\bm{r}_{j}))\cdot\big{(}\hat{\bm{\sigma}}_{j}\times\mathbb{E}_{z}(t)\big{)}+\big{(}\hat{\bm{\sigma}}_{j}\times\mathbb{E}_{z}(t)\big{)}\cdot(\hat{\bm{p}}_{j}-q\bm{A}(\bm{r}_{j}))\Big{]}$
(8) $\displaystyle=$
$\displaystyle-\sum_{j=1}^{N}\frac{q\hbar}{4m^{*2}c^{2}}\Big{[}\big{(}\sigma_{y}(\cos(\phi)\bm{e}_{r}-\sin(\phi)\bm{e}_{\phi})-\sigma_{x}(\sin(\phi)\bm{e}_{r}+\cos(\phi)\bm{e}_{\phi})\big{)}\Big{]}_{j}\cdot\hat{\bm{\Pi}}_{j}E_{z}(t)$
(9) $\displaystyle=:$ $\displaystyle-\hat{d}_{1}E_{z}(t)$ (10)

where the Darwin contribution $\propto\mathrm{div}\bm{E}$ vanishes due to
assuming a homogeneous electric field. From Eq. (9) we immediately notice that
the perturbing Hamiltonian starts to mix the two different spin channels by
$\sigma_{x},\sigma_{y}$. This has not been the case for our $\hat{H}_{0}$,
which solely depends on the identity Pauli matrix $\sigma_{z}$ instead. This
has important implications, since it ensures that the only radially induced
current $\langle\hat{j}^{r}(t)\rangle$ must be induced by the spin-orbit
current operator $\hat{\mathbb{j}}^{\rm SO}_{j}:=-\hbar
q/(4m^{*2}c^{2})(\hat{\bm{\sigma}}_{j}\times\bm{E})$[30] within Kubo’s linear
response formalism. In contrast, paramagnetic, diamagnetic and magnetization
currents cannot induce radial currents perturbatively, since they are all
spin-preserving operators. Notice, this property, i.e.,
$\hat{j}^{r}=\hat{\mathbb{j}}^{\rm SO}\cdot\bm{e}_{r}$, is universally true
for any radially symmetric magnetic fields with a superimposed homogeneous
electric field perturbation, which may simplify experimental realizations.
Applying the Kubo linear response relation, the total radial current can be
written as

$\displaystyle\langle\hat{j}^{r}(t)\rangle$ $\displaystyle=$
$\displaystyle\langle\hat{j}^{r}(t)\rangle_{0}-{i\mkern
1.0mu}\int_{0}^{t}dt^{\prime}\big{\langle}[\hat{j}^{r}(t),\hat{H}_{1}(t)]\big{\rangle}_{0}$
(11) $\displaystyle=$
$\displaystyle-\int_{-\infty}^{\infty}dt^{\prime}\hat{\sigma}^{rz}(t-t^{\prime})E_{z}(t^{\prime}),$
(12)

where the first term becomes exactly zero.[17] The $\langle\rangle_{0}$
indicates a ground state depending at temperature $T$. Additionally we
introduced the radial Hall conductivity,

$\displaystyle\sigma^{rz}(t-t^{\prime})=-{i\mkern
1.0mu}\theta(t-t^{\prime})\big{\langle}[\hat{j}^{r}(t),\hat{d}_{1}(t)]\big{\rangle}_{0},$
(13)

with unit step function $\theta(t)$. Taking the Laplace transform with respect
to time transforms the convolution integral into,

$\displaystyle j^{r}(\omega,E_{F},T)$ $\displaystyle=$
$\displaystyle-\sigma^{rz}(\omega,E_{F},T)E_{z}(\omega).$ (14)

The frequency dependent (AC) transversal conductivity can be written at
temperature $T$,[31, 32]

$\displaystyle\sigma^{rz}(\omega,E_{F},T)$ $\displaystyle=$
$\displaystyle-{i\mkern
1.0mu}\sum_{a,b}f(E_{a})(1-f(E_{b}))\bigg{[}\frac{\bra{a}\hat{j}^{r}\ket{b}\bra{b}\hat{d}_{1}\ket{a}}{E_{a}-E_{b}+\hbar\omega+{i\mkern
1.0mu}\Gamma}+\frac{\bra{a}\hat{d}_{1}\ket{b}\bra{b}\hat{j}^{r}\ket{a}}{E_{a}-E_{b}-\hbar\omega-{i\mkern
1.0mu}\Gamma}\bigg{]},$ (15)

with Fermi distribution $f(E_{a}):=1/(e^{(E_{a}-E_{F})/k_{B}T}+1)$ and
$\Gamma>0$. Notice that we implicitly restrict our subsequent analysis to
bound state currents / conductivties, since the continuum states giving rise
to free currents are not known. However, restricting the $\sum_{a,b}$ to bound
states will not affect the validity of our calculations, as we will see later.
The frequency dependent absorptive (imaginary) part of the spin-orbit
conductivity can be found by the identity $1/(x+{i\mkern
1.0mu}\Gamma)=P(1/x)-{i\mkern 1.0mu}\pi\delta(x)$,[32] for which we find

$\displaystyle\Im(\sigma^{rz}(\omega,E_{F},T))$ $\displaystyle=$
$\displaystyle\pi{i\mkern 1.0mu}\sum_{a,b}f(E_{a})(1-f(E_{b}))\cdot$ (16)
$\displaystyle\bigg{[}\frac{\bra{a}\hat{j}^{r}\ket{b}\bra{b}\hat{d}_{1}\ket{a}\Gamma}{(E_{a}-E_{b}+\hbar\omega)^{2}+\Gamma^{2}}-\frac{\bra{a}\hat{d}_{1}\ket{b}\bra{b}\hat{j}^{r}\ket{a}\Gamma}{(E_{b}-E_{a}+\hbar\omega)^{2}+\Gamma^{2}}\bigg{]}$

assuming a Lorentzian lineshape with a finite lifetime $\Gamma>0$ for the
$\delta$-distribution.

In a next step we compute the AC absorptive Hall conductivity under
experimentally plausible conditions according to Eq. (16). For this purpose we
focus on a 2D electron gas made of InSb, which possesses an extremely low
effective electron mass of $0.015$ [a.u.] that makes it an ideal candidate to
investigate spin-orbit effects in magnetic fields.[33]. For our calculations
we assume a temperature of $20$ [mK] and a static vector potential of
$A_{\phi}=-6\cdot 10^{-4}$, which gives rise to a $1/r$ decaying magnetic
field in z-direction on a mT-scale for radial distances from the origin in the
$\mu m$-regime. The homogeneous perturbing electric field along the z-axis is
chosen as $E_{z}=-6\cdot 10^{-6}$ [a.u] which corresponds to $-3.1$
[V/$\mu$m]. Furthermore, we assume relatively long lived states by setting
$\tau=1$ [ms] with corresponding sharp lineshape $\Gamma=1/\tau$ that
partially compensates for the small prefactors of spin-orbit effects. The
numerical evaluation of Eq. (16) simplifies considerably, when considering
that the matrix element $\bra{a}\hat{d}_{1}\ket{b}$ imposes that only spin-
flips can have a non-zero transition matrix elements, i.e.

$\displaystyle s_{a}=-s_{b},$ (17)

which leaves us with the spin-orbit current operator only,

$\displaystyle\hat{j}^{r}=\hat{j}^{\rm SO,r}=-\frac{\hbar
q}{4m^{*2}c^{2}}\bm{\sigma}_{j}\times\bm{E}_{z}(0)\cdot\bm{e_{r}}=-\frac{\hbar
q}{4m^{*2}c^{2}}(\sigma_{y}\cos\phi-\sigma_{x}\sin\phi)E_{z}.$ (18)

Eventually, Eqs. (17) & (18) jointly determine the angular selection rule

$\displaystyle l_{a}-l_{b}=\pm 1.$ (19)

Notice that those angular and spin selection rules remain valid for any radial
symmetric magnetic fields. Furthermore, it also ensures that our restriction
to bound states is valid, since any overlap with the (unknown) scattering
states of the continuum must vanish except for the smallest angular momentum
quantum numbers $\min_{l}(l+g^{*}s/2)>0$, which separate the bound states from
the continuum.[17] Eventually one can find a simplified expression for
$\Im(\sigma^{rz}(\omega,E_{F},T))$ as derived in App. A, which was implemented
for our numerical calculations. The resulting absorptive radial spin orbit
Hall conductivity spectra is shown in Fig (2) for different effective
$g^{*}$-factors. Careful inspection of the AC spectrum reveals that the main
contributions stem from localized electrons close to the origin, i.e., varying
the finite number of numerical states does not change the strongest absorption
peaks, which indicates that the overlap integrals become smaller, the more
delocalized the eigenstates become. Thanks to this localisation effect we can
converge our numerical calculations. This also suggests that the local /
atomic nature of the AC absorption is mainly related to the dacaying magnetic
field and the infinite ranging vector potential plays a minor role. We find
that for small $g^{*}=1$ significant absorption lines only occur for
relatively low frequencies, whereas around $g^{*}=2$ a characteristic pattern
occurs across the entire frequency regime. For the larger $g^{*}=3$ only a few
significant absorption peaks occur. However, the range of allowed energetic
transitions is significantly increased since the energetically lowest
eigenvalue can become negative, while the ionization energy $E_{A^{2}}$
remains constant independently of $g^{*}$.

At this point, we would also like to mention two additional properties of the
induced radial spin-orbit Hall current that are immediately evident from the
analytics. First, the orientation of the perturbing field does not influence
the radial current response, since

$\displaystyle
j^{r}=-\sigma^{rz}(E_{z},E_{F},T)E_{z}\propto\frac{E_{z}^{2}}{m^{*4}c^{4}}$
(20)

and furthermore it is suppressed by $m^{*4}c^{4}$, which suggests a tiny
effect overall. However, as we have seen in Fig. 2, a significant radial spin-
orbit conductivity may still emerge for small effective electron masses and
allowed transitions with long-lived states. Those aspects should be taken into
account for a potential verification of the predicted effects.

(a) $g^{*}=1$

(b) $g^{*}_{c}=2$

(c) $g^{*}=3$

Figure 2: Absorptive radial spin-orbit Hall conductivity spectra with respect
to different Fermi energies $E_{F}$ and driving frequencies $\omega$ of the
external $E_{z}$-field perturbation. The shown spectral features for our InSb
2D electron gas are mainly caused by the local discretization of the 2D
electron gas in the vicinity of the magnetic field divergence at the origin.
In contrast, the more delocalized electronic states, that are further away
from the origin, do not contribute significantly to the spectral features,
which is a consequence of vanishing overlap matrix elements.

##  III Divergent absorption in static limit $\omega\rightarrow 0$

In a next step, we investigate the zero frequency limit
$\Im(\sigma^{rz}_{0}(E_{F},T))=\lim_{\omega\rightarrow
0}\Im(\sigma^{rz}(\omega,E_{F},T))$ of our setup, which shows particularly
interesting features that originate from allowed transitions of degenerate
states $E_{a}=E_{b}$. In the static limit Eq. (16) reduces to,

$\displaystyle\Im(\sigma^{rz}_{0}(E_{F},T))$ $\displaystyle=$
$\displaystyle\pi{i\mkern
1.0mu}\sum_{a,b}f(E_{a})(1-f(E_{b}))\bigg{[}\frac{\bra{a}\hat{j}^{r}\ket{b}\bra{b}\hat{d}_{1}\ket{a}\Gamma}{(E_{a}-E_{b})^{2}+\Gamma^{2}}-\frac{\bra{a}\hat{d}_{1}\ket{b}\bra{b}\hat{j}^{r}\ket{a}\Gamma}{(E_{b}-E_{a})^{2}+\Gamma^{2}}\bigg{]}.,$
(21)

which is evaluated for different $g^{*}$-factors and Fermi energies at $T=20$
[mK] of our InSb 2D electron gas in Fig. (3). Numerics reveals that overall
the DC limit of the spin-orbit absorptive conductivity remains small, as one
expects from Eq. (20). However, close inspection of $g^{*}$-factors in the
vicinity of the critical non-relativistic $g^{*}_{c}:=2$ value reveal a phase
transition (i.e. sharp sign flip). In other words, by changing $g^{*}$, the
external electric field will be absorbed by either radially in-flowing or out-
flowing currents. Similarly to the AC case, we reach numerical convergence
only, with a finite number of states, thanks to the localization of the
significant overlap integrals. However, we notice that this is no longer true
for Fermi energies approaching the ionization energy
$E_{A^{2}}:=\lim_{n\rightarrow\infty}E_{n,l,s}=q^{2}A^{2}_{\phi}/(2m^{*})$,
where the conductivity starts to scale with the finite system size.

In a next step, we try to better understand the aforementioned numerical
absorption and scaling features by analytical considerations in the zero
temperature limit,

$\displaystyle\Im(\sigma^{rz}(\omega,E_{F},0))$ $\displaystyle=$
$\displaystyle\pi{i\mkern 1.0mu}\sum_{E_{a}<E_{F}\leq
E_{b}}\bigg{[}\frac{\bra{a}\hat{j}^{r}\ket{b}\bra{b}\hat{d}_{1}\ket{a}\Gamma}{(E_{a}-E_{b}+\hbar\omega)^{2}+\Gamma^{2}}-\frac{\bra{a}\hat{d}_{1}\ket{b}\bra{b}\hat{j}^{r}\ket{a}\Gamma}{(E_{b}-E_{a}+\hbar\omega)^{2}+\Gamma^{2}}\bigg{]}$
(22) $\displaystyle=$
$\displaystyle-\pi\Gamma\frac{\hbar^{2}q^{2}E_{z}}{16m^{*4}c^{4}}\sum_{E_{a}<E_{F}\leq
E_{b}}\bra{R_{a}}\ket{R_{b}}\cdot$
$\displaystyle\begin{cases}+\Big{[}\frac{\bra{R_{b}}\frac{-\hbar
l_{a}}{r}+qA_{\phi}-\hbar\partial_{r}\ket{R_{a}}}{(E_{a}-E_{b}+\hbar\omega)^{2}+\Gamma^{2}}+\frac{\bra{R_{a}}\frac{-\hbar
l_{b}}{r}+qA_{\phi}+\hbar\partial_{r}\ket{R_{b}}}{(E_{b}-E_{a}+\hbar\omega)^{2}+\Gamma^{2}}\Big{]}\delta_{l_{a}-l_{b},1}\delta_{s_{a},\downarrow}\delta_{s_{b},\uparrow}&\text{for
}0<g^{*}<2\\\ 0&\text{for }g^{*}=2\\\ -\Big{[}\frac{\bra{R_{b}}\frac{-\hbar
l_{a}}{r}+qA_{\phi}+\hbar\partial_{r}\ket{R_{a}}}{(E_{a}-E_{b}+\hbar\omega)^{2}+\Gamma^{2}}+\frac{\bra{R_{a}}\frac{-\hbar
l_{b}}{r}+qA_{\phi}-\hbar\partial_{r}\ket{R_{b}}}{(E_{b}-E_{a}+\hbar\omega)^{2}+\Gamma^{2}}\Big{]}\delta_{l_{a}-l_{b},-1}\delta_{s_{a},\uparrow}\delta_{s_{b},\downarrow}&\text{for
}g^{*}>2\end{cases},$

where the different cases with respect to $g^{*}$ follow the selection rules
of Eq. (A) in combination with an exchange of the energetic order, i.e.
$E_{a}\lesseqqgtr E_{b}\mapsto E_{a}\gtreqqless E_{b}$, when crossing the
critical value $g^{*}_{c}=2$. While Eq. (III) already suggests a phase
transition, further insights can be achieved by investigating the transition
matrix elements in the asymptotic limit at the ionization threshold, i.e. by
taking $n\rightarrow\infty$ (i.e. energies closest to $E_{A^{2}}$) and
considering states $E_{a}\overset{n\rightarrow\infty}{\rightarrow}E_{b}=E_{F}$
(see App. B):

$\displaystyle\bra{R_{a}}\ket{R_{b}}$ $\displaystyle=$
$\displaystyle\bra{R_{b}}\ket{R_{a}}\overset{n\rightarrow\infty}{=}-1$ (24)
$\displaystyle\bra{R_{a}}\frac{1}{r}\ket{R_{b}}$ $\displaystyle=$
$\displaystyle\bra{R_{b}}\frac{1}{r}\ket{R_{a}}\overset{n\rightarrow\infty}{=}0$
(25)
$\displaystyle\lim_{n\rightarrow\infty}\bra{R_{a}}\partial_{r}\ket{R_{b}}$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\bra{R_{b}}\partial_{r}\ket{R_{a}}=\frac{1}{2}.$
(26)

This suggests for the DC limit,

$\displaystyle|\lim_{n\rightarrow\infty}\Im(\sigma^{rz}_{0}(E_{F,n},0))|$
$\displaystyle>$
$\displaystyle\left|-\pi\Gamma\frac{\hbar^{2}q^{3}E_{z}A_{\phi}}{8m^{*4}c^{4}}\sum_{E_{a}<E_{F}=E_{b}}\cdot\begin{cases}\frac{1}{(E_{a}-E_{b})^{2}+\Gamma^{2}}\delta_{l_{a}-l_{b},1}\delta_{s_{a},\downarrow}\delta_{s_{b},\uparrow}&\text{for
}0<g^{*}<2\\\ 0&\text{for }g^{*}=2\\\
-\frac{1}{(E_{a}-E_{b})^{2}+\Gamma^{2}}\delta_{l_{a}-l_{b},-1}\delta_{s_{a},\uparrow}\delta_{s_{b},\downarrow}&\text{for
}g^{*}>2\end{cases}\right|,$

which implies

$\displaystyle\lim_{E_{F}\rightarrow E_{A^{2}}}\Im(\sigma^{rz}_{0}(E_{F},0))$
$\displaystyle=$
$\displaystyle-\pi\Gamma\frac{\hbar^{2}q^{3}E_{z}A_{\phi}}{8m^{*4}c^{4}}\left\\{\begin{array}[]{lr}\infty&\text{for
}0<g^{*}<2\\\ 0&\text{for }g^{*}=2\\\ -\infty&\text{for
}g^{*}>2\end{array}\right..$ (30)

To proof that the summation in Eq. (III) indeed diverges in the thermodynamic
limit, we need to find an infinite number of state pairs $\ket{a},\ket{b}$
that obey $E_{a}<E_{F}\leq E_{b}$. This can be seen by assuming
$g^{*}=2+2\epsilon,\ \epsilon>0$ setting $E_{F}=E_{b}$ and solving
$E_{n_{a}+k_{c},l_{a},1/2}=E_{b}=E_{n_{a},l_{a}+1,-1/2}=E_{F}\overset{n_{a}\rightarrow\infty}{\rightarrow}E_{A}^{2}$
for integer $k_{c}$ which yields

$\displaystyle
k_{c}=\frac{\epsilon(2n_{a}+1)}{2l+1-\epsilon}\overset{n_{a}\rightarrow\infty}{\rightarrow}\infty,\
0<\epsilon<2l+1,$ (31)

where an analogous argument holds for the case $\epsilon<0$. An unbounded
$k_{c}$ implies that the summation in Eq. (30) indeed diverges, i.e. the
number of contributing states grows at least with $k_{c}$, since they are
either strictly positive or negative. We consider the resulting divergent DC
absorption by spin-orbit interaction at the ionization threshold as one of the
main results of this work. In other words, Eq. (30) suggests that a (small)
static, perpendicular electric field will be screened entirely by spin-orbit
effects for Fermi energies approaching the ionization threshold! Depending on
the $g^{*}$-factor, radially in- or out-flowing bound state currents will
cause this effect, which can be re-interpreted as a longitudinal polarization
effect by looking at the Maxwell equations. From
$\bm{\nabla}\times\bm{B}=\mu_{0}(\bm{J}_{b}+\bm{J}_{b}+\epsilon_{0}\partial_{t}\bm{E})$
where $\bm{J}_{b}$ indicates bound current density and $\bm{J}_{f}$ the free
current density. The free currents can be safely discard for our setting as
mentioned previously. We then notice that the curl of the magnetic field
cannot account for the radial spin-orbit currents, which automatically implies
that the spin-orbit current density $j_{r}(r)$ effectively correspond to a
longitudinal polarization $P_{r}$ of the system in the macroscopic formulation
of the Maxwell equations, i.e, $\partial_{t}P_{r}=j_{r}=-\sigma^{rz}E_{z}$. In
other words, the divergent absorptive (imaginary) part of $\sigma^{rz}$
indicates that the external (perpendicular) electric field is entirely
screened, as it would be the case inside (!) a perfect conductor. This
screening effect emerges in our setup merely as a consequence of the constant
vector potential $A_{\phi}$, i.e. it is an Aharonov-Bohm-like behaviour for
infinite degeneracies in the thermodynamic limit. Notice that the contributing
states for $E_{F}\rightarrow E_{A^{2}}$ can be considered as radially de-
localized[17]. However, the observed divergence is not only an effect of
delocalization, but rather caused by the distorted Landau-level structure,
which is a consequence of delicate interplay between the vector potential and
the magnetic field (Zeeman interaction).[17] The resulting accumulation point
at the ionization energy threshold can be occupied by any state with finite
angular momentum in the thermodynamic limit (setting $n\rightarrow\infty$).
Hence, this highly degenerate state of matter will at least theoretically be
very sensitive towards all sorts of external DC perturbations. However,
clearly, for any experimental realization things will be much more complex.
For example, a perfect constant $A_{\phi}$-potential, i.e. a perfect
$1/r$-magnetic field is not feasible, which in practice means that states far
enough away from the origin will always belong to the continuum solution. In
other words, finite system sizes will reduce the effect and complicate the
experimental verification. Furthermore, beyond linear response effects may in
principle lift the contributing energetic degeneracies. In addition, finite
lifetimes and thermal effects for electronic fillings up to (almost) the
continuum limit (at least for our ideal setup, not involving any band gaps)
may also hamper an experimental realization. Nevertheless, we believe
distorted Landau levels should emerge in experiments with strong non-local
behaviour, provided that on can realize a radial symmetric setup, with
sufficiently localized magnetic field and long range vector potential.

Following this practical argument, it is tempting to ask if there are further
divergences present in our system that the finite state numerics may have
missed. While we have observed divergent absorptive DC behaviour for the
fillings arbitrarily close to the ionization energy, the question emerges if a
similar effect could also occur at lower Fermi energies. To shine further
light on this question, we search for degenerate eigenvalues obeying the
angular and spin selection rules as follows,

$\displaystyle E_{n_{a},l_{a},1/2}(g^{*})$ $\displaystyle=$ $\displaystyle
E_{n_{b},l_{a}+1,-1/2}(g^{*}).$ (32)

By assuming a positive rational $g^{*}=g_{1}/g_{2}>0,\
\\{g_{1},g_{2}\\}\in\mathbb{N}$ it turns out that we can find an infinite set
of quantum numbers $n_{a},l_{a},n_{b}$ fulfilling Eq. (32), given that there
is at least one solution. In more detail, we find a discrete set of infinitely
degenerate energy levels at (see App. C),

$\displaystyle
E^{\infty}(g^{*}\in\mathbb{Q}^{*}_{+}):=\begin{cases}E_{n,l,1/2},n\neq
l&\text{for }g^{*}=2\\\
\frac{q^{2}A_{\phi}^{2}}{2m^{*}}\Big{(}1-\Big{[}\frac{|2-g^{*}|}{2k}\Big{]}^{2}\Big{)}>0,\
k\in\mathbb{O}&\text{otherwise. }\end{cases}$

with $\mathbb{Q}^{*}_{+}$ being the strictly positive rational numbers and
$\mathbb{O}$ the positive odd integer numbers. The resulting energy pattern
resembles a squeezed Rydberg-like series that strongly depends on $g^{*}$, as
can be seen by the vertical green lines in Fig. 3. Nevertheless, they all
share an accumulation point at $E^{\infty}\rightarrow E_{A^{2}}$,
independently of $g^{*}$, as can be seen for $k\rightarrow\infty$, which is in
line with the previous considerations at the ionization threshold. Notice also
that the density of the degenerate bands will in practice delicately depend on
the chosen $g^{*}$-factor. For our ideal $1/r$-magnetic field setup, numerics
suggests that the Rydberg-like bands contribute only locally to the DC
conductivity $\Im(\sigma^{rz}_{0}(E_{F},0))$ as it was the case for the AC
response, since the matrix overlap seems to strongly decline for states
localized further away from the origin. This drastically limits the absorbance
for Fermi energies close to $E^{\infty}(g^{*})$ except for the previously
discussed divergent case at the ionization threshold. However, this strong
local suppression may no longer hold for different radial symmetric field
shapes, which in turn could facilitate the detection of strong spin-orbit
effects at specific Fermi energies, similarly to our ideal system at the
ionization threshold.

(a) $g^{*}=1$

(b) $g^{*}=3$

(c) $g^{*}=4-g^{*}_{\rm rel}=1.9977$

(d) $g^{*}=2$

(e) $g^{*}=g^{*}_{\rm rel}=2.0023$

Figure 3: Absorptive (imaginary) part of the DC ($\omega\rightarrow 0$) Hall
conductivity with respect to the Fermi energy $E_{F}$ for different
$g^{*}$-factors at $T=20$ [mK]. Quantum numbers were explored numerically up
to $n=20$, which seem to yield converged results for Fermi energies not
approaching the continuum limit at $E_{F}=E_{A^{2}}$. In this case only
analytical considerations become feasible that indeed suggest a diverging
absorptive Hall conductivity in the thermodynamic limit, i.e., for
$n\rightarrow\infty$.

##  IV Analytic solution and protected degeneracies for time-dependent
magnetic fields

We have seen so far that our system possesses intriguing linear response
features that emerge in the DC limit due to the presence of (infinitely)
degenerate states. This immediately rises the question of how robust those
degeneracies are. As already mentioned, they will overall be very delicate to
achieve in experiments for multiple reasons. However, surprisingly they seem
to be protected with respect to slow enough changes of the applied strength of
the $1/r$-magnetic field, as we subsequently will show. It turns out that we
can find an analytical solutions for the non-relativistic Hamiltonian expanded
up to order $1/(m^{*}c)^{2}$,[27] i.e. including the spin-orbit and Darwin
interaction, in a similar spirit to Ref. 17. In more detail we can solve the
eigenvalue problem for,

$\displaystyle\hat{\tilde{H}}(t)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\frac{1}{2m^{*}}\hat{\bm{\Pi}}_{j}^{2}-\frac{g^{*}q\hbar}{4m^{*}}\bm{\sigma}_{j}\cdot\bm{B}(\bm{r}_{j},t)$
$\displaystyle-\frac{q\hbar}{8m^{*2}c^{2}}\Big{[}\hat{\bm{\Pi}}_{j}\cdot\big{(}\bm{\sigma}_{j}\times\bm{E}(\bm{r}_{j},t)\big{)}+\big{(}\bm{\sigma}_{j}\times\bm{E}(\bm{r}_{j},t)\big{)}\cdot\hat{\bm{\Pi}}_{j}\Big{]}$
$\displaystyle-\frac{q\hbar}{8m^{*2}c^{2}}\mathrm{div}\bm{E}(\bm{r}_{j},t),$

as well, instead of only solving Eq. (1). We now assume slowly varying
external fields of the following form
$\bm{A}(\bm{r},t):=A_{\phi}(t)\bm{e}_{\phi}$ and
$\bm{B}(\bm{r},t)=A_{\phi}(t)/r\bm{e}_{z}$. The corresponding electric field
is given by
$\bm{E}(\bm{r},t)=-\nabla_{\bm{r}}V/e-(\partial/\partial_{t})\bm{A}=-\dot{A}_{\phi}(t)\bm{e}_{\phi}$,
which immediately removes the Darwin interaction (last term) in our
Hamiltonian formulation. By applying the method of Frobenius, we can determine
the bound state spectrum (see App. D),

$\displaystyle\tilde{E}_{n,l,s}(t)=\frac{q^{2}A_{\phi}^{2}}{2m^{*}}-\frac{16m^{*3}c^{4}}{q^{2}\hbar^{2}\dot{A}_{\phi}^{2}}\Bigg{[}1-\sqrt{1-\frac{q^{4}\hbar^{2}\dot{A}_{\phi}^{2}A_{\phi}^{2}}{16m^{*4}c^{4}}\bigg{[}\frac{2l+g^{*}s}{2n+1}\bigg{]}^{2}}\Bigg{]},\
n\geq l,\ l+g^{*}s/2>0.$ (34)

While on a first glance the spectrum looks much different compared to the
static Eq. (2), by taking the static limit one indeed recovers

$\displaystyle\lim_{\dot{A}\rightarrow 0}\tilde{E}_{n,l,s}(t)$
$\displaystyle=$ $\displaystyle E_{n,l,s},$ (35)

as one would expect. More remarkable, however, is that the spin-orbit
interaction, which now contributes due to the time-dependent vector potential,
does not change the structure of the spectrum, i.e., degeneracies are not
lifted, since one can reformulate Eq. (34) as follows,

$\displaystyle\tilde{E}(E_{n,l,s}(t),t)$ $\displaystyle=$
$\displaystyle\frac{q^{2}A_{\phi}^{2}}{2m^{*}}-\frac{16m^{*3}c^{4}}{q^{2}\hbar^{2}\dot{A}_{\phi}^{2}}\Bigg{[}1-\sqrt{1+\frac{q^{2}\hbar^{2}\dot{A}_{\phi}^{2}}{8m^{*3}c^{4}}(E_{n,l,s}(t)-E_{A^{2}})}\Bigg{]}.$
(36)

Consequently, the degenerate states seem to be protected with respect to slow
variations of $A_{\phi}$ at least up to order $1/(m^{*}c)^{2}$. However, the
overall squeezing of the energetic structure is extremely small as one can see
in Fig. 4 for moderate time-variations $\dot{A}$. Clearly, this picture might
change for jumps in the magnetic field, but then one automatically violates
the assumption of a slow field variation, i.e. one would need to solve the
time-dependent Schroedinger problem instead.

Figure 4: Deviation of the time-dependent energy eigenvalue $\tilde{E}$ for
the chosen InSb setup, with respect to the strictly static energy $E$ measured
by $\Delta E:=\tilde{E}(E)-E$ for different small variations of the externally
applied vector potential $A_{\phi}$. Notice, the extremely small shift due to
non-zero $\dot{A}_{\phi}$. However, as explained in the main text, the much
more striking feature is that the structure of the static energy eigenvalues
remains preserved exactly, i.e. $\tilde{E}(E)$.

We can not only determine analytic eigenvalues, but also the corresponding
orthonormal eigenstates in terms of generalized Laguerre polynomials
$L^{\nu}_{w}(x)$, see App. (D),

$\displaystyle\tilde{\Psi}_{n,l,s}(t)=\frac{1}{\sqrt{\tilde{N}_{n,l,s}}}e^{il\phi}e^{-{i\mkern
1.0mu}\gamma(r)}e^{-\frac{\tilde{x}(r)}{2}}\tilde{x}^{l}(r)L_{n-l}^{2l}(\tilde{x}(r))\chi(s),$
(37)

which are identical to the static solution given in Eq. (3), except for a
$r$-dependent complex phase factor,

$\displaystyle\gamma(r):=\frac{\delta s}{\sqrt{1-\delta^{2}}}x(r)$ (38)

and a different scaling in radial coordinates,

$\displaystyle\tilde{x}(r):=\sqrt{1-\delta^{2}}\frac{2qA_{\phi}}{\hbar}\frac{2l+g^{*}s}{2n+1}r,$
(39)

where
$\delta:=\frac{q\hbar^{2}}{4m^{*2}c^{2}}\dot{A}_{\phi}\sqrt{\frac{m^{*}|\tilde{E}^{-}|}{2\hbar^{2}}}$
with $\tilde{E}^{-}:=\tilde{E}-E_{A^{2}}$. The corresponding normalisation is
explicitly calculated as,

$\displaystyle\tilde{N}_{n,l,s}$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\int_{0}^{\infty}\tilde{\Psi}_{n,l,s}^{*}\tilde{\Psi}_{n,l,s}rdrd\phi$
(40) $\displaystyle=$ $\displaystyle
2\pi\frac{(n+l)!}{(n-l)!}(2n+1)\bigg{(}\sqrt{1-\delta^{2}}\frac{2qA_{\phi}}{\hbar}\frac{2l+g^{*}s}{2n+1}\bigg{)}^{-2}.$

Notice that for constant vector potential $\dot{A}_{\phi}=0$, we naturally
recover the previously known static solution as given in Eq. (3). The
emergence of a radial dependent phase factor opens interesting research
perspectives with regards to geometric phases and topological protection for
slow variations of $A_{\phi},\dot{A}_{\phi}$, which may substantiate
theoretically the protected energetic structure of Eq. (36). However, this
clearly goes beyond the scope of this work.

##  V Conclusion and outlook

We have investigated the effect of spin-orbit interaction in a 2D electron gas
with distorted Landau level structure that arises from an externally applied
radial symmetric, $1/r$-decaying magnetic field. By applying a weak
perpendicular electric field $\bm{E}_{z}$ with respect to the 2D electron gas
(and in parallel with the applied magnetic field), we find that radial Hall
currents emerge solely from spin-orbit interaction. The absorptive part of the
radial Hall conductivity shows spectral features that are of atomistic nature,
i.e., dominated by the discrete energy levels localized close to the origin.
While overall the magnitude of absorption is expected to be small, because of
the $1/(m^{*}c)^{4}$-prefactor and the quadratic dependency of the radial Hall
current on the $\bm{E}_{z}$ perturbation. Partially, this suppression can be
mediated by choosing a small effective electron mass of the host material
(InSb) and by assuming long-lived states. Surprisingly, things change when
approaching Fermi energies close to the ionization threshold, where we could
show analytically that a divergent static spin-orbit absorbance emerges in the
thermodynamic limit due to the distorted Landau level structure. This
divergent linear response implies that the external electric field is entirely
absorbed by induced radial spin-orbit currents, i.e. by radial polarization,
where the induced flow / polarization direction depends on the effective
$g^{*}$-factor of the material. In other words, the small electric field
perpendicular to the 2D electron gas is screened entirely, as it would be the
case inside a perfect conductor. Interestingly, the divergent bound state
currents also show a critical behaviour at $g^{*}_{c}=2$, where the radial
current response is reversed. In our setup, the physical origin of the
divergence is a consequence of the long-range (const) vector potential
(Aharonov-Bohm-like effect), which clearly in this form is experimentally out
of reach, due to the associated magnetic field divergence at the origin.
However, for other radial magnetic field, which induce differently distorted
Landau levels, we would still expect the generic emergence of some discrete
level structure with possibly strong linear response to external
perturbations. Probably in a similar fashion to our case, for which we found a
discrete, infinitely degenerate Rydberg-like band structure that obeys allowed
angular momentum and spin selection rules. While this structure turns out to
be of minor relevance for our setup, things may change for different (more
realistic) radial field dependencies. On should also keep in mind that the
proposed spin-orbit effects may depend non-trivially on many other external
influences (e.g. Coulomb interaction, decoherence, impurities). Nevertheless,
we could show analytically that the structure of the energy eigenvalues
remains preserved exactly with respect to slow variations of the applied
$1/r$-magnetic field strength, by solving the Dirac equation expanded up to
order $1/(m^{*}c)^{2}$. In particular, we find that the time-dependent field
variations cannot lift the static degeneracies, which suggests a protection
mechanism that will become the focus of future work. Clearly, there is still a
long way to go, with many obstacles to overcome, for an experimental
verification of our theoretical predictions. However, we believe the
experimental realization of rotationally distorted Landau levels would not
only be highly fruitful with respect to spin-orbit effects, but more generally
by opening novel pathways to engineer (distorted) Landau level physics from
local to non-local degeneracies.

###### Acknowledgements.

We thank Jerome Faist and Gregor Jotzu for inspiring discussions and helpful
comments. This work was made possible through the support of the RouTe Project
(13N14839), financed by the Federal Ministry of Education and Research
(Bundesministerium für Bildung und Forschung (BMBF)) and supported by the
European Research Council (ERC-2015-AdG694097), the Cluster of Excellence
“CUI: Advanced Imaging of Matter” of the Deutsche Forschungsgemeinschaft
(DFG), EXC 2056, project ID 390715994 and the Grupos Consolidados (IT1249-19).
The Flatiron Institute is a division of the Simons Foundation. Author
contributions: D.S. initiated the project, derived the analytic solutions,
performed numerical calculations and discovered the spin-orbit absorption
divergence. M.R. contributed to the mathematical accuracy and rigorousness.
All authors developed the physical interpretation and wrote the manuscript.

## Data Availability Statement

Python implementation available on reasonable request.

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##  Appendix A Radial spin-orbit Hall conductivity simplification

For our radial symmetric $1/r$-decaying magnetic field setup the radial spin-
orbit hall conductivity can be simplified by using the angular and spin-orbit
selection rules that enter as follows,

$\displaystyle\bra{s_{a},l_{a}}\sigma_{y}\cos\phi-\sigma_{x}\sin\phi\ket{s_{b},l_{b}}$
$\displaystyle=$ $\displaystyle\begin{cases}-{i\mkern 1.0mu},&\text{for
}l_{a}-l_{b}=-1,s_{a}=\uparrow,s_{b}=\downarrow\\\ {i\mkern 1.0mu},&\text{for
}l_{a}-l_{b}=1,s_{a}=\downarrow,s_{b}=\uparrow\\\
0,&\text{otherwise}\end{cases}$ (41)
$\displaystyle\bra{s_{a},l_{a}}-\sigma_{y}\sin\phi-\sigma_{x}\cos\phi\ket{s_{b},l_{b}}$
$\displaystyle=$ $\displaystyle\begin{cases}-1,&\text{for
}l_{a}-l_{b}=-1,s_{a}=\uparrow,s_{b}=\downarrow\\\ -1,&\text{for
}l_{a}-l_{b}=1,s_{a}=\downarrow,s_{b}=\uparrow\\\
0,&\text{otherwise}\end{cases}$ (42)

which eventually yields,

$\displaystyle\Im(\sigma^{rz}(\omega,E_{F},T))$ $\displaystyle=$
$\displaystyle\pi{i\mkern
1.0mu}\sum_{a,b}f(E_{a})(1-f(E_{b}))\bigg{[}\frac{\bra{a}\hat{j}^{r}\ket{b}\bra{b}\hat{d}_{1}\ket{a}\Gamma}{(E_{a}-E_{b}+\hbar\omega)^{2}+\Gamma^{2}}-\frac{\bra{a}\hat{d}_{1}\ket{b}\bra{b}\hat{j}^{r}\ket{a}\Gamma}{(E_{b}-E_{a}+\hbar\omega)^{2}+\Gamma^{2}}\bigg{]}$
(43) $\displaystyle=$ $\displaystyle\pi{i\mkern 1.0mu}\Gamma\frac{\hbar
q}{4m^{*2}c^{2}}\sum_{a,b}f(E_{a})(1-f(E_{b}))\bra{a}\sigma_{y}\cos\phi-\sigma_{x}\sin\phi\ket{b}$
$\displaystyle\bigg{[}\frac{\bra{b}\hat{d}_{1}\ket{a}}{(E_{a}-E_{b}+\hbar\omega)^{2}+\Gamma^{2}}+\frac{\bra{a}\hat{d}_{1}\ket{b}}{(E_{b}-E_{a}+\hbar\omega)^{2}+\Gamma^{2}}\bigg{]}$
$\displaystyle=$ $\displaystyle\pi{i\mkern
1.0mu}\Gamma\frac{\hbar^{2}q^{2}E_{z}}{16m^{*4}c^{4}}\sum_{a,b}f(E_{a})(1-f(E_{b}))\bra{a}\sigma_{y}\cos\phi-\sigma_{x}\sin\phi\ket{b}$
$\displaystyle\Big{[}\frac{\bra{b}(-\sigma_{y}\sin\phi-\sigma_{x}\cos\phi)\cdot(-\frac{{i\mkern
1.0mu}\hbar}{r}\partial_{\phi}-qA_{\phi})+(\sigma_{y}\cos\phi-\sigma_{x}\sin\phi)\cdot(-{i\mkern
1.0mu}\hbar\partial_{r})\ket{a}}{(E_{a}-E_{b}+\hbar\omega)^{2}+\Gamma^{2}}+$
$\displaystyle\frac{\bra{a}(-\sigma_{y}\sin\phi-\sigma_{x}\cos\phi)(-\frac{{i\mkern
1.0mu}\hbar}{r}\partial_{\phi}-qA_{\phi})+(\sigma_{y}\cos\phi-\sigma_{x}\sin\phi)(-{i\mkern
1.0mu}\hbar\partial_{r})\ket{b}}{(E_{b}-E_{a}+\hbar\omega)^{2}+\Gamma^{2}}\Big{]}$
$\displaystyle=$ $\displaystyle\pi{i\mkern
1.0mu}\Gamma\frac{\hbar^{2}q^{2}E_{z}}{16m^{*4}c^{4}}\sum_{a,b}f(E_{a})(1-f(E_{b}))\bigg{\\{}$
$\displaystyle-i\bra{R_{a}}\ket{R_{b}}\Big{[}\frac{\bra{R_{b}}(-1)\cdot(\frac{\hbar
l_{a}}{r}-qA_{\phi})+(i)\cdot(-{i\mkern
1.0mu}\hbar\partial_{r})\ket{R_{a}}}{(E_{a}-E_{b}+\hbar\omega)^{2}+\Gamma^{2}}+$
$\displaystyle\frac{\bra{R_{a}}(-1)(\frac{\hbar
l_{b}}{r}-qA_{\phi})+(-{i\mkern 1.0mu})(-{i\mkern
1.0mu}\hbar\partial_{r})\ket{R_{b}}}{(E_{b}-E_{a}+\hbar\omega)^{2}+\Gamma^{2}}\Big{]}\delta_{l_{a}-l_{b},-1}\delta_{s_{a},\uparrow}\delta_{s_{b},\downarrow}$
$\displaystyle+i\bra{R_{a}}\ket{R_{b}}\Big{[}\frac{\bra{R_{b}}(-1)\cdot(\frac{\hbar
l_{a}}{r}-qA_{\phi})+(-i)\cdot(-{i\mkern
1.0mu}\hbar\partial_{r})\ket{R_{a}}}{(E_{a}-E_{b}+\hbar\omega)^{2}+\Gamma^{2}}+$
$\displaystyle\frac{\bra{R_{a}}(-1)(\frac{\hbar l_{b}}{r}-qA_{\phi})+({i\mkern
1.0mu})(-{i\mkern
1.0mu}\hbar\partial_{r})\ket{R_{b}}}{(E_{b}-E_{a}+\hbar\omega)^{2}+\Gamma^{2}}\Big{]}\delta_{l_{a}-l_{b},1}\delta_{s_{a},\downarrow}\delta_{s_{b},\uparrow}\bigg{\\}}$
$\displaystyle=$
$\displaystyle-\pi\Gamma\frac{\hbar^{2}q^{2}E_{z}}{16m^{*4}c^{4}}\sum_{a,b}f(E_{a})(1-f(E_{b}))\bigg{\\{}$
$\displaystyle-\bra{R_{a}}\ket{R_{b}}\Big{[}\frac{\bra{R_{b}}\frac{-\hbar
l_{a}}{r}+qA_{\phi}+\hbar\partial_{r}\ket{R_{a}}}{(E_{a}-E_{b}+\hbar\omega)^{2}+\Gamma^{2}}+$
$\displaystyle\frac{\bra{R_{a}}\frac{-\hbar
l_{b}}{r}+qA_{\phi}-\hbar\partial_{r}\ket{R_{b}}}{(E_{b}-E_{a}+\hbar\omega)^{2}+\Gamma^{2}}\Big{]}\delta_{l_{a}-l_{b},-1}\delta_{s_{a},\uparrow}\delta_{s_{b},\downarrow}$
$\displaystyle+\bra{R_{a}}\ket{R_{b}}\Big{[}\frac{\bra{R_{b}}\frac{-\hbar
l_{a}}{r}+qA_{\phi}-\hbar\partial_{r}\ket{R_{a}}}{(E_{a}-E_{b}+\hbar\omega)^{2}+\Gamma^{2}}+$
$\displaystyle\frac{\bra{R_{a}}\frac{-\hbar
l_{b}}{r}+qA_{\phi}+\hbar\partial_{r}\ket{R_{b}}}{(E_{b}-E_{a}+\hbar\omega)^{2}+\Gamma^{2}}\Big{]}\delta_{l_{a}-l_{b},1}\delta_{s_{a},\downarrow}\delta_{s_{b},\uparrow}\bigg{\\}}.$

##  Appendix B Radial overlap integrals

For analytical accessiblity, we assume that
$E_{a}\overset{n\rightarrow\infty}{\rightarrow}E_{b}$, which means that the
radial scaling factor $x(r)=:\nu r$, defined from Eq. (4), becomes equal for
both states $\ket{a}$ and $\ket{b}$. Notice that $\nu\propto
1/(2n+1)\overset{n\rightarrow\infty}{\rightarrow}0$. The resulting radial
overlap integrals can then be solved analytically

$\displaystyle\bra{R_{a}}\ket{R_{b}}$
$\displaystyle\overset{\nu_{a}=\nu_{b}}{=}$
$\displaystyle\frac{1}{\sqrt{N_{l}N_{l+1}}}\int_{0}^{\infty}e^{-\nu r}(\nu
r)^{2l+1}L_{n-l}^{2l}(\nu r)L_{n-l-1}^{2l+2}(\nu r)rdr$ (48) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N_{l}N_{l+1}}\nu^{2}}\int_{0}^{\infty}e^{-\rho}(\rho)^{2l+2}L_{n-l}^{2l}(\rho)L_{n-l-1}^{2l+2}(\rho)d\rho$
(49) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N_{l}N_{l+1}}\nu^{2}}\int_{0}^{\infty}e^{-\rho}(\rho)^{2l+2}\big{[}L_{n-l}^{2l+2}(\rho)-2L_{n-l-1}^{2l+2}(\rho)+L_{n-l-2}^{2l+2}(\rho)\big{]}\cdot$
$\displaystyle L_{n-l-1}^{2l+2}(\rho)d\rho$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{(n-l)!}{(n+l)!(2n+1)}\frac{(n-l-1)!}{(n+l+1)!(2n+1)}}\frac{(n+l+1)!}{(n-l-1)!}$
(51) $\displaystyle\overset{n\rightarrow\infty}{=}$ $\displaystyle-1,$ (52)

where we have used the following well known 3-point rule and orthogonality
relation for the generalized Laguerre polynomials,[34]

$\displaystyle L_{n}^{\alpha}$ $\displaystyle=$ $\displaystyle
L_{n}^{\alpha+1}-L_{n-1}^{\alpha+1}$ (53)
$\displaystyle\int_{0}^{\infty}x^{\alpha}e^{-\alpha}L_{n}^{\alpha}(x)L_{m}^{\alpha}(x)dx$
$\displaystyle=$ $\displaystyle\frac{\Gamma(n+\alpha+1)}{n!}\delta_{n,m}.$
(54)

with,[17]

$\displaystyle N_{l}$ $\displaystyle:=$
$\displaystyle\nu^{-2}\frac{(n+l)!}{(n-l)!}(2n+1),$ (55)

and Mathematica [35] for the limiting procedure.

In a next step we similarly investigate,

$\displaystyle\bra{R_{a}}\frac{1}{r}\ket{R_{b}}$
$\displaystyle\overset{\nu_{a}=\nu_{b}}{=}$
$\displaystyle\frac{1}{\sqrt{N_{l}N_{l+1}}}\int_{0}^{\infty}e^{-\nu r}(\nu
r)^{2l+1}L_{n-l}^{2l}(\nu r)L_{n-l-1}^{2l+2}(\nu r)dr$ (56) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N_{l}N_{l+1}}\nu}\int_{0}^{\infty}e^{-\rho}(\rho)^{2l+1}L_{n-l}^{2l}(\rho)L_{n-l-1}^{2l+2}(\rho)d\rho$
(57) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N_{l}N_{l+1}}\nu}\int_{0}^{\infty}e^{-\rho}(\rho)^{2l+1}\big{[}L_{n-l}^{2l+1}(\rho)-L_{n-l-1}^{2l+1}(\rho)\big{]}$
$\displaystyle\big{[}L_{n-l-1}^{2l+1}(\rho)+L_{n-l-2}^{2l+1}(\rho)+...+L_{1}^{2l+1}(\rho)+1\big{]}d\rho$
$\displaystyle=$
$\displaystyle\nu\sqrt{\frac{(n-l)!}{(n+l)!(2n+1)}\frac{(n-l-1)!}{(n+l+1)!(2n+1)}}\cdot$
$\displaystyle\bigg{(}\frac{(n+l+1)!}{(n-l)!}-\frac{(n+l)!}{(n-l-1)!}\bigg{)}$
$\displaystyle\overset{n\rightarrow\infty}{=}$ $\displaystyle 0,$ (60)

and we find,

$\displaystyle\bra{R_{a}}\partial_{r}\ket{R_{b}}$
$\displaystyle\overset{\nu_{a}=\nu_{b}}{=}$
$\displaystyle\frac{1}{\sqrt{N_{l}N_{l+1}}}\int_{0}^{\infty}e^{-\frac{\nu
r}{2}}(\nu r)^{l+1}L_{n-l-1}^{2l+2}(\nu r)\big{[}\partial_{r}e^{-\frac{\nu
r}{2}}(\nu r)^{l}L_{n-l}^{2l}(\nu r)\big{]}rdr$ (61) $\displaystyle=$
$\displaystyle-\bra{R_{a}}\ket{R_{b}}+l\bra{R_{a}}\frac{1}{r}\ket{R_{b}}-\frac{1}{\sqrt{N_{l}N_{l+1}}}\int_{0}^{\infty}e^{-\nu
r}(\nu r)^{2l+1}L_{n-l-1}^{2l+1}(\nu r)L_{n-l-1}^{2l+2}(\nu r)rdr$ (62)
$\displaystyle=$
$\displaystyle-\bra{R_{a}}\ket{R_{b}}+l\bra{R_{a}}\frac{1}{r}\ket{R_{b}}$
$\displaystyle-\frac{1}{\sqrt{N_{l}N_{l+1}}\nu^{2}}\int_{0}^{\infty}e^{-\rho}(\rho)^{2l+2}\big{[}L_{n-l-1}^{2l+2}(\rho)-L_{n-l-2}^{2l+2}(\rho)\big{]}L_{n-l-1}^{2l+2}(\rho)d\rho$
$\displaystyle=$
$\displaystyle-\bra{R_{a}}\ket{R_{b}}+l\bra{R_{a}}\frac{1}{r}\ket{R_{b}}$
$\displaystyle-\sqrt{\frac{(n-l)!}{(n+l)!(2n+1)}\frac{(n-l-1)!}{(n+l+1)!(2n+1)}}\frac{(n+l+1)!}{(n-l-1)!}$
$\displaystyle\overset{n\rightarrow\infty}{=}$ $\displaystyle\frac{1}{2}.$
(65)

Analogous derivations can be applied to show that
$\bra{R_{b}}\partial_{r}\ket{R_{a}}\rightarrow 1/2$.

##  Appendix C Derivation squeezed Rydberg divergences

To solve the integer value problem imposed by the energy degeneracy relation
given in Eq. (32), three different regimes of rational $g^{*}:=g_{1}/g_{2}>0$
have to be considered separately. For $0<g^{*}<2$ we find,

$\displaystyle\frac{2l_{1}+g_{1}/(2g_{2})}{2n_{1}+1}$ $\displaystyle=$
$\displaystyle\frac{2l_{1}+g_{1}/(2g_{2})}{2n_{2}+1}+\frac{2(1-g_{1}/(2g_{2}))}{2n_{2}+1}\overset{!}{=}C$
(66) $\displaystyle n_{2}$ $\displaystyle:=$ $\displaystyle n_{1}+\Delta
n,\Delta n\in\mathbb{N}\ \mathrm{with}\ \Delta n\leq n_{1}$ (67)
$\displaystyle 2g_{2}(2(n_{1}+\Delta
n)+1)\frac{2l_{1}+g_{1}/(2g_{2})}{2n_{1}+1}$ $\displaystyle=$ $\displaystyle
4g_{2}l_{1}+g_{1}+2(2g_{2}-g_{1})$ $\displaystyle\Rightarrow$
$\displaystyle\Delta n$ $\displaystyle=$
$\displaystyle\frac{(2g_{2}-g_{1})(2n_{1}+1)}{4g_{2}l_{1}+g_{1}}\in\mathbb{N}$
$\displaystyle\Rightarrow$ $\displaystyle\frac{4g_{2}l_{1}+g_{1}}{2n_{1}+1}$
$\displaystyle=$ $\displaystyle\frac{2g_{2}-g_{1}}{k},\ k\in\mathbb{N}$
$\displaystyle\Rightarrow$ $\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{2g_{2}-g_{1}}{2g_{2}k}$ (71) $\displaystyle E^{\infty}$
$\displaystyle:=$
$\displaystyle\frac{q^{2}A_{\phi}^{2}}{2m^{*}}\bigg{(}1-\bigg{[}\frac{2g_{2}-g_{1}}{2g_{2}k}\bigg{]}^{2}\bigg{)}$
(72)

A similar argument applies for $2<g^{*}$, where we find an analogous
expression, by using $n_{2}:=n_{1}-\Delta_{n},\ \Delta n\in\mathbb{N}$
instead,

$\displaystyle C=\frac{|2g_{2}-g_{1}|}{2g_{2}k}$ (73)

which eventually yields,

$\displaystyle
E^{\infty}=\frac{q^{2}A_{\phi}^{2}}{2m^{*}}\bigg{(}1-\bigg{[}\frac{|2-g^{*}|}{2k}\bigg{]}^{2}\bigg{)},\
g^{*}\in\mathbb{Q}^{*}_{+}\setminus\\{2\\}.$ (74)

So far we have not said anything about the allowed $k$-numbers except that
they must be positive integers. However, there are additional constraints with
respect to $g_{1},g_{2}$ as well as the quantum numbers
$n_{1},l_{1},n_{2},l_{2}$ to be obeyed. From relation in Eq. (C), we notice
the following, depending on whether or not $g_{1}$ is an even
$e\in\mathbb{E}:=\\{2n,\ n\in\mathbb{N}\\}$ or odd $o\in\mathbb{O}:=\\{2n-1,\
n\in\mathbb{N}\\}$ number:

$\displaystyle g_{1}=o,\
2g_{2}=e\Rightarrow\frac{o_{1}}{o_{2}}=\frac{o_{3}}{k}\Rightarrow
k\in\mathbb{O}$ (75) $\displaystyle g_{1}=e,\
2g_{2}=e\Rightarrow\frac{e_{1}}{o_{2}}=\frac{e_{3}}{k}\Rightarrow
k\in\mathbb{O}$ (76)

Hence we have found that $k$ must be an odd number. In a last step, we look at
the special non-realtivistic case $g^{*}=2$ for which we find trivially,

$\displaystyle\frac{2l_{1}+1}{2n_{1}+1}$ $\displaystyle=$
$\displaystyle\frac{2l_{1}+1}{2n_{2}+1}$ (77) $\displaystyle n_{2}$
$\displaystyle=$ $\displaystyle n_{1},$ (78)

which implies that Eq. (32) holds almost everywhere expect for the lowest flat
band,[17] due to the condition $l+g^{*}s/2>0$ for the angular quantum numbers.
This inequality at the same time also imposes that

$\displaystyle C<1,$ (79)

which sets limits on the minimally allowed quantum numbers $k$, depending on
$g_{1}$ and $g_{2}$ according to Eqs. (71) & (73). The resulting expression
for the squeezed Rydberg like series, imposed by Eq. (32), is then given in
Eq. (III) of the main text.

##  Appendix D Derivation of an analytic solution for a time-dependent
magnetic field in the slowly varying limit

In order to solve the quasi-static eigenvalue problem of Eq. (IV), we follow
an analogous procedure to Ref. 17 and choose a cylindrical coordinate system,
which yields

$\displaystyle\hat{\tilde{H}}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\bigg{[}-\frac{\hbar^{2}}{2m^{*}}\bm{\nabla}^{2}_{j}+\frac{A_{\phi}q\hbar}{m^{*}}\bigg{(}{i\mkern
1.0mu}\frac{\partial}{r_{j}\partial\phi_{j}}-\frac{g_{s}^{*}\sigma_{z,j}}{4r_{j}}\bigg{)}-{i\mkern
1.0mu}\frac{q\hbar}{8m^{*2}c^{2}}\hbar\sigma_{z,j}\dot{A}_{\phi}\Big{[}\frac{1}{r_{j}}+2\frac{\partial}{\partial
r_{j}}\Big{]}\bigg{]}+N\frac{q^{2}A_{\phi}^{2}}{2m^{*}},$

assuming a strictly two-dimensional electron gas. Fortunately, the
contribution of the diamagnetic term
$E_{A^{2}}:=\frac{q^{2}A_{\phi}^{2}}{2m^{*}}$ remains constant for all $N$
electrons in radial coordinates, which reduces the complexity of our problem
considerably. In a next step, we introduce

$\displaystyle\alpha$ $\displaystyle:=$
$\displaystyle\frac{A_{\phi}q\hbar}{m^{*}}>0,$ (81) $\displaystyle\epsilon$
$\displaystyle:=$ $\displaystyle\frac{q\hbar^{2}}{4m^{*2}c^{2}}\dot{A}_{\phi}$
(82)

which allows a more compact notation and the resulting eigenvalue problem for
a single electron can be written as

$\displaystyle\bigg{[}-\frac{\hbar^{2}}{2m^{*}}\Big{(}\frac{\partial^{2}}{\partial
r^{2}}+\frac{\partial}{r\partial
r}+\frac{\partial^{2}}{r^{2}\partial\phi^{2}}\Big{)}+{i\mkern
1.0mu}\alpha\frac{\partial}{r\partial\phi}+\frac{\alpha
g^{*}\sigma_{z}}{4r}-{i\mkern 1.0mu}\epsilon
s\Big{[}\frac{1}{r}+2\frac{\partial}{\partial r}\Big{]}\bigg{]}\tilde{\Psi}$
$\displaystyle=$ $\displaystyle\tilde{E}^{-}\tilde{\Psi},$

where the constant $E_{A^{2}}$-term is neglected for the moment. The angular
and spin problem can trivially be solved by separation of variables as
$\tilde{\Psi}(r,\phi,s)=\tilde{R}(r)\Phi(\phi)\chi(s)$, with spin function
$\chi$ and $\Phi=e^{{i\mkern 1.0mu}l\phi}$ with $l\in Z$, $s=\pm\frac{1}{2}$,
since $[\hat{\tilde{H}},\frac{\partial}{\partial\phi_{j}}]=0$. This leaves us
with the radial problem

$\displaystyle\hat{\tilde{H}}_{l,s}R$ $\displaystyle:=$
$\displaystyle\bigg{[}-\frac{\hbar^{2}}{2m^{*}}\big{(}\frac{\partial^{2}}{\partial
r^{2}}+\frac{\partial}{r\partial
r}-\frac{l^{2}}{r^{2}}\big{)}-\alpha\frac{l+g^{*}s/2}{r}-{i\mkern
1.0mu}\epsilon s\Big{[}\frac{1}{r}+2\frac{\partial}{\partial
r}\Big{]}\bigg{]}\tilde{R}=\tilde{E}^{-}\tilde{R}.$ (84)

Bound state eigenvalues: To solve for the attractive eigenvalue problem, we
apply the method of Frobenius and match orders of a series expansion similar
to Ref. 17. Therefore, we define

$\displaystyle\rho$ $\displaystyle:=$
$\displaystyle\sqrt{\frac{8m^{*}|\tilde{E}^{-}|}{\hbar^{2}}}r,\geq 0$ (85)
$\displaystyle\lambda_{l,s}$ $\displaystyle:=$
$\displaystyle\alpha(l+g^{*}s/2)\sqrt{\frac{m^{*}}{2\hbar^{2}|\tilde{E}^{-}|}},$
(86) $\displaystyle\delta$ $\displaystyle:=$
$\displaystyle\epsilon\sqrt{\frac{m^{*}|\tilde{E}^{-}|}{2\hbar^{2}}},\
0\leq\delta^{2}<1$ (87)

for which our radial problem assumes a convenient form,

$\displaystyle\bigg{[}\frac{\partial^{2}}{\partial\rho^{2}}+\frac{\partial}{\rho\partial\rho}-\frac{l^{2}}{\rho^{2}}+\frac{\lambda_{l,s}}{\rho}-\frac{1}{4}-\frac{{i\mkern
1.0mu}\delta s}{\rho}-2{i\mkern 1.0mu}\delta
s\frac{\partial}{\partial\rho}\bigg{]}\tilde{R}(\rho)=0.$ (88)

To reach a simple closed form solution, we introduce the Ansatz

$\displaystyle\tilde{R}(\rho)=e^{-\beta\rho}f(\rho)$ (89)

with

$\displaystyle\beta:=-{i\mkern 1.0mu}\delta
s+\frac{1}{2}\sqrt{1-4\delta^{2}s^{2}}=-{i\mkern 1.0mu}\delta
s+\frac{1}{2}\sqrt{1-\delta^{2}}.$ (90)

This choice imposes an upper bound for $\delta^{2}$, i.e. $\delta^{2}<1$. This
ensures that we have a normalizable solution for $\tilde{R}(\rho)$, given in
Eq. (89), i.e. having an exponentially decaying real part for
$\rho\rightarrow\infty$. Notice that large $\delta^{2}$-values would also
imply a rapidly changing external vector potential, which certainly
contradicts the initial assumption of a slow (adiabatic) time-evolution.
Consequently, we find an ODE for $f(\rho)$ of the following form

$\displaystyle\bigg{[}\frac{\partial^{2}}{\partial\rho^{2}}-2\beta\frac{\partial}{\partial\rho}+\frac{\partial}{\rho\partial\rho}-\frac{l^{2}}{\rho^{2}}+\Big{(}\lambda_{l,s}-\beta\Big{)}\frac{1}{\rho}-\frac{{i\mkern
1.0mu}\delta s}{\rho}-2{i\mkern 1.0mu}\delta
s\frac{\partial}{\partial\rho}\bigg{]}f(\rho)=0.$ (91)

If we apply the series representation
$f(\rho)=\sum_{j=0}^{\infty}c_{j}\rho^{j}$ and match the different orders in
$\rho$, we find after an index shift $j\mapsto j+1$ with $c_{-1}=0$:

$\displaystyle\sum_{j=-1}^{\infty}c_{j+1}j(j+1)\rho^{j-1}-2\beta
c_{j}j\rho^{j-1}-2{i\mkern 1.0mu}\delta
sc_{j}j\rho^{j-1}+c_{j+1}(j+1)\rho^{j-1}$ (92) $\displaystyle-\beta
c_{j}\rho^{j-1}-l^{2}c_{j+1}\rho^{j-1}+\lambda_{l,s}c_{j}\rho^{j-1}-{i\mkern
1.0mu}\delta sc_{j}\rho^{j-1}$ $\displaystyle=0$

This gives rise to the indicial equation:

$\displaystyle
c_{j+1}[(j+1)^{2}-l^{2}]=c_{j}[\sqrt{1-\delta^{2}}j+\frac{\sqrt{1-\delta^{2}}}{2}-\lambda_{l,s}].$
(93)

It implies the ”series switches on” for $c_{j+1}$ when $(j+1)^{2}=l^{2}$, i.e.
$j+1=l$, and it can terminate only if
$\sqrt{1-\delta^{2}(|\tilde{E}^{-}|)}j+\sqrt{1-\delta^{2}(|\tilde{E}^{-}|)}/2-\lambda_{l,s}(\tilde{E}^{-})=0$.
Otherwise one would converge to a non-normalizable solution since
$c_{j+1}\rightarrow\frac{c_{j}}{j}$ for large $j$ and
$f\rightarrow\sum_{j=0}^{\infty}\frac{\rho^{j}}{j!}$. This also implies that
$\lambda>0$, i.e. we find a bound state solution only for $l+g^{*}s/2>0$
similar to the strictly static results.[17] Now, introducing quantum number
$n:=j$ leads to a quadratic equation in $\tilde{E}^{-}$:

$\displaystyle\Big{(}n+\frac{1}{2}\Big{)}^{2}=\frac{\lambda^{2}}{1-\delta^{2}}=\frac{\alpha^{2}(l+g^{*}s/2)^{2}m^{*}}{2\hbar^{2}}\frac{1}{|\tilde{E}^{-}|}\Bigg{[}\frac{1}{1-\frac{\epsilon^{2}m^{*}}{2\hbar^{2}}|\tilde{E}^{-}|}\Bigg{]},\
n\geq l.$ (94)

with a simple closed form solution for the energy eigenvalues

$\displaystyle\tilde{E}^{-}_{n,l,s}=\pm\frac{\hbar^{2}}{m^{*}\epsilon^{2}}\Bigg{[}1\pm\sqrt{1-\frac{\alpha^{2}\epsilon^{2}m^{*2}}{\hbar^{4}}\bigg{[}\frac{2l+g^{*}s}{2n+1}\bigg{]}^{2}}\Bigg{]},\
n\geq l,\ l+g^{*}s/2>0.$ (95)

We will subsequently see that the quantum numbers $n,l,s$ uniquely define an
eigenfunction of the PDE given in Eq. (D) with corresponding unique eigenvalue
$\tilde{E}$. To determine the correct sign, we notice that for
$\epsilon\rightarrow 0$ our PDE given in Eq. (D) reduces to the problem of a
static vector potential ($\dot{\bm{A}}_{\phi}=0$) with known solutions.[17]
The known eigenvalues are indeed recovered exactly by choosing the negative
signs twice, i.e., from

$\displaystyle E_{n,l,s}^{-}$ $\displaystyle\overset{!}{=}$
$\displaystyle\lim_{\epsilon\rightarrow
0}\tilde{E}^{-}_{n,l,s}=\lim_{\epsilon\rightarrow
0}\Bigg{\\{}-\frac{\hbar^{2}}{m^{*}\epsilon^{2}}\Bigg{[}1-\sqrt{1-\frac{\alpha^{2}\epsilon^{2}m^{*2}}{\hbar^{4}}\bigg{[}\frac{2l+g^{*}s}{2n+1}\bigg{]}^{2}}\Bigg{]}\Bigg{\\}}$
(96) $\displaystyle=$
$\displaystyle-\frac{q^{2}A_{\phi}^{2}}{2m^{*}}\bigg{[}\frac{2l+g^{*}s}{2n+1}\bigg{]}^{2},$
(97)

whereas a positive sign in front of the square-root would lead to a divergent
limit instead. Finally, we obtain Eqs. (34)-(36), as introduced in the main
part of the manuscript.

Eigenfunctions: After having identified the bound state energy eigenvalues for
$n\geq l,l+g^{*}s/2>0$, we can next find the corresponding eigenfunctions by
expressing $f(\rho)=\rho^{l}L(\rho)$.[36] This turns Eq. (91) into

$\displaystyle\rho\frac{d^{2}L}{d\rho^{2}}+(2l+1-\sqrt{1-\delta^{2}}\rho)\frac{dL}{d\rho}+\bigg{[}-\frac{\sqrt{1-\delta^{2}}}{2}-l\sqrt{1-\delta^{2}}+\lambda\bigg{]}L$
$\displaystyle=$ $\displaystyle 0,$ (98)

which can be further simplified by substituting
$\tilde{x}:=\sqrt{1-\delta^{2}}\rho$ and using Eq. (94) to,

$\displaystyle\tilde{x}\frac{d^{2}L}{d\tilde{x}^{2}}+(2l+1-\tilde{x})\frac{dL}{d\tilde{x}}+(n-l)L$
$\displaystyle=$ $\displaystyle 0,\ n,l\in\mathbb{N}_{0}.$ (99)

This PDE can be solved by the associated Laguerre polynomials $L_{n-l}^{2l}$
of degree $n-l$ and parameter $2l$. [37] The associated Laguerre polynomials
are given by Rodrigues’ formula,[37]

$\displaystyle
L_{w}^{\nu}(\tilde{x})=\frac{\tilde{x}^{-\nu}e^{\tilde{x}}}{w!}\frac{d^{w}}{d\tilde{x}^{w}}(e^{-\tilde{x}}\tilde{x}^{w+\nu}).$
(100)

Therefore, the radial solution of our problem given in Eq. (84) becomes

$\displaystyle\tilde{R}_{n,l,s}(\tilde{x})=e^{-{i\mkern 1.0mu}\frac{\delta
s}{\sqrt{1-\delta^{2}}}\tilde{x}}e^{-\frac{\tilde{x}}{2}}\bigg{(}\frac{\tilde{x}}{\sqrt{1-\delta^{2}}}\bigg{)}^{l}L_{n-l}^{2l}(\tilde{x}).$
(101)

Consequently, the orthonormal eigenfunctions of the full problem are found as
given in Eqs. (37) - (40) in the main section. The orthogonality of the
eigenfunctions can be shown identically to the argument given in Ref. 17,
where surprisingly the same degeneracy argument remains valid because of Eq.
(36).