# Satellite galaxies’ drag on field stars in the Milky Way

Xilong Liang School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing 100049, China  National Astronomical
Observatories, Chinese Academy of Sciences,Beijing 100101, China  Jifeng Liu
School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing 100049, China  National Astronomical Observatories, Chinese
Academy of Sciences,Beijing 100101, China  WHU-NAOC Joint Center for
Astronomy, Wuhan University, Wuhan, China  Jingkun Zhao National Astronomical
Observatories, Chinese Academy of Sciences,Beijing 100101, China  School of
Astronomy and Space Science, University of Chinese Academy of Sciences,
Beijing 100049, China  Kun Xu  School of Astronomy and Space Science,
University of Chinese Academy of Sciences, Beijing 100049, China  National
Astronomical Observatories, Chinese Academy of Sciences,Beijing 100101, China

###### Abstract

With Gaia EDR3 data, velocity dispersion of Milky Way field stars around
satellite galaxies have been investigated. We have fitted velocity dispersion
against distance to satellite galaxy and found the gradient of velocity
dispersion is related to the mass of satellite galaxy. With order-of-magnitude
approximations, a linear correlation has been fitted between the mass of
satellite galaxy and gradient of velocity dispersion caused by its
gravitational drag. Though our result is an observational qualitative result,
it shows better relation could be obtained with more observations in the
future.

Galaxies: Dwarf; Galaxies: Kinematics and Dynamics; Stellar kinematics

##  1 Introduction

The popular way to provide rough mass constraints for spheroidal galaxies is
using the virial theorem (Poveda, 1958). With spherical Jeans equation, former
researchers can estimate accurate masses for dispersion-supported stellar
systems (Wolf et al., 2010). Formalisms have been built between velocity
dispersions of stars in dwarf galaxy and mass within its half-light radius,
which can estimate accurate mass consistent with the result of full Jeans
analysis (Walker et al., 2009). As observational ability grows, more and more
individual stars can be discerned within dwarf galaxies, but it is still
easier to observe individual stars around the out skirt than those in the
center of dwarf galaxies. Stars around boundary are more sensitive to traces
of dynamical interaction between satellite galaxy and its host galaxy. Tidal
tails are not target of this research and we focus on those satellites that
show little evidence of Galactic cannibalism, namely no clear stellar streams
striped from satellites by the tidal force of host galaxy.

Former researchers have investigated interaction of galaxies by application of
dynamical friction formula (Chandrasekhar, 1943) or direct numerical
simulation (Lin & Tremaine, 1983; White, 1983), but they mainly focused on
orbital decay of satellite. Weinberg (1989) studied the response of a
spherical stellar system to a periodic perturbation of satellite. Carlberg &
Hartwick (1989) found that sinking satellites can make the velocity dispersion
horizontal to the disk larger than the vertical velocity dispersion, as is
observed. Lacey (1984) studied the influence of massive gas clouds on stellar
velocity dispersions in galactic discs, which is close to our work. When a
satellite galaxy passes through the halo of our Galaxy, impulse approximation
is adopted to treat its gravitational effect on field stars. The impulse
approximation (Öpik, 1932) was introduced to calculate stellar perturbations
on comets (Oort, 1950), but the situation is similar for satellite galaxies.

Section 2 introduces the data and sample adopted to do this research. Section
3 introduces the model adopted to fit velocity distributions. The final
section presents the result and discussion.

##  2 Data analysis

###  2.1 data

In the Gaia era (Gaia Collaboration et al., 2016), many stars around satellite
galaxies have information available regarding spatial positions and proper
motions. For every satellite listed in Table 1, we selected all sources in
Gaia EDR3 (Gaia Collaboration et al., 2021) that are within $5^{\circ}$ from
the center of each galaxy. When those data are downloaded from the Gaia
gaiaedr3.gaia_source catalogue, some quality criteria have been adopted, which
are respectively $\textrm{RUWE}\leq 1.4,$ $bp\\_rp\geq 0.2$ and $parallax-
parallax\\_error<0.03$, where RUWE is the renormalized unit weight error
(Riello et al., 2021). Parallaxes have been corrected using the prescription
suggested in Lindegren et al. (2021) and used to select sample around each
satellite galaxy. We did not use the parallax to calculate distance of each
star, since most satellite galaxy are too far away. For each satellite galaxy,
its surrounding stars have been selected by apply a parallax cut according to
its known distance to Earth. The upper limits of parallax are all relatively
tight, taken as about 10 to 30 kpc away from each different satellite galaxy.
The lower limits of parallax are relatively loose that most of them are
negative to include enough stars. Then all stars within the parallax cut are
assumed as the same distance as the satellite galaxies to Earth. Since we
focus only on the distribution of tangent velocities, the error caused by the
distance approximation will not be larger than an order of unity. Velocities
along right ascension and declination directions have been calculated by
projecting all stars in the same distance as the satellite galaxy, which will
no doubt induce huge uncertainties. The fitting of velocity distribution can
only be done in two dimensions without radial velocity and distance.
Unfortunately, the sum of velocity dispersion along two directions is not
necessarily two thirds of the total velocity dispersion in all three
dimensions, because velocity dispersions may be different from each
directions. We think using sum of velocity dispersion in two dimensions
representing that in three dimensions will only induces uncertainty with an
order of unity.

To show variations of velocity distribution along the radius of each satellite
galaxy, stars have been divided into spatial concentric annulus in the
projected plane. The left panel of figure 1 shows concentric annulus taken for
Aquarius2 dwarf galaxy as an example. These concentric annuluses have same
area for the same satellite galaxy, but different areas for different
galaxies, because the masses of different satellite galaxies and star
densities of their field stars are different. We assumed stars has
approximately homogeneous distribution in space and the same area makes sure
each annulus sample has similar numbers of stars for a same satellite galaxy.
Then the parallax approximation would has the same effect on all samples
around one satellite galaxy. Velocity distributions of stars in each annulus
sample have been fitted separately. The inner radius of smallest annulus of
each satellite galaxy are taken larger than $0.5^{\circ}$, so that the
velocity distribution is dominated by field stars belonging to the Milky Way.
For each single star, whether it belongs to the satellite galaxy or field
stars of the Milky Way is not important for us. What we want is the velocity
distribution of field stars. For each spatial annulus sample, we have divided
it in velocity coordinate system into square bins with size equal 0.2 mas yr-1
(about the median value of uncertainties of proper motion). Then took the star
counts in each bin to represent the density distribution and took the mean
value of uncertainties of proper motions of stars in each bin as uncertainty
of that bin. We have ignored the uncertainty of distance which should be the
dominating part of total uncertainty, but we think this way of treating
uncertainty can still represent relative size of uncertainties between each
bins. Then velocity distribution model described by equation 2 has been
adopted to fit the density distribution in velocity space. The EMCEE code
(Foreman-Mackey et al., 2013) has been used to obtain best fitted parameters
and uncertainties of them. We have used uniform priors on free parameters and
the logarithm of probability function. As for fitting steps, we first used the
Python Scipy package optimize module to obtain least-squares estimation of
parameters, then used them to initialize 100 random walkers. After threw away
the initial 100 steps as ”burn-in” steps, we run 1000 steps of MCMC. The
parameters and their uncertainties are taken from 16th, 50th, and 84th
percentiles of the samples in the marginalized distributions of MCMC results.

###  2.2 model

The equilibrium state of an infinite, homogeneous stellar system can be
described by Maxwellian function in phase space
($\overrightarrow{x},\overrightarrow{v}$) and the velocity distribution is
described by Schwarzschild distribution function. We used three components to
fit the velocity distributions of surrounding stars around satellite galaxies
of the Milky Way, which are respectively for field stars from the Milky Way,
stars from satellite galaxy itself and for possible observational artifact.
The right panel of figure 1 shows fitted three components of velocity
distributions of each annulus taken for Aquarius2 dwarf galaxy as an example.
The black star in the right panel shows the known velocity of Aquarius2 dwarf
galaxy from literature (McConnachie & Venn, 2020). Grey dots in the background
show the velocity distribution of the whole sample. Three eclipses of each
color represent three components fitted for stars in each annulus sample.
Though we do not know which star is actually from satellite galaxy, they are
assumed having a distinct velocity distribution from field stars. With main
part of stars from satellite galaxy already been fitted by one component,
small number of stars from satellite galaxy mixed in the main part of field
stars have negligible effect on the velocity distribution of field stars.

The Schwarzschild distribution function used for fitting velocity distribution
is

$f(v_{1},v_{2})=Cexp(-\frac{((v_{1}-v_{1c})cost+(v_{2}-v_{2c})sint)^{2}}{a^{2}}-\frac{-((v_{1}-v_{1c})sint+(v_{2}-v_{2c})cost)^{2}}{b^{2}})$
(1)

in which $a$ and $b$ are two axes of the velocity ellipse, while $t$ is
angular separation between axes of the velocity ellipse and axes of coordinate
system. Coordinate $v=(v_{1},v_{2})$ is observed velocity of each star, while
$(v_{1c},v_{2c})$ is the center coordinate of the velocity ellipse. The same
form of function has been applied to all three components and all parameters
are free for fitting. We take the one fitted component with largest velocity
dispersion as velocity distribution of field stars. The square of velocity
dispersion is taken as $\sigma_{v}^{2}=a^{2}+b^{2}$. We obtained velocity
dispersions of the Milky Way field star component for every spatial concentric
annulus. Then we fitted parameters for equation 7 with velocity dispersions
against distance $r$ to the center of satellite galaxy. The EMCEE code has
been used to obtain best fitted parameters and their uncertainties. Though
there are so many approximation treatment of uncertainties in velocity
treatment, we think our result can still reveal some main tendencies.

Figure 1: The concentric annuluses taken for Aquarius2 dwarf galaxy and
corresponding velocity eclipses of velocity distributions fitted for stars in
each annulus. Each dot represent a star and colors are corresponding to each
other in two panels. These grey dots in right panel are all stars in all
annuluses, while the black star in the right panel is the known velocity of
Aquarius2 dwarf galaxy from literature.

The dynamical friction between a satellite galaxy and field stars of the Milky
Way will systematically transfer energy from their relative orbital motion
into random motions of their constituent stars. The process considered here is
a satellite galaxy with mass $M$ passing through field stars with mass $m\ll
M$. Though satellite galaxies are large complicated systems, we assume them as
point masses for simplicity. Moreover, the spatial distribution of field stars
has been roughly approximated as infinite and homogeneous, because the Galaxy
is much larger than its satellite galaxies. While the satellite galaxy is much
larger than field stars, the dominant effect of the encounters is to exert
dynamical friction for the satellite galaxy. For those stars with impact
parameters smaller than radius of satellite galaxy, we assume they have been
accreted by satellite galaxy or switch from one circular orbit to another and
do not increase stars’ random velocity. For those with impact parameters
similar to radius of satellite galaxy, they experience strong encounters and
we assume they have obtain much energy and distribute far away from the main
part of stars in velocity coordinate system. Thus they do not affect the
velocity distribution of the main part of field stars.

The impulse approximation is valid only if the encounter time is short
compared to the crossing time. According to page 655, chapter 8.2 of Binney &
Tremaine (2008), the crossing time (sometimes also called the dynamical time)
can be estimated as $r/\sigma$, where $r$ is the Galactic radius and $\sigma$
is the velocity dispersion at the radius. While the encounter time can be
estimated as $\sim l/v$, where $l$ is the size of region passed through by
satellite galaxy. Table 1 lists our estimation of crossing time and encounter
time of each satellite galaxy in our sample. $\sigma$ is calculated directly
by square root of $\sigma_{v_{0}}^{2}$ listed in table 1. The size of region
we are studying around satellite galaxy is around 10 kpc, which is used to
calculate encounter time for the whole sample. To obtain relative velocity,
fitted center of velocity distribution of Galaxy field star component is taken
as tangential velocity of the Milky Way and its radial velocity is taken as
zero. While for satellite galaxies, proper motions and radial velocities has
been taken from Li et al. (2021) and Simon (2019). Coordinates module of
Astropy package has been used to transform observable into velocities. The
impulse approximation is not a good approximation for some satellite galaxies
that not have significantly shorter encounter time than crossing time and they
have been removed from our sample. With impulse approximation, the velocity
perturbation in the encounter for each field star is

$\Delta v_{\perp}=\frac{GM}{bv}$

, where G is the gravitational constant; $M$ is the mass of a satellite
galaxy, which has been assumed as a point mass; $b$ is the distance of closest
approach; while $v$ the velocity of a field star relative to the satellite
galaxy. Supposing the direction of velocity of each field star does not change
much during the impact, then the line of $b$ is approximately perpendicular to
$\vec{v}$, thus

$b\thickapprox\frac{|\vec{r}\times\vec{v}|}{|\vec{v}|}=\frac{|(rcos\theta,rsin\theta)\times(v_{1},v_{2})|}{|\vec{v}|}.$
(2)

Where $\vec{r}=(rcos\theta,rsin\theta)$ is star’s spatial coordinate with
origin at the satellite galaxy, while $\vec{v}=(v_{1},v_{2})$ is star’s
velocity coordinate. We made a simplified approximation that the encounter
only adds energy along the direction perpendicular to the original velocity
direction. Since stars has been assumed homogeneous distribution in space, the
mean of velocity perturbation is zero, but the velocity dispersion becomes
larger. The variance of perturbed velocity component $v_{\perp}$ can be
calculated by integrating over entire phase space:

$\displaystyle\overline{(\Delta v_{\perp})^{2}}$ $\displaystyle=$
$\displaystyle\int(\frac{GM}{bv})^{2}f(v)dvdrrd\theta$ (3) $\displaystyle=$
$\displaystyle\int\frac{G^{2}M^{2}}{(v_{2}rcos\theta-
v_{1}rsin\theta)^{2}}f(v_{1},v_{2})drdv_{1}dv_{2}rd\theta$ (4)
$\displaystyle=$
$\displaystyle\int\frac{G^{2}M^{2}}{r}dr\frac{f(v_{1},v_{2})dv_{1}dv_{2}d\theta}{(v_{1}sin\theta-
v_{2}cos\theta)^{2}}$ (5)

The $M$ is constant, while

$\int G^{2}\frac{f(v_{1},v_{2})dv_{1}dv_{2}d\theta}{(v_{1}sin\theta-
v_{2}cos\theta)^{2}}$

integrated over the whole velocity space is only a function of velocity
dispersion and has little dependency on $r$. If denoting the integral by $C$,
the equation 5 can be expressed as

$\overline{(\Delta v_{\perp})^{2}}=M^{2}C\int\frac{dr}{r}$ (6)

For an annulus with small width, the change of velocity square is
approximately:

$\overline{(\Delta v_{\perp})^{2}}=(\ln\frac{r+\Delta
r}{r})M^{2}C=\sigma_{v}^{2}-\sigma_{v0}^{2}$ (7)

where $C$ is a constant to $r$ under condition ignoring possible correlation
between velocity distribution and spatial distribution of stars in an annulus.
The $\sigma_{v}^{2}$ is the present total velocity dispersion, while
$\sigma_{v_{0}}^{2}$ is the total velocity dispersion before velocity
perturbation. Fitting $\sigma_{v}^{2}$ against $(\ln\frac{r+\Delta r}{r})$, we
have obtained $M^{2}C$ and $\sigma_{v_{0}}^{2}$ at the same time. The first
eight panels of figure 2 show corner distributions in parameter space of two
axes of the velocity ellipse for each annulus sample of AquaII. While the rest
panels show corner distributions of $M^{2}C$ versus $\sigma_{v_{0}}^{2}$ of
each satellite galaxy in our sample. Though only a few points have been used
to fit equation 7 for each satellite galaxy, we think it is enough to reveal
the main relation.

Figure 2: The corner plots of fitted two axes of velocity ellipses of annulus
samples of AquaII and $M^{2}C$ versus $\sigma_{v_{0}}^{2}$ for each satellite
galaxy.

##  3 Result and Discussion

Figure 3: Fitted $M^{2}C$ versus known mass $M_{0}^{2}$.

Figure 3 shows fitted $M^{2}C$ versus known mass $M_{0}^{2}$ in logarithmic
coordinate which is for aesthetics since mass square spans several orders of
magnitude. $M^{2}C$ is fitted coefficients from equation 7, while $M_{0}^{2}$
is known masses of satellite galaxies from literatures listed in table 1. The
blue straight line is only a line fitted between $M^{2}C$ and $M_{0}^{2}$ and
does not indicate $C$ has to be a constant. The error bars symbolically show
three times of uncertainties obtained by former steps, since uncertainty
brought by distance should be much larger than uncertainty brought by proper
motion. The spearman correlation coefficient between $M^{2}C$ and $M_{0}^{2}$
is 0.81, which means their correlation is close to linear relation. The
spearman correlation coefficient between $\sigma_{v0}^{2}$ and distance from
Earth is 0.96, which means their correlation is also very close to linear
relation. Our result at least qualitatively shows that gradient of velocity
dispersion of field stars around a satellite galaxy is strongly related to the
total mass of the satellite galaxy. Without full three dimensional velocity
information, it is meaningless to use more complicated functions to describe
the correlation than linear relation.

Since gravity is always an attractive force, a equilibrium gravitational
system must be inhomogeneous. Our relation is obtained under the assumption of
an infinite, homogeneous stellar system of field stars which is only an
approximation of the real observed spatial distribution, but can illuminate
much of the response behavior of real stellar system. We think the total
uncertainty should not be more than an order of magnitude larger than
parameter itself. The gravitational mass of satellite galaxy is no doubt
related to the enlargement of velocity dispersion of field star caused by its
gravitational drag.

Our result is only an order-of-magnitude argument, but in the future, large
surveys like CSST and LSST will cover more stars in local universe and they
can provide radial velocity and even chemical abundances information. With
more proper motions and parallaxes information from Gaia future data release,
it is possible that much more stars will have full 6 dimensional information.
Even then distinguishing single stars in the dense center of dwarf galaxies is
still very hard, more precise analysis about field stars around satellite
galaxy is promising.

We thank the anonymous referee for his/her constructive suggestions. This work
is supported by Strategic Priority Program of the Chinese Academy of Sciences
under grant number XDB41000000, the Fundamental Research Funds for the Central
Universities and Project funded by China Postdoctoral Science Foundation No.
2021M703168. This study is also supported by the National Natural Science
Foundation of China under grant No. 11973048, 11927804, 11890694 and National
Key R&D Program of China No. 2019YFA0405502. We acknowledge the support from
the 2m Chinese Space Station Telescope project: CMS-CSST-2021-A10, CMS-
CSST-2021-B03, CMS-CSST-2021-B05. The authors also acknowledge all the open-
source software involved in this study, specially TOPCAT, Python and R soft.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular, the institutions
participating in the Gaia Multilateral Agreement. Table 1 lists fitted
parameters in equation 7, distances, known masses of satellite galaxies and
referenced literatures. Column $ct$ and $et$ are respectively crossing time
and encounter time in unit of million year.

Table 1: Parameters. Name | $M^{2}C$ | $\sigma_{v0}^{2}$ | $dis$ | $M_{0}$ | $ct$ | $et$ | reference
---|---|---|---|---|---|---|---
unit | $10^{4}$ |  $10^{4}$ km2s-2 | kpc | $10^{6}M_{sun}$ | Myr | Myr |
Aquarius2 | $267^{+26}_{-37}$ | $1332^{+28}_{-23}$ | 105 | 2.7 | 28.8 | 4.0 | Torrealba et al. (2016)
Carina1 | $1390^{+134}_{-117}$ | $363^{+123}_{-126}$ | 107 | 6.3 | 55.8 | 5.1 | McConnachie (2012)
Canes Venatici1 | $3561^{+337}_{-305}$ | $5995^{+412}_{-321}$ | 218 | 27.0 | 28.1 | 2.7 | Simon & Geha (2007)
Canes Venatici2 | $480^{+58}_{-44}$ | $2811^{+49}_{-41}$ | 161 | 0.91 | 30.2 | 4.0 | McConnachie (2012)
Coma Berenices | $378^{+48}_{-37}$ | $197^{+39}_{-33}$ | 45 | 0.94 | 32.1 | 8.6 | McConnachie (2012)
Crater2 | $397^{+38}_{-37}$ | $1871^{+43}_{-39}$ | 116 | 5.5 | 26.7 | 4.5 | Ji et al. (2021)
Draco1 | $539^{+53}_{-51}$ | $409^{+54}_{-47}$ | 76 | 11.0 | 37.5 | 8.8 | McConnachie (2012)
Draco2 | $75^{+10}_{-7}$ | $71^{+8}_{-7}$ | 24 | 0.32 | 28.7 | 12.4 | Martin et al. (2016)
Eridanus2 | $1551^{+146}_{-148}$ | $9261^{+146}_{-162}$ | 382 | 12.0 | 39.6 | 2.6 | Li et al. (2017)
Grus1 | $755^{+65}_{-80}$ | $2546^{+75}_{-71}$ | 116 | 2.5 | 23.0 | 4.9 | Walker et al. (2016)
Hercules | $103^{+82}_{-140}$ | $1081^{+99}_{-121}$ | 126 | 2.6 | 38.4 | 3.3 | McConnachie (2012)
Hydra2 | $54^{+9}_{-6}$ | $2575^{+8}_{-5}$ | 148 | 0.8 | 29.1 | 3.1 | Kirby et al. (2015)
Leo1 | $2550^{+229}_{-256}$ | $7270^{+202}_{-194}$ | 258 | 12.0 | 30.2 | 1.4 | McConnachie (2012)
Leo2 | $734^{+76}_{-71}$ | $4770^{+67}_{-79}$ | 236 | 4.6 | 34.0 | 1.6 | McConnachie (2012)
LeoT | $393^{+34}_{-43}$ | $7647^{+37}_{-40}$ | 422 | 3.9 | 48.2 | 1.5 | McConnachie (2012)
Phoenix1 | $205^{+22}_{-18}$ | $25783^{+22}_{-21}$ | 409 | 0.89 | 25.4 | 0.9 | McConnachie (2012)
Pisces2 | $841^{+48}_{-94}$ | $3889^{+95}_{-90}$ | 181 | 1.6 | 29.0 | 3.2 | Kirby et al. (2015)
Segue1 | $18^{+30}_{-2}$ | $84^{+2}_{-5}$ | 28 | 0.26 | 30.3 | 5.9 | McConnachie (2012)
Triangulum2 | $192^{+19}_{-20}$ | $53^{+18}_{-21}$ | 36 | 0.89 | 49.4 | 20.6 | Kirby et al. (2015)
Tucana2 | $183^{+15}_{-17}$ | $619^{+52}_{-54}$ | 54 | 2.7 | 21.8 | 10.2 | Walker et al. (2016)
Tucana5 | $184^{+16}_{-18}$ | $385^{+16}_{-18}$ | 52 | 1.7 | 26.4 | 9.7 | Simon et al. (2020)
Ursa Major1 | $1276^{+166}_{-110}$ | $519^{+134}_{-126}$ | 97 | 15 | 42.5 | 6.3 | Simon & Geha (2007)
Ursa Major2 | $574^{+53}_{-64}$ | $10^{+67}_{-47}$ | 32 | 4.9 | 97.8 | 28.5 | Simon & Geha (2007)

## References

  * Binney & Tremaine (2008) Binney, J. & Tremaine, S. 2008, Galactic Dynamics: Second Edition, by James Binney and Scott Tremaine. ISBN 978-0-691-13026-2 (HB). Published by Princeton University Press, Princeton, NJ USA, 2008.
  * Carlberg & Hartwick (1989) Carlberg, R. G. & Hartwick, F. D. A. 1989, ApJ, 345, 196. doi:10.1086/167895
  * Chandrasekhar (1943) Chandrasekhar, S. 1943, ApJ, 97, 255. doi:10.1086/144517
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., et al. 2013, PASP, 125, 306. doi:10.1086/670067
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1. doi:10.1051/0004-6361/201629272
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1. doi:10.1051/0004-6361/202039657
  * Ji et al. (2021) Ji, A. P., Koposov, S. E., Li, T. S., et al. 2021, ApJ, 921, 32. doi:10.3847/1538-4357/ac1869
  * Kirby et al. (2015) Kirby, E. N., Cohen, J. G., Simon, J. D., et al. 2015, ApJ, 814, L7. doi:10.1088/2041-8205/814/1/L7
  * Kirby et al. (2015) Kirby, E. N., Simon, J. D., & Cohen, J. G. 2015, ApJ, 810, 56. doi:10.1088/0004-637X/810/1/56
  * Lacey (1984) Lacey, C. G. 1984, MNRAS, 208, 687. doi:10.1093/mnras/208.4.687
  * Li et al. (2017) Li, T. S., Simon, J. D., Drlica-Wagner, A., et al. 2017, ApJ, 838, 8. doi:10.3847/1538-4357/aa6113
  * Li et al. (2021) Li, H., Hammer, F., Babusiaux, C., et al. 2021, ApJ, 916, 8. doi:10.3847/1538-4357/ac0436
  * Lin & Tremaine (1983) Lin, D. N. C. & Tremaine, S. 1983, ApJ, 264, 364. doi:10.1086/160604
  * Lindegren et al. (2021) Lindegren, L., Klioner, S. A., Hernández, J., et al. 2021, A&A, 649, A2. doi:10.1051/0004-6361/202039709
  * Martin et al. (2016) Martin, N. F., Geha, M., Ibata, R. A., et al. 2016, MNRAS, 458, L59. doi:10.1093/mnrasl/slw013
  * McConnachie (2012) McConnachie, A. W. 2012, AJ, 144, 4. doi:10.1088/0004-6256/144/1/4
  * McConnachie & Venn (2020) McConnachie, A. W. & Venn, K. A. 2020, AJ, 160, 124. doi:10.3847/1538-3881/aba4ab
  * Oort (1950) Oort, J. H. 1950, Bull. Astron. Inst. Netherlands, 11, 91
  * Öpik (1932) Öpik, E. 1932, Proceedings of the American Academy of Arts and Sciences, 67, 169. doi:10.2307/20022899
  * Palmer & Papaloizou (1985) Palmer, P. L. & Papaloizou, J. 1985, MNRAS, 215, 691. doi:10.1093/mnras/215.4.691
  * Poveda (1958) Poveda, A. 1958, Boletin de los Observatorios Tonantzintla y Tacubaya, 2, 3
  * Riello et al. (2021) Riello, M., De Angeli, F., Evans, D. W., et al. 2021, A&A, 649, A3. doi:10.1051/0004-6361/202039587
  * Simon & Geha (2007) Simon, J. D. & Geha, M. 2007, ApJ, 670, 313. doi:10.1086/521816
  * Simon (2019) Simon, J. D. 2019, ARA&A, 57, 375. doi:10.1146/annurev-astro-091918-104453
  * Simon et al. (2020) Simon, J. D., Li, T. S., Erkal, D., et al. 2020, ApJ, 892, 137. doi:10.3847/1538-4357/ab7ccb
  * Taylor (2005) Taylor Mark B., 2005, Astronomical Data Analysis Software and Systems XIV, 347, 29
  * Torrealba et al. (2016) Torrealba, G., Koposov, S. E., Belokurov, V., et al. 2016, MNRAS, 463, 712. doi:10.1093/mnras/stw2051
  * Walker et al. (2016) Walker, M. G., Mateo, M., Olszewski, E. W., et al. 2016, ApJ, 819, 53. doi:10.3847/0004-637X/819/1/53
  * Walker et al. (2009) Walker, M. G., Mateo, M., Olszewski, E. W., et al. 2009, ApJ, 704, 1274. doi:10.1088/0004-637X/704/2/1274
  * Weinberg (1989) Weinberg, M. D. 1989, MNRAS, 239, 549. doi:10.1093/mnras/239.2.549
  * White (1983) White, S. D. M. 1983, ApJ, 274, 53. doi:10.1086/161425
  * Wolf et al. (2010) Wolf, J., Martinez, G. D., Bullock, J. S., et al. 2010, MNRAS, 406, 1220. doi:10.1111/j.1365-2966.2010.16753.x


11institutetext: Simon Fraser University, Burnaby, Canada
11email<EMAIL_ADDRESS>
# Graph Neural Network Approach to Semantic Type Detection in Tables ††thanks:
Published at Pacific-Asia Conference on Knowledge Discovery and Data Mining
2024

Ehsan Hoseinzade Corresponding author Ke Wang

###### Abstract

This study addresses the challenge of detecting semantic column types in
relational tables, a key task in many real-world applications. While language
models like BERT have improved prediction accuracy, their token input
constraints limit the simultaneous processing of intra-table and inter-table
information. We propose a novel approach using Graph Neural Networks (GNNs) to
model intra-table dependencies, allowing language models to focus on inter-
table information. Our proposed method not only outperforms existing state-of-
the-art algorithms but also offers novel insights into the utility and
functionality of various GNN types for semantic type detection. The code is
available at https://github.com/hoseinzadeehsan/GAIT

###### Keywords:

graph neural networks language model semantic types .

##  1 Introduction

Accurately identifying (or tagging) the semantic types of columns inside a
table is crucial for different information retrieval tasks like data cleaning
[16], schema matching [17], and data discovery [6]. One emerging application
is automatically tagging sensitive columns in a table, such as personal
information, before deciding what information can be released. Previous works
showed that machine learning approaches outperform traditional methods in
predicting semantic types [9, 26, 1, 2]. Sherlock [9], a single-column
prediction framework, feeds various features of a column to a deep feed-
forward neural network to get the prediction. This method ignores the global
context and the dependencies between columns, making it difficult to
distinguish the semantic types in cases like in Figure 1. SATO [26] improves
upon Sherlock by adding a topic modeling module and a structured prediction
module on top of Sherlock, to jointly predict semantic types of all the
columns in a table by leveraging the topic of a table and the dependencies
between columns in a table.

Building on the trend of applying machine learning to tabular data,
researchers have started using language models like BERT [5]. By feeding
tables to BERT, which was originally designed for textual data, they exploit
its extensive pre-training. This adaptation has created new frameworks that
fine-tune BERT for column type annotation [4]. In addition to the values of
the target column, two other sources of information can be used to improve the
accuracy of semantic type annotation: intra-table information refers to other
columns in the same table and inter-table information refers to other tables
in the data. Given that language models have a small limit on the number of
input tokens (BERT takes a maximum of 512 tokens), column type annotation
models are developed to handle only one of the two mentioned sources of
information.

Incorporating intra-table information has led to multi-column prediction
approaches [25] that are designed to address the limitation of single-column
prediction models in situations like Fig 1 by accounting for broader table
context, specifically column relationships and information. TABBIE [10]
encodes rows and columns of a table respectively to get a better understanding
of tables. The most prominent work in this category is Doduo [18] where BERT
is modified to receive the whole columns of a table and predict their semantic
types together.

Having inter-table information [21] can be a huge help in cases where the
target column does not have enough high-quality data to make a good semantic
prediction. For example, if a table has a column with entries like ’Orange’
and ’Peach’, the semantic type is ambiguous. However, by identifying and
augmenting this column with columns of similar tables that have entries like
’Red’ and ’Blue’, the semantic type becomes clearer, indicating that this
column is likely about colors rather than fruits. The most recent work, RECA
[19], in addition to the values of the target column, identifies values of the
most useful similar tables and feeds them to BERT to get the semantic type of
the target column.

However, due to the small limit on the number of input tokens of language
models like BERT, Doduo and RECA have the following drawbacks:

Figure 1: The two tables on the right both have the column containing the
values “Paris”, “Ottawa” and “London”. Without considering information coming
from other columns it is difficult for a single-column prediction model to
detect the actual semantic types of these columns. The multi-column prediction
will label these columns correctly by jointly predicting all columns in a
table.

  1. 1.

Doduo feeds the whole table to BERT and because of that, it poses difficulties
in handling wide tables. For instance, the average number of columns of tables
in Open Data is 16, but there’s a large variance with some tables having
hundreds of columns. Furthermore, Doduo is not designed in a way that can
incorporate inter-table context information and ignores this useful source of
information [19].

  2. 2.

RECA incorporates inter-table but not intra-table information, predicting the
semantic type of each column individually. This approach enables RECA to
handle wide tables within the language model’s small token limit, as it
doesn’t need to model the entire table at once. However, this means it
overlooks the valuable information in column relationships, crucial in complex
scenarios like Fig 1, where semantic types are difficult to distinguish.

Thus, a question is whether it is possible to incorporate both inter-table and
intra-table information without suffering from the difficulty of handling wide
tables as in Doduo and to benefit from the generalization power of the
language model based approach. Inspired by recent developments in computer
vision such as visual reasoning, object detection, and scene graph generation
[3, 14] where a main task is tagging the objects inside an image by leveraging
the relations among the objects in the image, we propose to augment any
single-column prediction framework, especially those incorporating inter-table
information like RECA (addressing the drawback of Doduo) by a graph neural
network (GNN) module to model the whole dependencies between columns. Thus,
our framework incorporates both inter-table and intra-table information. In
particular, we model each table by a graph with the nodes representing the
columns and the edges representing the dependencies between columns through
Message Passing of GNN. By considering the dependencies between all pairs of
columns, this approach, called GAIT (Graph bAsed semantIc Type detection),
addresses the above drawback of RECA, making it a multi-column prediction
framework. The challenge is how to represent the features of a column by a
node so that Message Passing can leverage the dependencies among columns.

GAIT stands out in efficiently handling wide tables and benefiting from a
language model based approach by building on top of models like RECA that are
single-column predictions and language model based. While Doduo’s
effectiveness decreases in scenarios with minimal column dependency, and RECA
faces challenges when similar tables are scarce, GAIT’s integration of both
inter-table and intra-table information makes it a competitive model in these
diverse scenarios. This dual-data approach enables GAIT to maintain its
performance and effectively address the limitations encountered by models
focusing on either inter-table or intra-table information alone.

##  2 Related works

Column type prediction methods are typically grounded into two categories,
i.e., deep learning based frameworks and language model based frameworks.

Deep learning based models. ColNet [1] uses DBpedia cell value lookups to
create examples and trains a CNN with Word2Vec embeddings. HNN [2], models
intra-column semantics, enhancing Colnet. Sherlock [9] employs column
statistics, paragraph, word, and character embeddings to predict column types
through a neural network. SATO [26] builds on Sherlock, adding topic modeling
features and adjusting predictions for column dependencies using a CRF.

Language model based models. Language models, like BERT [5], have been adopted
for table tasks [23] including column type prediction. TaBERT [25] utilizes
BERT as a base model to capture the table content features. TURL [4] is pre-
trained unsupervisedly, using a visibility matrix for row and column context,
and then fine-tuned for table-related tasks. TABBIE [10] separately processes
the rows and columns of tables to give a better understanding of them. Doduo
[18] predicts all the columns of a table together by feeding the whole table
to BERT. RECA [19] incorporates inter-table context information by finding and
aligning relevant tables. TCN [21] suggests using both intra-table and inter-
table information for column type prediction. However, it needs table schema
and page topic, which many datasets, like Webtables and Semtab don’t have.

Summary: Most of the previous works [25, 18, 4, 10], except for TCN and RECA,
do not incorporate inter-table information for prediction. TCN [21] requires
having table schema and page topic, which does not exist in many datasets.
RECA does not incorporate intra-table dependencies. Our GAIT predicts semantic
types of columns exclusively based on the content of the tables by integrating
both intra-table and inter-table information.

##  3 Problem Definition

We aim to predict the semantic types of the columns of a given table with
missing column headings. This problem is called table annotation. To learn to
predict semantic types, a collection of labeled tables is given as the
training data $D$, where each table $t(c_{1},c_{2},...,c_{n})$ consists of $n$
columns and each column is labeled as one of the $k$ pre-defined semantic
types, also called classes, e.g., Age, Name, Country (note that semantic types
are different from atomic types like integer and string). Note that the number
of columns $n$ and rows can differ for different tables. Typically, the first
step is to extract a feature vector (embedding) to represent a column $c_{i}$.
After applying a feature extractor function $\phi$ to the values of a column
$c_{i}$ and potential inter-table information related to $c_{i}$, an
$m$-dimension feature (embedding) vector $\psi_{i}$ is generated for column
$c_{i}$. The rest of the task is to learn a mapping $f$ that, given
$\psi=<\psi_{1},...,\psi_{n}>$ of a table of $n$ unlabeled columns, predicts
the classes for the $n$ columns in the table.

Figure 2: The framework of GAIT: GAIT adds a GNN learning on top of the
single-column prediction module, which is RECA in this work. The output of
RECA is a class distribution for each column in a table, which provides the
initial hidden state of the node representing that column in the GNN. For a
table with $n$ columns, RECA is performed $n$ times. Then, the GNN learns the
best representations of the hidden states of all nodes to minimize a loss
function, through Message Passing that models the dependencies between
columns.

##  4 GAIT

Figure 2 shows the framework of GAIT. It is built on a single-column
prediction provided by RECA. We opted for RECA due to its high performance in
column type annotation as a result of incorporating useful inter-table
information from other relevant tables. That said, GAIT’s design is versatile.
While we utilize RECA, other single-column prediction modules can be
integrated. GAIT employs a GNN in which each graph represents a table with the
nodes representing the columns in the table. The initial representation of
each node is the logits outputted by RECA for the represented column. Once fed
with such class distributions as input, the training of GNN is responsible for
capturing the dependencies of classes among columns and does not further
involve the lower-level single-column prediction module. This approach treats
the preliminary prediction of the single-column prediction of RECA as the node
features for training GNN, which is more efficient than concatenating the
networks of low-level modules into a giant neural network. Our method stacks a
GNN as a meta-learner on top of RECA (i.e., two classifiers) instead of
concatenating RECA and GNN into one classifier, which improves the overall
performance according to stacked generalization technique [24]. We now present
more details.

###  4.1 Single-column Prediction

The single-column prediction is responsible for generating the preliminary
prediction of each column. We use RECA [19] for this task. In the RECA
process, the primary goal is to improve the understanding of a target column
in the main table by integrating relevant data from other tables. The process
begins by identifying named entities across all tables. Each entity is
assigned a type from a predefined set, with the most common type within a
column being selected as its representative named entity type. Following this,
RECA constructs the named entity schema for each table, which includes the
named entity types of all its columns. The next step is to find the topical
relevance of other tables to the main table. This is done by calculating the
Jaccard similarity between the words in the main table and other tables.
Tables that are similar enough are chosen as candidate tables for further
analysis. Among these candidates, tables with the same named entity schema as
the main table are labeled as relevant tables. Additionally, tables with
similar, but not identical, named entity schemas are called sub-related
tables. The final step involves combining the data from the target column in
the main table with data from columns having the same named entity type in
both related and sub-related tables and feeding it to a language model, BERT,
to find the semantic type of the target column.

###  4.2 Graph-based Prediction

Graph Modeling of a Table The training data for GNN is a collection of graphs
organized into several mini-batches, where each graph corresponds to a table
in the original training data. For a table with $n$ columns, we create a graph
of $n$ nodes where each node represents a column in the table and create an
edge between each pair of columns. Initially, each node $u$ of a graph has the
representation $h^{0}_{u}$ initialized to the logits $<o_{1},...,o_{k}>$
outputted by RECA for the corresponding column, which has one value for each
class. This initial state represents the class bias of single-column
prediction. In addition, each node is associated with the true class of the
represented column.

Message Passing Subsequently, the representation of all the nodes in a mini-
batch of graphs is updated through the Message Passing mechanism of GNN along
edges. For this purpose, we consider three different types of GNNs, graph
convolutional network (GCN) [11], gated graph neural network (GGNN) [15], and
graph attention network (GAT) [20], with the following $UPDATE$ functions
where $\sigma$ is the activation function, $N(u)$ is a list of nodes connected
to node $u$, $h_{u}^{s}$ is the representation (also called embedding) of node
$u$ at step $s\geq 0$, $W^{(s)}$ is a model parameter:

  * •

GCN: assigns equal weights to all the neighbor nodes while updating the
embedding of each node (Eq 1).

$h_{u}^{(s+1)}=\sigma(\sum_{v\in N(u)\cup
u}\frac{W^{(s)}h_{v}^{(s)}}{\sqrt{|N(u)||N(v)|}})$ (1)

  * •

GGNN: uses gated recurrent unit (GRU) to evaluate messages coming from
adjacent nodes while updating the embedding of each node (Eq 2).

$h_{u}^{s+1}=GRU(h_{u}^{s},\sum_{v\in N(u)}W^{(s)}h_{v}^{s})$ (2)

  * •

GAT: updates node embedding (Eq 3) according to the multi-head attention
weights (Eq 4), where $K$ is the number of attention heads, $a^{(s,k)}$ and
$W^{(s,k)}$ are model parameters for attention head $k$, and $\oplus$ is
concatenation.

$h_{u}^{(s+1)}=\oplus_{k=1}^{K}(\sigma\sum_{v\in
N(u)\cup\\{u\\}}\alpha^{(s,k)}_{u,v}W^{(s,k)}h_{v}^{(s)})$ (3)

$\alpha^{s,k}_{u,v}=\frac{\exp(ReLU(a^{(s,k)^{T}}(W^{(s,k)}h_{u}^{(s)}\oplus
W^{(s,k)}h_{v}^{(s)})))}{\sum_{v^{\prime}\in
N(u)\cup\\{u\\}}\exp(ReLU(a^{(s,k)^{T}}(W^{(s,k)}h_{u}^{(s)}\oplus
W^{(s,k)}h_{v^{\prime}}^{(s)})))}$ (4)

The $UPDATE$ function is applied to each node $u$ in the mini-batch of graphs
for $S$ steps, where $S$ is the number of hidden layers and output layer.
$h^{S}_{u}$ has one unit for each of the $k$ classes and serves the final
output for the $k$ classes. The class prediction for the node $u$ is given by
applying softmax to $h^{S}_{u}$.

The main difference among GCN, GGNN, and GAT is in their treatment of adjacent
nodes (columns). GAT uses an attention mechanism to assign varying weights to
these nodes based on their importance. In contrast, GCN averages the features
of neighbor nodes, while GGNN processes these features through a GRU to
determine their relevance before updating the node embeddings.

Loss Function Given the logit vector $h^{S}_{u}$ for a node $u$ with the true
class $class_{u}$, the loss for this node is computed by the negative log-
likelihood. The loss for a mini-batch of graphs is the sum of the loss of all
the nodes inside the mini-batch of graphs (Eq 5). We update the model
parameters to minimize the loss of a mini-batch by performing stochastic
gradient descent.

$loss=\sum_{u=1}^{\\#node}-\log(\frac{\exp(h_{u}^{S}[class_{u}])}{\sum_{m=1}^{k}\exp(h_{u}^{S}[m])})$
(5)

###  4.3 Overall Prediction

After training the GNN, to classify columns in a new table $t$, we first get
the output of RECA (before softmax) for each column in $t$. These outputs are
the initial representation $h^{0}_{u}$ of nodes $u$ in the graph representing
the table $t$. Then, message passing is done using the learned parameters in
the training phase to get the predicted class for each node, which is the
predicted class for the corresponding column in $t$. Fig 2 shows how GAIT
predicts labels of a table.

##  5 Evaluation

###  5.1 Evaluation Method

Performance metrics. Like previous works [9, 26, 18, 19], we collect weighted
f-score and macro f-score on the test data. The former is the average of
f-score of all classes, weighted by class frequencies, and the latter is the
average of treating all classes equally, regardless of their frequencies. The
macro f-score better reflects the model performance on infrequent classes. We
evaluate model performance using 5-fold cross-validation, reporting the mean
and standard deviation of the above f-scores from the test split of each fold.

Datasets. We use two datasets summarized in Table 1.

Webtables [26, 18, 19]: This dataset contains 32262 tables and 78 unique
classes extracted from the Webtables directory of VizNet [8]. We use exactly
the same 5-fold cross-validation split as in [19] , which splits the tables
(not columns) into a train set and test set in 5-folds. So, we copy directly
the f-scores of the baseline algorithms (more details on the baselines below)
except for RECA from [19].

Semtab2019 [19]: It contains 3045 tables and 275 unique classes. While this
dataset covers wider tables (an average of 4.5 columns per table), only 7603
columns are annotated. The split proposed in RECA [19] randomly divided
columns (not tables) into train, validation, and test sets. Although the
column-wise splitting of data makes sense for RECA due to the column-wise
prediction of RECA, GAIT requires having a full table to model the
dependencies between columns in the table. Therefore, our 5-fold validation
splits tables (instead of columns) into the train set and test set for this
dataset. At each fold, we further split the train set into 80% for training
and 20% for validation.

Table 1: Datasets used and the number of tables with the specified number of
columns.

Dataset | #types | #tables | #Col | avg col
---|---|---|---|---
Semtab | 275 | 3045 | 7603 | 4.5
Webtable | 78 | 32262 | 74141 | 2.3

Algorithms for comparison. All experiments were conducted with Tesla V100s. We
used the publicly available source code of
RECA111https://github.com/ysunbp/RECA-paper for the single-column prediction
module of GAIT. The GNN module of GAIT was implemented using the deep graph
library [22] , Adam optimization with a learning rate of $1e-3$ and weight
decay of $5e-4$ for training. We trained GCN, GGNN, and GAT for 100, 200, and
100 epochs respectively. To optimize the GAT structure, we tested various #
attention heads ([1, 2, 4, 8, 12]) and update steps $S$ ([1, 2, 3, 4]),
selecting the best model from the validation set as default. Similarly, for
GGNN and GCN, we determined the default model by experimenting with different
update steps $S$ ([1, 2, 3, 4]). Three algorithms for GAIT were finally
chosen: $\textnormal{{GAIT}}_{\textnormal{{GAT}}}$ (GAT with $S=2$), GAITGGNN
(GGNN with $S=3$), and GAITGCN (GCN with $S=2$).

Since GAIT incorporates RECA as its single-column prediction module, naturally
we evaluate GAIT against the baseline methods outlined in RECA’s paper and
RECA itself. These baselines are described below and their source codes are
publicly available and are used for our evaluation:

  * •

Sherlock [9]: Sherlock is a deep learning model that extracts character-level,
word-level, paragraph-level and global-level statistical features from tables
to form vector representations for table columns.

  * •

TaBERT [25]: TaBERT simultaneously analyzes queries and a table, selecting
three crucial rows to create table content snapshots. It then uses BERT to
develop representations for each table column, aiding in classification.

  * •

TABBIE [10]: improves TaBERT by separately processing the rows and columns of
tables. The embedding of the target column is used for prediction.

  * •

Doduo [18]: Modifies BERT to feed the whole columns of a table to BERT and
predicts the semantic types of all of the columns in a table together.

  * •

RECA [19]: RECA finds relevant tables for the target table and uses the
information coming from these tables and the values of the target column to
predict the semantic type of the target column.

We do not compare with SATO [26] and TURL [4] as Doduo outperformed them.
Since TCN [21] requires having table schema and page topic [19], it cannot be
applied to our datasets.

###  5.2 Results

Table 2 Shows the performance of GAIT and the baseline algorithms. GAIT
outperforms Sherlock by a large margin. The main reason behind the poor
performance of Sherlock compared with other models is its simplicity. While
other models including GAIT utilize language models for semantic type
prediction, Sherlock relies on simple semantic features to do so. Furthermore,
Sherlock does not use intra-table or inter-table information for prediction.

Table 2: Macro f-score and weighted f-score. | Semtab | Webtables
---|---|---
Model | Weighted f-score | Macro f-score | Weighted f-score | Macro f-score
sherlock [9] | 0.638$\pm$0.009 | 0.417$\pm$0.017 | 0.844$\pm$0.001 | 0.670$\pm$0.010
TaBERT [25] | 0.756$\pm$0.011 | 0.401$\pm$0.025 | 0.896$\pm$0.005 | 0.650$\pm$0.011
TABBIE [10] | 0.798$\pm$0.012 | 0.542$\pm$0.022 | 0.929$\pm$0.003 | 0.734$\pm$0.019
Doduo [18] | 0.819$\pm$0.010 | 0.565$\pm$0.021 | 0.928$\pm$0.001 | 0.742$\pm$0.012
RECA [19] | 0.825$\pm$0.015 | 0.583$\pm$0.019 | 0.935$\pm$0.032 | 0.783$\pm$0.017
$\textnormal{GAIT}_{\textnormal{GGNN}}$ | 0.844$\pm$0.003 | 0.606$\pm$0.018 | 0.936$\pm$0.003 | 0.797$\pm$0.022
$\textnormal{GAIT}_{\textnormal{GCN}}$ | 0.845$\pm$0.006 | 0.622$\pm$0.020 | 0.939$\pm$0.004 | 0.794$\pm$0.017
$\textnormal{GAIT}_{\textnormal{GAT}}$ | 0.852$\pm$0.004 | 0.643$\pm$0.017 | 0.940$\pm$0.003 | 0.799$\pm$0.019

Among language model based models TaBERT shows the worst performance because
it was initially developed for table semantic parsing and column embeddings
generated by TaBERT are not suitable for column type annotation [19]. RECA,
single-column prediction module of GAIT, outperforms both TABBIE and Doduo.
TABBIE and Doduo use the limited input tokens of language models to process
intra-table context while ignoring the inter-table context information when
generating the embeddings of the target columns. However, RECA mainly focuses
on extracting useful inter-table context information to enhance the embeddings
of the target columns [19].

GAIT with different GNNs outperforms RECA, and by extension TABBIE and Doduo
in both datasets. In particular, GAIT shows about 6% and 2.7% improvement in
macro and weighted f-scores over RECA in the Semtab-dataset. These results
prove that modeling the dependencies between columns in a table, which is the
main advantage of GAIT over RECA, is useful. GAIT successfully applies a GNN
on top of RECA to do so. Among different variations of GAIT, GAT shows the
best performance. Assigning different weights to adjacent nodes (columns)
according to their importance when updating representation of a node is the
key to the superior performance of GAT compared to GCN and GGNN.

In both datasets, GAIT’s improvement is larger on the macro f-score than the
weighted f-score. This means that infrequent classes that label fewer columns
benefit more from the whole dependency approach of the GNN approach. Such
classes have less presence in the data and their learning tends to rely on the
dependencies on other columns in a table. GAIT provides a mechanism to
leverage such dependencies. This also explains why GAIT shows a better
enhancement in the Semtab dataset compared to Webtables dataset. The 3045
tables of Semtab have 275 semantic types for columns while the 32262 tables of
Webtables are limited to 78 semantic types. Consequently, many more infrequent
classes in Semtab can benefit from modeling the whole dependencies of GAIT.

To provide a better insight into this improvement, we divide the 275 classes
of Semtab dataset into three equally sized bins of High, Medium, and Low
frequencies (about 92 classes in each bin) according to the columns labeled by
classes and show the macro f-score of the classes in each bin for
$\textnormal{GAIT}_{\textnormal{GAT}}$ (best GAIT) and RECA (best baseline) in
Fig 3. While $\textnormal{GAIT}_{\textnormal{GAT}}$ improves RECA in all the
three bins, the bigger improvements happen in the low-frequency bins, for
example, the absolute improvement of 11% or the relative improvement of 96.5%
for the Low bin. Fig 3 also demonstrates that the real challenge in developing
column type annotation models is how to have a reliable prediction for medium
and low-frequency classes as the performance for high-frequency classes is
already good enough. The large improvements of
$\textnormal{GAIT}_{\textnormal{GAT}}$ over RECA in low-frequency classes is a
clear sign of its superiority in handling such classes.

Figure 3: The macro f-score of $\textnormal{GAIT}_{\textnormal{GAT}}$ and RECA
on Semtab dataset, for High, Medium, and Low-frequency classes.

We also study the impact of the number of columns in a table on both
$\textnormal{GAIT}_{\textnormal{GAT}}$ and RECA. Table 3 shows the improvement
of $\textnormal{GAIT}_{\textnormal{GAT}}$ on macro and weighted f-scores over
RECA separately for tables of a different number of columns. As the number of
columns in a table increases, the performance of RECA, which is also the
single-column prediction module of GAIT, increases. Having more columns in a
table better reveals context of that table, so RECA can find more relevant
inter-table information which is beneficial to both RECA and GAIT. Thus, the
need for dependencies between columns in
$\textnormal{GAIT}_{\textnormal{GAT}}$ decreases. The column dependency method
GAIT improves RECA mainly for low-frequency classes and tables of 2 to 4
columns, as in case of Semtab.

Table 3: The f-score improvement of $\textnormal{GAIT}_{\textnormal{GAT}}$
over RECA by tables of different number of columns.

| Semtab | Webtables
---|---|---
| macro f-score | weighted f-score | macro f-score | weighted f-score
#col | RECA | $\textnormal{GAIT}_{\textnormal{GAT}}$ | RECA | $\textnormal{GAIT}_{\textnormal{GAT}}$ | RECA | $\textnormal{GAIT}_{\textnormal{GAT}}$ | RECA | $\textnormal{GAIT}_{\textnormal{GAT}}$
2 | 0.563 | 0.603 (+4.0%) | 0.798 | 0.828 (+3.0%) | 0.738 | 0.758 (+2.0%) | 0.932 | 0.936 (+0.4%)
3 | 0.545 | 0.616 (+7.1%) | 0.797 | 0.827 (+3.0%) | 0.743 | 0.762 (+1.9%) | 0.927 | 0.930 (+0.3%)
4 | 0.566 | 0.618 (+5.2%) | 0.865 | 0.880 (+1.5%) | 0.727 | 0.746 (+1.9%) | 0.960 | 0.961 (+0.1%)
5 | 0.664 | 0.682 (+1.8%) | 0.862 | 0.862 (+0.0%) | 0.540 | 0.548 (+0.8%) | 0.978 | 0.978 (+0.0%)

##  6 Conclusion

Language model based approaches recently showed promising results in column
type annotation thanks to the semantic knowledge preserved in them. This paper
addresses some drawbacks of previous language model-based approaches, namely,
failing to incorporate inter-table and intra-table information simultaneously
due to the input token limit of language models. Our solutions, GAIT, employ
graph neural networks to model the intra-table dependencies, letting language
models focus on handling inter-table information. Experiments on different
datasets provide evidence of the effectiveness of our solutions. Looking
ahead, considering the recent advancements in large language models (LLMs) for
column type annotation [7, 12, 27, 13] exploring alternative LLMs beyond BERT
to address inter-table information could be a promising future research.

Acknowledgement. The work of Ke Wang is supported in part by a discovery grant
from Natural Sciences and Engineering Research Council of Canada.

## References

  * [1] Chen, J., Jiménez-Ruiz, E., Horrocks, I., Sutton, C.: Colnet: Embedding the semantics of web tables for column type prediction. In: AAAI (2019)
  * [2] Chen, J., Jiménez-Ruiz, E., Horrocks, I., Sutton, h.: Learning semantic annotations for tabular data. In: IJCAI. vol. 33, pp. 2088–2094 (2019)
  * [3] Chen, X., Li, L.J., Fei-Fei, L., Gupta, A.: Iterative visual reasoning beyond convolutions. In: CVPR. pp. 7239–7248 (2018)
  * [4] Deng, X., Sun, H., Lees, A., Wu, Y., Yu, C.: Turl: Table understanding through representation learning. ACM SIGMOD Record 51(1), 33–40 (2022)
  * [5] Devlin, J., Chang, M.W., Lee, K., Toutanova, K.: Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv:1810.04805 (2018)
  * [6] Fernandez, R.C., Abedjan, Z., Koko, F., Yuan, G., Madden, S., Stonebraker, M.: Aurum: A data discovery system. In: ICDE. pp. 1001–1012. IEEE (2018)
  * [7] Feuer, B., Liu, Y., Hegde, C., Freire, J.: Archetype: A novel framework for open-source column type annotation using large language models. arXiv (2023)
  * [8] Hu, K., Gaikwad, S., Hulsebos, M., Bakker, M.A., Zgraggen, E., Hidalgo, C., Kraska, T., Li, G., Satyanarayan, A., Demiralp, Ç.: Viznet: Towards a large-scale visualization learning and benchmarking repository. In: CHI. pp. 1–12 (2019)
  * [9] Hulsebos, M., Hu, K., Bakker, M., Zgraggen, E., Satyanarayan, A., Kraska, T., Demiralp, Ç., Hidalgo, C.: Sherlock: A deep learning approach to semantic data type detection. In: SIGKDD. pp. 1500–1508 (2019)
  * [10] Iida, H., Thai, D., Manjunatha, V., Iyyer, M.: Tabbie: Pretrained representations of tabular data. arXiv preprint arXiv:2105.02584 (2021)
  * [11] Kipf, T.N., Welling, M.: Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907 (2016)
  * [12] Korini, K., Bizer, C.: Column type annotation using chatgpt. arXiv (2023)
  * [13] Li, P., He, Y., Yashar, D., Cui, W., Ge, S., Zhang, H., Fainman, D.R., Zhang, D., Chaudhuri, S.: Table-gpt: Table-tuned gpt for diverse table tasks. arXiv (2023)
  * [14] Li, Y., Ouyang, W., Zhou, B., Wang, K., Wang, X.: Scene graph generation from objects, phrases and region captions. In: Proceedings of the IEEE International Conference on Computer Vision. pp. 1261–1270 (2017)
  * [15] Li, Y., Tarlow, D., Brockschmidt, M., Zemel, R.: Gated graph sequence neural networks. arXiv preprint arXiv:1511.05493 (2015)
  * [16] Limaye, G., Sarawagi, S., Chakrabarti, S.: Annotating and searching web tables using entities, types and relationships. VLDB 3(1-2), 1338–1347 (2010)
  * [17] Rahm, E., Bernstein, P.A.: A survey of approaches to automatic schema matching. the VLDB Journal 10(4), 334–350 (2001)
  * [18] Suhara, Y., Li, J., Li, Y., Zhang, D., Demiralp, Ç., Chen, C., Tan, W.C.: Annotating columns with pre-trained language models. In: SIGMOD (2022)
  * [19] Sun, Y., Xin, H., Chen, L.: Reca: Related tables enhanced column semantic type annotation framework. VLDB 16(6), 1319–1331 (2023)
  * [20] Veličković, P., Cucurull, G., Casanova, A., Romero, A., Lio, P., Bengio, Y.: Graph attention networks. arXiv preprint arXiv:1710.10903 (2017)
  * [21] Wang, D., Shiralkar, P., Lockard, C., Huang, B., Dong, X.L., Jiang, M.: Tcn: Table convolutional network for web table interpretation. In: WWW (2021)
  * [22] Wang, M., Zheng, D., Ye, Z., Gan, Q., Li, M., Song, X., Zhou, J., Ma, C., Yu, L., Gai, Y., et al.: Deep graph library: A graph-centric, highly-performant package for graph neural networks. arXiv preprint arXiv:1909.01315 (2019)
  * [23] Wang, Z., Dong, H., Jia, R., Li, J., Fu, Z., Han, S., Zhang, D.: Tuta: Tree-based transformers for generally structured table pre-training. In: SIGKDD (2021)
  * [24] Wolpert, D.H.: Stacked generalization. Neural networks 5(2), 241–259 (1992)
  * [25] Yin, P., Neubig, G., Yih, W.t., Riedel, S.: Tabert: Pretraining for joint understanding of textual and tabular data. arXiv preprint arXiv:2005.08314 (2020)
  * [26] Zhang, D., Suhara, Y., Li, J., Hulsebos, M., Demiralp, Ç., Tan, W.C.: Sato: Contextual semantic type detection in tables. arXiv preprint arXiv:1911.06311 (2019)
  * [27] Zhang, H., Dong, Y., Xiao, C., Oyamada, M.: Jellyfish: A large language model for data preprocessing. arXiv (2023)


# The Huygens Principle of Angle-Resolved Photoemission

Simon Moser  Physikalisches Institut and Würzburg-Dresden Cluster of
Excellence ct.qmat, Universität Würzburg, 97074 Würzburg, Germany
<EMAIL_ADDRESS>
###### Abstract

Angle-resolved photoemission spectroscopy (ARPES) measures the interference of
dipole allowed Coulomb wavelets from the individual orbital emitters that
contribute to an electronic band. If Coulomb scattering of the outgoing
electron is neglected, this Huygens view of ARPES simplifies to a Fraunhofer
diffraction experiment, and the relevant cross-sections to orbital
Fouriertransforms. This plane wave approximation (PWA) is surprisingly
descriptive of photoelectron distributions, but fails to reproduce kinetic
energy dependent final state effects like dichroism. Yet, Huygens principle of
ARPES can be easily adapted to allow for distortion and phase shift of the
outgoing Coulomb wave. This retains the strong physical intuition and low
computational cost of the PWA, but naturally captures momentum dependent
interference effects in systems that so far required treatment at the ab
initio level, such as linear dichroism in Rashba systems BiAg2 and AgTe.

Introduction.—In 1678, Dutch physicist Christiaan Huygens proposed his famous
principle of wave mechanics, stating that every point on a wavefront is itself
the source of a spherical wavelet. Combined with Augustin-Jean Fresnel’s
1818ies insight that these secondary wavelets all mutually interfere to form
the actual wavefront, this intuitive picture provides an appropriate
explanation of (near and far field) wave-propagation, reflection and
refraction, and most importantly: diffraction Miller (1991).

For angle-resolved photoemission spectroscopy (ARPES), a well established
technique to map electronic structure of adsorbed molecules and ordered solid
state, such an intuitive interpretation in terms of simple wave mechanics
remains elusive. This is remarkable, as fundamentally, the electronic
structure contrast produced by ARPES relies on the coherent interference of
photoelectron wavelets emitted from individual orbital emitters that are
phase-locked through their atomic arrangement, i.e., the structure of a
particular molecule, or the lattice properties of an ordered solid.

In 2009, this very insight along with the availability of efficient
photoelectron detectors, pioneered a novel imaging technique of real space
molecular orbitals based on the ARPES response of adsorbed organic molecules
Puschnig  _et al._ (2009); Lueftner  _et al._ (2014). This orbital tomography
technique assumes that the photoelectrons transition into plane waves that
freely propagate to the detector, and that the ARPES intensity distribution is
determined by the real space orbital’s Fourier transform (see Ref. Moser
(2017) and references therein). ARPES hence intuitively maps onto a Fraunhofer
diffraction experiment and the quest for a Huygens principle of ARPES thus
seems to be complete.

The tempting use of plane wave final states is flawed, however, as it neglects
scattering of the outgoing photoelectron in the Coulomb potential of the ion
it leaves behind Bradshaw and Woodruff (2015), and thus inherently fails to
describe photon energy dependent final state interference such as dichroism.
In fact, it produces an ubiquitous
$\boldsymbol{\epsilon}\cdot\boldsymbol{k_{f}}$ polarization term that
genuinely suppresses outgoing photoelectron momenta $\boldsymbol{k_{f}}$ that
move perpendicular to the polarization vector $\boldsymbol{\epsilon}$; a model
artifact that is rarely observed with this stringency in experiments.

Model.—Yet, such discrepancies can be overcome without loss of intuition or
computational ease, taking into account the appropriate scattering state of
the outgoing photoelectron, i.e., a partial wave expansion

$\displaystyle\chi_{\eta}(\boldsymbol{r})$ $\displaystyle=$ $\displaystyle
4\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{l}i^{l}~{}e^{i\sigma_{l}}R_{\eta
l}(r)Y_{l}^{m}(\boldsymbol{\Omega}_{k_{f}})Y_{l}^{m*}(\boldsymbol{\Omega}_{r})$

in terms of Coulomb wavelets built from spherical harmonics $Y_{l}^{m}$,
radial wave functions $R_{\eta l}$ and Coulomb phase shifts $\sigma_{l}$
Messiah (1961); Cooper (1962). The Coulomb distortion of $\chi_{\eta}$ with
respect to the free electron is described by the Sommerfeld parameter
$\eta=Z/a_{0}k_{f}$, which in the limit of small ion charge $Z$ and large
photoelectron momenta $k_{f}\equiv|\boldsymbol{k}_{f}|$ yields
$\eta\rightarrow 0$, $R_{\eta l}(r)\rightarrow j_{l}(k_{f}r)$ and
$\sigma_{l}\rightarrow 0$, and thus naturally retrieves the plane wave
expansion $\chi_{\eta}(\boldsymbol{r})\rightarrow
e^{i\boldsymbol{k}_{f}\cdot\boldsymbol{r}}$ Messiah (1961); Abramowitz  _et
al._ (1988).

Computing dipole transitions from a hydrogen like atomic orbital $\Phi_{nlm}$
into scattering states $\chi_{\eta}$, we find

$\displaystyle\boldsymbol{M}^{\eta}_{nlm}$ $\displaystyle\propto$
$\displaystyle\langle\chi_{\eta}|\boldsymbol{\nabla}|\Phi_{nlm}\rangle$
$\displaystyle=$
$\displaystyle\underbrace{\widetilde{f}(k_{f})~{}\boldsymbol{Y}_{l,l+1,m}(\boldsymbol{\Omega}_{k_{f}})}_{\text{dipole
transition }l\rightarrow
l+1}+\underbrace{\widetilde{g}(k_{f})~{}\boldsymbol{Y}_{l,l-1,m}(\boldsymbol{\Omega}_{k_{f}})}_{\text{dipole
transition }l\rightarrow l-1}~{},$

where we introduced the complex-valued radial cross-sections
$\widetilde{f}(k_{f})$ and $\widetilde{g}(k_{f})$, whose atomic limit is given
more explicitly in the Suppl. Info.

Based on this expression, we can now formulate the Huygen’s principle of
ARPES: Every atomic orbital participating in the photoemission process is the
source of two dipole allowed Coulomb wavelets, and the Coulomb wavelets
emanating from all these orbital emitters mutually interfere. More explicitly,
the vector spherical harmonics $\boldsymbol{Y}_{l,l\pm 1,m}$ describe the
orbital symmetry of the two dipole allowed emission channels $l\rightarrow
l\pm 1$ that are reached by a given polarization vector
$\boldsymbol{\epsilon}$ Arfken  _et al._ (2013), whose interference is
determined by their $k_{f}$-dependent radial cross-section ratio
$|\widetilde{f}|/|\widetilde{g}|$ and relative phase
$\Delta\sigma=\arg(\widetilde{f}/\widetilde{g})$.

While in the PWA, both $|\widetilde{f}|/|\widetilde{g}|=\text{const}$ and
$\Delta\sigma=0$ are independent of $k_{f}$ (see Suppl. Info), it is precisely
their Coulomb induced $k_{f}$-dependence that produces kinetic energy
dependent final state interferences – and thus bears the potential to describe
photon energy dependent dichroism as we will see in more detail later.

We note in passing, that this $k_{f}$-dependence of
$|\widetilde{f}|/|\widetilde{g}|$, i.e., the fact that individual
photoemission channels can be suppressed or enhanced by an appropriate choice
of photon energy, is a direct consequence of the dipole operator’s velocity
form
$\mathcal{H}_{\text{int}}\propto\boldsymbol{\epsilon}\cdot\boldsymbol{\nabla}$
that we use here, but is not reproduced by its length form
$\mathcal{H}_{\text{int}}\propto\boldsymbol{\epsilon}\cdot\boldsymbol{r}$ Day
_et al._ (2019) (see Suppl. Info). In particular, while the length form is
unbounded and not well defined for extended (infinite) systems Pendry (1976);
Drake and Cassar (2006); Lebech  _et al._ (2012), the velocity form is
translation invariant and thus directly applicable to Bloch states, whose
Wannier representation in turn can be expanded in terms of an atomic orbital
basis

$\Psi_{\boldsymbol{k}}(\boldsymbol{r})=\sum_{\boldsymbol{R}}e^{i\boldsymbol{k}\cdot\boldsymbol{R}}\sum_{nlm}c^{\boldsymbol{k}}_{nlm}\Phi_{nlm}(\boldsymbol{r})~{},$

and the first sum runs over all lattice sites $\boldsymbol{R}$ participating
in the photoemission process. In the independent center approximation, i.e.,
ignoring scattering of outgoing electrons in the Coulomb potential of adjacent
atoms, the ARPES intensity now compactly reads

$I\propto|\langle\Psi_{\boldsymbol{k}_{f}}|\boldsymbol{\epsilon}\cdot\boldsymbol{\nabla}|\Psi_{\boldsymbol{k}}\rangle|^{2}=\delta(\boldsymbol{k}-\boldsymbol{k}_{f})~{}|\boldsymbol{\epsilon}\cdot\boldsymbol{\mathcal{M}_{\boldsymbol{k}}}\cdot\boldsymbol{c}_{\boldsymbol{k}}|^{2}~{},$
(1)

with the $N\times 3$ dimensional dipole transition matrix
$\boldsymbol{\mathcal{M}_{\boldsymbol{k}}}$ coupling the $N$-dimensional
initial state vector
$\boldsymbol{c}_{\boldsymbol{k}}=(c^{\boldsymbol{k}}_{n_{1}l_{1}m_{1}},...,c^{\boldsymbol{k}}_{n_{N}l_{N}m_{N}})^{\top}$
to the three dimensional polarization vector $\boldsymbol{\epsilon}$, and
$\delta(\boldsymbol{k}-\boldsymbol{k}_{f})$ representing momentum conservation
(we thus set $\boldsymbol{k}_{f}\equiv\boldsymbol{k}$ from now).

Dichroism.—Analyzing matrix equation 1, we immediately note that exploiting
full polarization control in an experiment can yield a maximum of six linear
independent equations to retrieve at most three complex eigenvector components
of $\boldsymbol{c}_{\boldsymbol{k}}$ from ARPES intensity measurements.
Taking, e.g., a linear combination
$|\Psi\rangle=c_{x}|p_{x}\rangle+c_{y}|p_{y}\rangle+c_{z}|p_{z}\rangle$ of
$p$-orbitals, we find
$\boldsymbol{\mathcal{M}}_{\boldsymbol{k}}\propto\widetilde{f}(k)~{}\boldsymbol{\mathcal{M}}_{p\rightarrow
d}(\boldsymbol{\Omega}_{k})+\widetilde{g}(k)~{}\boldsymbol{\mathcal{M}}_{p\rightarrow
s}(\boldsymbol{\Omega}_{k})$,

where the $3\times 3$ matrices $\boldsymbol{\mathcal{M}}_{p\rightarrow d}$ and
$\boldsymbol{\mathcal{M}}_{p\rightarrow s}$ describe dipole transitions from
the initial state $\boldsymbol{c}=(c_{x},c_{y},c_{z})^{\top}$ into $d$ and $s$
channels at a given polarization $\boldsymbol{\epsilon}$. Developing these
matrices in terms of small photoelectron emission angles $\theta_{k}$ Arfken
_et al._ (2013), we find (in Cartesian coordinates)

$\displaystyle\underset{p\rightarrow
d}{\boldsymbol{\mathcal{M}}(\boldsymbol{\Omega}_{k})}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2\pi}}\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&-2\\\
\end{array}\right)+\frac{3}{2\sqrt{2\pi}}\left(\begin{array}[]{ccc}0&0&-\cos\phi_{k}\\\
0&0&\sin\phi_{k}\\\ \cos\phi_{k}&\sin\phi_{k}&0\\\
\end{array}\right)\theta_{k}+\mathcal{O}(\theta_{k}^{2})~{};$ (8)
$\displaystyle\underset{p\rightarrow
s}{\boldsymbol{\mathcal{M}}(\boldsymbol{\Omega}_{k})}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\pi}}\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&1\\\ \end{array}\right)~{}.$ (12)

Figure 1: (a-d) ARPES data measured along the $k_{x}k_{z}$ mirror plane of
BiAg2/Ag(111), reproduced from Ref. Bentmann  _et al._ , 2017. (a) Measurement
with $s$-polarized light and $h\nu=26$ eV. Both spin-orbit split surface
states $\Psi^{\pm}$ appear with equal intensities $I_{s}^{+}\sim I_{s}^{-}$
irrespective of photon energy $h\nu$ (see Ref. Bentmann  _et al._ , 2017 for
more data). In contrast, ARPES taken with $p$-polarized light exhibits a
$h\nu$-dependent intensity swap between $\Psi^{\pm}$: (b) $I_{p}^{+}\sim
I_{p}^{-}$ at $h\nu=30$ eV; (c) $I_{p}^{+}/I_{p}^{-}\sim 0$ at $h\nu=22$ eV;
(d) $I_{p}^{-}/I_{p}^{+}\sim 0$ at $h\nu=26$ eV. (e) Illustration of phase
dependent photoemission $s$\- ($s_{x/z}$, black dashed) and $d$-channel
($d_{x/z}$, magenta) interference, individually shown for $\Psi^{-}$ (blue,
top) and $\Psi^{+}$ (red, bottom) in the complex plane. Channels are
represented by unit vectors for clarity. For $\Delta\sigma=\pm\pi$ and $0$,
the absolute phase between channels $s=s_{x}+s_{z}$ and $d=d_{x}+d_{z}$ is
$\pi/2$ for bands $\Psi^{\pm}$, and their intensities
$I^{+}=|M^{+}|^{2}=|M^{-}|^{2}=I^{-}$are equal. For $\Delta\sigma=\pm\pi/2$,
however, $s$-and $p$-channels of $\Psi^{\pm}$ are in phase and interfere
constructively, while they are in antiphase and interfere destructively for
$\Psi^{\mp}$. Figure 2: Phase dependent ARPES intensity of bands $\Psi^{\pm}$
in AgBi2 calculated for the explicit experimental geometry
$\tan\alpha=\epsilon_{z}/\epsilon_{x}=3$ used in Fig. 1 (b-d) Bentmann  _et
al._ (2017). As detailed out in Suppl. Info, we find a unique value set
$|\widetilde{f}|/|\widetilde{g}|=0.72$ and $\Delta\sigma\sim\mp\pi/9$ that
reproduces the complete intensity suppression of $\Psi^{\pm}$ in Figs. 1 (c)
and (d).

Clearly, the $d$-channel mixes in and out-of plane orbital and polarization
components and only diagonalizes right at normal emission $\theta_{k}=0$,
where the contributions from $p_{z}$ are twice as large and in antiphase to
the contributions from $p_{x}/p_{y}$. (As we will see later, this has
important consequences for bands carrying orbital angular momentum OAM.) In
contrast, the $s$-channel is isotropic and diagonal for any emission angle
$\boldsymbol{\Omega}_{k}=(\theta_{k},\phi_{k})$, and thus provides a one-to-
one mapping of eigenvector- onto light polarization components. This implies
that ARPES at photoelectron momenta where the $d$-channel is suppressed, i.e.,
$\widetilde{f}(k)=0$, is a direct probe of the band’s orbital character. In
particular, linear polarizations $\boldsymbol{\epsilon}_{x}$ and
$\boldsymbol{\epsilon}_{y}$ then directly probe the eigenvector amplitudes
$|c_{x}|\propto\sqrt{I_{x}}$ and $|c_{y}|\propto\sqrt{I_{y}}$ (the $c_{z}$
component is fixed by the normalization condition
$c_{x}^{2}+c_{y}^{2}+c_{z}^{2}=1$), while linear (LD) and circular dichroism
(CD) in the $xy$-plane probe their mutual interference

$\displaystyle I_{\text{LD}}$ $\displaystyle\propto$
$\displaystyle\frac{I_{x-y}-I_{x+y}}{I_{x-y}+I_{x+y}}=\frac{2\Re(c_{x}^{*}c_{y})}{|c_{x}|^{2}+|c_{y}|^{2}}~{};$
$\displaystyle I_{\text{CD}}$ $\displaystyle\propto$
$\displaystyle\frac{I_{x-iy}-I_{x+iy}}{I_{x-iy}+I_{x+iy}}=\frac{2\Im(c_{x}^{*}c_{y})}{|c_{x}|^{2}+|c_{y}|^{2}}=\frac{1}{\hbar}\frac{\langle
L_{z}\rangle}{|c_{x}|^{2}+|c_{y}|^{2}}~{},$

and retrieve OAM component $\langle
L_{z}\rangle=\langle\Psi|L_{z}|\Psi\rangle$ and phase relation
$\arg(c_{x}/c_{y})=\arctan(I_{\text{CD}}/I_{\text{LD}})$.
In analogy, additional LD and CD experiments within the orthogonal $xz$ and
$yz$ planes will further provide access to the angular momenta $\langle
L_{y}\rangle$ and $\langle L_{x}\rangle$ as well as phase relations
$\arg(c_{x}/c_{z})$ and $\arg(c_{y}/c_{z})$, respectively, in principle
allowing for a full reconstruction of the eigenstate vector $\boldsymbol{c}$
from ARPES intensity measurements Schüler  _et al._ (2020, 2021). Note,
however, that this generally requires the photoemission $d$-channel and
consequent final state interferences to be reliably suppressed, i.e., a
photon-energy where $\widetilde{f}(k)\sim 0$.

Application.—Let us illustrate this corollary in a well studied model system
whose large $Z$ constituents give rise to elevated final state scattering and
strong spin orbit coupling (SOC): the surface alloy BiAg2/Ag(111) Bentmann
_et al._ (2017). Density functional theory (DFT) finds its low energy surface
electronic structure to be of primarily Bi $6p$ and Ag $5s$ orbital character,
with two Rashba bands $\Psi^{\pm}$ whose SOC shaped wave-functions along the
system’s $k_{x}k_{z}$ mirror plane are well described by
$|\Psi^{\pm}\rangle=\frac{1}{\sqrt{2}}|p_{z},\uparrow\downarrow\rangle\mp\frac{i}{2}|p_{x},\uparrow\downarrow\rangle\pm\frac{1}{2}|p_{y},\downarrow\uparrow\rangle$,
with spinors $|\uparrow\downarrow\rangle$ quantized along the $y$-axis and
orbital angular momenta $\langle L_{y}\rangle^{\pm}=\mp\hbar/\sqrt{2}$ (Fig. 1
a) Mirhosseini  _et al._ (2009); Zhang  _et al._ (2013).
ARPES experiments with $s$-polarized light and the sample mirror- and ARPES
scattering planes coinciding (Fig. 1 a), display both the $\Psi^{+}$ and
$\Psi^{-}$ Rashba bands with equal intensity, irrespective of photon energy
(See detailed data in Ref. Bentmann  _et al._ (2017)). In contrast, ARPES
experiments with $p$-polarized light find a photon energy dependent swap of
intensity between $\Psi^{+}$ and $\Psi^{-}$ (Fig. 1 b-d) Meier  _et al._
(2009); Bentmann  _et al._ (2017).

Based on our model, these observations can be easily understood in terms of
photon energy-dependent $s$ and $d$ channel interference: From Eq. 8, ARPES
close to normal emission with $s$-polarized light $\boldsymbol{\epsilon}_{y}$
mostly projects out the $p_{y}$-orbital contributions, leading to an equal
intensity distribution
$I_{s}^{\pm}\propto|\pm\frac{1}{2}(\sqrt{2}\widetilde{g}+\widetilde{f})|^{2}$
among both bands $\psi^{\pm}$, modulating synchronously with the $k$-, i.e.
kinetic- or photon energy dependent interference of the $s$\- and $d$\-
channels.

In contrast, the $p$-polarized geometry is receptive to both the $p_{x}$ and
$p_{z}$ orbital contributions. The intensity distribution is given by
$I_{p}^{\pm}\propto|\mp\frac{i}{2}(\sqrt{2}\widetilde{g}+\widetilde{f})\cos\alpha+\frac{1}{\sqrt{2}}(\sqrt{2}\widetilde{g}-2\widetilde{f})\sin\alpha|^{2}$,
where $\alpha$ is the angle of light incidence that quantifies the ratio of
in- and out of plane polarization $\tan\alpha=\epsilon_{z}/\epsilon_{x}$.
Interestingly, we now find a disparity
$I_{p}^{+}-I_{p}^{-}\propto|\widetilde{f}||\widetilde{g}|\sin
2\alpha\sin\Delta\sigma$ between bands $\Psi^{\pm}$ that scales with the
Coulomb phase shift $\Delta\sigma=\arg(\widetilde{f}/\widetilde{g})$ between
$s$\- and $d$-channels, but vanishes for $|\widetilde{f}||\widetilde{g}|=0$
(either channel suppressed) and $\alpha=0$ or $\pi/2$
($\boldsymbol{\epsilon}_{x}$ and $\boldsymbol{\epsilon}_{z}$ not mixed). This
is a direct consequence of the interference of both the $s$ and $d$-channels
resulting from both the $p_{x}$ and $p_{z}$ orbitals. In particular, the
pertinent $\pi$-phase shift between the $p_{x}$ and $p_{z}$ derived
$d$-channels (the minus sign in the $d$-channel entry ‘-2’ of expression 8)
reverses their chirality with respect to the $s$-channels. This along with the
opposite OAM $\langle L_{y}\rangle^{\pm}=\mp\hbar/\sqrt{2}$ of bands
$\Psi^{\pm}$ (the $\pm\pi/2$ phase between $p_{x}$ and $p_{z}$ orbitals)
results in a band dependent phase difference between $s$\- and $d$-waves that
is controlled by the Coulomb phase shift $\Delta\sigma$.

We visualize this effect in Fig. 1 (e), where the $d_{x/z}$-channels emitted
from $p_{x}$ and $p_{z}$ orbitals (for clarity represented by unit vectors in
the complex plane) are rotated by $\Delta\sigma$ around their corresponding
$s_{x/z}$-channels. For $\Delta\sigma=\pm\pi$ and $0$, the absolute phase
between $s=s_{x}+s_{z}$ and $d=d_{x}+d_{z}$ is $\pi/2$ for both bands
$\Psi^{\pm}$, and their ARPES intensities
$I^{\pm}\propto|M^{\pm}|^{2}=|s_{x}+d_{x}+s_{z}+d_{z}|^{2}$ are consequently
identical. For $\Delta\sigma=\pm\pi/2$, however, $s$\- and $d$-channels are in
phase and interfere constructively for $\Psi^{\pm}$, while they are out of
phase and interfere destructively for $\Psi^{\mp}$. This mutual exchange of
intensity thus results from the interplay between the opposite chiralities of
bands $\Psi^{\pm}$ (their OAMs; marked by the sign change in $s_{x}$) in
concert with the opposite chiralities of their photoemission $s$\- and
$d$-channels (marked by the sign change in $d_{z}$). According to Eq. 8, these
arguments also hold for systems carrying OAM along $x$ and light polarized in
the $yz$-plane, while the effect vanishes for OAM along $z$ and $xy$-polarized
light, where $s$\- and $d$-channel chiralities are equal (Suppl. info).

Returning to AgBi2 and examining $I_{p}^{\pm}$ within the experimental
geometry $\epsilon_{z}/\epsilon_{x}\sim 3$ used in Ref. Bentmann  _et al._
(2017) in more detail (Suppl. Info), we identify a unique cross-section ratio
$|\widetilde{f}|/|\widetilde{g}|\sim 0.72$ that produces the total band
suppression $I_{p}^{\pm}/I_{p}^{\mp}\sim 0$ observed in Fig. 1 (c,d) for a
phase shift of $\Delta\sigma\sim\pm\pi/9$ (Fig. 2). For these particular
photon energies, the model thus allows us to extract detailed information on
the photoemission final state.

Let us further study the angular dependence of linear dichroism in a system
with similar out- and in plane orbital mixing: the 2D honeycomb monolayer
AgTe/Ag(111) Ünzelmann  _et al._ (2020). Its occupied low energy electronic
structure is described by two distinct bands: band
$|\alpha\rangle=\sin\phi_{k}|p_{x}\rangle-\cos\phi_{k}|p_{y}\rangle$, of
tangential orbital character with zero angular momentum; and Rashba split band
$|\beta\rangle=\cos\phi_{k}|p_{x}\rangle+\sin\phi_{k}|p_{y}\rangle+ik_{\|}\delta_{sp}|sp_{z}\rangle$
111The splitting is not resolved in the data shown., of primarily radial
orbital character, but with in- and out of plane orbital mixing
$\delta_{sp}\sim 4.15$ Å producing a rotating in plane orbital angular
momentum
$\langle\boldsymbol{L}\rangle^{\beta}=2k_{\|}\delta_{sp}\hbar~{}(\sin\phi,\cos\phi,0)^{\top}$
that is governed – in contrast to BiAg2 where SOC is decisive Bentmann  _et
al._ (2017) – by inversion symmetry breaking at the surface Ünzelmann  _et
al._ (2020); Ünzelmann (2021).

Like in our previous discussion, $s$-polarized light close to normal emission
projects out the $p_{y}$ character and delivers
$I_{s}^{\alpha}\propto\cos^{2}\phi_{k}$ and
$I_{s}^{\beta}\propto\sin^{2}\phi_{k}$ (Fig. 3 a,b). In analogy, $p$-polarized
light projects out the $p_{x}$ character of band $|\alpha\rangle$ and delivers
$I_{p}^{\alpha}\propto\sin^{2}\phi_{k}$. In band $|\beta\rangle$, however,
$s$-and $d$-channel interference again yields a $k$-dependent final state
intensity
$I_{p}^{\beta}\propto|\cos\phi_{k}(\sqrt{2}\widetilde{g}+\widetilde{f})\cos\alpha+ik_{\|}\delta_{sp}(\sqrt{2}\widetilde{g}-2\widetilde{f})\sin\alpha|^{2}$,
which breaks the twofold rotational symmetry of the $\cos^{2}\phi_{k}$ if
photo emission $s$\- and $d$-channels are out of phase. This produces the
oppositely oriented half moon structures in Fig. 3 (c,e) Ünzelmann (2021);
Ünzelmann  _et al._ , whose angular intensity distributions are fitted to the
model in panels (d,f), and provide the relevant cross-section ratios and
Coulomb phase shifts annotated to the figure.

Figure 3: (a) AgTe/Ag(111) ARPES constant energy maps at 1.3 eV binding
energy, reproduced from Ref. Ünzelmann  _et al._ , 2020. As described in the
text, band $\beta$ exhibits a two-fold rotational symmetry when measured with
$s$-polarized light at $h\nu=65$ eV (a,b), but shows a distinct half moon
signature with $p$-polarized light at $25$ eV (c,d), which is flipped at $58$
eV (e,f). Black curves in (b,d,f) show a best fit to angular intensity
distributions extracted from (a,c,e) at $k_{\|}=0.12$ Å-1, employing the
experimental geometry $\tan\alpha=1/\sqrt{2}$ (magic angle light incidence)
and the orbital mixing parameter $\delta_{sp}\sim 4.15$ Å of Ref. Ünzelmann
_et al._ , 2020

Discussion.—Finally, let us discuss why – despite the preceding arguments –
the PWA has served so well in describing ARPES intensity distributions, in
particular also in orbital tomography Bradshaw and Woodruff (2015): The latter
is typically applied to small organic molecules, whose main element carbon is
light, Coulomb final state effects fade out quickly with increasing photon
energy ($\eta\propto Z/k\rightarrow 0$,
$\widetilde{f}/\widetilde{g}\sim\text{const}$) and at least the angular part
of the PWA holds. The investigated orbital character is almost entirely C
$2p$, spin-orbit effects and OAM can thus be neglected and proper light
polarization further limits orbital interference to a minimum – ideally even
suppresses one of the two photo emission channels altogether.

However, it is exactly these reasons that render the PWA problematic in more
complex systems. Intermixing of any additional orbital character will
introduce two additional Coulomb wavelets that participate in final state
interference, and if hybridization to heavy elements, e.g., to a metallic
substrate is involved, Coulomb scattering will become significant. Orbital
details obtained from a Fourier reconstruction in orbital tomography might
then be meaningless.

Albeit it is not obvious if and how a similar reconstruction based on Coulomb
waves could be implemented without bias (the outgoing photoelectron exit wave
is deformed by a per se unknown potential), a detailed quantitative
confrontation of model and experiment beyond the conceptional discussion in
this work might still be feasible. This, however, crucially relies on well
constructed initial states, both what concerns the angular part (that can be
routinely obtained by downfolding of a Kohn-Sham eigenbasis obtained from
density functional theory Mostofi  _et al._ (2008)), but in particular what
concerns the radial part, which depends on a (less obvious) realistic
description of Coulomb potentials close to the nuclei. If such an approach
will turn out predictive for complex single- or even many electron systems has
to be seen. What we have already shown so far, however, is that this simple
Huygens principle of ARPES has the potential to deliver ballpark figures of
final state interference effects that so far required one-step photoemission
calculations at the ab initio level Scholz  _et al._ (2013); Dauth  _et al._
(2016); Bentmann  _et al._ (2021), while it maintains the computational ease
and the priceless intuition of the PWA.

Acknowledgements.—I thank Henriette Maaß, Maximilian Ünzelmann, Hendrik
Bentmann and Friedel Reinert of EP7, Würzburg, for raising the problem of
final state interference in BiAg2 and AgTe, and for sharing their experimental
data. Further, I thank Philipp Eck, Jonas Erhardt, Hans Kirschner, Peter
Puschnig, Ralph Claessen and Phil Woodruff for helpful discussions and
valuable feedback on this work. Funding support came from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy through the Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter ct.qmat (EXC 2147, Project ID
390858490) as well as through the Collaborative Research Center SFB 1170
ToCoTronics (Project ID 258499086).

## References

  * Miller (1991) D. A. B. Miller, Optics Letters 16, 1370 (1991).
  * Puschnig  _et al._ (2009) P. Puschnig, S. Berkebile, A. J. Fleming, G. Koller, K. Emtsev, T. Seyller, J. D. Riley, C. Ambrosch-Draxl, F. P. Netzer, and M. G. Ramsey, Science 326, 702 (2009).
  * Lueftner  _et al._ (2014) D. Lueftner, T. Ules, E. M. Reinisch, G. Koller, S. Soubatch, F. S. Tautz, M. G. Ramsey, and P. Puschnig, Proceedings of the National Academy of Sciences 111, 605 (2014).
  * Moser (2017) S. Moser, Journal of Electron Spectroscopy and Related Phenomena 214, 29 (2017).
  * Bradshaw and Woodruff (2015) A. M. Bradshaw and D. P. Woodruff, New Journal of Physics 17, 013033 (2015).
  * Messiah (1961) A. Messiah,  _Quantum Mechanics_ (North-Holland Publishing, Amsterdam, 1961).
  * Cooper (1962) J. W. Cooper, Physical Review 128, 681 (1962).
  * Abramowitz  _et al._ (1988) M. Abramowitz, I. A. Stegun, and R. H. Romer, American Journal of Physics 56, 958 (1988).
  * Arfken  _et al._ (2013) G. B. Arfken, H. J. Weber, and F. E. Harris,  _Mathematical Methods for Physicists_, 7th ed. (Elsevier Academic Press, 2013).
  * Day  _et al._ (2019) R. P. Day, B. Zwartsenberg, I. S. Elfimov, and A. Damascelli, npj Quantum Materials 4, 54 (2019), arXiv:1909.12081 .
  * Pendry (1976) J. Pendry, Surface Science 57, 679 (1976).
  * Drake and Cassar (2006) G. W. F. Drake and M. M. Cassar,  _SciencesNew York_, edited by G. Drake (Springer New York, New York, NY, 2006) p. 1504.
  * Lebech  _et al._ (2012) M. Lebech, J. C. Houver, G. Raseev, A. S. dos Santos, D. Dowek, and R. R. Lucchese, The Journal of Chemical Physics 136, 094303 (2012).
  * Bentmann  _et al._ (2017) H. Bentmann, H. Maaß, E. E. Krasovskii, T. R. F. Peixoto, C. Seibel, M. Leandersson, T. Balasubramanian, and F. Reinert, Physical Review Letters 119, 106401 (2017), arXiv:1507.04664 .
  * Schüler  _et al._ (2020) M. Schüler, U. De Giovannini, H. Hübener, A. Rubio, M. A. Sentef, and P. Werner, Science Advances 6, 1 (2020), arXiv:1905.09404 .
  * Schüler  _et al._ (2021) M. Schüler, T. Pincelli, S. Dong, T. P. Devereaux, M. Wolf, L. Rettig, R. Ernstorfer, and S. Beaulieu,  , 1 (2021), arXiv:2103.17168 .
  * Mirhosseini  _et al._ (2009) H. Mirhosseini, J. Henk, A. Ernst, S. Ostanin, C.-T. Chiang, P. Yu, A. Winkelmann, and J. Kirschner, Physical Review B 79, 245428 (2009).
  * Zhang  _et al._ (2013) H. Zhang, C.-X. Liu, and S.-C. Zhang, Physical Review Letters 111, 066801 (5 pp.) (2013).
  * Meier  _et al._ (2009) F. Meier, J. H. Dil, and J. Osterwalder, New Journal of Physics 11, 125008 (2009).
  * Ünzelmann  _et al._ (2020) M. Ünzelmann, H. Bentmann, P. Eck, T. Kißlinger, B. Geldiyev, J. Rieger, S. Moser, R. C. Vidal, K. Kißner, L. Hammer, M. A. Schneider, T. Fauster, G. Sangiovanni, D. Di Sante, and F. Reinert, Physical Review Letters 124, 176401 (2020), arXiv:1912.05210 .
  * Ünzelmann (2021) M. Ünzelmann,  _Interplay of Inversion Symmetry Breaking and Spin-Orbit Coupling - From the Rashba Effect to Weyl Semimetals_ , Ph.D. thesis (2021).
  * (22) M. Ünzelmann, H. Bentmann, and F. Reinert, unpublished .
  * Mostofi  _et al._ (2008) A. A. Mostofi, J. R. Yates, Y. S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Computer Physics Communications 178, 685 (2008), arXiv:0708.0650 .
  * Scholz  _et al._ (2013) M. R. Scholz, J. Braun, D. Marchenko, A. Varykhalov, M. Lindroos, Y. J. Wang, H. Lin, A. Bansil, H. Ebert, A. Volykhov, L. V. Yashina, O. Rader, J. Sánchez-Barriga, J. Braun, D. Marchenko, A. Varykhalov, M. Lindroos, Y. J. Wang, H. Lin, A. Bansil, J. Minár, H. Ebert, A. Volykhov, L. V. Yashina, and O. Rader, Physical Review Letters 110, 216801 (2013).
  * Dauth  _et al._ (2016) M. Dauth, M. Graus, I. Schelter, M. Wießner, A. Schöll, F. Reinert, and S. Kümmel, Physical Review Letters 117, 1 (2016).
  * Bentmann  _et al._ (2021) H. Bentmann, H. Maaß, J. Braun, C. Seibel, K. A. Kokh, O. E. Tereshchenko, S. Schreyeck, K. Brunner, L. W. Molenkamp, K. Miyamoto, M. Arita, K. Shimada, T. Okuda, J. Kirschner, C. Tusche, H. Ebert, J. Minár, and F. Reinert, Physical Review B 103, L161107 (2021).