# Testing the Galactic Centre potential with S-stars

Alexander F. Zakharov, 1,2
1 Bogoliubov Laboratory for Theoretical Physics, JINR, 141980 Dubna, Russia
2 National Research Nuclear University MEPhI (Moscow Engineering Physics
Institute),
Kashirskoe highway 31, Moscow, 115409, Russia
E-mail<EMAIL_ADDRESS>
(Accepted 2021 October 1. Received 2021 September 29; in original form 2021
August 27)

###### Abstract

Two groups of astronomers used the large telescopes Keck and VLT for decades
to observe trajectories of bright stars near the Galactic Centre. Based on
results of their observations the astronomers concluded that trajectories of
the stars are roughly elliptical and foci of the orbits are approximately
coincide with the Galactic Centre position. In a last few years a self-
gravitating dark matter core–halo distribution was suggested by Ruffini,
Argüelles, Rueda (RAR) and this model was actively used in consequent studies.
In particular, recently it has been claimed that the RAR-model provides a
better fit of trajectories of bright stars in comparison to the conventional
model with a supermassive black hole. The dark matter distribution with a
dense core having a constant density as it was suggested in the RAR-model
leaves trajectories of stars elliptical like in Kepler’s two-body problem.
However, in this case not the foci of the ellipses coincide with the Galactic
Center but their centers while the orbital periods do not depend on semi-major
axes. These properties are not consistent with the observational data for
trajectories of bright stars.

###### keywords:

Dark matter – Supermassive black holes – Sgr A*– The Galactic Centre

††pubyear: 2021††pagerange: Testing the Galactic Centre potential with
S-stars–Testing the Galactic Centre potential with S-stars

##  1 Introduction

For decades astronomers observed bright stars which are moving very closely to
the Galactic Centre. Analyzing trajectories of these stars one can deduce the
functional form of the gravitational potential there. Observations of Keck and
GRAVITY (VLT) groups showed that in the first approximation the stars are
moving along elliptical orbits and foci of these orbits are roughly coinciding
with the Galactic Centre (Ghez et al., 2003, 2005, 2008; Gillessen et al.,
2009a; Genzel et al., 2010; Ciurlo et al., 2020). If we apply the simplest
approach based on laws of classical mechanics we could follow the Newton’s
derivation of the gravitational potential from the Kepler’s laws (Kopeikin et
al., 2011). For the conventional model with the supermassive black hole and
for a dense core model considered below such an approach looks reasonable
since for these orbits radial coordinates $r>>r_{g}$ ($r_{g}$ is the
Schwarzschild radius for supermassive black hole) and general relativistic
effects could be evaluated as perturbations of Newtonian model. Recently,
Gravity Collaboration et al. (2018a, 2019a) and Keck group (Do et al., 2019)
has reported discovery of the relativistic redshift of spectral lines for S2
star near its periapsis passage in May 2018, the redshift was consistent with
theoretical estimates done in the first post-Newtonian approximation of
general relativity. These results are confirming the universality of gravity
laws and did not support the alternative theories of gravity predicting
different value of redshift in the weak and strong gravitational field
(Capozziello et al., 2006, 2007; Capozziello & De Laurentis, 2011). In the
last year Gravity Collaboration et al. (2020) reported a discovery of
relativistic (Schwarzschild) precession for S2 star orbit.

If there is a supermassive black hole along with a bulk distribution of
ordinary matter forming a stellar cluster and a dark matter inside the orbits
of bright stars, the bulk distribution of matter causes precession of the
stellar orbits in the direction which is opposite to that caused by
relativistic effects of the black hole (Rubilar, Eckart, 2001; Nucita, De
Paolis, Ingrosso et al., 2007; Zakharov et al., 2007).

Observational data for trajectories of bright stars can be used to test
predictions of general relativity and to constrain parameters of alternative
theories of gravity such as $f(R)$ (Borka et al., 2012), Yukawa potential
(Borka et al., 2013), theories with massive graviton (Zakharov et al., 2016,
2018; Hees et al., 2017), black holes with a tidal charge (Zakharov et al.,
2018), and others (Gravity Collaboration et al., 2019b). Constraints on
variations of a fine structure constant were imposed by observations of bright
stars near the supermassive black hole at the Galactic Centre as shown in
(Hees et al., 2020), where other astrophysical ways to constrain the fine
structure constant variations are mentioned as well.

Many theoretical models for the Galactic Centre have been proposed in the past
and some of them were rejected as being inconsistent with the analysis of
observational data and at the time being the model consisting of a single
supermassive black hole looks the most preferable. Nonetheless, several years
ago Ruffini, Argüelles & Rueda (2015) proposed that the gravitational
potential at the central part of our galaxy is better approximated by a dark
matter distribution having a dense core and a diluted halo. Later, the dark
matter distribution was called the RAR-model. Recently Becerra-Vergara et al.
(2021) declared that this model provides a better fit of trajectories of the
bright stars in comparison with the supermassive black hole model. Below we
consider properties of bright star trajectories in gravitational field of a
dense core in the RAR-model and conclude that these properties are not
consistent with the existing observational data.

##  2 Central field

We assume that a gravitational potential for the Galactic Centre is
spherically symmetric. At this stage we use the laws of the Newtonian
mechanics. This is sufficient for our goal and the general relativistic
effects have a post-Newtonian order of magnitude and do not change our
conclusions. Here we introduce basic notations for the central field following
the textbooks by Whittaker (1917); Arnold (1989); Goldstein, Poole & Safko
(2002). Let’s assume, that the potential $U=U({r})$ does not depend on vector
$\bf{r}$, it depends only on the radial distance $r=|{\bf r}|$. Then, motion
of a freely-falling test particle with mass $m$ obeys the Newtonian law of
gravity

$m\ddot{\bf{r}}=-\dfrac{\partial U}{\partial\mathbf{r}}=-\dfrac{\partial
U}{\partial{r}}\mathbf{e}_{r},$ (1)

where $\mathbf{e}_{r}$ is an orthonormal vector in a radial direction.
Therefore, the conservation law for angular momentum can be written in the
form

$h=m\dot{\phi}(t)r^{2}(t)=const,$ (2)

where $\phi$ is the angle in the orbital plane and the dot denotes a time
derivative. Introducing potential $V(r)=U(r)+\dfrac{h^{2}}{2mr^{2}}$, one has

$m\ddot{r}=-\dfrac{\partial V}{\partial{r}}.$ (3)

The total energy of the particle in one-dimensional case is

$E=\dfrac{m\dot{r}^{2}}{2}+V(r).$ (4)

From Eq. (4) one has

$\dot{r}=\sqrt{\dfrac{2}{m}\left[E-V(r)\right]},$ (5)

therefore,

$t=\int\dfrac{dr}{\sqrt{\dfrac{2}{m}\left[E-V(r)\right]}}.$ (6)

Integrating Eq. (2) we get

$\phi=\int_{r_{0}}^{r}\dfrac{dr}{r^{2}\sqrt{\dfrac{2mE}{h^{2}}-\dfrac{2mU}{h^{2}}-\dfrac{1}{r^{2}}}}.$
(7)

If we introduce a new variable $u=1/r$, we have

$\phi=-\int_{u_{0}}^{u}\dfrac{du}{\sqrt{\dfrac{2mE}{h^{2}}-\dfrac{2mU}{h^{2}}-u^{2}}}.$
(8)

From inequality $V(r)\leq E$, we have one or several intervals (Arnold, 1989)

$0\leq r_{min}\leq r\leq r_{max}\leq\infty,$ (9)

where $V(r_{min})=V(r_{max})=E$. If

$0<r_{min}<r_{max}<\infty,$ (10)

the corresponding motion is bounded and it is located inside a ring with an
inner radius $r_{min}$ and an outer radius $r_{max}$.

For bounded orbits we could introduce the following angle characterizing
precessional motion of the orbit, see for instance, the textbooks by Landau &
Lifshitz (1993) and Arnold (1989)

$\Phi=\int_{r_{min}}^{{r_{max}}}\dfrac{hdr}{mr^{2}\sqrt{\dfrac{2}{m}\left[E-V(r)\right]}}.$
(11)

For a central potential $U(r)$ Bertrand (1873) proved the following theorem
(see also the available English translation by Santos (2011)): there are only
two potentials of the central field which yield closed elliptical orbits.
These potentials are the $U_{HO}=ar^{2}$ and $U_{N}=-k/r$ (the harmonic
oscillator and the Newtonian potentials). This result is also discussed by
Landau & Lifshitz (1993) and Arnold (1989), while the book by Arnold (1990)
presents a historical background leading to the discovery of Bertrand’s
theorem.

##  3 Closed orbits in harmonic oscillator potential

As soon as a particular potential is known we can calculate the angle $\Phi$
in Eq. (11). For instance, for the harmonic oscillator and for this case
$\Phi_{HO}=\pi/2$ (Arnold, 1989) while it is well-known that for the Newtonian
potential one has $\Phi_{N}=\pi$. It is known that the orbit is closed if the
angle $\Phi$ commensurable with $2\pi$, or in other words, if $\Phi=2\pi m/n$,
where $m$ and $n$ are integers (Arnold, 1989). Thus, computation of the angle
$\Phi$ immediately tells us whether the orbit is closed or not. Clearly, they
are closed in both cases of the harmonic oscillator and the Newtonian
potential. What remains is to understand where the geometric center of the
elliptical orbit is located with respect to the center of gravity. It is well
known that for the Newtonian potential the center of gravity is not at the
center of the ellipse but in its focal point. This is not the case for the
harmonic oscillator potential as we demonstrate below.

Substituting $U_{HO}=a/u^{2}$ in Eq. (8), we obtain (since $u=1/r$)

$\phi=-\int_{u_{0}}^{u}\dfrac{du}{\sqrt{\dfrac{2mE}{h^{2}}-\dfrac{2ma}{u^{2}h^{2}}-u^{2}}}.$
(12)

Following Whittaker (1917) we introduce notations $\beta=\dfrac{2mE}{h^{2}}$,
$\mu=2ma$, $v=u^{2}$ and showed that

$\phi=-\dfrac{1}{2}\int_{v_{0}}^{v}\dfrac{dv}{\sqrt{\beta
v-\dfrac{\mu}{h^{2}}-v^{2}}}.$ (13)

The integral in Eq. (13) is evaluated with Euler’s substitutions and if
$\gamma$ is an integration constant, we obtain (Whittaker, 1917)

$2(\phi-\gamma)=\arccos{\dfrac{v-\dfrac{\beta}{2}}{\sqrt{\dfrac{\beta^{2}}{4}-\dfrac{\mu}{h^{2}}}}},$
(14)

therefore, following Whittaker (1917)

$\dfrac{1}{r^{2}}=\dfrac{\beta}{2}+\sqrt{\dfrac{\beta^{2}}{4}-\dfrac{\mu}{h^{2}}}\cos(2\phi-2\gamma),$
(15)

This equation describes the ellipse whose center is at the origin of
coordinates that coincides with the center of gravity. The angular frequency
and the period of harmonic oscillator potential are $\omega=\sqrt{2a}$ and
$T_{HO}=2\pi/\omega$.

##  4 Central part of dark matter distribution in the RAR model: core
potential

Recently, many alternative models have been proposed for the Galactic Center
in addition to the conventional model of the supermassive black hole with mass
around $4\times 10^{6}M_{\odot}$. There are two main reasons to develop such
alternative models. First, some theorists hope that alternative theories of
gravity could explain dark matter phenomenon and properties of compact objects
in alternative theories may be different from properties of black holes in
general relativity. Second, other researchers hope that proposed dark matter
distributions could provide better fits for observational data and many
authors argued that instead of the supermassive black hole the Galactic Centre
contains a dense core with constant density, see for instance, Chavanis &
Sommeria (1998); Bilic et al. (2002a, b); Ruffini, Argüelles & Rueda (2015);
Argüelles et al. (2016, 2018, 2019, 2020); Becerra-Vergara et al. (2020, 2021)
and references therein. These authors considered various dark matter
distributions with a low density halo and a high density core. The most
popular model of this type was proposed by Ruffini, Argüelles & Rueda (2015)
and it is called the RAR-model. In the RAR model the central part of the
galaxy represents a ball of constant density $\rho_{0}$ having a radius $R$.
Let us consider motion of a test particle (a star) in the potential field of
the RAR model. We assume that the orbit of the particle is bounded and has a
pericenter $r_{p}$ and an apocenter $r_{a}$.

In this case, one could analyze three different conditions for density and
related potentials.

Condition A. $r_{a}\leq R$ and star is moving inside the ball of the constant
density $\rho_{0}$. Then we have a harmonic oscillator potential $U_{HO}(r)$
along the entire orbit, since according to the law of the Newtonian gravity
the potential is formed only by the matter contained inside the radius $r$

$\displaystyle 4\pi G\int_{0}^{r}\rho_{0}rdr=U_{HO}(r)=ar^{2},$ (16)

where $a=2\pi G\rho_{0}$, $G$ is the gravitational constant.

Condition B. $r_{p}\leq R<r_{a}$ and the particle moves both inside and
outside of the dense core.

Condition B1. If we assume that the mass of the dark matter contained inside
of the pericenter of the particle’s orbit is equal to the mass of the presumed
BH

$\displaystyle 4\pi\int_{0}^{r_{p}}\rho_{0}r^{2}dr\approx M_{BH}$ (17)

and the mass of the dark matter outside of the ball of radius $r_{p}$ is much
smaller than the mass of the BH

$\displaystyle 4\pi\int_{r_{p}}^{r}\rho(r)r^{2}dr<<M_{BH},$ (18)

then this case is very similar to the standard approach consisting of a
supermassive black hole and perhaps an extended matter distribution with a
negligibly small mass inside a quasi-elliptical orbit of a star. As we noted
earlier for our needs the Newtonian approximation looks reasonable since
currently the observed bright stars move very far away from the black hole
horizon.

Condition B2. If

$\displaystyle 4\pi G\int_{0}^{r}\rho(r)rdr=\tilde{U}_{HO}(r)\approx
U_{HO}(r)=ar^{2},\quad\forall r\in(R,r_{a}),$ (19)

then one could use $U_{HO}(r)$ as a suitable approximation for the
gravitational potential. Eq. (19) is valid for very small density gradients
for all $r$ in the interval $R<r<r_{a}$ (or in other words, for these $r$ we
have $\rho(r)\approx\rho_{0}$). This case is similar to the case with
Condition A, where $U_{HO}(r)$ determines the exact expression for the
gravitational potential, however, trajectories in the harmonic oscillator
potential are not exact solutions but suitable approximations for potential
$\tilde{U}_{HO}(r)$.

Condition B3. If conditions B1 and B2 are not valid, then the theoretical
model for gravitational potential can not be approximated with $U_{HO}(r)$ and
$U_{N}(r)$. Therefore, in principle there is an opportunity that there are
bounded orbits which are not elliptical and not closed. One should note that
observed bounded orbits at the Galactic Center are closed and quasi-elliptical
in the first approximation. This case deserves a separate detailed
investigation but we do not pursue it in this Letter.

Condition C. In this case $R<r_{p}$ and relations (17,18) are valid then
$U_{N}(r)$ could be chosen as a good approximation for a gravitational
potential. Otherwise, similarly to case B3, gravitational potential can not be
approximated with $U_{HO}(r)$ and $U_{N}(r)$.

##  5 Conclusions

High-precision astrometric monitoring of quasi-elliptical trajectories of
stars with high eccentricities is instrumental for distinguishing $U_{HO}(r)$
and $U_{N}(r)$ potentials since the centers of the elliptic orbits of the
stars should coincide with the Galactic Center in the case of the RAR
potentials for a dense core while in the case of the Newtonian potential star
foci of the ellipses coincide with the Center. As it was noted in Section 3
the orbital periods of stars moving in the field of harmonic oscillator
potential are constant and they do not depend on semi-major axes. Assume that
we have to choose a suitable potential for the Galactic Centre from two
options $U_{HO}(r)$ and $U_{N}(r)$. An inspection of a set of trajectories
with high eccentricities (see, for instance, Fig. 16 in paper by Gillessen et
al. (2009a), Fig. 16 (right panel) from paper by Genzel et al. (2010) and
Fig.8 in paper by Gillessen et al. (2017)) clearly showed that stars are
moving around the joint focus but not the centre.

Therefore, currently, we can state with a rather good confidence that all
bounded observed orbits of bright stars near the Galactic Centre are
elliptical and their foci practically coincide with the Galactic Centre.
Therefore, we have to conclude that the central potential inside the spherical
shell where these observed bounded orbits are located must be the Newtonian
$U_{N}(r)$. In Section 3 it was shown that in the case of a harmonic
oscillator potential $U_{HO}(r)$ which corresponds to the RAR dark matter
model with a dense core inside a ball with a constant density and ellipse
centers for different orbits should coincide with the Galactic Centre. Periods
of trajectories in $U_{HO}(r)$ potential do not depend on semi-major axis.
These properties are not supported by observational data reported by Keck and
GRAVITY (VLT) teams which means that the conventional model with a
supermassive black hole generating approximately the Newtonian potential in a
spherical shell where bright star orbits are located, is a preferable model
for physical interpretation of the observed trajectories.

## Acknowledgements

The author thanks G. A. Alekseev for useful discussions. The author also
appreciates a referee for numerous useful remarks which helped to improve the
Letter.

## Data Availability

The astrometric data used in this letter were obtained from papers by
Gillessen et al. (2009a, 2017); Genzel et al. (2010); Do et al. (2019).

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Present address: ]Lincoln Laboratory, Massachusetts Institute of Technology,
Lexington, MA 02421-6426, USA.

# Visualizing the atomic-scale origin of metallic behavior in Kondo insulators

Harris Pirie  Department of Physics, Harvard University, Cambridge, MA 02138,
USA  Clarendon Laboratory, University of Oxford, Oxford, OX1 3PU, UK  Eric
Mascot  Department of Physics, University of Illinois at Chicago, Chicago, IL
60607, USA  Christian E. Matt  Yu Liu  Pengcheng Chen  M. H. Hamidian
Department of Physics, Harvard University, Cambridge, MA 02138, USA  Shanta
Saha  Xiangfeng Wang  Johnpierre Paglione  Maryland Quantum Materials Center,
Department of Physics, University of Maryland, College Park, MD 20742, USA
Graeme Luke  Department of Physics and Astronomy, McMaster University,
Hamilton, ON L8S 4M1, Canada  David Goldhaber-Gordon  Department of Physics,
Stanford University, Stanford, CA 94305, USA  Stanford Institute for Materials
and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA
94025, USA  Cyrus F. Hirjibehedin  [  London Centre for Nanotechnology,
University College London (UCL), London WC1H 0AH, UK  Department of Physics
and Astronomy, UCL, London WC1E 6BT, UK  Department of Chemistry, UCL, London
WC1H 0AJ, UK  J. C. Séamus Davis  Clarendon Laboratory, University of Oxford,
Oxford, OX1 3PU, UK  Department of Physics, University College Cork, Cork T12
R5C, Ireland  LASSP, Department of Physics, Cornell University, Ithaca, NY
14850, USA  Max-Planck Institute for Chemical Physics of Solids, D-01187
Dresden Germany  Dirk K. Morr  Department of Physics, University of Illinois
at Chicago, Chicago, IL 60607, USA  Jennifer E. Hoffman
<EMAIL_ADDRESS>Department of Physics, Harvard University,
Cambridge, MA 02138, USA

###### Abstract

A Kondo lattice is often electrically insulating at low temperatures. However,
several recent experiments have detected signatures of bulk metallicity within
this Kondo insulating phase. Here we visualize the real-space charge landscape
within a Kondo lattice with atomic resolution using a scanning tunneling
microscope. We discover nanometer-scale puddles of metallic conduction
electrons centered around uranium-site substitutions in the heavy-fermion
compound URu2Si2, and around samarium-site defects in the topological Kondo
insulator SmB6. These defects disturb the Kondo screening cloud, leaving
behind a fingerprint of the metallic parent state. Our results suggest that
the mysterious 3D quantum oscillations measured in SmB6 could arise from these
Kondo-lattice defects, although we cannot rule out other explanations. Our
imaging technique could enable the development of atomic-scale charge sensors
using heavy-fermion probes.

Figure 1: Expected disruption of the screening cloud around Kondo holes. (A)
In a uniform Kondo lattice, magnetic moments at each site (gray arrows) are
coherently screened by itinerant conduction electrons (blue cloud) to form a
spinless ground state of heavy fermions (orange line), characterized by the
wavevector $k_{\mathrm{F}}^{\mathrm{h}}$. (B) If one moment is removed to
create a Kondo hole, the conduction electrons previously screening it can
redistribute themselves. (C) The redistributed screening cloud causes
oscillations of the local conduction electron density
$n_{\mathrm{c}}(\mathbf{r})$, interaction strength $\nu(\mathbf{r})$, and
magnetic susceptibility $\chi_{\mathrm{m}}(\mathbf{r})$ at the conduction-band
wavevector $k_{\mathrm{F}}^{\mathrm{c}}$, as shown schematically. (D) In a
uniform Kondo lattice, the Kondo resonance creates a peak-dip feature in the
calculated d$I$/d$V$, caused by the quantum interference between tunneling
into the conduction band and the $f$-electron states with respective
amplitudes $t_{c}$ and $t_{f}$. The energy position of the peak (black
triangles) shifts linearly according to the local conduction-electron density
$n_{\mathrm{c}}$. (E) The calculated rectification $R(V)=|I(+V)/I(-V)|$
acquires a strong peak because d$I$/d$V$ is asymmetric around the Fermi level
$E_{F}$ (which occurs at $V=0$). The $R(V)$ peak amplitude depends on the
d$I$/d$V$ peak energy. These changes are almost linear over the small range of
local doping expected around a Kondo hole (inset). (F) The calculated
oscillations in d$I(\mathbf{r},V)$/d$V$ at the Fermi level around a Kondo hole
match the hybridized Fermi surface ($k_{\mathrm{F}}^{\mathrm{h}}$, orange line
in inset). (G) In contrast, the calculated $n_{c}(\mathbf{r})$ varies
according to the circular wavevector of the unhybridized Fermi surface
($k_{\mathrm{F}}^{\mathrm{c}}$, blue line in inset) as it mainly reflects the
disturbance to the screening cloud. (H) Calculated $R(\mathbf{r},V)$ is
dominated by unhybridized electrons for biases within the hybridization gap.
The calculations in (D)-(H) are based on a Kondo-Heisenberg model with
nearest-neighbor hopping strength $t$, Kondo coupling $J=2t$,
antiferromagnetic exchange $I=0.002t$, and tunneling amplitudes
$t_{f}/t_{c}=-0.025$. In (D) and (E), the hybridization strength is fixed at
$\nu=0.1t$ and the antiferromagnetic correlation strength is fixed at
$\chi=0.0003t$. The Fermi wavelength is $\lambda_{F}^{c}=8a_{0}$ in (F)-(H),
where $a_{0}$ is the lattice spacing.  Figure 2: Thorium dopants induce Kondo-
hole behavior in URu2Si2. (A) Schematic band structure of URu2Si2 showing the
onset of heavy fermion bands (gray solid lines) at temperatures below
$T_{\mathrm{o}}=17.5$ K, as itinerant conduction electrons (blue line)
hybridize with a renormalized 5$f$ level (gray dashed line), reducing the
Fermi wavevector from $k_{\mathrm{F}}^{\mathrm{c}}$ to
$k_{\mathrm{F}}^{\mathrm{h}}$. (B) Experimental measurement of an asymmetric
Fano lineshape in the tunneling conductance at temperatures below $T_{o}$ on
the U termination (gray curve). This feature shifts towards the Fermi level
near a thorium dopant (black triangles), consistent with an expected change in
local charge density. (C) For a fixed bias, the $R(V)$ peak amplitude (black
triangle) is highly sensitive to the d$I$/d$V$ peak position. The spectra in
(B) and (C) are averaged over the 18 well-isolated thorium dopants marked in
(D). (D) The measured $R(\mathbf{r},V)$ exhibits clear oscillations that
manifest as a ring in (E) the 4-fold-symmetrized Fourier transform. (F) These
oscillations match the high-temperature Fermi wavevector of
$2k_{\mathrm{F}}^{\mathrm{c}}\approx 0.3\ (2\pi/a)$, both above and below
$T_{\mathrm{o}}$. (G) In contrast, a conventional d$I$/d$V$ measurement
couples to the temperature-dependent Fermi surface, which changes dramatically
from 18.6 K to 5.9 K. For clarity, the 18.6 K data have been scaled in (F) and
offset in (G).

1 When the electrons in a material interact strongly with one another, they
often produce unexpected behavior. Above a characteristic temperature $T_{K}$,
a lattice of local $f$ moments within a conducting Fermi sea behaves like an
ordinary magnetic metal, with a Curie-Weiss susceptibility. But below $T_{K}$,
the competition between antiferromagnetic ordering of the local moments and
their screening by conduction electrons leads to a rich phase diagram,
exhibiting quantum criticality (1), unconventional superconductivity (2), and
heavy fermions (3, 4)—quasiparticles with $f$-electron character (Fig. 1A). A
Kondo insulator forms if the spectral gap opened by hybridization between the
conduction band and the renormalized $f$ band spans the Fermi level.
Mysteriously, some Kondo insulators seem to remember their metallic parent
state long after this gap is fully developed. For example, the topological
Kondo insulator SmB6 displays a sizable bulk optical conductivity at terahertz
frequencies (5) and a finite electronic specific heat at low temperatures (6,
7, 8, 9). A complete 3D Fermi surface matching its high-temperature metallic
state was reconstructed from quantum oscillation (10, 9) and Compton
scattering (11) measurements performed in the insulating regime. Notably,
these metallic properties persist even as the bulk resistivity of SmB6
increases by 10 orders of magnitude (12). This discrepancy led to several
theoretical proposals: some argue that the metallic behavior is intrinsic,
either a consequence of the small hybridization gap in Kondo insulators (13),
or arising from exotic charge-neutral quasiparticles (14, 15); others suggest
an extrinsic origin (16, 17, 18, 19), which implies the presence of
microscopic metallic pockets.

2 Charge inhomogeneity is commonplace at nanometer length scales, especially
in materials with strong electron interactions that promote competing orders
(20). In a Kondo lattice, defects that substitute or remove the
$f$-contributing moment, called Kondo holes, have a widespread impact on the
nearby electronic structure (21, 22, 23). First, these defects locally
untangle the hybridized wavefunction, leaving puddles of unhybridized
conduction electrons behind (Fig. 1B). In theory, these charge puddles should
have the same itinerant character as the metallic parent state (22), but they
have not been imaged directly. Additionally, the excess conduction electrons
released from hybridization adjust the strength of their interactions with the
remaining $f$ moments (22, 24), leading to enhanced local magnetism (25) (Fig.
1C). For example, Sm1-xLaxB6 samples with nonmagnetic La dopants are known to
display increased specific heat and magnetic susceptibility compared to
undoped samples (26, 27, 28). More recently, the existence of local metallic
puddles around Gd dopants in Sm1-xGdxB6 was inferred from electron spin-
resonance measurements (29). Meanwhile, an increased concentration of Sm
vacancies in Sm1-xB6 was shown to globally inhibit the development of the
hybridization gap (30), eventually leading to bulk conduction (12, 31). All of
these findings suggest that Sm-site defects manifest as Kondo holes in SmB6,
yet their key signature—the accompanying charge oscillations relating to the
parent metallic Fermi surface (22)—remain undetected by any microscopic probe.

3 Directly imaging the metallic puddles around Kondo holes is difficult,
because the inherent screening strongly renormalizes the bare charge
distribution. However, there are a few promising approaches (32, 33, 34). The
most common is to decorate the tip of a Kelvin probe force microscope with a
single atom or molecule (35, 36). This technique was used to image the charge
variations within an adsorbed molecule (37). However, it becomes inaccurate
for small tip-sample separations because of the influence of short-range
forces (38, 39), complicating further improvements to its spatial resolution
(40). Meanwhile, a scanning tunneling microscope (STM) routinely achieves the
sub-nanometer spatial resolution, cryogenic temperatures, and sub-meV energy
resolution required to access atomic charge distributions, but existing
methods to extract the electrostatic potential from the STM vacuum decay
length contain significant artifacts (41). Consequently, simultaneously
achieving the high charge precision and high spatial resolution required to
measure the charge environment around a Kondo hole is not possible using
existing techniques.

Figure 3: Kondo holes nucleate metallic puddles in SmB6. (A) Schematic band
structure of SmB6 showing the hybridization between conduction electrons (blue
dashed line) and localized 4$f$ moments (gray dashed line), which leads to an
inverted band structure (gray solid line) hosting emergent heavy Dirac surface
states with a reduced Fermi wavevector (orange). (B) STM topography of the
$(2\times 1)$-reconstructed Sm surface of lightly Fe-doped SmB6. Both the Fe
dopant and Sm vacancy in this image are expected to act as Kondo holes because
they each displace a 4$f$ moment. (C-D) Near the Fe dopant, the measured
d$I$/d$V$ peak changes energy position (black triangles), leading to large
variations in $R(\mathbf{r},V)$ peak amplitude. The spectra in (C) and (D)
have been offset for clarity. (E) $R(\mathbf{r},V)$ in the same area as shown
in (B) contains clear oscillations around the two impurities. (F) Linecut of
$R(\mathbf{r},V)$ along the white dashed line in (B). (G) $R(\mathbf{r},V)$
oscillations appear as a sharp ring in the 2-fold-symmetrized Fourier
transform (taken from a larger $65\times 80$-nm2 area for enhanced
$\mathbf{q}$ resolution), which matches the unhybridized 5$d$ Fermi surface
inferred from ARPES experiments (54) (dashed line). The surface reconstruction
creates a sharp peak in $R(\mathbf{r},V)$ at $Q_{\mathrm{Bragg}}=(0,\pi/a)$.

4 Here we develop a dedicated STM modality to image the charge environment
within a Kondo lattice with sub-ångström resolution. At temperatures below
$T_{K}$, we image charge oscillations matching the parent Fermi surface
centered around spinless thorium atoms in the Kondo metal URu2Si2, and around
three separate Sm-site defects in the Kondo insulator SmB6. Importantly, the
charge puddles we image in SmB6 exhibit the same metallic wavevector seen in
recent quantum oscillation experiments (10, 9), suggesting Kondo-lattice
defects are the source of those oscillations.

##  .1 Measuring local charge density in a Kondo lattice

5 To visualize the conduction-electron density $n_{c}(\mathbf{r})$ in a Kondo
lattice (and hence local charge $-en_{c}(\mathbf{r})$, where $-e$ is the
electron charge), we first show theoretically that $n_{c}(\mathbf{r})$
determines the energy position of the Kondo resonance
$\tilde{\varepsilon}_{f}(\mathbf{r})$, which forms near the Fermi level as the
magnetic $f$ moments are screened by conduction electrons. Then, we establish
an experimental metric capable of detecting the sub-meV variations in
$\tilde{\varepsilon}_{f}(\mathbf{r})$ around a Kondo hole. Our technique takes
advantage of how the many-body Kondo resonance responds to local doping. In
the Abrikosov fermion representation for local moments,
$\tilde{\varepsilon}_{f}$ is the Lagrange multiplier that enforces uniform
$f$-electron density, typically $n_{f}=1$ at each site. As additional charge
carriers $\Delta n_{c}$ enter a uniform Kondo lattice, the hybridized Fermi
surface reshapes to accommodate them, leading to a corresponding change in
$\tilde{\varepsilon}_{f}$ in order to maintain $n_{f}=1$ (see black triangles
in Fig. 1D, and Fig. S2E). The magnitude and direction of the shift in
$\tilde{\varepsilon}_{f}$ depend on the details of the band structure. But the
relationship between $n_{c}$ and $\tilde{\varepsilon}_{f}$ is linear over a
wide range of band parameters and charge doping (see Fig. S2), implying that
the charge density at position $\mathbf{r}$, can usually be inferred by
measuring $\tilde{\varepsilon}_{f}(\mathbf{r})$. In fact, the linear
dependence of $\tilde{\varepsilon}_{f}(\mathbf{r})$ on $n_{c}(\mathbf{r})$ was
recently verified experimentally by micron-scale angle-resolved photoemission
(ARPES) measurements in Eu-doped SmB6 (42).

6 In STM measurements, the Kondo resonance normally appears as a peak-dip
feature in the tunneling conductance d$I$/d$V$ (43) (where $I$ is the sample-
to-tip tunneling current at applied sample bias $V$), because of the presence
of multiple tunneling channels (44, 45) (see calculation in Fig. 1D). In
simple cases, $\tilde{\varepsilon}_{f}$ can be estimated by fitting d$I$/d$V$
to a Fano-like model (46, 47). However, the exact value of
$\tilde{\varepsilon}_{f}$ depends on the model used, so this approach is not
immediately suitable for detecting the small, sub-meV energy shifts in
$\tilde{\varepsilon}_{f}(\mathbf{r})$ expected around a Kondo hole. Instead,
we track the ratio of forward-to-backward tunneling current, i.e. the local
rectification $R(\mathbf{r},V)=|I(\mathbf{r},+V)/I(\mathbf{r},-V)|$. This
ratio is insensitive to STM setup artifacts, and it was previously used to
track charge inhomogeneity from the spectral weight transfer at high biases in
hole-doped cuprates (48). Here, we focus on low biases, typically $V\lesssim
10$ mV, where the small shifts in $\tilde{\varepsilon}_{f}(\mathbf{r})$
generate large variations in $R(\mathbf{r},V)$ owing to the energy asymmetry
of d$I$/d$V$ about the Fermi level at $V=0$ (see calculations in Fig. 1, D and
E, and Fig. S2). To demonstrate this effect locally, we self-consistently
calculated d$I(\mathbf{r},V)$/d$V$, $n_{c}(\mathbf{r})$, and $R(\mathbf{r},V)$
around a Kondo hole in a metallic Kondo lattice, as shown in Fig. 1, F to H.
The calculated d$I(\mathbf{r},V)$/d$V$ at $V=0$ tracks the local Fermi-level
density of states, so it reveals the hybridized Fermi surface of heavy
fermions with a wavevector $2k_{\mathrm{F}}^{\mathrm{h}}$. In contrast, both
$n_{c}(\mathbf{r})$ and $R(\mathbf{r},V)$ are dominated by static oscillations
at the unhybridized wavevector $2k_{\mathrm{F}}^{\mathrm{c}}$, associated with
the Friedel-like redistribution of the Kondo screening cloud. The correlation
between $n_{c}(\mathbf{r})$ and $R(\mathbf{r},V)$ establishes
$R(\mathbf{r},V)$ as a qualitative probe of local charge, except at very short
distances from a Kondo hole ($|\mathbf{r}|\sim a$) likely because $n_{f}=1$ is
not enforced at that site.

##  .2 Kondo holes in URu2Si2

7 To test our technique, we first studied the Kondo metal URu2Si2 with 1%
thorium dopants, which are known to induce Kondo-hole behavior (49, 24).
Previous STM measurements mapped a metal-like Fermi surface in URu2Si2 for
temperatures above $T_{\mathrm{o}}=17.5$ K, consisting of a single conduction
band with wavevector $k_{\mathrm{F}}^{\mathrm{c}}\approx 0.3\ \pi/a$, where
$a$ is the lattice constant ((46), see Fig. 2A). The onset of coherent heavy
fermion bands below $T_{\mathrm{o}}$ (50) is accompanied by the appearance of
a peak-dip feature in d$I$/d$V$, i.e. the Kondo-Fano resonance (see Fig. 2B).
Close to a thorium dopant, this feature shifts upwards in energy, towards the
Fermi level. This energy shift—and even the barely perceptible shifts 2 nm
away from the dopant—are easily detected in the amplitude of $R(\mathbf{r},V)$
(see Fig. 2C). For biases within the hybridization gap $|V|<\Delta/e\approx 5$
mV (where $\Delta$ is the gap magnitude), $R(\mathbf{r},V)$ displays
widespread spatial oscillations emanating from thorium dopants, as shown in
Fig. 2, D to F. Their wavevector of $0.29\pm 0.01\ (2\pi/a)$ agrees perfectly
with the hybridization oscillations previously measured around Kondo holes in
this compound (24). It matches the URu2Si2 parent metallic Fermi surface
detected above $T_{\mathrm{o}}$ from our measured quasiparticle interference
patterns in d$I(\mathbf{r},V)$/d$V$ at $V=0$, but it is distinct from the
heavy bands that we measured below $T_{\mathrm{o}}$ (see Fig. 2G). As a final
check, we independently extracted $\tilde{\varepsilon}_{f}(\mathbf{r})$ by
fitting d$I(\mathbf{r},V)$/d$V$ curves to a Fano model (see Fig. S3). The
excellent agreement between $\tilde{\varepsilon}_{f}(\mathbf{r})$ and
$R(\mathbf{r},V)$ corroborates the existence of charge oscillations at
$2k_{\mathrm{F}}^{\mathrm{c}}$ in URu2Si2, indicating that some electrons
retain their itinerant character around Kondo holes, even below
$T_{\mathrm{o}}$.

Figure 4: Kondo holes backscatter heavy Dirac fermions. (A) Topography of an
SmB6 region that contains 15 well-isolated Kondo holes (position indicated by
red and green triangles) on several $(2\times 1)$\- or $(1\times
2)$-reconstructed domains (dotted lines). (B) For energies within the Kondo-
insulating gap, the Fourier-transformed d$I$/d$V$ along $q_{\mathrm{y}}$
(perpendicular to Sm rows) contains a linearly dispersing signal (black dashed
line) corresponding to quasiparticle interference from backscattered heavy
Dirac fermions. The Fourier transform from the $(1\times 2)$ domains was
rotated by 90∘ before being averaged with that from the $(2\times 1)$ domains.
(C) The intensity of backscattering from topological states, calculated from
Fourier-filtering d$I$/d$V$ at the $y$ component of the backscattering
wavevector $q_{y}=2k_{y}^{\mathrm{ss}}$, is strongly peaked around each Kondo
hole. This map is computed only for ordered patches of the sample, as marked
in (A), and excludes step edges.

##  .3 Metallic puddles in SmB6

8 In our Kondo insulating SmB6 samples, any atomic defect that replaces a Sm
atom to alter the $4f$ moment could generate metallic puddles like those seen
in URu2Si2. We searched for these puddles in flux-grown samples lightly doped
with Fe, which contain two clear Sm-site defects: Sm vacancies and Fe
substitutions (see Fig. 3B). We focused on the $(2\times 1)$ Sm termination,
as its charge environment most closely represents that of the bulk (51). As in
URu2Si2, we noticed that the d$I$/d$V$ peak attributed to the Kondo resonance
changes its energy position near candidate Kondo holes (Fig. 3C), strongly
impacting the $R(\mathbf{r},V)$ peak amplitude (Fig. 3D). Similar shifts in
d$I$/d$V$ peak position were previously linked to the buildup of charge around
boron clusters on the Sm $(1\times 1)$ termination (52). For biases within the
hybridization gap $|V|<\Delta/e\approx 10$ mV, $R(\mathbf{r},V)$ reveals
prominent oscillations around Sm-site defects (see Fig. 3, E and F). These
oscillations create a sharp ellipse in the Fourier transform of
$R(\mathbf{r},V)$, as shown in Fig. 3G. The wavevectors of the
$R(\mathbf{q},V)$ ellipse are larger than those of the surface state detected
by quasiparticle interference imaging (53) and they do not disperse for biases
within the hybridization gap, indicating a different origin (see Fig. S4). On
the other hand, the size and shape of the ellipse matches the unhybridized
$5d$ band found by extrapolating ARPES data (54) to the Fermi level (i.e. it
matches the SmB6 metallic parent state), after accounting for band folding on
the $(2\times 1)$ surface (see Fig. 3G and Fig. S5). Our observation of this
$5d$ wavevector within the Kondo insulating gap is direct evidence of atomic-
scale metallicity around Kondo holes. This metallicity is supported by the
large residual d$I$/d$V$ at $V=0$ mV that we measured around Kondo holes
(green curve in Fig. 3C), indicating a sizable Fermi-level density of states
even when the metallic surface states are suppressed (55). We confirmed this
discovery by checking for $R(\mathbf{r},V)$ oscillations around a third type
of Kondo hole, Gd dopants, as detailed in Fig. S6.

##  .4 Magnetic fluctuations at Kondo holes in SmB6

9 Our $R(\mathbf{r},V)$ maps show the real-space structure of the metallic
puddles around Kondo holes in SmB6. For these puddles to contribute to the
measured de Haas-van Alphen oscillations in magnetization, they must have a
finite magnetic susceptibility. Several Sm-site defects are already suspected
to be locally magnetic based on their impact on bulk susceptibility (26, 27,
56, 7) and their influence on the topologically emergent surface states (53,
55). In general, topological surface states can provide a test of local
magnetism because they are protected against backscattering from non-magnetic
defects, but not from magnetic defects that locally break time-reversal
symmetry (57). This additional magnetic backscattering was previously imaged
around Fe dopants in two Bi-based topological insulators (58, 59). Here we
visualize the intensity of magnetic fluctuations at Sm-site defects in SmB6 by
identifying spatial regions where its surface states backscatter. For biases
within the hybridization gap, we measured large-area d$I(\mathbf{r},V)$/d$V$
maps that contain clear quasiparticle interference patterns at the
backscattering wavevector
$\mathbf{q}\equiv\mathbf{k}_{\mathrm{f}}-\mathbf{k}_{\mathrm{i}}=2\mathbf{k}^{\mathrm{ss}}$
(Fig. 4B), consistent with our previous report (53). We determined the spatial
origin of this signal by Fourier-filtering d$I(\mathbf{r},V)$/d$V$ at the
wavevector $2\mathbf{k}^{\mathrm{ss}}$ to create an image of the local
backscattering strength (Fig. 4C). Most of the peaks in this image align with
the positions of Sm vacancies or Fe dopants, indicating that these Kondo holes
harbor the necessary magnetic fluctuations to backscatter topological states.

##  .5 Discussion and Outlook

10 The charge puddles around Kondo holes present an alternative yet compelling
origin for many of the strange observations of metallic behavior in SmB6.
First, the detection of de Haas-van Alphen (magnetic) oscillations without
accompanying Shubnikov-de Haas (resistivity) oscillations (10, 9, 8) is
expected for electrically isolated metallic puddles, provided they do not meet
the percolation threshold (which could be unreachable (17)). Second, the large
Fermi surface size and light effective mass extracted by bulk probes (10, 9,
11) is in excellent agreement with our observation of itinerant $5d$
electrons. Third, the magnetic length of the high-frequency (large-$k_{F}$)
quantum oscillations that onset above 35 T (10, 9) is comparable to the
$R(\mathbf{r})$ decay length of $\gamma=2.6$ nm, such that a Landau orbit
could fit inside a metallic puddle. Additionally, many of the metallic
properties were detected in floating-zone-grown samples (10, 6, 9, 11, 5),
which are known to have higher concentrations of Sm vacancies than samples
grown with an aluminum flux (60). Floating-zone samples also contain a higher
concentration of dislocations (31), which may similarly disrupt the Kondo
screening cloud and thus further enhance the quantum oscillation amplitude
beyond that expected from Sm vacancies alone. In contrast, the quantum
oscillations completely disappear in flux-grown samples once embedded aluminum
is removed (8).

11 Atomic-scale charge inhomogeneity has a profound impact on many interacting
quantum materials, but it has typically not been possible to measure. In
Kondo-lattice systems, $R(\mathbf{r},V)$ provides a peek at the ground-state
charge landscape, which is strongly perturbed by Kondo holes. These Kondo
holes nucleate nanometer-scale metallic puddles that could explain many of the
strange phenomena detected by bulk probes. More broadly, the sensitivity to
local charge within a Kondo lattice may enable atomic-scale charge imaging
using STM tips decorated with a Kondo impurity (61) or fabricated from heavy-
fermion materials (62).

12 Acknowledgements We thank An-Ping Li, Brian Skinner, Christian Wagner,
Felix Lüpke, Stefan Ulrich, Yun Suk Eo, and Zachary Fisk, for helpful
conversations. We thank Anjan Soumyanarayanan, Michael Yee, and Yang He for
their help measuring Gd-doped SmB6. Funding: This project was supported by the
Gordon and Betty Moore Foundation’s EPiQS Initiative through grants GBMF4536,
GBMF9071, and GBMF9457. The experiments at Harvard were supported by the US
National Science Foundation grant DMR-1410480. The data interpretation
received support from AFOSR grant FA9550-21-1-0429. The work of E.M. and
D.K.M. was supported by the US Department of Energy, Office of Science, Basic
Energy Sciences, under Award DE-FG02-05ER46225. C.E.M. is supported by the
Swiss National Science Foundation under fellowship P400P2_183890. Work at the
University of Maryland was supported by AFOSR FA9550-22-1-0023. Research at
McMaster University was supported by the Natural Sciences and Engineering
Research Council. J.C.S.D. acknowledges support from the Science Foundation of
Ireland under Award SFI 17/RP/5445, and from the European Research Council
under Award DLV-788932. This project received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement 893097. Author contributions: H.P., C.E.M.,
Y.L., P.C., and M.H.H. carried out the STM experiments. S.S., X.W., J.P., and
G.L. synthesized the samples. E.M. and D.K.M. developed the theoretical model.
D.G.G., C.F.H., and J.C.S.D. contributed to the understanding of the results.
H.P. and J.E.H. analyzed the data and wrote the manuscript with contributions
from E.M., C.F.H., J.C.S.D., and D.K.M. Competing interests: The authors have
no competing interests. Data and materials availability: All data and analysis
presented in this paper are deposited in Zenodo (63).

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problem  Input: Question: [intoc] 1st 08 2022

# Conjunctive Queries for Logic-Based
Information Extraction

Sam M. Thompson

##  Chapter 1 Abstract

This thesis offers two logic-based approaches to conjunctive queries in the
context of information extraction. The first and main approach is the
introduction of conjunctive query fragments of the logics $\mathsf{FC}$ and
$\mathsf{FC}[\mathsf{REG}]$, denoted as $\mathsf{FC\mathchar 45\relax CQ}$ and
$\mathsf{FC[REG]\mathchar 45\relax CQ}$ respectively. $\mathsf{FC}$ is a
first-order logic based on word equations, where the semantics are defined by
limiting the universe to the factors of some finite input word.
$\mathsf{FC}[\mathsf{REG}]$ is $\mathsf{FC}$ extended with regular
constraints.

Our first results consider the comparative expressive power of
$\mathsf{FC[REG]\mathchar 45\relax CQ}$ in relation to document spanners (a
formal framework for the query language AQL), and various fragments of
$\mathsf{FC[REG]\mathchar 45\relax CQ}$ – some of which coincide with well-
known language generators, such as patterns and regular expressions.

Then, we look at decision problems. We show that many decision problems for
$\mathsf{FC\mathchar 45\relax CQ}$ and $\mathsf{FC[REG]\mathchar 45\relax CQ}$
(such as equivalence and regularity) are undecidable. The model checking
problem for $\mathsf{FC\mathchar 45\relax CQ}$ and $\mathsf{FC[REG]\mathchar
45\relax CQ}$ is $\mathsf{NP}$-complete even if the $\mathsf{FC\mathchar
45\relax CQ}$ is acyclic – under the definition of acyclicity where each word
equation in an $\mathsf{FC\mathchar 45\relax CQ}$ is an atom. This leads us to
look at the “decomposition” of word equations into binary word equations (i.
e., of the form $x\mathbin{\dot{=}}y\cdot z$). If a query consists of only
binary word equations and the query is acyclic, then model checking is
tractable and we can enumerate results efficiently. We give an algorithm that
decomposes an $\mathsf{FC\mathchar 45\relax CQ}$ into an acyclic
$\mathsf{FC\mathchar 45\relax CQ}$ consisting of binary word equations in
polynomial time, or determines that this is not possible.

The second approach is to consider the _dynamic complexity_ of $\mathsf{FC}$.
This uses the common way of encoding words in a relational structure using a
universe with a linear order along with symbol predicates. Then, each element
of the universe can carry a symbol if the predicate for said symbol holds for
that element. Instead of the “usual way” (looking at $\mathsf{FO}$ over these
structures), we study the dynamic complexity, where symbols can be modified.
As each of these modifications only changes one position, the result of a
query before and after the modification is likely to be related. This gives
rise to dynamic descriptive complexity classes based on the logic needed to
incrementally _maintain_ a query. For such an approach, conjunctive queries
are sufficient to maintain the so-called core spanners. In fact, dynamic
conjunctive queries (DynCQ ) are actually more expressive than core spanners,
and dynamic first-order logic (DynFO ) is more expressive than generalized
core spanners.

##  Chapter 2 Acknowledgements

First and foremost, I’d like to thank my supervisor, Dominik D. Freydenberger
for his unwavering support, guidance, and feedback. His endless enthusiasm has
made the three years spent on this project truly enjoyable. I could not have
asked for a better supervisor and mentor.

I’d also like to thank the numerous people at Loughborough university who have
helped me during my time here, both as an undergraduate student and a PhD
student. I am grateful to Daniel Reidenbach for being my second supervisor, as
well as for sparking my interest in theoretical computer science with his
undergraduate module. Thank you to Andrea Soltoggio and Szymon Łopaciuk for
all the interesting discussions over lunch. An additional thanks to Szymon for
numerous conversations in the office about research and more. I wish to
express my gratitude to Parisa Derakhshan, Iain Phillips, Robert Mercaş,
Manfred Kufleitner, Dominik D. Freydenberger, and Yanning Yang for allowing me
to teach their modules. Doing so has made me a more well-rounded computer
scientist.

I am grateful to Joel D. Day and Ana Sălăgean for agreeing to be my internal
examiners, and Anthony W. Lin for being my external examiner.

I would also like to thank Justin Brackemann for the helpful feedback and list
of typos in the proof of LABEL:polytime.

Last, but not least, I’d like to thank my family. Without all their
encouragement, I would not have been able to finish this project. I can’t put
into words how thankful I am for my parents, my sister, and my grandparents.

###### Contents

  1. 1 Abstract
  2. 2 Acknowledgements