# Anderson localization of a Rydberg electron

Matthew T. Eiles<EMAIL_ADDRESS>Max-Planck-Institut für Physik komplexer
Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany  Alexander Eisfeld
<EMAIL_ADDRESS>Max-Planck-Institut für Physik komplexer Systeme,
Nöthnitzer Str. 38, D-01187 Dresden, Germany  Jan M. Rost<EMAIL_ADDRESS>Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187
Dresden, Germany

###### Abstract

Highly excited Rydberg atoms inherit their level structure, symmetries, and
scaling behavior from the hydrogen atom. We will demonstrate that these
fundamental properties enable a thermodynamic limit of a single Rydberg atom
subjected to interactions with nearby ground state atoms. The limit is reached
by simultaneously increasing the number of ground state atoms and the level of
excitation of the Rydberg atom, for which the Coulomb potential supplies
infinitely many and highly degenerate excited states. Our study reveals a
surprising connection to an archetypal concept of condensed matter physics,
Anderson localization, facilitated by a direct mapping between the Rydberg
atom’s electronic spectrum and the spectrum of a tight-binding Hamiltonian.
The hopping amplitudes of this tight-binding system are determined by the
arrangement of ground state atoms and can range from nearest-neighbor to
power-law-tailed to effectively infinite-range, giving rise to different
localization scenarios. For arrangements yielding nearest-neighbor hopping
amplitudes we identify clear signatures of the Anderson localization of the
Rydberg electron.

The origin of quantum mechanics is inextricably linked to the bound state
spectrum of hydrogen bohrConstitution1913 ; pauliUeber1926 ;
schrodingerUndulatory1926 . The Coulomb potential supports an infinite series
of discrete energy levels labeled by an integer-valued principle quantum
number $\nu$. Because of hydrogen’s underlying $SO(4)$ symmetry, these levels
are $\nu^{2}$-fold degenerate banderGroup1966 ; gallagherRydberg2005 . This
enhances the effect of external perturbations, as evinced by the response of
hydrogen atoms to electric and magnetic fields friedrichHydrogen1989a  or to
electron scattering gailitisInfluence1963 ; sadeghpourDominant1990 . The study
of these aspects exposes deep connections between the excited electronic
structure of hydrogen and seemingly disparate physical arenas. Compelling
examples include the hydrogen atom in a strong magnetic field, which is
fundamental to quantum chaos and non-linear dynamics delandeQuantum1986 ;
wintgenIrregular1989 , and the organization of doubly-excited H- states into
multiplets, a phenomenon akin to the symmetry classifications ubiquitous in
elementary particle physics tannerTheory2000a . In this article we forge a
connection between the hydrogen atom and condensed matter via the concept of
Anderson localization.

Hydrogen’s properties are shared by the highly excited Rydberg states of more
complicated atomic species since the influence of the multielectron core
essentially vanishes for these exaggerated states characterized by large $\nu$
values, almost millisecond lifetimes, and nearly micron-scale orbits
seatonQuantum1983 ; gallagherRydberg2005 . Localized perturbations to the
Rydberg atom, caused by the scattering of its electron off of one or more
ground state atoms – denoted scatterers in the following – can be described
via an effective interaction encapsulated by Fermi’s zero-range contact
pseudopotential fermiSopra2008 . By mixing the degenerate states within each
$\nu$ manifold, the otherwise weak interaction of the scatterers can have a
surprisingly strong effect greeneCreation2000 ; shafferUltracold2018a .
Recently, nearly arbitrarily shaped optical tweezer arrays have become
available bernienProbing2017 ; browaeysManybody2020 ;
bluvsteinControlling2021a . These allow for the possibility of perturbing a
Rydberg atom with a predetermined configuration of point-like impurities
hunterRydberg2020a .

Figure 1a illustrates the modified level structure resulting from the
immersion of $M$ scatterers within the Rydberg wave function. Many states in
each $\nu$ manifold are not perturbed, but a subspace of dimension $M$ splits
away and possesses a density of states which depends non-trivially on the
scatterer arrangement hunterRydberg2020a . The spectrum of this perturbed
subspace coincides identically with that of a tight-binding Hamiltonian
eilesRing2020

$\hat{H}=\sum_{q=1}^{M}E_{q}|{q}\rangle\langle{q}|+\sum_{q=1}^{M}\sum_{q^{\prime}\neq
q}^{M}V_{qq^{\prime}}|{q}\rangle\langle{q^{\prime}}|,$ (1)

where $\\{|{q}\rangle\\}$ is a basis of wave functions localized on individual
scatterers (sites). The on-site potentials $E_{q}$ and hopping amplitudes
$V_{qq^{\prime}}$ arise from the Rydberg electron’s motion in the confluence
of infinite-ranged Coulomb and zero-ranged electron-scatterer potentials.
Equation 1 creates an unexpected conceptual link between a Rydberg atom
interacting with many ground state atoms and the dynamics of a particle
hopping through a lattice. Not only are the spectra identical, but the
eigenstates in both representations share many important features, as
illustrated in Fig. 1.

Figure 1:  Energy landscape and wave functions of the perturbed Rydberg atom
a, The level structure of the perturbed Rydberg atom. The Coulomb potential
$V(r)=-1/r$ supports an infinite bound spectrum $E_{\nu}=-1/2\nu^{2}$, denoted
with blue lines. The length of each line represents the level degeneracy
$D_{\nu}=\nu^{2}$ and similarly the typical size of the electronic states,
$\langle r\rangle\sim\nu^{2}$. The inset highlights the densities of states
(DoS) of three Rydberg levels when the atom is perturbed by a ring of $M$
scatterers with radius $2\nu^{2}$. A highly structured DoS consisting of $M$
perturbed states (orange) forms, shifted away from the unaffected $M-\nu^{2}$
states (blue). In the thermodynamic limit, $M,\nu\to\infty$, the bandwidth and
center of mass of the shifted DoS are (within an overall scaling factor)
independent of $M$ and $\nu$. b, and c, The absolute values of the amplitudes
of the electronic state (blue) and site representation (black spheres) for
eigenstates located at the marked positions in the DoS for both periodic and
disordered arrays. The scatterers are shown as orange spheres. Both
representations exhibit the same behavior in the vicinity of the scatterers.
d, An exemplary Rydberg basis state, which is spherically symmetric and
delocalized over the scatterers. e, An exemplary trilobite state
$|{T_{q}}\rangle$ associated with the scatterer $|{q}\rangle$, marked in red.
The trilobite’s amplitude at $\vec{R}_{q}$ determines the on-site potential
$E_{q}$, while its amplitude at $\vec{R}_{q^{\prime}}$ determines the hopping
amplitude $V_{qq^{\prime}}$.

We exploit this link to demonstrate that the Rydberg electron can undergo
Anderson localization: the entire spectrum of electron eigenstates
exponentially localizes in the presence of arbitrarily weak disorder in the
thermodynamic limit of infinite system size andersonAbsence1958 ;
thoulessElectrons1974 ; leeAnderson1981 ; abrahamsScaling1979 ;
eversAnderson2008 . To realize the thermodynamic limit in the Rydberg system
we determine a relationship between $M$ and $\nu$ such that increasing them in
tandem – relying on the infinite series and scaling relations of Rydberg
levels – leads to a well-defined Hamiltonian whose matrix elements are
independent of its size. We study effectively one-dimensional localization by
placing the scatterers on a ring around the Rydberg atom’s core, and then
randomly disordering either their radial or angular positions. We show that
different ring radii lead to different hopping amplitudes, ranging from the
nearest-neighbor interactions conventionally studied to more unusual long-
range and sign-changing interactions. This flexibility gives rise to a variety
of Anderson models and provides insight into the circumstances under which the
Rydberg electron localizes fully.

The Perturbed Rydberg Atom
To demonstrate the equivalency between the spectrum of equation 1 and the
Rydberg spectrum, we investigate the Hamiltonian of the Rydberg atom perturbed
by $M$ scatterers. The perturbation is too weak to couple different $\nu$
manifolds. Thus, we consider only the $\nu^{2}$ degenerate states of a given
manifold, labeling these with a collective index $i=\\{l,m\\}$, where $0\leq
l\leq\nu-1$ and $|m|\leq l$ are the angular momentum quantum numbers. The
Hamiltonian matrix elements

$\mathcal{H}_{ii^{\prime}}=-\frac{1}{2\nu^{2}}\delta_{ii^{\prime}}+2\pi\sum_{q=1}^{M}\mathcal{W}_{iq}\mathcal{W}^{\dagger}_{qi^{\prime}}$
(2)

are obtained by summing the Rydberg energy of the isolated atom and the
interaction potential, a sum over the $M$ contact pseudopotentials. Here, and
throughout, we use atomic units. The $\nu^{2}\times M$ rectangular matrix
$\mathcal{W}$ is composed of Rydberg wave functions evaluated at each of the
scatterers, i.e. $\mathcal{W}_{iq}=\phi^{*}_{\nu i}(\vec{R}_{q})$. More
details about the derivation and validity of this Hamiltonian are provided in
Supplementary Section 1. We have set the $s$-wave atom-electron scattering
length to unity since it is identical for all atoms on the ring.

To make the relationship between the perturbed Rydberg Hamiltonian
$\mathcal{H}$ and the tight-binding Hamiltonian of equation 1 apparent, we
introduce the so-called “trilobite” states greeneCreation2000 ;
boothProduction2015a . The trilobite
$|{T_{q}}\rangle=\sum_{i=1}^{\nu^{2}}\mathcal{W}_{qi}^{\dagger}|{i}\rangle$ is
the perturbed eigenstate of the system having only one scatterer at
$\vec{R}_{q}$ liuPolyatomic2006 ; eilesUltracold2016 ; eilesTrilobites2019 .
Unlike the individual Rydberg eigenstates $|{i}\rangle$, which are spherically
symmetric and extend over the entire scatterer array (Fig 1c), the state
$|{T_{q}}\rangle$ is peaked – but not completely localized – at the
scatterer’s position $\vec{R}_{q}$ (Fig. 1d). We expand $\mathcal{H}$ into
these trilobite states, and after simplifying the resulting generalized
eigenvalue equation accounting for their non-orthogonality, we obtain a
standard eigenvalue equation $H|{\Psi_{k}}\rangle=E_{k}|{\Psi_{k}}\rangle$.
The matrix elements of $H$,

$H_{qq^{\prime}}=-\frac{1}{2\nu^{2}}\delta_{qq^{\prime}}+2\pi\sum_{i=1}^{\nu^{2}}\mathcal{W}_{qi}^{\dagger}\mathcal{W}_{iq^{\prime}},$
(3)

exactly correspond to those in the Hamiltonian represented in the site basis
given by equation 1. Fig. 1b,c displays, for both a periodic and a disordered
system, exemplary eigenstates using this site basis (black spheres) and the
full position-space representation of the electronic eigenstate (surface
plots).

This “trilobite” representation has additional conceptual and numerical
advantages. Since the matrix element $H_{qq^{\prime}}/(2\pi)$ is the amplitude
of the trilobite state $|{T_{q}}\rangle$ at the position of a different
scatterer, $\vec{R}_{q^{\prime}}$, it can be estimated pictorially. For
example, the trilobite in Fig. 1d shows that hopping is restricted to nearest-
neighbor sites in this ring configuration. A further advantage is that the
trilobite functions can be expressed using only $s$-wave Rydberg wave
functions eilesUltracold2016 ; eilesTrilobites2019 . This simplifies
calculations at large $\nu$ values and facillitates asymptotic expansions.

Scaling to the thermodynamic limit
In a typical solid-state system described by a tight-binding Hamiltonian (eq.
1), the elements $E_{q}$ and $V_{qq^{\prime}}$ are independent of $M$ and the
thermodynamic limit is reached by increasing the system’s size, i.e.
$M\to\infty$. However, the matrix elements $H_{qq^{\prime}}$ of equation 3
depend strongly on both $\nu$ and $M$: the Rydberg atom’s size and energy
scales are $\nu$-dependent, and the hopping amplitudes depend on the distance,
inversely proportional to $M$, between scatterers.

As an initial step in separating these scales, we accomodate the overall size
of the Rydberg wave function by parameterizing the ring’s radius as
$2\nu^{2}R$, where $R\in[0,1]$. This parametrization ensures that systems with
different $\nu$ but identical $R$ values have similar properties eilesRing2020
, and the range of $R$ keeps the scatterers within the classically allowed
region. We will discuss the specific cases $R=1$, $R=0.75$, and $R=0.5$ in
detail.

In a subsequent step, we eliminate the $M$-dependence at a coarse-graining
level by fixing $M$ as a function of $\nu$ such that the inter-scatterer
distance, and hence the hopping amplitudes, are invariant with respect to
changes in $\nu$. The functional form of $M(\nu)$ hinges on the resolving
power of the Rydberg wave functions. A useful heuristic is that Rydberg states
can resolve as many in-plane scatterers as they have available azimuthal
nodes, requiring a linear relationship $M(\nu)=\nu$ for most allowed $R$
values. We use this for $R=0.5$ and $R=0.75$. For $R\to 1$ those Rydberg
states possessing the many azimuthal nodes needed to resolve scatterers become
exponentially small. Thus, fewer scatterers can be resolved and a sublinear
relationship is required. In particular, for the case $R=1$, we set
$M(\nu)\sim\nu^{2/3}$ (specifically, $M=\lfloor 3\nu^{2/3})\rfloor$, where
$\lfloor x\rfloor$ is the integer part of $x$).

Finally, we extract the residual $\nu$-dependence of the matrix elements
$H_{qq^{\prime}}$, which also depends on $R$. For $R=1$ we find that the
matrix elements are proportional to $\nu^{-13/3}$, but for $R=0.75$ they are
proportional to $\nu^{-4}$. The matrix elements of the $R=0.5$ case do not
simultaneously possess a global $\nu$-dependence, as discussed in further
detail below. This gives rise to interesting finite size effects.

Figure 2:  Characteristic energies and scaling laws for R = 1.  a, Hopping
amplitudes as a function of angle around the ring. The angular positions of
the scatterers are marked with points. b,c, Dispersion relations for
$30\leq\nu\leq 500$ in increments of 5. As $\nu$ increases these discrete
spectra tend towards the continuous analytic dispersion relation obtained from
the model Hamiltonian, shown as the dashed black curve in b.  Figure 3:
Localization behavior of the R = 1 scenario.  a,b, the minimum, mean, and
maximum values of the normalized participation ratios for radial and angular
disorder, respectively. The dashed lines show the asymptotic behavior
$\mathcal{P}\sim\nu^{\gamma}$, labeled by the numerical fit values for
$\gamma$. c,d, the energy-resolved normalized participation ratio values for
$30\leq\nu\leq 500$, using the exact model, and $10^{3}\leq\nu\leq 10^{5}$
(blue curves) using the asymptotic model. Note that the equivalent $M$ values
are used as labels in d.

Now, we are in a position to factor out an overall $\nu$-dependence such that
the matrix elements $H_{qq^{\prime}}$, for fixed $R$ and $M(\nu)$, are
independent of $\nu$. Taking advantage of the infinite series of Rydberg
levels, the thermodynamic limit of a Rydberg atom is realized with
$\nu\to\infty$. In Fig. 2 we illustrate this analysis for $R=1$. Fig. 2 (a)
shows the angular dependence of the trilobite state for three different $\nu$
values. The appropriately scaled eigenspectra, shown in Fig. 2(b) for
$\nu\in[30,500]$, are independent of $\nu$. Here, we have shown the
eigenspectra for periodic and disordered scatterer arrangements, where
disorder was introduced by random variation in the radial positions of the
scatterers. The disorder scaling requires additional analysis since it is not
clear a priori that the disorder in position has the same $\nu$-dependence as
the resulting disorder in the matrix elements. For example, although angular
disorder leads to first-order energy disorder shifts with the same
$\nu$-scaling for all considered $R$ values, in the $R=1$ and $R=0.5$ cases
radial disorder leads to additional $\nu$-dependencies that must be removed by
scaling the positional disorder with $\nu$. For $R=1$ the radial disorder
strength must be diminished as $\nu^{-2/3}$. These details are discussed
further in Supplementary Section 6.

Figure 4:  Characteristics of the perturbed Rydberg atom at R = 0.75 and R =
0.5.  a,  Eigenspectra of the $R=0.75$ ring for $30\leq\nu\leq 500$ plotted as
a function of wave number (the mirror-image $k>0$ spectra are not shown). The
black curve shows the approximate spectrum obtained in the $\nu\to\infty$
limit, taking the asymptotic form of the hopping to be
$V_{qq^{\prime}}\sim\nu^{-4}\text{sinc}[\pi\sqrt{3}(q-q^{\prime})]$. b, The
$R=0.75$, $\nu=30$ trilobite state. c, $\mathcal{P}$ distributions for
$\nu=200$, $350$, and $500$ with fixed radial disorder. d, Eigenspectra of the
$R=0.5$ ring for $30\leq\nu\leq 500$, plotted as a function of wave number.
Since the spectra are symmetric about $k=0$, spectra for even $\nu$ are
plotted only for negative $k$ values and odd $\nu$ values are plotted only for
positive $k$ values. The black lines give the flat band and on-site energies
for $\nu=500$. e, The $R=0.5$ trilobite state for $\nu=30$ and two exemplary
eigenstates of the disordered system, in both cases using $\nu=30$. f,
$\mathcal{P}$ distributions for several $\nu$ values with fixed angle
disorder.

Transition from extended to localized states
To quantify the extent of localization and systematically show that all
eigenstates localize in the thermodynamic limit, one typically examines
statistical properties of the eigenspectrum oganesyanLocalization2007 ;
shklovskiiStatistics1993  or, as we do here, the eigenstates directly
kramerLocalization1993 ; mirlinStatistics2000 . The normalized participation
ratio, defined for the eigenstate
$|{\Psi_{k}}\rangle=\sum_{q=1}^{M}c_{q}^{(k)}|{q}\rangle$ as

$\mathcal{P}(k)=\left(M\sum_{q=1}^{M}|c_{q}^{(k)}|^{4}\right)^{-1},$ (4)

is a good indicator of the localization length. In a maximally localized
(delocalized) state, $\mathcal{P}\to 1/M$ ($\mathcal{P}\to 1$). Perfectly
delocalized states with strictly real coefficients are characterized by
$\mathcal{P}(k)=2/3$, and therefore we consider states with $\mathcal{P}\geq
2/3$ to be extended. As we show in Supplemental Section 3, the participation
ratio computed in the site basis is equivalent to a spatial participation
ratio measured at the scatterer positions, and thus localization occurs
simultaneously in both representations.

Anderson localization in the R = 1 ring
As implied by the nearest-neighbor hopping terms revealed by Fig. 2, the $R=1$
case allows for a direct comparison with the standard Anderson model. Using
exact diagonalization, we compute the eigenspectrum and participation ratios
for both radial and angular disorder, averaged over $\mathcal{N}$ disorder
realizations (see Supplementary section 9 for details). To extrapolate our
numerical results to the thermodynamic limit it is necessary to study very
high $\nu$. The tight-binding formalism provides a clear numerical advantage
over brute-force diagonalization of the Rydberg Hamiltonian (Eq. 2), since the
matrix dimension $M(\nu)$ is always smaller than the $\nu^{2}$ size of the
Rydberg subspace. For the largest $\nu$ studied here we diagonalize a matrix
of dimension $10^{5}$, which in the Rydberg representation has dimension
$10^{11}$.

In addition to the exact spectrum of $H$ computed for $\nu\leq 500$, we also
compute the spectrum of a model Hamiltonian containing only nearest and next-
nearest neighbor hopping amplitudes extracted from the asymptotic
$\nu\to\infty$ limit of $H$. These spectra agree excellently, even for the
smallest considered $\nu$ values. Based on this favorable comparison, we use
the model Hamiltonian for $\nu>500$ where the exact Hamiltonian becomes
numerically cumbersome.

The key results of this study are displayed in Fig. 3. In Fig. 3 (a) and (b)
we characterize the extent of localization by using the minimum, mean, and
maximum values of $\mathcal{P}$ as a function of $\nu$. The fixed disorder
strength is sufficiently weak such that extended states having
$\mathcal{P}>2/3$ are still present for the lowest $\nu$ values. Numerical
power-law fits of this data show that
$\langle\mathcal{P}\rangle\sim\nu^{-2/3}\sim M^{-1}$, where $\langle\rangle$
denotes an average over the entire spectrum and disorder realizations. This
numerical evidence clearly indicates that all eigenstates of the $R=1$ ring
localize in the thermodynamic limit, confirming the existence of Anderson
localization.

The energy-resolved participation ratios provide insight into the role of
correlations in the disorder distributions and the distinction between on- and
off-diagonal disorder titovNonuniversality2005 ; kuhlEnhancement2008 ;
izrailevAnomalous2012 ; soukoulisOffdiagonal1981 . The positively correlated
off-diagonal radial disorder manifests itself in the pronounced asymmetry in
the distribution in Fig. 3 (c), especially in contrast to the case in Fig. 3
(d) which has anti-correlated off-diagonal disorder. The residual asymmetry
still visible in Fig. 3 arises from the negative next-nearest-neighbor hopping
term. A sharp feature in the middle of the band depends on the parity of
$M$:when $M$ is odd(even) there is a minimum(maximum). A state with infinite
localization length is predicted to occur at the exact band middle in one-
dimensional models with off-diagonal disorder soukoulisOffdiagonal1981 ;
theodorouExtended1976 ; brouwerDensity2000 ; this could be the source of this
feature, which is further modified by the correlated disorder.

Long-ranged hopping when R < 1
To illustrate the diversity of localization scenarios possible with a
perturbed Rydberg atom, we briefly discuss two other ring sizes, $R=0.75$ and
$R=0.5$. As seen in the trilobite functions plotted in Fig. 4 (b) and (e), the
hopping terms for these cases extend over some ($R=0.75$) and all ($R=0.5$)
sites. We will first contrast the disorder-free properties of these two
systems before discussing their responses to the presence of disorder.

For $R=0.75$, the hopping terms oscillate as a function of $|q-q^{\prime}|$
before decaying rapidly around $|q-q^{\prime}|\approx M/10$. At
$\nu\to\infty$, the continuous form of the hopping amplitudes tends
asymptotically toward a sinc function truncated at a finite range, as detailed
in Supplementary Section 7. As shown in Fig. 4(a) this results in an
eigenspectrum closely approximated by a box function, whose flat bands are
broadened by the deviations from the asymptotic form of the hopping
amplitudes. Note that the spectra are only shown for half the range of allowed
wave numbers, since they are symmetric about $k=0$.

The $R=0.5$ hopping amplitudes oscillate over the entire ring, rising to a
maximum at the opposite side (see Fig. 4(e)). The effect is particularly
strong for even values of $\nu$, leading to a dimerization of the system
phillipsLocalization1991  and strongly impacting the observed disorder-free
eigenspectra shown in Fig. 4(c). They condense into two relatively flat bands
separated by a wide band gap when $\nu$ is even, or a single band when $\nu$
is odd. We find that the dominant hopping amplitude $V_{qq+M/2}$ scales as
$\nu^{-13/3}$, while the other hopping amplitudes scale as $\nu^{-5}$. When
$\nu$ is even, a coupled-dimer model shows that the width of the band gap
scales as $\nu^{-1/3}$ and thus closes in the thermodynamic limit. The
strongly split levels around $k=0$ are manifestations of the all-to-all
coupling, and survive in in the thermodynamic limit, as shown for a simplex
model ossipovAnderson2013 .

Finally, we analyze which phenomena in the disordered system arise because of
the different properties described above. Fig. 4(f) shows three $\mathcal{P}$
distributions for the radial-disordered $R=0.75$ system. The regions with
nearly flat bands localize uniformly. The levels lying in the band gap are
well-separated in energy, impeding localization, but as $\nu$ increases the
gaps between these levels is found numerically to close approximately as
$\nu^{-0.27}$. This causes the band of extended states visible in Fig. 4(f)
around $i/M=0.75$ to shrink as $\nu$ increases, suggesting that the boundaries
of this region are not mobility edges but rather finite size effects.

The $\mathcal{P}$ distributions for the angular-disordered $R=0.5$ system are
shown in Fig. 4(g). As in the previous cases, localization occurs most rapidly
at band edges: the band gap present in the even-$\nu$ spectrum leads to a
pronounced valley in the participation ratio that is absent in the odd $\nu$
case. Fig. 4(e) shows two exemplary $\nu=30$ eigenstates from this valley.
These are approximately symmetric under reflection and localize on two
opposite sites due to the dominant opposite-neighbor coupling. Although the
overall $\mathcal{P}$ distributions shrink to lower values as $\nu$ increases,
we find that states near $k=0$, for this disorder strength and range of $\nu$,
appear to remain extended. This is akin to the behavior of systems with
sufficiently long-range power-law interactions, which have an extended state
at the band edge in the thermodynamic limit rodriguezAnderson2003a ;
demouraLocalization2005a ; mirlinTransition1996 . However, these results
cannot be applied so simply to the Rydberg system for which long-range
correlation and off-diagonal disorder can enhance localization
nosovCorrelationinduced2019a .

Outlook
By uncovering and exploiting the surprising relationship between the
electronic eigenstates of a perturbed Rydberg atom and those of a tight-
binding Hamiltonian, we have connected two paradigmatic concepts in atomic and
condensed matter physics, showing that the Rydberg electron of a hydrogen-like
atom can undergo Anderson localization. This mapping is contingent on two
atypical conditions in a single-particle system: high degeneracy and an
infinite spectrum of bound states. Bertrand’s theorem states that the only
central force potentials in which all bound orbits are closed are the Coulomb
and harmonic oscillator potentials bertrand1873theoreme ; quantum
mechanically, this implies that these are unique in providing both the
requisite degeneracy and infinite spectrum. We expect that the states of a
quantum harmonic oscillator will localize under similar conditions as
discussed here, which may also further elucidate the supersymmetric links
between these systems alankosteleckySupersymmetry1985 . The study of the two-
dimensional hydrogen atom or elliptical harmonic oscillators could reveal the
role of inherent symmetry properties of the underlying structure in the
localization properties keski-rahkonenQuantum2019 .

The ring of ground state atoms is not the only interesting implementation of a
perturbed Rydberg atom. A variety of quasi two-dimensional systems could be
constructed with scatterers arranged into a spherical shell, staggered,
stacked, or intersecting rings, or a helix. It is impossible to realize an
analogue of a three-dimensional lattice since scatterers at different
distances from the radial core would have wildly different on-site energies
and hopping amplitudes. A random three-dimensional system, such as an
ultracold gas, corresponds to a tight-binding Hamiltonian characterized by
strong on-site disorder and a complicated set of strongly disordered hopping
amplitudes. Such a system will exhibit both localized states arising from
strongly coupled spatial clusters of scatterers and delocalized states
resulting from the very long-ranged coupling between sites
luukkoPolyatomic2017 ; abumwisExtended2020 .

The experimental realization of this concept involves tradeoffs between the
challenges of preparing and manipulating very highly excited atoms and the
difficulty of positioning ground state atoms. Experimental signatures of the
localization length are provided by observable properties such as the
photoionization rate or dipole moments of the eigenstates, which will differ
dramatically depending on the wave function extent.

###### Acknowledgements.

Acknowledgements: The authors are grateful for numerous valuable discussions
with P. Giannakeas and A. Hunter. M.T.E. and A.E. thank I. Khaymovich for
useful discussions regarding long-range hopping. M.T.E acknowledges partial
support from the Alexander von Humboldt Stiftung. A.E. acknowledges support
from the DFG via a Heisenberg fellowship (Grant No EI 872/5-1)

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##  I  Introduction and contents

This supplementary material aims at facilitating access to details of the main
paper on several issues and is accordingly structured into different parts.
Section 1 provides more detailed background for the Hamiltonian and its matrix
elements in the Rydberg state representation. In particular, various
approximations used in defining this Hamiltonian as well as its matrix
elements (equation 2) are addressed. Section 2  reveals the explicit steps
regarding the transformation into the trilobite basis, leading from 2 in the
main text to 3. Section 3 derives several important relationships between the
eigenvectors in the two representations and measurables constructed from these
eigenvectors, proving that the localization of the wave function in the site
basis corresponds to localization in position space.  Section 4 re-examines
the matrix elements, specializing to the ring geometry studied here, and
Section 5 derives the eigenspectrum of that Hamiltonian for the reader’s
convenience. Sections 6-8 provide additional details on the properties of the
$R=1,R=0.75$ and $R=0.5$ systems, respectively, in each case with and without
disorder. Section 9 contains some numerical details.

##  II (1) Hamiltonian in the Rydberg wave function basis

We write the Hamiltonian of the perturbed Rydberg Hamiltonian in anticipation
of its expansion into the basis of Rydberg states:

$H=-\sum_{lm}\frac{|{\nu lm}\rangle\langle{\nu
lm}|}{2(\nu-\mu_{l})^{2}}+2\pi\sum_{q=1}^{M}a_{s}[k(R_{q})]|{\vec{R}_{q}}\rangle\langle{\vec{R}_{q}}|.$
(5)

The first term of Eq. 5 is the energy spectrum of the isolated Rydberg atom,
here including also the effect of short-range non-hydrogenic interactions via
the quantum defects $\mu_{l}$ These are typically non-zero only for $l<3$ in
alkali atoms, and for simplicity we just set all $\mu_{l}=0$. The effect of
non-zero quantum defects can be included by using the protocol of Ref.
eilesTrilobites2019  or approximately neglected by removing states with $l<3$
from the basis $\\{i\\}=\\{l,m\\}$ in the text. The second term of Eq. 5
describes the electron-scatterer interaction using the Fermi pseudopotential
fermiSopra2008 . We include only $s$-wave scattering for two reasons. First,
the low kinetic energies characteristic of the Rydberg electron suppress the
influence of higher order partial waves. Second, the angular momentum quantum
numbers $L$ characterizing these partial waves are approximately good quantum
numbers of the whole system over a wide range of $\nu$ and $R$ values
eilesTrilobites2019 , and thus to a good approximation the effects of higher
order scattering terms can be treated independently. If desired, higher order
partial waves can be included via the incorporation of terms developed by
Omont omontTheory1977 ) and again using the protocol developed in Ref.
eilesTrilobites2019 .

The partial wave expansion used in deriving these pseudopotentials requires
that energy-dependent phases be used. The $s$-wave scattering length
$a_{s}[k(R_{q})]$ thus has an implicit dependence on the Rydberg-scatterer
distance $R_{q}$ via the semi-classical momentum,
$k^{2}=-\frac{1}{\nu^{2}}+\frac{2}{R_{q}}$. In the ring geometry, the
scattering length is identical for all scatterers, and scales out of the
problem completely. When the ring is not perfect, this no longer holds
strictly, but the scattering length varies very slowly as a function of $R$.
It is a therefore a fine approximation to continue to assume that it is
constant. As a result we have set this to unity in the main text of the paper.
However, in Section II, we show how to include different scattering lengths
within the same formalism. Starting in Section III, we again

Since the effect of these Fermi pseudopotentials is generally weak, we
truncate the Hilbert space to a single $\nu$ manifold. Note that, in the
thermodynamic limit and with the geometries considered in the main text, this
is guaranteed since the energy separation between Rydberg manifolds scales as
$\nu^{-3}$ while the width and center of mass of the perturbed subspace scale,
at most, as $\nu^{-4}$. Truncation of the basis to a single degenerate
manifold implies that the first term of equation 5 contributes only an
irrelevant energy offset, which we set to zero in the following. It also
determines the set of Rydberg basis states having the same principal quantum
number $\nu$ but different angular momentum quantum numbers $l$ and $m$ with
which to represent the perturbation potential. We utilize the shorthand
$|{i}\rangle=|{\nu lm}\rangle$ and $|{i^{\prime}}\rangle=|{\nu
l^{\prime}m^{\prime}}\rangle$ to describe the Rydberg basis, whose position-
space representation is given by the three-dimensional hydrogenic wave
functions

$\langle{\vec{r}}|{\nu lm}\rangle=\phi_{\nu lm}(\vec{r})=\frac{u_{\nu
l}(r)}{r}Y_{lm}(\hat{r}).$ (6)

The matrix elements of the Hamiltonian in the Rydberg basis $|{i}\rangle$ are

$H_{ii^{\prime}}=2\pi\sum_{q=1}^{M}a_{s}[k(R_{q})]\phi_{i}^{*}(\vec{R}_{q})\phi_{i^{\prime}}(\vec{R}_{q}).$
(7)

As stated in the text, diagonalization of this $\nu^{2}\times\nu^{2}$
Hamiltonian leads to $M$ shifted eigenvalues and $M-\nu^{2}$ unshifted and
degenerate levels.

##  III (2) Transformation into the trilobite basis

In this section, we see how to obtain only the non-zero $M$ eigenvalues
directily. We absorb a factor $\sqrt{a_{s}[k(R_{q})]}$ into each wave function
$\phi_{i}(\vec{R}_{q})$, and define the rectangular $\nu^{2}\times M$ matrix

$\mathcal{W}_{iq}=\sqrt{a_{s}[k(R_{q})]}\phi_{i}^{*}(\vec{R}_{q}).$ (8)

Note that care must be taken in the following equations if $a_{s}[k(R)]<0$,
since then it is vital to not complex conjugate $\sqrt{a_{s}[k(R_{q})]}$ in
$\mathcal{W}^{\dagger}$. To avoid this minor complication, in the following we
simply assume $a_{s}[k(R_{q})]>0$, i.e. the system is never constructed such
that none of the perturbers passes through the distance to the Rydberg core
such that the Ramsauer-Townsend minimum energy condition is met.

The $\mathcal{W}$ matrices allow us to write the Hamiltonian as a matrix
product, using Einstein notation when summing over repeated indices Keep in
mind that sums over $q$ or $p$ range from $1$ to $M$ while sums over $i$ range
from $1$ to $\nu^{2}$. The Hamiltonian in the Rydberg basis representation is,
in this more compact notation,

$H_{ii^{\prime}}=2\pi\mathcal{W}_{iq}\mathcal{W}^{\dagger}_{qi^{\prime}}.$ (9)

This is a separable matrix. Its rank is not equal to its dimension $\nu^{2}$;
in fact, it has the same rank and eigenspectrum as the $M\times M$ matrix

$\tilde{H}_{qq^{\prime}}=2\pi\mathcal{W}_{qi}^{\dagger}\mathcal{W}_{iq^{\prime}}.$
(10)

We can also derive this by considering the eigenfunctions of a single
scatterer, known as trilobite functions. In position space, the trilobite
associated with the scatterer at position $\vec{R}_{q}$ is eilesTrilobites2019

$\langle\vec{r}|T_{q}\rangle=\phi_{i}(\vec{r})\mathcal{W}_{iq}.$ (11)

Note that this is not normalized, nor is it orthogonal to other trilobites.
The overlap of a trilobite at $R_{q}$ with one at $R_{q^{\prime}}$ is

$\langle
T_{q}|T_{q^{\prime}}\rangle=\mathcal{W}^{\dagger}_{qi}\left(\int\phi_{i}^{*}(\vec{r})\phi_{i^{\prime}}^{*}(\vec{r})\mathrm{d^{3}}{r}\right)\mathcal{W}_{i^{\prime}q^{\prime}}=\mathcal{W}^{\dagger}_{qi}\delta_{ii^{\prime}}\mathcal{W}_{i^{\prime}q^{\prime}}=\mathcal{W}^{\dagger}_{qi}\mathcal{W}_{iq^{\prime}}.$
(12)

Now, let us define a set of trilobite quasiparticles $\\{|{T_{q}}\rangle\\}$
associated with the set of scatterer positions $\\{\vec{R}_{q}\\}$ into which
we can expand the $M$-scatterer Hamiltonian. We have (in the first line the
sum over $p$ is included explicitly)

$\displaystyle\langle{T_{q}}|H|{T_{q^{\prime}}}\rangle$
$\displaystyle=\sum_{p=1}^{M}\int\phi_{i}^{*}(\vec{r})\mathcal{W}_{qi}^{\dagger}2\pi\sqrt{a_{s}[k(R_{p})]}\delta^{3}(\vec{r}-\vec{R}_{p})\sqrt{a_{s}[k(R_{p})]}\phi_{i^{\prime}}(\vec{r})\mathcal{W}_{i^{\prime}q^{\prime}}\mathrm{d^{3}}{r}$
$\displaystyle=2\pi\mathcal{W}^{\dagger}_{qi}\mathcal{W}_{ip}\mathcal{W}_{pi^{\prime}}^{\dagger}\mathcal{W}_{i^{\prime}q^{\prime}}.$
(13)

From this we obtain a generalized eigenvalue equation,

$\displaystyle
2\pi\mathcal{W}^{\dagger}_{qi}\mathcal{W}_{ip}\mathcal{W}_{pi^{\prime}}^{\dagger}\mathcal{W}_{i^{\prime}q^{\prime}}{v_{q^{\prime}}^{(k)}}=\epsilon^{(k)}\mathcal{W}^{\dagger}_{qi}\mathcal{W}_{iq^{\prime}}{v_{q^{\prime}}^{(k)}}.$
(14)

Multiplying both sides by the inverse of the overlap matrix leads to

$2\pi\mathcal{W}_{qi}^{\dagger}\mathcal{W}_{iq^{\prime}}{v_{q^{\prime}}^{(k)}}=\epsilon^{(k)}{v_{q}^{(k)}},$
(15)

which is the same as when using Eq. (10). We therefore see that this
transformation of the original Hamiltonian is equivalent to expanding the
Hamiltonian in the basis of “trilobite" wave functions (Eq. 14, followed by
reducing the problem from a generalized eigenvalue equation to a standard
eigenvalue equation by inverting the overlap matrix. The Hamiltonian can then
be written in a transparent way as

$\displaystyle\tilde{H}$
$\displaystyle=\sum_{q}E_{q}|{q}\rangle\langle{q}|+\sum_{q}\sum_{q^{\prime}\neq
q}V_{qq^{\prime}}|{q}\rangle\langle{q^{\prime}}|,$ (16)

where $E_{q}$ and $V_{qq^{\prime}}$ are the probability function of the
trilobite state and the overlap between different trilobite states,
respectively. This overlap between different trilobites $|{T_{q}}\rangle$ and
$|{T_{q^{\prime}}}\rangle$ is, in turn, equal to the trilobite state
$|{T_{q}}\rangle$ evaluated at the position of scatterer $q^{\prime}$. These
overlaps can be evaluated using

$\displaystyle V_{qq^{\prime}}$ $\displaystyle=2\pi\sum_{lm}\phi_{\nu
lm}^{*}(\vec{R}_{q})\phi_{\nu lm}(\vec{R}_{q^{\prime}})$ (17) $\displaystyle
E_{q}$ $\displaystyle=V_{qq}.$ (18)

This trilobite transformation has therefore transformed the Hamiltonian $H$
into a type of tight-binding Hamiltonian $\tilde{H}$ describing a particle
moving in a lattice of sites $q$ with on-site potentials $E_{q}$ and hopping
amplitudes $V_{qq^{\prime}}$. Note that
$\langle{q}|{q^{\prime}}\rangle=\delta_{qq^{\prime}}$, although
$\langle{T_{q}}|{T_{q^{\prime}}}\rangle\neq\delta_{qq^{\prime}}$.

##  IV (4) Comparison of observables computed in the two representations

Clearly, no matter which Hamiltonian representation (Rydberg or trilobite), we
diagonalize, the eigenspectrum is identical. However, the eigenvectors differ;
the eigenvectors of Eq. 9 give the coefficients needed to build the position-
space wave function as a linear combination of degenerate Rydberg states,
while the eigenvectors of Eq. 10 give the decomposition into the tight-binding
site basis. To connect these eigenvectors, which for the eigenvalue
$\epsilon^{(k)}$ are denoted $v_{i}^{(k)}$ in the Rydberg basis and
$\tilde{v}_{q}^{(k)}$ in the trilobite basis, we explicitly transform from the
Rydberg hamiltonian to the trilobite hamiltonian in the eigenvalue equation,

$\displaystyle
2\pi\mathcal{W}_{iq}\mathcal{W}^{\dagger}_{qi^{\prime}}v_{i^{\prime}}^{(k)}$
$\displaystyle=\epsilon^{(k)}v_{i}^{(k)}$ $\displaystyle\implies
2\pi\mathcal{W}^{\dagger}_{pi}\mathcal{W}_{iq}\left[\mathcal{W}^{\dagger}_{qi^{\prime}}v_{i^{\prime}}^{(k)}\right]$
$\displaystyle=\epsilon^{(k)}\mathcal{W}^{\dagger}_{pi}v_{i}^{(k)}$
$\displaystyle\implies
2\pi\mathcal{W}^{\dagger}_{pi}\mathcal{W}_{iq}\tilde{v}_{q}^{(k)}$
$\displaystyle=\epsilon^{(k)}\tilde{v}_{p}^{(k)},$

in other words, $\tilde{v}_{p}^{(k)}=\mathcal{W}^{\dagger}_{pi}v_{i}^{(k)}$.
Going the other direction, we have

$\displaystyle\mathcal{W}_{i^{\prime}p}\tilde{v}_{p}^{(k)}$
$\displaystyle=\mathcal{W}_{i^{\prime}p}\mathcal{W}^{\dagger}_{pi}v_{i}^{(k)}$
$\displaystyle\implies 2\pi\mathcal{W}_{i^{\prime}p}\tilde{v}_{p}^{(k)}$
$\displaystyle=\epsilon^{(k)}v_{i^{\prime}}^{(k)},$

since on the right hand side we have the eigenvalue equation for
$v_{i}^{(k)}$. Therefore,
$v_{i}^{(k)}=\frac{2\pi\mathcal{W}_{ip}\tilde{v}_{p}^{(k)}}{\epsilon^{(k)}}$.

In practical calculations, we obtain the eigenvectors $\tilde{v}$ via
numerical diagonalization of Eq. 10. These are normalized. To obtain the
position representation of the wave function we need the transformation
between the trilobite and Rydberg representations developed above as well as
the normalization. This is given by evaluating

$\displaystyle\delta_{kk^{\prime}}=\tilde{v}_{q}^{(k)*}\tilde{v}_{q}^{(k^{\prime})}$
$\displaystyle=\frac{v_{i^{\prime}}^{(k)*}\mathcal{W}_{i^{\prime}q}\mathcal{W}^{\dagger}_{qi}v_{i}^{(k)^{\prime}}}{N_{k}^{2}}$
$\displaystyle=\frac{\delta_{kk^{\prime}}\epsilon^{(k)}}{2\pi N_{k}^{2}},$

where $N_{k}$ is the desired normalization constant. We therefore see that the
normalization constant is $N_{k}=\sqrt{\epsilon^{(k)}/2\pi}$. From the
normalized eigenvectors $\tilde{v}_{p}^{(k)}$ obtained numerically, we get the
normalized eigenvectors in the Rydberg basis,
$v_{i}^{(k)}=(\epsilon^{(k)}/2\pi)^{-1/2}\mathcal{W}_{ip}\tilde{v}_{p}^{(k)}$.

In addition to the wave functions and eigenvalues, which we use to analyze the
structure and behavior of the system, we will study the normalized
participation ratio (NPR, denoted $\mathcal{P}$), which characterizes the
localization of the sytem. Since the wave functions in the trilobite basis
give the close connection between our system and the tight-binding hamiltonian
of the Anderson model, we will primarily study localization in this
representation. Here, the normalized participation ratio of the $k$th
eigenstate is defined

$\mathcal{P}(k)=\left(M\sum_{q}^{M}|\tilde{v}_{q}^{(k)}|^{4}\right)^{-1}.$
(19)

When $\tilde{v}_{q}^{(k)}$ is localized on a single scatterer $\vec{R}_{p}$,
or in a single trilobite state $|{p}\rangle$, then
$\tilde{v}_{q}^{(k)}=\delta_{qp}$ and $\mathcal{P}(k)=M^{-1}$. However, when
$\tilde{v}_{q}^{(k)}$ is localized in a plane wave with amplitudes $e^{2\pi
ikq}/\sqrt{M}$, then $\mathcal{P}(k)=(M\cdot M\cdot M^{-2})^{-1}=1$. Finally,
if a real representation of the delocalized Bloch states is used (as sometimes
occurs in the presence of degenerate states, as on the ring), then
$\mathcal{P}(k)=(4M^{-1}\sum_{q}\cos^{4}(2\pi
kq))^{-1}=(4M^{-1}(3M/8))^{-1}=2/3$.

A necessary condition is that localization in this representation remains
physically meaningful. We therefore should compute also the NPR in real space,
in order to see if the electron localizes spatially on the scatterers,
according to the eigenvector $\tilde{v}^{(k)}$. We define the spatial NPR,
$\mathcal{P}_{spatial}(k)$, as a participation ratio of the electronic wave
function evaluated at and summed only over scatterer positions,

$\mathcal{P}_{spatial}(k)=\left(M\sum_{P}\left|\sum_{i}v_{i}^{(k)}\phi_{i}(\vec{R}_{P})\right|^{4}\right)^{-1}=\left(M\sum_{P}\left|\sum_{i}\mathcal{W}^{\dagger}_{Pi}v_{i}^{(k)}\right|^{4}\right)^{-1}.$
(20)

We next write this in terms of the trilobite eigenvector,

$\mathcal{P}_{spatial}(k)=\left(M\sum_{P}\left|\sum_{i,q}\mathcal{W}^{\dagger}_{Pi}\frac{\mathcal{W}_{iq}\tilde{v}_{q}^{(k)}}{[\epsilon^{(k)}]^{1/2}}\right|^{4}\right)^{-1}=\left(M\sum_{P}\left|\frac{\epsilon^{(k)}\tilde{v}_{P}^{(k)}}{[\epsilon^{(k)}]^{1/2}}\right|^{4}\right)^{-1}.$
(21)

Thus, we find that
$\mathcal{P}_{spatial}(k)=[\epsilon^{(k)}]^{-2}\mathcal{P}(k)$; localization
in the trilobite basis implies spatial localization, albeit with a
normalization factor given by the eigenenergy.

This normalization factor can be removed by considering relative spatial
probabilities in the formulation of the spatial participation ratio, since the
most relevant localization measure is not localization relative to the entire
allowed volume (which our previous measure characterizes) but instead
localization within the spatial volume of interest, namely the 1D line.

The probability of finding the electron on site $p$ is

$\text{Prob}(p)=\left|\sum_{i}v_{i}^{(k)}\phi_{i}(\vec{R}_{p})\right|^{2}.$
(22)

The probability of finding the electron at the position of one scatterer
relative to the total probability of finding it at any perturber is then

$P(p)=\frac{\text{Prob}(p)}{\sum_{P}\text{Prob}(p)}.$ (23)

Note that this probability is normalized so that there is unit probability to
find the electron on the ring of scatterers, i.e. $\sum_{p}P(p)=1$. Prob$(p)$
can be rewritten in terms of $\mathcal{W}$, which facillitates a series of
simplifications,

$\text{Prob}(p)=\left|\sum_{i}\mathcal{W}_{Pi}^{\dagger}v_{i}^{(k)}\right|^{2}=\left|\sum_{i,q}\mathcal{W}^{\dagger}_{Pi}\frac{\mathcal{W}_{iq}\tilde{v}_{q}^{(k)}}{[\epsilon^{(k)}]^{1/2}}\right|^{2}=\left|\frac{\epsilon^{(k)}\tilde{v}_{P}^{(k)}}{[\epsilon^{(k)}]^{1/2}}\right|^{2}=|\epsilon^{(k)}|\left|\tilde{v}_{P}^{(k)}\right|^{2},$
(24)

where in the second step we transformed the eigenvector into the trilobite
representation, and in the third step we recognized the appearance of the
Hamiltonian matrix acting on the trilobite eigenvector.

Using this in the spatial participation ratio definition gives

$\mathcal{P}_{spatial}(k)=\left(M\sum_{P}\left|\frac{\text{Prob}(P)}{\sum_{P^{\prime}}\text{Prob}(P^{\prime})}\right|^{2}\right)^{-1}=\left(M\frac{\sum_{P}\left|\tilde{v}_{P}^{(k)}\right|^{4}}{\left|\sum_{P^{\prime}}|\tilde{v}_{P^{\prime}}^{(k)}|^{2}\right|^{2}}\right)^{-1}=\left(M\sum_{P}|\tilde{v}_{P}^{(k)}|^{4}\right)^{-1}=\mathcal{P}(k)$
(25)

Thus, we see that the two participation ratios are equivalent.

##  V IV. Matrix elements in the ring geometry

Here we return to the matrix elements of the Rydberg hamiltonian in the
trilobite representation, i.e. equation 17. In the ring geometry we study
here, where the scatterers all lie in a plane and are equidistantly spaced,
the on-site potentials $E_{q}$ and hopping amplitudes $V_{qq^{\prime}}$ can
also be written

$\displaystyle E_{q}$ $\displaystyle=\sum_{lm}\mathcal{R}_{lm}(R_{q},R_{q})$
(26) $\displaystyle V_{qq^{\prime}}$
$\displaystyle=\sum_{lm}\mathcal{R}_{lm}(R_{q},R_{q^{\prime}})e^{-\frac{im2\pi}{M}(q-q^{\prime})},$
(27)

where

$\mathcal{R}_{lm}(R_{q},R_{q^{\prime}})=\frac{(l+\frac{1}{2})(l-m)!(l+m)!}{\left[\left(\frac{l+m}{2}\right)!\left(2^{(l+1)}\frac{l-m}{2}\right)!\right]^{2}}\left[\frac{u_{\nu
l}(2\nu^{2}R_{q})}{R_{q}\nu^{2}}\right]\left[\frac{u_{\nu
l}(2\nu^{2}R_{q^{\prime}})}{R_{q^{\prime}}\nu^{2}}\right]$ (28)

The expressions in 26 can be analytically summed, yielding

$\displaystyle E_{q}$ $\displaystyle=\frac{(R_{q}^{-1}-1)[u_{\nu
0}(2\nu^{2}R_{q})]^{2}+\nu^{2}[u_{\nu
0}^{\prime}(2\nu^{2}R_{q})]^{2}}{2\nu^{2}}$ (29) $\displaystyle
V_{qq^{\prime}}$ $\displaystyle=\frac{u_{\nu 0}^{\prime}(t_{-})u_{\nu
0}(t_{+})-u_{\nu 0}(t_{-})u_{\nu 0}^{\prime}(t_{+})}{2(t_{+}-t_{-})},$ (30)

with

$\displaystyle t_{\pm}$
$\displaystyle=\nu^{2}\left(R_{q}+R_{q^{\prime}}\pm\sqrt{R_{q}^{2}+R_{q^{\prime}}^{2}-2R_{q}R_{q^{\prime}}\cos(2\pi(q-q^{\prime})/M)}\right).$
(31)

and where $u_{\nu l}(r)$ are the reduced hydrogen radial functions and
$u^{\prime}_{\nu l}(r)=\frac{du_{\nu l}(r)}{dr}$. It is curious to note that
only the $s$-wave radial wave function needs to be evaluated, making these
expressions useful computationally, since only one (out of $\nu$ possible
radial wave functions) function, along with its derivative, must be evaluated.
It is also useful in determining asymptotic properties, since these are
determined by the behavior of only a single function and its derivative.

##  VI (5) Eigenspectrum in the ring geometry

Due to the periodicity of the ring, the trilobite representation eigenvectors
are

$\tilde{v}_{q}^{(k)}=\frac{1}{\sqrt{M}}e^{-\frac{2\pi ikq}{M}},$ (32)

where $q=1,...,M$ and $k=-M/2,-M/2+1,...,0,...,M/2-1$ when $M$ is even and
$k=-(M-1)/2,...,0,...,(M-1)/2$ when $M$ is odd. The corresponding eigenvalues
are

$\displaystyle\epsilon(k)$ $\displaystyle=E+\cos\left(\pi
k\right)V_{1\frac{M}{2}+1}\delta_{0,M(\text{mod}2)}+2\sum_{q^{\prime}=1}^{\tilde{M}}\cos\left(\frac{2\pi
kq^{\prime}}{M}\right)V_{1q^{\prime}+1},$ (33)

where $E$ is the on-site energy, $\tilde{M}$ is $M/2-1$ for even $M$ and
$(M-1)/2$ for odd $M$. Alternatively, we can compute the eigenvalues directly
from the Rydberg functions,

$\displaystyle c_{kj}H_{jj^{\prime}}c_{k^{\prime}j^{\prime}}$
$\displaystyle=\sum_{jj^{\prime}}\sum_{lm}\frac{\mathcal{R}_{lm}(R_{1},R_{1})}{M}e^{\frac{2\pi
i}{M}\left(kj-m(j-j^{\prime})-k^{\prime}j^{\prime}\right)}.$ (34)

After rearranging the sums,

$\displaystyle c_{kj}H_{jj^{\prime}}c_{k^{\prime}j^{\prime}}$
$\displaystyle=\sum_{lm}\frac{\mathcal{R}_{lm}(R_{1},R_{1})}{M}\sum_{jj^{\prime}}e^{\frac{2\pi
i}{M}\left(kj-m(j-j^{\prime})-k^{\prime}j^{\prime}\right)},$ (35)

we can compute the sum over $j,j^{\prime}$, which is only over the complex
exponential,

$\displaystyle\sum_{jj^{\prime}}\dots$ $\displaystyle=\exp\left(\frac{2\pi
i}{M}(k+m-k^{\prime}M-mM)\right)$ (36)
$\displaystyle\times\frac{(e^{2ik\pi}-e^{2im\pi})(e^{2im\pi}-e^{2ik^{\prime}\pi})}{(e^{2ik\pi/M}-e^{2im\pi/M})(e^{2im\pi/M}-e^{2ik^{\prime}\pi/M})}.$

This gives $M^{2}\delta_{kk^{\prime}}\delta_{(k-m)\text{mod}M,0}$. Thus,

$\displaystyle\epsilon(k)$
$\displaystyle=M(\nu)\sum_{lm}\mathcal{R}_{lm}(R_{1},R_{1})\delta_{(m-k)\text{mod}M,0}.$
(37)

##  VII (6) Properties of the $R=1$ system

The $R=1$ system is the simplest and most straightforward to connect to known
theoretical results, as its hopping terms die off rapidly with
$|q-q^{\prime}|$ and can be very well approximated with only a nearest
neighbor model. Two important asymptotic limits of the $s$-wave radial
functions are

$\displaystyle\lim_{\nu\to\infty}u_{\nu 0}(2\nu^{2})$
$\displaystyle=a\nu^{-5/6},a\approx-0.56355$ (38)
$\displaystyle\lim_{\nu\to\infty}u^{\prime}_{\nu 0}(2\nu^{2})$
$\displaystyle={b}{\nu^{-13/6}},b\approx 0.326.$ (39)

The on-site potentials are therefore

$\lim_{\nu\to\infty}E_{q}=\frac{[u^{\prime}_{\nu
0}(2\nu^{2})]^{2}}{2}\approx\frac{b^{2}\nu^{-13/3}}{2}=a_{1}\nu^{-13/3},\,\,a_{1}\approx
0.053138.$ (40)

From a numerical experiment, we find that setting $M=\text{Floor}(3\nu^{2/3})$
results in a consistent scaling of the hopping terms with an overall factor
$\nu^{-13/3}$.

The hopping terms can be determined numerically:

$\displaystyle\lim_{\nu\to\infty}V_{qq+1}$
$\displaystyle=b_{1}{\nu^{-13/3}},\,\,b_{1}\approx{0.01355}$ (41)
$\displaystyle\lim_{\nu\to\infty}V_{qq+2}$ $\displaystyle-
c_{1}\nu^{13/3},\,\,c_{1}\approx{0.0004}.$ (42)

With these, we can construct a model Hamiltonian which very closely matches
the numerical Rydberg results. The small and negative next-nearest-neighbor
hopping terms result in a slight asymmetry in the eigenspectrum.

To analytically treat the influence of disorder, we expand the Hamiltonian
matrix elements to first order in the positional disorder and perform the same
asymptotic analysis as in the periodic case. This gives

$\displaystyle\nu^{13/3}E_{q}$ $\displaystyle\approx
a_{1}-g_{1}\nu^{2/3}\delta_{q}$ (43) $\displaystyle\nu^{13/3}V_{qq+1}$
$\displaystyle\approx
b_{1}-f_{1}(\overline{\delta}_{q^{\prime}}-\overline{\delta}_{q})-e_{1}\nu^{2/3}(\delta_{q}+\delta_{q^{\prime}})$
(44) $\displaystyle\nu^{13/3}V_{qq+2}$ $\displaystyle\approx-c_{1},$ (45)

where $a_{1}$, $b_{1}$, and $c_{1}$ were reported above, and $e_{1}\approx
0.015$, $f_{1}\approx 0.04$, and $g_{1}\approx 0.1519$.

From this we make three conclusions:

  * •

The radial positional disorder in the $R=1$ case must be rescaled by a factor
$\nu^{-2/3}$ in order to provide a constant energetic disorder as $\nu$
increases; this is a necessary step for obtaining the propert thermodynamic
limit.

  * •

Radial disorder leads to on-site disorder and, roughly an order of magnitude
smaller, positively correlated hopping disorder.

  * •

Angle disorder leads to anti-correlated off-diagonal disorder; there is no on-
site disorder.

##  VIII (7) Properties of the $R=0.75$ system

This $R$ value is more challenging to study since there are more non-zero
hopping elements to consider. We therefore proceed primarily numerically. We
find that the diagonal elements are

$E_{q}=\frac{1}{2\nu^{2}}\left[\frac{1}{3}(u_{\nu
0}(3/2\nu^{2}))^{2}+\nu^{2}(u^{\prime}_{\nu
0}(3/2\nu^{2})\right]^{2}\approx\frac{a_{0.75}}{\nu^{4}},\,\,a_{0.75}=\frac{0.3667}{2}$
(46)

In the asymptotic limit, $\nu\to\infty$, the hopping terms for relatively
small $|q-q^{\prime}|<\nu/20$, approach the surprisingly simple functional
form

$V_{qq^{\prime}}\sim\frac{a_{0.75}}{\nu^{4}}\frac{\sin(\omega(q-q^{\prime})}{\omega|q-q^{\prime}|},$
(47)

where $\omega=\pi\sqrt{3}$. For $\nu/20<|q-q^{\prime}|<\nu/10$
(approximately), the hopping terms continue to oscillate, but with mostly
constant amplitude. At $|q-q^{\prime}|\approx\nu/10$, the hopping terms
rapidly decay to zero.

Figure 5:  Hopping amplitudes for $\nu=1000$ for the $R=0.75$ system (blue)
compared with the asymptotic form of equation 47 (red)

Taking a ’truncated’ sinc function for the hopping gives, its only asymptotic
and approximate character, a very good qualitative prediction of the
eigenspectrum. The cutoff length (around $\nu/10$) affects the bandwidth of
the two individual bands, which become flatter and narrower as as the cutoff
length increases. The proportion of states in the lower and upper band depend
on the frequency of the oscillations.

Performing the same disorder analysis as in the $R=1$ case, we find that the
energy disorder stemming from radial disorder is proportional to
$\overline{\delta}\nu^{-4}$, and therefore the radial disorder in this case
needs no further scaling as in the $R=1$ case. The same holds for the angle-
disorder, which remains purely off-diagonal, and the correlation (anti-
correlation) of the hopping disorder in the radial (angular) disorder cases
remains as well.

##  IX (8) Properties of the $R=0.5$ system

When the ring is positioned at half the radius of the Rydberg orbit, $R=0.5$,
the Hamiltonian’s diagonal elements are straightforward to evaluate:

$E_{q}=\frac{1}{2\nu^{2}}\left[[u_{\nu
0}(\nu^{2})]^{2}+\nu^{2}[u^{\prime}_{\nu
0}(\nu^{2})]^{2}\right]\approx\frac{0.6366}{2\nu^{4}}=a_{0.5}\nu^{-4}.$ (48)

The largest hopping element, due to the shape of the trilobite orbitals at
this ring radius, is between site $q$ and site $q+M/2$, or, if $M$ is odd,
between site $q$ and sites $q+(M\pm 1)/2$. The parity of $M$ therefore plays a
key role in overall form of the eigenspectrum, in contrast to the previous $R$
values where it was irrelevant. When $M$ is even, the hopping term between $q$
and $q+M/2$ is

$V_{q,q+M/2}=\frac{u_{\nu 0}^{\prime}(0)u_{\nu
0}(2\nu^{2})}{4\nu^{2}}=\frac{u_{\nu
0}(2\nu^{2})}{2\nu^{7/2}}\approx\frac{-0.5635}{2\nu^{13/3}}=-c_{0.5}\nu^{-13/3}$
(49)

When $M$ is odd, the two neighboring particles on the opposite side of the
ring have identical amplitudes, also scaling like $\nu^{-13/3}$. Note that the
appearance of fractional exponents in these scaling relations stems again from
the peculiar behavior of the $R=1$ radial wave function. For other hopping
terms the expression $V_{qq^{\prime}}$ no longer depends on the exceptional
scaling of the radial wave function at $R=1$ and we can use the naive scaling
behavior $u_{\nu 0}(R)\sim\nu^{-1}$ and $u^{\prime}_{\nu 0}(R)\sim\nu^{-2}$
safely. Together this gives a $\nu^{-5}$ scaling for the off-diagonal
elements.

This case is therefore distinct from the other two that we have considered in
that its matrix elements scale with $\nu$ in different ways. To understand the
resulting eigenspectrum in the even parity case, we construct an approximate
Hamiltonian. This has $a_{0.5}$ on the diagonal and a constant hopping
$b_{0.5}/\nu$ to all sites except for the opposite site, which has a hopping
amplitude of $-c_{0.5}/\nu^{1/3}$. The resulting matrix can be diagonalized
analytically to gain some insight into the Rydberg composite’s spectrum. Its
eigenvalues are $a_{0.5}+c_{0.5}\nu^{-1/3}$ ($\times\nu/2$ degeneracy),
$a_{0.5}-2b_{0.5}\nu^{-1}-c_{0.5}\nu^{-1/3}$ ($\times\nu/2-1$ degeneracy), and
a single eigenvalue $a_{0.5}+b_{0.5}(\nu-2)\nu^{-1}-c_{0.5}\nu^{-1/3}$. For
large $\nu$ the eigenspectrum consists of two flat bands separated from
$a_{0.5}$ by $c_{0.5}\nu^{-1/3}$, and a single state lying at $b_{0.5}$. In
the thermodynamic limit the band gap closes completely and the system
condenses to a flat band with a single shifted state. In this limit, the
system qualitatively resembles the odd-parity $M$ state. Although this is a
highly simplified qualitative picture of the $R=0.5$ Rydberg eigenspectra, the
basic features exist also in the real case.

The dominant hopping term, connecting opposite sites on the ring, scales as
$\nu^{-13/3}$, as seen above. Angle disorder shifts this hopping term by a
term second order in the positional disorder strength, but overall having a
$\nu^{-13/3}$ dependence as well. Under radial disorder, this hopping term
shifts by a term first order in the positional disorder strength, but having a
$\nu^{-11/3}$ scaling. Like the $R=1$ case, this also requires a rescaling of
the positional disorder, $\overline{\delta}\to\nu^{-2/3}\overline{\delta}$, to
obtain the proper scaling. In turn, this means that the diagonal disorder
decreases with larger $\nu$, since this can be shown to scale in the normal
way, as $\nu^{-4}$.

Due to the reduction in on-site disorder necessary in order to avoid
increasingly strong off-diagonal disorder in this system in the thermodynamic
limit, in the main text we consider only angle disorder.

##  X (9) numerical details

Positions of the scatterers In the ring geometry the scatterers are placed at
angles $\phi_{q}=\frac{2\pi q}{M}$ in a plane centered around the Rydberg
core. We introduce disorder either by shifting the angles of the scatterers,
$\phi_{q}\to\frac{2\pi}{M}\left[q+\delta_{q}(\nu)\right]$, or their positions,
$R_{q}\to[1+\overline{\delta}_{q}(\nu)]R_{q}$. The former case leads to anti-
correlated off-diagonal disorder (i.e. the energetic disorder in a hopping
term is proportional to $\delta_{q}-\delta_{q^{\prime}}$), while the latter
leads to uncorrelated diagonal disorder and a weaker, correlated off-diagonal
disorder (i.e. the energetic disorder in a hopping term is proportional to
$\overline{\delta}_{q}+\overline{\delta_{q^{\prime}}}$). We take
$\delta_{q},\overline{\delta}_{q}$ to be independent Gaussian random variables
with variance $\sigma^{2}$ and mean zero.
Details of the disorder averaging In the text, we use several different
disorder strengths. For the $R=1$ angle disorder case, $\sigma=17\times
10^{-3}$. For the $R=1$ radial disorder case, $\sigma=(2\times
10^{-3})\cdot(30^{2/3})\approx 0.01931$. Each random number drawn from this
distribution $\overline{\delta}_{q}$ was divided by $\nu^{2/3}$ to ensure that
the energy disorder remains constant as $\nu$ increases. For the $R=0.75$
radial disorder case, $\sigma=1.33\times 10^{-3}$. For the $R=0.5$ angle
disorder case, $\sigma=22\times 10^{-3}$ was used. In all cases we averaged
over 1000 disorder realizations.