# Bound states without potentials: localization at singularities

Eric He<EMAIL_ADDRESS>University of California, Berkeley, California
94720, USA  Department of Physics, Brock University, St. Catharines, Ontario
L2S 3A1, Canada  R. Ganesh<EMAIL_ADDRESS>Department of Physics, Brock
University, St. Catharines, Ontario L2S 3A1, Canada

###### Abstract

Bound state formation is a classic feature of quantum mechanics, where a
particle localizes in the vicinity of an attractive potential. This is
typically understood as the particle lowering its potential energy. In this
article, we discuss a paradigm where bound states arise purely due to kinetic
energy considerations. This phenomenon occurs in certain non-manifold spaces
that consist of multiple smooth surfaces that intersect one another. The
intersection region can be viewed as a singularity where dimensionality is not
defined. We demonstrate this idea in a setting where a particle moves on $M$
spaces ($M=2,3,4,\ldots$), each of dimensionality $D$ ($D=1,2$ and $3$). The
spaces intersect at a common point, which serves as a singularity. To study
quantum behaviour in this setting, we discretize space and adopt a tight-
binding approach. We generically find a ground state that is localized around
the singular point, bound by the kinetic energy of ‘shuttling’ among the $M$
surfaces. We draw a quantitative analogy between singularities on the one hand
and local attractive potentials on the other. To each singularity, we assign
an equivalent potential that produces the same bound state wavefunction and
binding energy. The degree of a singularity ($M$, the number of intersecting
surfaces) determines the strength of the equivalent potential. With $D=1$ and
$D=2$, we show that any singularity creates a bound state. This is analogous
to the well known fact that any attractive potential creates a bound state in
1D and 2D. In contrast, with $D=3$, bound states only appear when the degree
of the singularity exceeds a threshold value. This is analogous to the fact
that in three dimensions, a threshold potential strength is required for bound
state formation. We discuss implications for experiments and theoretical
studies in various domains of quantum physics.

##  I Introduction

A free particle, in its quantum mechanical ground state, typically spreads
uniformly to occupy all available space. This allows the particle to lower its
kinetic energy. However, in the presence of an attractive potential, it may
localize into a bound state to lower its potential energy. This phenomenon
reflects competition between kinetic and potential energies. In this article,
we discuss a paradigm where bound states form without any potentials. Rather,
the particle moves on a singular space consisting of multiple surfaces that
intersect at a ‘junction’. This allows for a new type of kinetic energy that
favours localization of the particle. We present a detailed analysis of this
phenomenon, focussing on the role of dimensionality.

Figure 1: Self-intersecting spaces. Left: Three wires (one-dimensional spaces)
that intersect at a point. Centre: Two sheets (two-dimensional spaces) that
intersect at a point. Right: Three cubes (three-dimensional spaces) that
intersect at a point.

We consider a class of spaces consisting of multiple surfaces that intersect
one another. The locus of intersection represents a singularity. In the
discussion below, we characterize each singularity by two quantities: co-
dimension and degree. Fig. 1 shows some examples. On the left, we have three
one-dimensional channels that intersect at a point-like junction. A generic
point in this space lies on one of the channels; its local neighbourhood is
one-dimensional. In contrast, the junction is a zero-dimensional region (a
point) whose local neighbourhood does not have a well-defined dimensionality.
The difference between these two dimensionalities is the co-dimension, which
is unity here. In addition, we assign a degree of three – representing the
number of channels that meet at this junction. In Fig. 1 (centre), we see two
sheets that intersect at a point. This represents a singularity of co-
dimension two. The space is generically two-dimensional, while the singularity
is point-like. As we have two sheets, the degree is two. In Fig. 1 (right), we
see three cubes which are understood to share a common point. The common point
is a zero-dimensional singularity. The co-dimension and the degree are three
in this case.

For comparison, we will also discuss bound states induced by attractive
potentials on smooth spaces. There is extensive literature available on bound
states induced by various potentials. It is well known that dimensionality
plays a key role. In 1D and 2D, an infinitesimal potential suffices to create
a bound state. However, in three and higher dimensions, a threshold potential
strength is requiredSimon (1976); Yang and de Llano (1989); Nieto (2002);
Lapicki and Geltman (2011). In this article, we use a tight binding approach
that can handle both singularities and potentials on the same footing.
Naively, the problem of a potential appears to be very different from that of
a singularity. However, our results bring out a deep connection. As we show
below, any singularity is quantitatively equivalent to a potential, in the
sense that it produces the same bound state.

##  II The tight-binding approach

The traditional approach to quantum mechanics is to construct a Hamiltonian
operator and find its eigenfunctions. This cannot be carried out on singular
spaces as the Hamiltonian cannot be written down in the vicinity of a
singularity. For example, a gradient operator cannot be defined near a
junction of two wires. Solutions can still be found using ad hoc methods. For
example, eigenfunctions can be found on each smooth segment, with a suitable
boundary condition imposed at the singularity. The choice of boundary
condition can affect the resultGratus et al. (1994); Kostrykin and Schrader
(1999); Andrade et al. (2016). We take an alternative approach using a tight-
binding discretization of the space. We take the space to consist of distinct
points, with the particle hopping between neighbouring points. The local
geometry of the space is reflected by the set of neighbours at each point.

The tight-binding problem lends itself to an ambiguity-free procedure for a
one-dimensional problem, such as the one shown in Fig. 1 (left). In two
dimensions and higher, there can be multiple ways of discretizing a smooth
surface. For example, a smooth two-dimensional region can be discretized into
a square or a triangular grid. Once this choice is made, there is no further
ambiguity in the procedure or the solutions obtained. In our calculations, we
choose a square (cubic) discretization for two (three) dimensions.

The Hamiltonian of the particle is represented as

$\displaystyle H=-t\sum_{\langle
mn\rangle}\big{\\{}c_{m}^{\dagger}c_{n}+c_{n}^{\dagger}c_{m}\big{\\}},$ (1)

where the hopping amplitude, $t$, sets the energy scale. The sum runs over
nearest-neighbour bonds, with $m$ and $n$ representing sites at the end of
each bond. The operator $c_{m}^{\dagger}$ creates a particle at site $m$,
while $c_{n}$ annihilates a particle at site $n$. Away from the singularity,
each site has $2D$ neighbours, where $D$ is the dimensionality of the surface.
As shown in Fig. 1, the singularity is a single site with $(2MD)$ neighbours,
where $M$ is the degree of the singularity. We have $2D$ neighbours per
surface, with $M$ surfaces in total. Operationally, we take each surface to
have linear dimension $L$ with periodic boundaries. The total number of sites
in the problem is then $N=ML^{D}-M+1$. The resulting Hamiltonian is an
$N\times N$ symmetric matrix. Its eigenvectors represent stationary states,
while the eigenvalues yield the corresponding energies. For small system
sizes, we carry out full diagonalization to find all eigenvectors and
eigenvalues. For large systems, we take advantage of the sparse character of
the tight binding Hamiltonian and employ Krylov-space-based routines to find
the lowest few eigenstates.

In the tight binding setup, an eigenstate satisfies the following relation at
every site:

$\displaystyle-t\sum_{m(n)}\psi_{m}=E\psi_{n},$ (2)

where $n$ represents any given site. The index $m(n)$ runs over the neighbours
of $n$, $\psi_{p}$ represents the eigenvector component at site $p$ and $E$
represents the eigenvalue.

A bound state can be identified in two ways: from the eigenvector or from the
eigenvalue. The eigenvector must be peaked at the singularity, decaying to
zero as we move away. The eigenvalue must lie below a threshold value, $-2Dt$,
representing the lowest value possible for a delocalized state on a
$D$-dimensional surface. This can be expressed in terms of a binding energy,
$E_{binding}=-2Dt-E_{state}$. A bound state must have a positive binding
energy. The higher the binding energy, more bound is the state.

For the one-dimensional tight binding problem with a setup as shown in Fig. 1
(left), bound state(s) can be found using analytic arguments. They are
exponentially localized around the singularity, as we show below. More
generally, the eigenvalues and eigenvectors can be found numerically.

##  III $D=1$: Intersecting wires

###  III.1 Analytic solution to the tight-binding problem

We first discuss the case of $M$ one-dimensional wires intersecting at a
point-like junction. The arguments in this section were first presented in
Ref. Khatua et al., 2021 in the context of a certain magnetic model. To
represent the wavefunction, we denote the junction site as $j=0$. To all other
sites, we assign an integer value that encodes distance from the junction. For
example, the immediate neighbours of the junction are assigned $j=1$. This
tight binding problem produces one bound eigenstate, described by the ansatz

$\displaystyle\psi_{j}=\frac{1}{\mathcal{N}}e^{-\alpha_{M}j};~{}~{}~{}\psi_{0}=\frac{1}{\mathcal{N}}.$
(3)

Here, $\mathcal{N}$ is a normalization constant. We demand that this be an
eigenstate with eigenvalue $E_{M}$. At a site away from the junction, the
eigenstate condition of Eq. 2 yields

$\displaystyle-t(e^{-\alpha_{M}}+e^{\alpha_{M}})=E_{M}.$ (4)

At the junction site, the same condition yields

$\displaystyle-2Mte^{-\alpha_{M}}=E_{M}.$ (5)

From Eqs. 4 and 5, we solve for $\alpha_{M}$ and $E_{M}$,

$\displaystyle\alpha_{M}=\frac{1}{2}\ln\\{2M-1\\},~{}~{}E_{M}=-2Mt/\sqrt{2M-1}.$
(6)

Note that for $M=1$, there is no singularity as we only have one wire. In this
limit, the bound state vanishes as $\alpha_{M=1}=0$ and $E_{M=1}=-2t$. For
$M\geq 2$, $\alpha_{M}$ represents a decay constant. The binding energy is
given by $E_{binding,D=1}=(-2t-E_{M})$. The normalization constant can be
explicitly found, $\mathcal{N}=\sqrt{\frac{2M-1}{M-1}}$.

For $M\geq 2$, $\alpha_{M}$ monotonically increases with $M$. In parallel,
$E_{M}$ monotonically decreases or equivalently, the binding energy
monotonically increases. This shows that the state becomes progressively more
bound as the degree of the singularity increases. For very large $M$, the
bound state is entirely localized at the singularity.

###  III.2 Comparison with bound states induced by a potential

The bound state at the singularity can be compared to one induced by a local
attractive potential. Within the tight-binding approach, we consider a smooth
one-dimensional chain with sites labelled by $j$, a coordinate that runs over
all integers. At $j=0$, we have an on-site attractive potential of strength
$g$. The Hamiltonian is given by

$\displaystyle
H_{1D,g}=-t\sum_{j}\big{\\{}c_{j}^{\dagger}c_{j+1}+c_{j+1}^{\dagger}c_{j}\big{\\}}-gc_{0}^{\dagger}c_{0}.$
(7)

This problem also generates a bound state, with the wavefunction

$\displaystyle\psi_{j}=\frac{1}{\mathcal{N}^{\prime}}e^{-\alpha_{g}|j|}.$ (8)

where $\mathcal{N}^{\prime}$ is a normalization constant. For $j\neq 0$, the
eigenstate condition takes the same form as Eq. 4. At $j=0$, the condition is
modified by the potential to give

$\displaystyle-2te^{-\alpha_{g}}-g=E_{g},$ (9)

where $E_{g}$ is the eigenvalue. It is convenient to express the potential
strength and the energy eigenvalue as dimensionless quantities, using
$\tilde{g}\equiv g/2t$ and $\tilde{E}_{g}=E_{g}/2t$. In terms of these
quantities, we find

$\displaystyle\alpha_{g}=\ln\\{\tilde{g}+\sqrt{\tilde{g}^{2}+1}\\};~{}~{}\tilde{E}_{g}=\frac{-\tilde{g}^{2}-\tilde{g}\sqrt{\tilde{g}^{2}+1}-1}{\tilde{g}+\sqrt{\tilde{g}^{2}+1}}.$
(10)

The normalization constant comes out to be
$\mathcal{N}^{\prime}=\sqrt{1+\frac{1}{\tilde{g}\\{\tilde{g}+\sqrt{\tilde{g}^{2}+1}\\}}}$.
These values describe a bound state induced by a potential on a smooth one-
dimensional space with no singularities. In contrast, those in Eq. 6 describe
a bound state created at a singularity of co-dimension 1 and degree $M$, with
no potential involved. Remarkably, the wavefunctions have the same form in
both cases as given by Eqs. 3 and 8. This allows us to draw a precise
equivalence, $M\leftrightarrow\tilde{g}_{M}$, where $\tilde{g}_{M}$ satisfies

$\displaystyle
M=\Big{[}\tilde{g}_{M}^{2}+\tilde{g}_{M}\sqrt{\tilde{g}_{M}^{2}+1}+1\Big{]}.$
(11)

The equivalence can be stated as follows. On the one hand, we consider a
singularity of co-dimension 1 and degree $M$, with no potential. On the other
hand, we consider a potential of strength $\tilde{g}_{M}$ on a smooth one-
dimensional chain. These two situations produce bound states with precisely
the same decay constant and binding energy.

The equivalent potential, $\tilde{g}_{M}$, increases monotonically with $M$.
For large $M$, we see that $\tilde{g}_{M}\sim\sqrt{M}$.

###  III.3 Mechanism for binding

A bound state has lower energy than the continuum of delocalized states. The
underlying mechanism provides some way for the bound state to lower its
energy. What is the mechanism in the case of a potential or in the case of a
singularity? This question can be directly addressed within the tight binding
approach where the Hamiltonian is a sum of local terms. We have one term for
each bond, representing the kinetic energy of hopping between two sites. In
the case of a potential, we also have an on-site potential energy. Given the
ground state wavefunction, we may evaluate the contribution of each term
separately to its energy.

For the case of a potential-induced bound state, the Hamiltonian is given by
Eq. 7 with the wavefunction given by Eq. 8. Bound state formation is driven by
the potential energy term. This can be seen by examining the potential energy
contribution as a fraction of the bound state’s energy,

$\displaystyle\frac{\langle-
gc_{0}^{\dagger}c_{0}\rangle}{{E}_{g}}=\left(\frac{\tilde{g}^{2}+\tilde{g}\sqrt{\tilde{g}^{2}+1}}{\tilde{g}^{2}+\tilde{g}\sqrt{\tilde{g}^{2}+1}+1}\right)^{2}=\left(\frac{-\tilde{g}}{\tilde{E}_{g}}\right)^{2}.$
(12)

For very small $\tilde{g}$, this quantity approaches zero as we have a state
that is well spread. At $\tilde{g}=1$, it becomes half. For large $\tilde{g}$,
it approaches unity; the energy of the bound state comes almost entirely from
the potential energy term. This reveals that the mechanism for bound state
formation is potential-energy-lowering.

We now compare with the case of a bound state at a singularity. The
Hamiltonian can be written in the form of Eq. 1 with the wavefunction given by
Eq. 3. Binding is driven by the $M$ bonds that connect outward from the
singularity. We evaluate

$\displaystyle\frac{-t\sum_{j(0)}\langle\big{\\{}c_{j}^{\dagger}c_{0}+c_{0}^{\dagger}c_{j}\big{\\}}\rangle}{E_{M}}=\frac{2M-2}{2M-1}.$
(13)

The index $j(0)$ runs over the 2M sites that are directly connected to the
singularity. When $M=1$, this ratio vanishes as we have a delocalized ground
state. For $M=2$, this ratio yields $2/3$, i.e., two-thirds of the ground
state energy arises from the immediate vicinity of the singularity. For large
$M$, this ratio approaches unity. That is, all of the ground state’s energy
comes from the $2M$ bonds that are connected to the junction. This represents
a kinetic energy contribution, arising from the particle’s ‘shuttling’ motion
from one wire to another. This is a new form of kinetic energy that is not
present on smooth surfaces. The ground state is bound due to shuttling-
kinetic-energy that can only be gained in the vicinity of the singularity.

##  IV $D=2$: Intersecting sheets

We next consider singularities of co-dimension 2, with an example shown in
Fig. 1 (centre). We consider the space of $M$ sheets that share a common
point. We discretize the space using a square grid for each sheet, assuming
that they intersect at the origin. This setup generates a single bound state
for any $M\geq 2$. However, the wavefunction cannot be expressed as a simple
analytic form. Instead, we present numerical solutions and fit them to a
suitable functional form.

###  IV.1 Numerical solution to the tight-binding problem

Figure 2: The ground state on a space with $M$ 2D sheets intersecting at a
point. The plots correspond to $M=2$ (top left), $M=3$ (top centre) and $M=5$
(top right). Each 2D sheet is taken to be a 100 $\times$ 100 grid with
periodic boundaries. The origin is common to all the sheets. The plots show
the spatial variation in the probability (amplitude squared). These
wavefunctions fit well to a modified Bessel form of order 0 (bottom left). As
the degree of the singularity increases, the decay constant obtained from the
fit increases (bottom centre) as does the binding energy (bottom right). The
binding energy is shown in units of $t$.

Fig. 2 (top row) shows examples of the bound state wavefunction – the
probability of finding the particle at each site. The panels, from left to
right, correspond to $M=2$, $3$ and $5$. In each case, the wavefunction on one
of the intersecting sheets is shown – the same wavefunction appears on every
sheet. From the plots, we immediately see localized character, with
probability peaked at the singularity and decaying as we move away. With
increasing degree of the singularity (increasing $M$), the ground state
becomes more tightly bound.

The same information is shown as a two-dimensional plot in Fig. 2 (bottom
left), with the wavefunction amplitude plotted against the radial coordinate
(distance from the singularity). The solution has circular symmetry: sites
with the same radial coordinate have the same wavefunction amplitude. Note
that the phase is uniform at all points. As shown in Fig. 2 (bottom left), the
amplitude is well fit by a function of the form

$\displaystyle f(r)=a~{}k_{0}(\gamma r+b).$ (14)

Here, $k_{\nu}$ is the modified Bessel function of the second kind, of order
$\nu$. This form is known from the continuum problem of a bound state induced
by an attractive local potential in two dimensions, e.g., a square well
potential (see Ref. Nieto, 2002; Lapicki and Geltman, 2011). The bound
wavefunction takes this form in the external region (outside the well). For
each value of $M$, we obtain $a$, $b$ and $\gamma$ as fitting parameters. The
coefficient $b$ represents a horizontal shift. For any $M\geq 2$, the best-fit
value of $b$ is less than the lattice spacing. The quantity $\gamma$ encodes a
horizontal stretch. It can be viewed as a decay constant – the higher the
value of $\gamma$, the more tightly bound is the wavefunction. As shown in
Fig. 2 (bottom centre), $\gamma$ increases monotonically with $M$. Finally,
Fig. 2 (bottom right) shows the binding energy vs. $M$. This encodes the
energy difference between the bound state and the lowest delocalized state
($E_{min.delocalized,2D}=-4t$). As $M$ increases, the binding energy
increases. This supports the contention that the ground state becomes more
tightly bound.

We summarize these findings as follows. A bound state is formed for any
singularity of co-dimension $2$. The higher the degree of the singularity, the
more tightly bound the state.

###  IV.2 Comparison with bound states induced by a potential

For comparison, we consider a smooth two-dimensional surface with a local
attractive potential. The Hamiltonian for this system is similar to Eq. 7. An
attractive potential of strength $g$ is introduced at one of the sites.
Analytic solutions cannot be found, but a single bound state is seen in the
numerics for any attractive potential. We describe this state below and
compare with the continuum problem of a local attractive potential in two
dimensionsNieto (2002).

Figure 3: The ground state wavefunction on a 2D sheet with an on-site
attractive potential. The panels on top correspond to $g=t$ (top left),
$g=2.032~{}t$ (top centre) and $g=2.911~{}t$ (top right). They show the
spatial variation of the probability (amplitude squared). The 2D sheet is
taken to be a 100 $\times$ 100 grid with periodic boundaries. The wavefunction
profile is shown at bottom left, with the amplitude plotted against the radial
coordinate. The data are fit to modified Bessel functions of order 0. At
bottom centre, we plot the decay constant vs. $g$. At bottom right, we plot
the binding energy (in units of $t$) vs. the strength of the potential. The
best fit curve to the form $E_{binding}=Ae^{-B/g}$ is shown.

Fig. 3 (top) shows the bound state wavefunction for $g/t=1,~{}2.032$ and
$2.911$. The latter two values are equivalent to singularities with $M=2$ and
$3$ respectively, as we discuss below. Fig. 3 (bottom left) shows the same
wavefunctions, fitted to the form given in Eq. 14. We find good agreement with
the Bessel function form, especially at large distances. As shown in Figs. 3
(bottom centre, bottom right), the decay constant ($\gamma$) and the binding
energy increase monotonically with $g$. The larger the potential, the tighter
is the bound state. For small $g$, the binding energy is exponentially weak.
As shown in the figure, the dependence on $g$ is well fit by the function
$E_{binding}=Ae^{-B/g}$. This form is known from the continuum problem of a
bound state induced by a local potential (say, of the delta-function form).
For example, it is invoked in the discussion of the Cooper instabilityCooper
(1956); Esebbag et al. (1992), where electron-pairs that are constrained to
live on a two-dimensional space experience a weak attraction.

Figure 4: Top: Binding energy vs. decay constant in two dimensions. Blue
circles represent bound states induced by a local potential on a smooth two-
dimensional sheet. From left to right, the points correspond to increasing
potential strength, with $g$ increasing from $0$ to $20~{}t$ in steps of $t$.
Red squares represent bound states induced by singularities of co-dimension
two. From left to right, the degree of the singularity ($M$) increases in
steps of $5$ from $0$ to $100$. The two data sets collapse onto the same
curve. Bottom: $g_{M}$ vs. $M$, where $g_{M}$ is the potential that is
equivalent to a singularity of degree $M$. The data fit well to a function of
the form $g_{M}\sim\sqrt{M}$.

We now compare the bound states induced by a singularity with that of a
potential. We treat $M$ and $g$ as tunable handles in the two cases. In both,
we obtain localized ground states that fit well to the same functional form.
In gross terms, the solutions are described by two parameters: the decay
constant and the binding energy. These are both monotonically increasing
functions of the tuning handle in each case. Remarkably, they are not
independent. The decay constant immediately determines the binding energy and
vice versa. This is shown in Fig. 4 which plots the binding energy vs. decay
constant for both singularity-induced and potential-induced bound states. The
data points collapse onto the same curve. This leads us to conclude that
potentials and singularities lead to the same bound states. For a singularity
of degree $M$, we can find an equivalent potential $g_{M}$ that generates a
bound state with the same decay constant and binding energy. Fig. 3 (bottom
right) shows the variation in $g_{M}$ with $M$. As $M$ increases, the
equivalent potential grows in strength. For large $M$, we find
$g_{M}\sim\sqrt{M}$.

We have verified that the equivalence goes beyond the decay constant and
binding energy. It holds even for the precise form of the wavefunction, up to
a change in the normalization to account for multiple sheets.

##  V $D=3$: Intersecting three-dimensional spaces

We proceed to singularities of co-dimension 3, with an example shown in Fig. 1
(right). We consider $M$ three-dimensional spaces that share a common point.
We discretize each space using a cubic $L\times L\times L$ grid, assuming that
the origin is common to all spaces. As with co-dimension-2, analytic solutions
cannot be found. We present numerical solutions and fit them to functional
forms that are inspired by the continuum problem.

###  V.1 Numerical solution to the tight-binding problem

The co-dimension-3 solutions present a remarkable difference when compared
with co-dimensions 1 and 2. A bound state forms only when the degree of the
singularity exceeds a threshold value. For example, as shown in Fig. 5 (left),
we find no bound state for $M=3$ (three cubes intersecting a point). However,
there is a clear bound state when $M=5$ (five cubes intersecting at a point),
as shown in Fig. 5 (right). This can be seen in various ways as we describe
below. Crucially, any such analysis requires a systematic approach to the
thermodynamic limit by increasing $L$. A true bound state will remain bound
with a constant ‘width’ as $L$ increases. In contrast, a delocalized state
will expand with increasing system size.

Figure 5: The ground state of a particle on a space of $M$ cubes intersecting
at a point, where $M=3$ (top left), $M=4$ (top centre) and $M=5$ (top right).
Each cube is taken to be a 20 $\times$ 20 $\times$ 20 grid with periodic
boundaries. The corner point is shared by all cubes, representing a point of
intersection. In each plot, the size and colour of markers both represent the
probability of finding the particle (amplitude squared) at a certain site. We
only show sites on a single cube as the wavefunction is the same across cubes.
At bottom left, we plot $r_{avg}/L$ vs. $1/L$ (see text). The data are fit to
the form $y=a+b/L+c/L^{2}$ in order to extrapolate to $L\rightarrow\infty$. At
bottom centre, we plot the binding energy vs. $1/L$ with fits to the form
$y=a+b/L+c/L^{2}$. Finally, at bottom right, we plot wavefunction amplitude
against the radial coordinate for $M=4,6,8$. The data are fit to a modified
Bessel form (see text).

We introduce a quantitative measure for localization,

$\displaystyle
r_{avg}=\sum_{j}|\vec{r}_{j}-\vec{r}_{0}|\times|\psi_{j,0}|^{2},$ (15)

where $j$ runs over all sites in the tight-binding setup. The distance between
the origin and site $j$ is denoted as $|\vec{r}_{j}-\vec{r}_{0}|$. Note that
distances are calculated as on the usual cubic lattice: a point with
coordinates $(x,y,z)$ is at a distance of $\sqrt{x^{2}+y^{2}+z^{2}}$ from the
origin. The probability amplitude of the ground state at this site is denoted
as $|\psi_{j,0}|$. Assuming that the particle resides in the ground state,
$r_{avg}$ denotes its average separation from the singularity. If the ground
state represents a true bound state, $r_{avg}/L$ will extrapolate to zero as
$L\rightarrow\infty$. Instead, if the ground state were delocalized,
$r_{avg}/L$ will extrapolate to a non-zero value. Fig. 5 (bottom left)
compares $r_{avg}/L$ vs. $1/L$ for various $M$ values. We see a qualitative
shift between $M=3$ and $M=4$, with $M\geq 4$ showing bound state formation.

We next examine the binding energy. It is defined as the energy separation
between the lowest state and the bottom of the delocalized continuum
($E_{min.delocalized,3D}=-6t$). If the ground state were truly bound,
$E_{binding}$ will approach a non-zero value as $L\rightarrow\infty$. In a
delocalized state, $E_{binding}$ will vanish for large $L$. Fig. 5 (bottom
centre) shows $E_{binding}$ vs. $1/L$ for various $M$ values. Once again, we
find behaviour that is consistent with bound state formation only when $M\geq
4$.

To characterize the wavefunction in a systematic manner, we fit it to the form

$\displaystyle h(r)=c~{}r^{-1/2}k_{1/2}(\gamma r+b).$ (16)

Here, $k_{1/2}$ represents a modified Bessel function of the second kind of
order $1/2$. This form is known from the continuum problem of a 3D attractive
square well. When a bound state is produced, its wavefunction follows this
form in the external region (outside the well)Nieto (2002); Lapicki and
Geltman (2011). We find a good fit to the modified Bessel form as long as
$M\geq 4$, as shown in Fig. 5 (bottom right). The wavefunction amplitude at
each site depends solely on the radial distance from the singularity. The
phase is ignored as it is uniform at all sites.

From the fit, we obtain the decay constant $\gamma$. For $M\geq 4$, the decay
constant increases with increasing $M$, as does the binding energy, as we
discuss below. The ground state becomes progressively more bound.

###  V.2 Comparison with bound states induced by a potential

As before, we compare our results with a smooth three-dimensional surface with
a local attractive potential. We model this as a tight binding problem on a
cubic grid. We place an attractive on-site potential of strength $g$ at the
origin. For small values of $g$, the ground state is not localized. A bound
state is formed only when $g$ exceeds a threshold value. The ground state is
plotted in the top panels of Fig. 6 for $g=4~{}t$, $4.657~{}t$ and
$5.246~{}t$. The first clearly shows a delocalized ground state, while the
latter two are bound. In fact, we will argue below that the latter two values
are equivalent to $M=4$ and $5$.

We approach the question of bound state formation in the same manner as with
the singularity above. Figs. 6 (bottom left) and (bottom centre) show
$r_{avg}/L$ and the binding energy vs. $1/L$ for a few potential strengths. We
see two clear regimes, $g\lesssim 4t$ and $g\gtrsim 4.05t$. In the former,
$r_{avg}/L$ extrapolates to a non-zero value as $L\rightarrow\infty$. At the
same time, the binding energy extrapolates to zero. In the latter, $r_{avg}/L$
extrapolates to zero, while the binding energy extrapolates to a non-zero
value. This suggests a threshold value, $g_{critical}$, that separates bound
and unbound behaviour. This result is the tight-binding analogue of a well-
known result in quantum mechanics: in three dimensions, a critical potential
strength is required for bound state formation.

The precise location of the critical point is difficult to pinpoint due to
system size limitations. For example, near the transition, the fitting curves
to $r_{avg}/L$ vs. $1/L$ cannot be reliably extrapolated to $1/L\rightarrow 0$
within accessible system sizes. The binding energy curves of Fig. 6 (bottom
centre) are somewhat clearer: $g_{critical}$ appears to lie between
$g=4.00~{}t$ and $4.05~{}t$.

Fig. 6 (bottom left) shows the evolution of $r_{avg}/L$ with $g$. We see a
qualitative change between two regimes, one where $r_{avg}/L$ extrapolates to
zero as $L\rightarrow\infty$ and the other where it tends to a non-zero value.
The boundary between these regimes cannot be clearly discerned within
accessible system sizes. In Fig. 6 (bottom centre), we see the evolution of
binding energy with system size, for various values of $g$. Based on the
values extrapolated to $L\rightarrow\infty$, we conclude that the critical
potential strength lies between $g=4.00~{}t$ and $4.05~{}t$. Fig. 6 (bottom
left) shows the evolution of $r_{avg}/L$ with $g$. The critical value of $g$
cannot be discerned within accessible system sizes.

When the potential exceeds $g_{critical}$, the resulting bound state fits well
to the modified Bessel form of Eq. 16. This is shown in Fig. 6 (bottom right).
The best-fit value of the decay constant, $\gamma$, increases monotonically
with increasing $g$. Likewise, the binding energy increases with $g$. A bound
state forms when $g$ exceeds $g_{critical}$, becoming progressively more bound
as $g$ increases further.

Figure 6: The ground state on a 3D space with a local attractive potential.
The panels on top correspond to $g=4~{}t$ (top left), $g=4.657~{}t$ (top
centre) and $g=5.246~{}t$ (top right). The space is taken to be a 20 $\times$
20 $\times$ 20 cubic grid with periodic boundaries with the on-site potential
at the corner. In each plot, the size and colour of markers both represent the
probability of finding the particle (amplitude squared) at a certain site. At
bottom left, we plot $r_{avg}/L$ vs. $1/L$ for various potential strengths.
The data are fit to the form $y=a+b/L+c/L^{2}$. At bottom centre, we plot the
binding energy (see text) vs. $1/L$, with fits to the form $y=a+b/L^{2}$. At
bottom right, the wavefunction amplitude is plotted as a function of the
radial distance from the potential. The data are fit to a modified Bessel
function form (see text).

Figure 7: Top: Binding energy vs. decay constant in three dimensions. Blue
circles represent bound states induced by a local potential. From left to
right, points correspond to increasing potential strength, with $g$ increasing
from $4.5~{}t$ to $20~{}t$ in steps of $0.5~{}t$. Red squares represent bound
states induced by singularities of co-dimension three. From left to right, the
degree of the singularity ($M$) increases in steps of $5$ from $5$ to $60$.
The two data sets collapse onto the same curve. Bottom: $g_{M}$ vs. $M$, where
$g_{M}$ is the potential that is equivalent to a singularity of degree $M$.

We now compare results for a singularity with those for a potential. Bound
states fit well to the same analytic form in both cases. As with the two-
dimensional case, the decay constant and the binding energy are not
independent. Fig. 7 (top) plots the variation of these two parameters for
singularity-driven and potential-driven bound states. Data from both cases
collapse onto a single curve. This brings out a quantitative equivalence
between singularities and potentials. For a singularity of degree $M$, we
assign an equivalent potential, $g_{M}$. The degree-$M$ singularity produces a
bound state with the same decay constant and binding energy as an attractive
potential of strength $g_{M}$. Fig. 7 (bottom) shows a plot of $g_{M}$ vs.
$M$. For large $M$, $g_{M}$ approximately scales as $\sqrt{M}$. The
equivalence is not restricted to the decay constant and binding energy. We
have verified that it holds for the precise forms of the wavefunction, up to a
change in the normalization constant.

##  VI Discussion

We have demonstrated that singularities arising from intersections produce
bound states in the same way as attractive potentials. This mapping is
quantitative in nature, where each singularity can be assigned an effective
potential strength. In singularities, the binding mechanism is the kinetic
energy of shuttling, where the particle moves back and forth across surfaces.
This can be viewed as ‘quantum indecision’ – the particle remains frozen at a
crossroads as it is unable to pick a direction of propagation. A bound state
allows the system to sample all surfaces to small distances, while rapidly
shuttling among surfaces. This notion may underlie the observation of bound
states in semiconductor junctions. Electrons and holes are known to form bound
states in architectures such as T-junctions or X-junctionsSols et al. (1989);
Berggren and Ji (1991); Ji and Berggren (1992). Likewise, collective excitonic
excitations are known to be bind to junctionsSchult et al. (1989); Goñi et al.
(1992); Hasen et al. (1997). Building upon these studies, experiments can be
designed to test our findings, e.g., with junctions of $M$ channels engineered
in a semiconductor. The binding phenomenon described here may hold in
classical wave mechanics as well, e.g., in junctions of photonic or phononic
waveguidesMaksimov and Sadreev (2006); Nakarmi et al. (2021).

Intersection-induced bound states play a role in quantum magnets. At low
energies, the physics of a quantum magnet resembles that of a particle moving
on the space of classical ground statesKhatua et al. (2019); Khatua (2021). In
certain magnetic clusters, frustration leads to complex spaces that contain
intersecting wires/sheets. At very low energies, the magnet freezes at the
intersection, i.e., it orders in a particular classical configuration. This
effect has been called ‘order by singularity’, as a special case of the well
known order-by-disorder phenomenonKhatua et al. (2019); Srinivasan et al.
(2020); Khatua et al. (2021). Co-dimension-1 singularities have been found and
argued to host bound statesKhatua et al. (2019); Srinivasan et al. (2020);
Khatua et al. (2021). Co-dimension-3 singularities have been found and argued
not to host bound statesKhatua et al. (2018); Khatua (2021). Magnetic clusters
can help to further test the results presented here. Frustrating interactions
may lead to classical ground states spaces with multiple wires, sheets or even
three-dimensional spaces. The degree of the singularity may also be tuned. The
Kitaev spin-$S$ chain serves as an example, with classical ground states that
form a network-like space. Each node of the network is an intersection of $M$
wires, where $M$ can be tuned by changing the length of the chainKhatua et al.
(2021).

The analogy between singularities and potentials highlights the role of
dimensionality in bound state formation. In higher dimensions, the tendency of
a particle to spread is stronger as the space available for spreading is
larger. As a result, a stronger potential or a singularity of higher degree is
required. A similar idea is invoked in Anderson localizationEconomou and
Soukoulis (1983); Economou et al. (1984). In one or two dimensions, an
infinitesimal amount of disorder suffices to localize a particle. However, a
threshold disorder strength is required in three and higher dimensions. This
has been related to the problem of a random walker and the mean time spent in
a neighbourhoodBerger et al. (2008). The higher the dimensionality, the
smaller is the time spent in a neighbourhood. The particle may show localizing
tendencies, which upon quantization, manifest as bound states.

We have based our arguments on a tight binding framework where potentials and
intersections can be handled on the same footing. Tight binding realizations
of intersecting spaces (e.g., as shown in Fig. 1) can be directly realized in
ultracold atomic gases by using optical lattices. More generally, the same
problem can also be addressed in the continuum. There is extensive literature
on quantum graphs where eigenfunctions of the Schrödinger operator can be
found on each link, with suitable boundary conditions enforced at junctions.
Studies have explored various choices for boundary conditions and their
consequencesKottos and Smilansky (1999); Znojil (2012); Andrade et al. (2016).
An open question is to determine the boundary conditions that reproduce the
tight binding results presented here.

Bound state formation may play a role in physical problems on abstract spaces.
For example, many metals are known to have sheet-like Fermi surfaces that
intersect. A Cooper pair in such a system has an abstract configuration space,
which also contains intersections. Our results suggest that a Cooper pairing
instability may occur even in the absence of attractions. This may be relevant
to 1/4-doped graphene, where the Fermi level reaches the van Hove singularity
at the edge centres of the Brillouin zone. Several studies have predicted
superconducting behaviour in this regimeBlack-Schaffer and Honerkamp (2014).
An exciting future direction is to explore superconductivity in the absence of
interactions, driven by singularity-induced Cooper-pair formation.

###### Acknowledgements.

We thank Diptiman Sen, Kirill Samokhin, Jean-Sébastien Bernier, Subhankar
Khatua and Abhiram Soori for insightful discussions. We thank Eric Tan for
discussions on technical aspects. This work was supported by a Discovery Grant
2022-05240 from the Natural Sciences and Engineering Research Council of
Canada.

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