# Universal scattering with general dispersion relations

Yidan Wang  Joint Quantum Institute, NIST/University of Maryland, College
Park, Maryland 20742 USA  Michael J. Gullans  Joint Center for Quantum
Information and Computer Science, NIST/University of Maryland, College Park,
Maryland 20742 USA  Xuesen Na  Department of Mathematics, University of
Maryland, College Park, Maryland 20742, USA  Seth Whitsitt Joint Quantum
Institute, NIST/University of Maryland, College Park, Maryland 20742 USA
Joint Center for Quantum Information and Computer Science, NIST/University of
Maryland, College Park, Maryland 20742 USA  Alexey V. Gorshkov  Joint Quantum
Institute, NIST/University of Maryland, College Park, Maryland 20742 USA
Joint Center for Quantum Information and Computer Science, NIST/University of
Maryland, College Park, Maryland 20742 USA

###### Abstract

Many synthetic quantum systems allow particles to have dispersion relations
that are neither linear nor quadratic functions.  Here, we explore single-
particle scattering in general spatial dimension $D\geq 1$ when the density of
states diverges at a specific energy. To illustrate the underlying principles
in an experimentally relevant setting, we focus on waveguide quantum
electrodynamics (QED) problems (i.e. $D=1$) with dispersion relation
$\epsilon(k)=\pm|d|k^{m}$, where $m\geq 2$ is an integer. For a large class of
these problems for any positive integer $m$, we rigorously prove that when
there are no bright zero-energy eigenstates, the $S$-matrix evaluated at an
energy $E\to 0$ converges to a universal limit that is only dependent on $m$.
We also give a generalization of a key index theorem in quantum scattering
theory known as Levinson’s theorem—which relates the scattering phases to the
number of bound states—to waveguide QED scattering for these more general
dispersion relations.  We then extend these results to general integer
dimensions $D\geq 1$, dispersion relations $\epsilon(\bm{k})=|\bm{k}|^{a}$ for
a $D$-dimensional momentum vector $\bm{k}$ with any real positive $a$, and
separable potential scattering.

The quantum mechanical scattering of few-body systems remains a challenging
theoretical problem. Even at low incoming energies, nonperturbative effects
render a general solution out of reach. A common workaround is based on
effective field theory whereby low-energy scattering is described in terms of
a few parameters such as the scattering length $a_{0}$ and the effective range
$r_{0}$ [1, 2, 3]. When $a_{0}\gg r_{0}$, the system is in the unitarity limit
where the universal physics of Efimov states [4, 2, 3] and unitary Fermi gases
[5, 6, 7] can emerge. Another approach where general results can be obtained
is by studying the analytic structure of the $S$-matrix at low energies. One
striking result in this context is the simple effect of dimensionality on
scattering theory. Two particles with short-range interactions perfectly
reflect off each other at the threshold in one dimension (1D), while they
transmit without seeing each other in higher dimensions. This effect arises
because the density of states diverges at the threshold as $1/\sqrt{E}$ in 1D,
but stays finite in higher dimensions.

Recent experimental progress in synthetic quantum matter allows for broad
control of dispersion relations. One class of such systems consists of tunable
periodic structures, including photonic crystal waveguides [8, 9, 10, 11, 12,
13, 14], twisted bilayer graphene [15, 16], superconducting qubit arrays [17,
18, 19], atomic arrays [20, 21, 22], and trapped-ion spin chains [23, 24].
Another class is polaritonic [25, 26] or spin-orbit coupled [27, 28] systems,
where the dispersion relation can be tuned in situ by external fields [29, 30,
31, 32]. In principle, the density of states at the scattering threshold can
be tuned to diverge faster than it does for quadratic dispersion relations.
This opens up the door to studying the implications of a more general density
of states without changing the dimension of the system. Recently, there is a
growing interest in the study of general dispersion relations in condensed
matter systems, where divergent electronic density of states is referred to as
a high-order Van Hove singularity [33, 34, 35]. In particular, power-law-
divergent density of states near the Fermi level leads to nontrivial metallic
states termed supermetals [34].

In this Letter, we explore the physics of divergent density of states from the
perspective of scattering theory. We illustrate that, when a particle has a
divergent density of states at a certain energy, its scattering matrix has a
nontrivial universal limit that depends on the rate of the divergence.  In the
main text of this Letter, we study single-particle scattering of photon-
emitter models in 1D ($D=1$) with a dispersion relation
$\epsilon(k)=\pm|d|k^{m}$, where $m$ is a positive integer. Notably,  when $m$
is even, these emitter scattering models describe scattering for incoming
frequencies near the band edge of photonic crystal waveguides coupled to atoms
[10] or quantum dots [8].  We discover that the $S$-matrix can take different
universal limits $\lim_{E\rightarrow 0}S(E)$ for different values of $m$. The
total reflection at the threshold for a quadratic dispersion relation is an
example of such universal behavior corresponding to $m=2$. In general, there
may be multiple classes of universal behaviors in the $S$-matrix corresponding
to each $m$, depending on the properties of interactions at $k=0$. In this
Letter, we consider a physically natural class of interactions and
characterize the universal behavior for each $m$. We also extend a key index
theorem in scattering theory known as Levinson’s theorem—which relates the
scattering phases to the number of bound states [36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46]—to the class of models considered in this Letter with these
more general dispersion relations.  To demonstrate the generality of our
methodology, in the Supplemental Material [47], we extend our discussions to
separable potential scattering, general integer dimensions $D\geq 1$ and
dispersion relations $\epsilon(k)=|\bm{k}|^{a}$, where $\bm{k}$ is a
$D-$dimensional momentum vector and $a$ is any positive real number. The
extension of our single-emitter results to spin-boson models is given in an
upcoming work [48]. These spin-boson models generalize the waveguide quantum
electrodynamics (QED) models introduced below by including emitter-photon
interaction terms beyond the rotating wave approximation—thereby, illustrating
the relevance of our results in the many-body regime of waveguide QED.

_Waveguide QED._ —In many synthetic quantum systems, particles propagating in
a 1D channel are scattered by emitters such as atoms, quantum dots, or
superconducting qubits. The emitters are often coupled to the environment,
which adds dissipation to the system composed of the emitters and the 1D
channel. Such models are broadly referred to as waveguide QED models. Since we
are interested in the scattering processes with a single photon coming in and
a single photon going out, it suffices to use a non-Hermitian effective
quadratic Hamiltonian

$\displaystyle H$ $\displaystyle=H_{0}+V,$ (1) $\displaystyle H_{0}$
$\displaystyle=\int_{-\infty}^{+\infty}dk\
\epsilon(k)C^{\dagger}(k)C(k)+\sum_{i,j=1}^{N}K^{R}_{ij}b_{i}^{\dagger}b_{j},$
(2) $\displaystyle V$
$\displaystyle=\int_{-\infty}^{+\infty}dk\left[\sum_{i=1}^{N}V_{i}(k)C(k)b^{\dagger}_{i}+\text{h.c}\right],$
(3)

where the bare Hamiltonian $H_{0}$ consists of the freely propagating
particles, while the interacting emitters are indexed by $i=1,2,\dots N$. $V$
describes the quadratic interaction between the particles and the emitters.
Through controlling the lattice structures of the photonic crystal waveguide,
the rate at which the density of states diverges at a particular energy can be
fine-tuned. Since we are discussing single-particle scattering with bounded-
strength interactions, only local spectral properties of the dispersion
relation matter, and our results are insensitive to the detailed behavior of
the dispersion far away from the threshold energy. In this Letter, we focus on
the dispersion relation $\epsilon(k)=\sigma|d|k^{m}$, where $\sigma=\pm 1$,
$|d|$ is a positive constant, and $m$ is a positive integer.  The case of
$m=1$ corresponds to a linear dispersion relation and has a non-universal
scattering matrix in the limit of zero energy 111Note, for dispersions
relations in 1D of the form $\epsilon(k)=|k|^{a}$, the $S$-matrix obtains a
universal value for any positive real $a$ [47]. These non-analytic dispersion
relations have a trivial universal limit for the $S$-matrix when $a\leq 1$.
For $a>1$, they have similar universal behavior of the $S$-matrix as the
positive even integer $m$ cases of $\epsilon(k)=\sigma|d|k^{m}$ studied in the
main text.. For this reason, we assume $m\geq 2$ in the discussion below.
When $\sigma=\pm 1$ and $m$ is even, $\epsilon(k)$ can be understood as the
lowest-order approximation of a dispersion relation around its local
minima/maxima, after a change of reference points for both energy and
momentum. Depending on whether we are considering bosons scattered by bosonic
emitters or fermions scattered by fermionic emitters, we have either
commutation or anti-commutation relations:
$[C(k),C^{\dagger}(k^{\prime})]_{\pm}=\delta(k-k^{\prime}),[b_{i},b^{\dagger}_{j}]_{\pm}=\delta_{ij}$.
$K^{R}_{ij}$ represents the matrix element of the $N\times N$ matrix
$\bm{K}^{R}$; $\bm{K}^{R}$ is the only non-Hermitian term in the Hamiltonian:
the Hermitian $\bm{A}$ and anti-Hermitian $i\bm{B}$ components of
$\bm{K}^{R}=\bm{A}+i\bm{B}$ represent, respectively, the coherent and
incoherent interactions among the emitters. $\bm{K}^{R}$ is dissipative when
$\bm{B}$ is non-positive and nonzero.

For convenience, we introduce a vector function
$|v_{k}\rangle=[V_{1}(k),\dots,V_{N}(k)]^{T}$, with corresponding basis states
given by the emitter excitations
$\\{b^{\dagger}_{1}\ket{0,g},\dots,b^{\dagger}_{N}\ket{0,g}\\}$, where
$\ket{0,g}$ is the ground state with zero excitation. In the most generic
scenario, $V_{i}(k)$ for different emitters are independent of each other.
Here, we consider the case where $\ket{v_{k}}$ can be written as
$\ket{v_{k}}=V(k)\ket{u}$. We further assume $V(k)$ is continuous at $0$ and
$V(0)\neq 0$. Under this constraint, the only relevant vector around $k=0$ is
$\ket{u}$, and effectively, there is only a single relevant “degree of
freedom” in the emitter vector space at $k=0$. We then show that the zero-
energy scattering behavior for multiple emitters can be reduced to the
behavior for $N=1$. As a result, we are able to obtain a complete
classification of the universal low-energy scattering behavior in these
models.

_Universal scattering._ —We start with a discussion that applies to the case
of general $\ket{v_{k}}$. The $S$-matrix for a single particle is defined
through the incoming and outgoing scattering eigenstates
$|\psi_{k}^{\pm}\rangle$, where the superscript $\pm$ specifies the boundary
conditions of the scattering states. The $S$-matrix element from one single-
particle scattering state $k$ to another $k^{\prime}$ is
$\mathcal{S}(k,k^{\prime})=\langle\psi^{-}_{k^{\prime}}|\psi^{+}_{k}\rangle$.
To explain the universal behavior of the $S$-matrix, it is useful to write
down its relation to the on-shell T-matrix:

$\displaystyle\mathcal{S}(k,k^{\prime})$
$\displaystyle=\delta(k\\!-\\!k^{\prime})-2\pi
i\delta[\epsilon(k)\\!-\\!\epsilon(k^{\prime})]T(E\\!+\\!i0^{+},k,k^{\prime}),$
(4)

where $0^{+}/0^{-}$ represents an infinitesimal positive/negative real number
and $E=\epsilon(k)$. For dispersion relation $\epsilon(k)=\sigma|d|k^{m}$ with
even $m$, there are two degenerate momenta $k_{1}(E),k_{2}(E)$ corresponding
to any energy $E>0$ ($E<0$) for $\sigma=+1$ ($\sigma=-1$). We can define a
$2\times 2$ matrix $\bm{S}(E)$ by picking out the scattering amplitudes
between degenerate momenta:

$\displaystyle S_{\alpha\beta}(E)=\delta_{\alpha\beta}-2\pi
i\,\frac{T[E\\!+\\!i0^{+},k_{\alpha}(E),k_{\beta}(E)]}{\sqrt{|\epsilon^{\prime}[k_{\alpha}(E)]\epsilon^{\prime}[k_{\beta}(E)]|}},$
(5)

where $\alpha,\beta\in\\{1,2\\}$ and  the prefactor
$|\epsilon^{\prime}[k_{\alpha}(E)]\epsilon^{\prime}[k_{\beta}(E)]|^{-1/2}$
comes from $\delta[\epsilon(k)-\epsilon(k^{\prime})]$ in Eq. (4). When $m$ is
odd, we can define $\bm{S}(E)=S(E)$ as a single complex number, given by Eq.
(5) when $k_{\alpha}(E)=k_{\beta}(E)=k(E)$ is the momentum corresponding to
energy $E$. If the Hamiltonian is Hermitian, $S(E)$ is unitary.

In 1D scattering, the matrix $\bm{S}(E)$ directly describes the transmission
and reflection between degenerate momenta and is often used instead of the
function $\mathcal{S}(k,k^{\prime})$. When $E\rightarrow 0$,
$|\epsilon^{\prime}[k_{\alpha}(E)]\epsilon^{\prime}[k_{\beta}(E)]|^{-1/2}$
diverges. Since $|S_{\alpha\beta}(E)|\leq 1$, $T(E+i0,k_{\alpha},k_{\beta})$
in Eq. (5) must approach zero to cancel the divergence, which is the key
behind the universal behavior of $\bm{S}(E)$.

To proceed further, we note that the Lippmann-Schwinger equations for this
emitter scattering model have a simple analytic structure. As a result, we can
write down the single-particle $T$-matrix $T(\omega,k,k^{\prime})$ in terms of
the Green’s function of the emitters $\bm{G}(\omega)$, which is a finite-
dimensional matrix [50]:

$\displaystyle T(\omega,k,k^{\prime})$ $\displaystyle=\langle
v_{k^{\prime}}|\bm{G}(\omega)|v_{k}\rangle,$ (6) $\displaystyle\bm{G}(\omega)$
$\displaystyle=\frac{1}{\omega\mathbb{1}_{N}-\bm{K}^{R}-\bm{K}(\omega)},$ (7)
$\displaystyle\bm{K}(\omega)$
$\displaystyle=\int_{-\infty}^{+\infty}dk\frac{|v_{k}\rangle\langle
v_{k}|}{\omega-\epsilon(k)},$ (8)

where $\mathbb{1}_{N}$ is an $N\times N$ identity matrix. Equations (6)-(8)
hold for general photon-emitter couplings where $V_{i}(k)$ are independent
functions for different emitters. There are two mathematical conditions on
$V_{i}(k)$ that are necessary for the integral in Eq. (8) to be well-defined
at any complex $\omega\neq 0$ outside the continuum spectrum 222 In emitter
scattering, it is natural to define the continuum spectrum to not include $0$.
First, we require that $V_{i}(k)$ is a locally square-integrable complex
function on the real line. Second, to ensure that no ultraviolet divergences
are present in the model, we impose a restriction on the large-$k$ behavior of
$V_{i}(k)$: when $k\rightarrow\pm\infty$, there exist $\gamma>1$ such that
$|V_{i}(k)|^{2}=o(|k|^{m-\gamma})$. Each element of the $N\times N$ matrix
$\bm{K}(\omega)$ is an analytic function on the complex plane with a branch
cut along the continuum spectrum. $\bm{K}(\omega=E+i0^{+})$ can be understood
as describing effective interactions between emitters induced by the 1D
channel.

To understand the properties of $T(E+i0)$ close to $E=0$, we need to
understand the behavior of $\bm{K}(\omega)$ around $\omega=0$. We can show
that the value of $\bm{K}(\omega)$ around $\omega=0$ is decided by the
dispersion relation and $V(0)\ket{u}$. Define $L(\omega)$ as the integral over
the free-particle propagator:

$\displaystyle L(\omega)$
$\displaystyle=\int_{-\infty}^{+\infty}dk\frac{1}{\omega-\epsilon(k)}.$ (9)

We see that, when $\omega\rightarrow 0$,
$L(\omega)^{-1}\frac{1}{\omega-\epsilon(k)}$ as a function of $k$ diverges at
$k=0$ and vanishes everywhere else. In addition, $\int_{-\infty}^{+\infty}dk\
L(\omega)^{-1}\frac{1}{\omega-\epsilon(k)}=1$ by definition of $L(\omega)$.
Hence, it follows from a standard result in functional analysis attributed to
Toeplitz [52] that $\lim_{\omega\rightarrow
0}L(\omega)^{-1}\frac{1}{\omega-\epsilon(k)}=\delta(k)$. Using the condition
that $\ket{v_{k}}=V(k)\ket{u}$ is continuous at $k=0$ and the definition of
$\bm{K}(\omega)$ in Eq. (8), we have

$\displaystyle\lim_{\omega\rightarrow
0}L^{-1}(\omega)\bm{K}(\omega)=|V(0)|^{2}\ket{u}\bra{u}.$ (10)

When the emitter region consists of a single site, $\bm{K}^{R}=K^{R}$ is a
complex number and Eq. (10) becomes $\lim_{\omega\rightarrow
0}L^{-1}(\omega)\bm{K}(\omega)=|V(0)|^{2}$. Using Eqs. (6) and (7), we then
have

$\displaystyle\lim_{\omega\rightarrow
0}L(\omega)T(\omega,k,k^{\prime})=-\frac{V^{*}(k^{\prime})V(k)}{|V(0)|^{2}},$
(11)

which is no longer dependent on $K^{R}$ because $\lim_{\omega\rightarrow
0}L^{-1}(\omega)K^{R}=0$. Although Eq. (11) is derived for the case of $N=1$,
we show through a rigorous mathematical analysis in the Supplemental Material
[47] that Eq. (11) holds as long as the Hamiltonian does not support a
“bright” zero-energy eigenstate, defined as a zero-energy eigenstate that has
a non-zero emitter and photonic amplitude. These bright states are
distinguished from “dark” states that have only a nonzero photonic amplitude
and rather generically arise at zero-energy in these models. The proof of Eq.
(11) for $N>1$ is the main technical result of this Letter as it underlies
both the universal scattering results and our proof of Levinson’s theorem.

When we evaluate the $S$-matrix in the limit $E\rightarrow 0$ using Eq. (5),
$k_{\alpha}(E),k_{\beta}(E)$ in the T-matrix are both sent to $0$. Using Eq.
(11) and the condition that $V(k)$ is continuous at $k=0$, we have

$\displaystyle\lim_{E\rightarrow
0}L(E+i0^{+})T[E+i0^{+},k_{\alpha}(E),k_{\beta}(E)]=-1,$ (12)

which shows that the on-shell $T$-matrix in the zero-energy limit is
independent of the details of the interaction and fully determined by the
dispersion relation; this is the reason behind the universal limit of the
$S$-matrix when $E\rightarrow 0$. In the Supplemental Material [47], we
evaluate Eq. (9) and obtain the $m$-dependent value of $L(\omega)$:

$\displaystyle L(\omega)=-\pi
i\kappa_{m}\rho(|\omega|)\exp\left(-i\theta\frac{m-1}{m}\right),$ (13)

where the complex frequency $\omega$ is parameterized in polar coordinates as
$\omega=\sigma\exp(i\theta)|\omega|$, and
$\rho(|\omega|)=\frac{2}{m|d|^{1/m}}|\omega|^{-1+1/m}$ corresponds to the
density of states at energy $E=|\omega|$. For even $m$,
$\kappa_{m}=\frac{2}{1-\mu^{2}}$ with $\mu=\exp(i\pi/m)$, while $L(\omega)$
has a branch cut along the continuum spectrum $(0,+\infty)$ for $\sigma=+1$ or
$(-\infty,0)$ for $\sigma=-1$. For odd $m$, $\kappa_{m}=-\frac{1}{\mu-1}$ for
$\theta\in(0,\pi)$ and $\kappa_{m}=-\frac{1}{\mu(\mu-1)}$ for
$\theta\in(\pi,2\pi)$, while $L(\omega)$ has a branch cut along the real line.
For both even and odd $m$, $L(\omega)$ diverges at the rate of density of
states $\rho(|\omega|)$ when $\omega$ approaches $0$.

(a) odd $m$

(b) even $m$

Figure 1:  Illustration of 1D scattering ($z$ is a spatial coordinate) near
zero energy for dispersion relation $\epsilon(E)=\sigma|d|k^{m}$ with
$\sigma=\pm 1$. Panel (a) is for odd $m$, where the scattering matrix is a
single transmission coefficient dependent on $m$ and the sign of energy is
$E=0^{\pm}$. Panel (b) is for even $m$, where the scattering matrix is a
$2\times 2$ matrix. The eigenstates of the scattering matrix are the symmetric
and antisymmetric incoming states, with eigenphases $\exp(2\pi i\sigma/m)$ and
$1$, respectively.

Now, we are ready to evaluate the limit of the $S$-matrix at zero energy. When
$m$ is odd, energy $E$ can approach $0$ from both above and below:
$E\rightarrow 0^{\pm}$. When $m$ is even, $E$ can only approach $0$ from one
side: $E\rightarrow 0^{+}$ when $\sigma=+1$ or $E\rightarrow 0^{-}$ when
$\sigma=-1$. Taking the limit $E\rightarrow 0^{\pm}$ in Eq. (5) for the
respective cases properly, we have

$\displaystyle\lim_{E\rightarrow
0^{\pm}}S_{\alpha\beta}(E)=\delta_{\alpha\beta}+\lim_{E\rightarrow
0^{\pm}}2\pi i\rho(|E|)L^{-1}(E+i0^{+}),$ (14)

where we have used Eq. (12) and the observation that $\lim_{E\rightarrow
0}|\epsilon^{\prime}(k_{\alpha}(E))\epsilon^{\prime}(k_{\beta}(E))|^{1/2}\rho(E)=1$.
Using Eqs. (13) and (14), we find for odd $m$

$\displaystyle\lim_{E\rightarrow 0^{\pm}}S(E)=\exp(\pm\pi i/m),$ (15)

as illustrated in Fig. LABEL:sub@subfig:odd. For even $m$, we find that the
$S$-matrix

$\displaystyle\lim_{E\rightarrow\sigma 0^{+}}\bm{S}(E)=\exp(\sigma
i\pi/m)\begin{bmatrix}\cos(\pi/m)&\sigma i\sin(\pi/m)\\\ \sigma
i\sin(\pi/m)&\cos(\pi/m)\end{bmatrix},$ (16)

is symmetric in the basis of degenerate momenta
$\\{|k_{1}=0^{+}\rangle,|k_{2}=0^{-}\rangle\\}$. The symmetric eigenstate
$|\psi_{\text{s}}\rangle=\frac{1}{\sqrt{2}}(1,1)^{T}$ has an eigenphase
$\exp(i\pi\sigma/m)$, while the antisymmetric eigenstate
$|\psi_{\text{a}}\rangle=\frac{1}{\sqrt{2}}(1,-1)^{T}$ has a trivial
eigenphase $1$. The scattering of the symmetric and antisymmetric incoming
states near zero energy is illustrated in Fig. LABEL:sub@subfig:even. For
quadratic dispersion $\epsilon(k)=|d|k^{2}$, we recover the well-known total
reflection:

$\displaystyle\lim_{E\rightarrow 0^{+}}\bm{S}(E)=\begin{bmatrix}0&-1\\\
-1&0\end{bmatrix}.$ (17)

The relation between the universal behavior of the S-matrix and the dispersion
relation also applies to other types of interactions. In the Supplemental
Material, we show that Eqs. (15) and (16) also hold for separable potential
scattering. In addition, we generalize our results to arbitrary integer
dimension $D\geq 1$ and dispersion relations $\epsilon(k)=|\bm{k}|^{a}$, where
$a>0$ is not required to be an integer. In these cases, we demonstrate that
the determinant of the S-matrix reaches a universal limit dependent only on
$a/D$.

_Levinson’s theorem.—_ Levinson’s theorem relates the quantized scattering
phase to the number of bound states in the system. In the literature, the
theorem has been discussed in various Hermitian systems and various dimensions
[36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], where the dispersion relation
close to the scattering threshold is always quadratic. In our recent work, we
generalized Levinson’s theorem to 1D emitter scattering, where dissipation is
present and the dispersion relation is linear at all $k$ [53]. In that case,
there is no well-defined scattering threshold. When we consider  dispersion
relations $\epsilon(k)=\sigma|d|k^{m}$ with the class of photon-emitter
couplings $\ket{v_{k}}=V(k)\ket{u}$, the $S$-matrix can take different
universal limits at zero energy, dependent on the value of the integer $m\geq
2$ [see Eqs. (15) and (16)]. This leads to a modification to Levinson’s
theorem, as we illustrate in the remainder of this Letter.

(a) $\epsilon=\pm|d|k$

(b) $\epsilon=|d|k^{2}$

(c) $\epsilon=|d|k^{6}$

(d) $\epsilon=\pm|d|k^{5}$

Figure 2:  Illustrations of the trajectories of $\det[\bm{S}(E)]$ of a
dissipative system in the complex plane when $E$ is increased from $E_{min}$
to $E_{max}$ for (a) $\epsilon(k)=\pm|d|k$, where the trajectory starts and
ends at $1$. (b) $\epsilon=|d|k^{2}$, where the trajectory starts at $-1$ and
ends at $1$. (c) $\epsilon=|d|k^{6}$, where the trajectory starts at
$\exp(i\pi/3)$ and ends at $1$. (d) $\epsilon=\pm|d|k^{5}$, where the
trajectory for $E\in(-\infty,0)$ (solid yellow) starts at $1$ and ends at
$S(0^{-})=\exp(i\pi/5)$ , while the trajectory for $E\in(0,+\infty)$ (solid
black) starts at $S(0^{+})=\exp(i\pi/5)$ and ends at $1$.

For simplicity, we assume that there are no bright zero-energy eigenstates and
no bound states in the continuum in the system. Before discussing general $m$,
we summarize the theorem for quadratic ($m=2$) and linear ($m=1$) dispersion
relations. When energy $E$ is increased from the lower end of the continuum
spectrum $E_{min}$ (which can be $-\infty$) to the upper end $E_{max}$ (which
can be $+\infty$), $\det[\bm{S}(E)]$ traces a trajectory in the complex plane.
In the case of $\epsilon(k)=k$, the $S$-matrix is an identity matrix at both
ends of the continuum spectrum. The trajectory of $\det[\bm{S}(E)]$ in these
cases forms a closed loop starting and ending at $1$, as illustrated in Fig.
LABEL:sub@subfig:lineardetS. For illustration purposes, we assume that the
system is dissipative, so the trajectory is not confined to the unit circle.
Levinson’s theorem states that the winding number of this loop around the
origin is equal to the decrease in the number of bound states $\Delta N_{B}$
after the interaction is turned on [38, 40]. For emitter scattering, the
number of bound states for the bare Hamiltonian $H_{0}$ is equal to the number
of emitters $N$; hence, $\Delta N_{B}=N-N_{B}$, where $N_{B}$ is the number of
bound states for the full Hamiltonian [53]. If we define the scattering phase
$\delta(E)$ of $\det[\bm{S}(E)]\equiv|\det[\bm{S}(E)]|\exp(2i\delta(E))$ as a
continuous function of $E$ 333 For dissipative systems, we assume that
$\det[\bm{S}(E)]\neq 0$ for any $E$ within the continuum spectrum, the theorem
can be stated as $\Delta\delta\equiv\delta(E_{max})-\delta(E_{min})=\pi\Delta
N_{B}$. For a quadratic dispersion relation $\epsilon(k)=k^{2}$, the
trajectory of $\det[\bm{S}(E)]$ starts at $\lim_{E\rightarrow
0}\det[\bm{S}(E)]=-1$ and ends at
$\lim_{E\rightarrow+\infty}\det[\bm{S}(E)]=1$, as illustrated in Fig.
LABEL:sub@subfig:quadraticdetS. As compared to the closed-loop case of Fig.
LABEL:sub@subfig:lineardetS, Levinson’s theorem is modified to
$\Delta\delta=\pi\Delta N_{B}+\pi/2$ 444For potential scattering, see
Levinson’s theorem for quadratic dispersion relation in 1D in Ref. [45]. For
emitter scattering, Levinson’s theorem for quadratic dispersion relation is,
to our knowledge, first presented in this Letter.

Next, we give our results on Levinson’s theorem for emitter scattering with
dispersion relation $\epsilon(k)=\sigma|d|k^{m}$ with $\sigma=\pm 1$ and
photon-emitter couplings $\ket{v_{k}}=V(k)\ket{u}$. First, consider the case
of even $m$. When $\sigma=+1$, the trajectory of $\det[\bm{S}(E)]$ starts at
$\lim_{E\rightarrow 0^{+}}\det[\bm{S}(E)]=\exp(2\pi i/m)$ [see Eq. (16)] and
ends at $\lim_{E\rightarrow+\infty}\det[\bm{S}(E)]=1$, as illustrated in Fig.
LABEL:sub@subfig:evendetS for $m=6$. When $\sigma=-1$, the trajectory of
$\det[\bm{S}(E)]$ starts at $\lim_{E\rightarrow-\infty}\det[\bm{S}(E)]=1$ and
ends at $\lim_{E\rightarrow 0^{-}}\det[\bm{S}(E)]=\exp(-2\pi i/m)$. In the
Supplemental Material [47], we prove that, for both cases,

$\displaystyle\Delta\delta=\pi(N-N_{B})+\pi\frac{m-1}{m}.$ (18)

When $m$ is odd, the continuum spectrum is $(-\infty,0)\cup(0,+\infty)$, and
the trajectory of $S(E)$ is discontinuous across $0$, as illustrated in Fig.
LABEL:sub@subfig:odddetS. When $E$ increases from $-\infty$ to $0$, the
trajectory starts from $1$ and ends at $\exp(-i\pi/m)$ [see Eq. (15)]. When
$E$ increases from $0$ to $+\infty$, the trajectory starts at $\exp(+i\pi/m)$
and ends at $1$. If we define $\Delta\delta$ as the sum of the winding phases
of the two continuous trajectories, $\Delta\delta$ satisfies Eq. (18), as we
show in the Supplemental Material [47].

_Outlook._ — In this Letter, we have illustrated how a divergent density of
states results in a wide variety of universal scattering behaviors.  An
immediate next step is to generalize our results to arbitrary photon-emitter
interactions and non-separable short-range potentials. Although our results
rigorously apply only in the zero-energy limit, our work establishes the
foundation for the development of a universal low-energy theory for general
dispersion relations. Similar to the case of quadratic dispersion relations,
we expect the scattering to be primarily determined by the scattering length
when the de Broglie wavelengths of the particles are large compared to the
range of the interaction. It will be interesting to explore how other well-
studied problems for massive particles—such as Efimov physics [4, 2, 3],
renormalization for the effective field theory [56, 57], and the $N$-body
scale [58]—are modified in the presence of these more general dispersion
relations.

Our work also motivates new directions in many-body physics. The fact that
bosons with quadratic dispersion relations form a Tonks–Girardeau gas at low-
temperature in 1D and a Bose-Einstein condensate in 3D is closely related to
the different behaviors of two-body scattering at the scattering threshold
(total reflection vs. no interaction). Our discovery of new nontrivial
universal behaviors of the S-matrix may lead to predictions of new phases of
dilute gases for systems with a divergent density of states. Furthermore, it
remains an outstanding challenge to describe emitter scattering when both
dissipation and coherent driving are present.

###### Acknowledgements.

We thank Sarang Gopalakrishnan, Zhen Bi, Darrick Chang, Abhinav Deshpande,
Chris Baldwin, and Simon Lieu for discussions. Y.W. and A.V.G. acknowledge
support by ARO MURI, AFOSR, AFOSR MURI, U.S. Department of Energy Award No.
DE-SC0019449, DoE ASCR Quantum Testbed Pathfinder program (award No. DE-
SC0019040), DoE ASCR Accelerated Research in Quantum Computing program (award
No. DE-SC0020312), and NSF PFCQC program.

## References

  * Bethe [1949] H. Bethe, Phys. Rev. 76, 38 (1949).
  * Braaten and Hammer [2006] E. Braaten and H.-W. Hammer, Phys. Rep. 428, 259 (2006).
  * Braaten and Hammer [2001] E. Braaten and H.-W. Hammer, Phys. Rev. Lett. 87, 160407 (2001).
  * Efimov [1973] V. Efimov, Nucl. Phys. A 210, 157 (1973).
  * Thomas  _et al._ [2005] J. Thomas, J. Kinast, and A. Turlapov, Phys. Rev. Lett. 95, 120402 (2005).
  * Nishida and Tan [2008] Y. Nishida and S. Tan, Phys. Rev. Lett. 101, 170401 (2008).
  * Nascimbène  _et al._ [2010] S. Nascimbène, N. Navon, K. Jiang, F. Chevy, and C. Salomon, Nature 463, 1057 (2010).
  * Hughes [2004] S. Hughes, Opt. Lett. 29, 2659 (2004).
  * Joannopoulos  _et al._ [2008] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade,  _Photonic Crystals: Molding the Flow of Light_ , 2nd ed. (Princeton University Press, 2008).
  * Hung  _et al._ [2013] C. L. Hung, S. M. Meenehan, D. E. Chang, O. Painter, and H. J. Kimble, New J. Phys. 15, 083026 (2013).
  * Goban  _et al._ [2014] A. Goban, C. L. Hung, S. P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, Nat. Commun. 5, 3808 (2014).
  * Goban  _et al._ [2015] A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, Phys. Rev. Lett. 115, 063601 (2015).
  * Hood  _et al._ [2016] J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S. P. Yu, D. E. Chang, and H. J. Kimble, Proc. Natl. Acad. Sci. U. S. A. 113, 10507 (2016).
  * Lodahl  _et al._ [2015] P. Lodahl, S. Mahmoodian, and S. Stobbe, Rev. Mod. Phys. 87, 347 (2015).
  * Tarnopolsky  _et al._ [2019] G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Phy. Rev. Lett. 122, 106405 (2019).
  * Cao  _et al._ [2018] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Nature 556, 43 (2018).
  * van Loo  _et al._ [2013] A. F. van Loo, A. Fedorov, K. Lalumiere, B. C. Sanders, A. Blais, and A. Wallraff, Science 342, 1494 (2013).
  * Devoret and Schoelkopf [2013] M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013).
  * Sundaresan  _et al._ [2019] N. M. Sundaresan, R. Lundgren, G. Zhu, A. V. Gorshkov, and A. A. Houck, Phy. Rev. X 9, 011021 (2019).
  * Endres  _et al._ [2016] M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz, A. Krajenbrink, C. Senko, V. Vuletic, M. Greiner, and M. D. Lukin, Science 354, 1024 (2016).
  * Barredo  _et al._ [2016] D. Barredo, S. De Léséleuc, V. Lienhard, T. Lahaye, and A. Browaeys, Science 354, 1021 (2016).
  * Madjarov  _et al._ [2019] I. S. Madjarov, A. Cooper, A. L. Shaw, J. P. Covey, V. Schkolnik, T. H. Yoon, J. R. Williams, and M. Endres, Phys. Rev. X 9, 041052 (2019).
  * Porras and Cirac [2004] D. Porras and J. I. Cirac, Phys. Rev. Lett. 92, 207901 (2004).
  * Debnath  _et al._ [2018] S. Debnath, N. Linke, S.-T. Wang, C. Figgatt, K. Landsman, L.-M. Duan, and C. Monroe, Phys. Rev. Lett. 120, 073001 (2018).
  * Kittel [1976] C. Kittel,  _Introduction to solid state physics_ , Vol. 8 (Wiley New York, 1976).
  * Band [2006] Y. B. Band,  _Light and matter: electromagnetism, optics, spectroscopy and lasers_ , Vol. 1 (John Wiley & Sons, 2006).
  * Campbell  _et al._ [2011] D. L. Campbell, G. Juzeliūnas, and I. B. Spielman, Phys. Rev. A 84, 025602 (2011).
  * Lin  _et al._ [2011] Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, Nature 471, 83 (2011).
  * Fleischhauer and Lukin [2000] M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. 84, 5094 (2000).
  * Mahan [2013] G. D. Mahan,  _Many-particle physics_ (Springer Science & Business Media, 2013).
  * Peyronel  _et al._ [2012] T. Peyronel, O. Firstenberg, Q. Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, Nature 488, 57 (2012).
  * Firstenberg  _et al._ [2013] O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletić, Nature 502, 71 (2013).
  * Isobe and Fu [2019] H. Isobe and L. Fu, Phys. Rev. Res. 1, 033206 (2019).
  * Yuan  _et al._ [2019] N. F. Yuan, H. Isobe, and L. Fu, Nat. Commun. 10, 1 (2019).
  * Yuan and Fu [2020] N. F. Yuan and L. Fu, Phys. Rev. B 101, 125120 (2020).
  * Levinson [1949] N. Levinson, K. Danske Vidensk. Selsk. K. Mat.-Fys. Medd. 25 (1949).
  * Jauch [1957] J. M. Jauch, Helv. Phys. Acta 30 (1957).
  * Ida [1959] M. Ida, Prog. of Theor. Phys. 21, 625 (1959).
  * Wright [1965] J. A. Wright, Phys. Rev. 139, B137 (1965).
  * Atkinson and Morgan [1966] D. Atkinson and D. Morgan, II Nuovo Cimento A (1971-1996) 41, 559 (1966).
  * Ma and Ni [1985] Z.-Q. Ma and G.-J. Ni, Phys. Rev. D 31, 1482 (1985).
  * Barton [1985] G. Barton, J. Phys. A Math. Gener. 18, 479 (1985).
  * Poliatzky [1993] N. Poliatzky, Phys. Rev. Lett. 70, 2507 (1993).
  * Dong  _et al._ [1998] S.-H. Dong, X.-W. Hou, and Z.-Q. Ma, Phys. Rev. A 58, 2160 (1998).
  * Dong and Ma [2000] S.-H. Dong and Z.-Q. Ma, Int. J. of Theor. Phys. 39, 469 (2000).
  * Ma [2006] Z.-Q. Ma, J. of Phys. A: Math. Gener. 39, R625 (2006).
  * [47] See the supplemental material at X.
  * [48] S. Whitsitt, Y. Wang, and A. V. Gorshkov, (in preparation) .
  * Note [1] Note, for dispersions relations in 1D of the form $\epsilon(k)=|k|^{a}$, the $S$-matrix obtains a universal value for any positive real $a$ [47]. These non-analytic dispersion relations have a trivial universal limit for the $S$-matrix when $a\leq 1$. For $a>1$, they have similar universal behavior of the $S$-matrix as the positive even integer $m$ cases of $\epsilon(k)=\sigma|d|k^{m}$ studied in the main text.
  * Suhl [1965] H. Suhl, Phys. Rev. 138, A515 (1965).
  * Note [2] In emitter scattering, it is natural to define the continuum spectrum to not include $0$.
  * Lax [2002] P. D. Lax,  _Functional Analysis_ (Wiley, 2002).
  * Wang  _et al._ [2018] Y. Wang, M. J. Gullans, A. Browaeys, J. Porto, D. E. Chang, and A. V. Gorshkov, arXiv:1809.01147 (2018).
  * Note [3] For dissipative systems, we assume that $\mathop{det}\displaylimits[\bm{S}(E)]\neq 0$ for any $E$ within the continuum spectrum.
  * Note [4] For potential scattering, see Levinson’s theorem for quadratic dispersion relation in 1D in Ref. [45]. For emitter scattering, Levinson’s theorem for quadratic dispersion relation is, to our knowledge, first presented in this Letter.
  * Adhikari  _et al._ [1995] S. K. Adhikari, T. Frederico, and I. Goldman, Phys. Rev. Lett. 74, 487 (1995).
  * Bedaque  _et al._ [1999] P. F. Bedaque, H.-W. Hammer, and U. Van Kolck, Phys. Rev. Lett. 82, 463 (1999).
  * Bazak  _et al._ [2019] B. Bazak, J. Kirscher, S. König, M. Pavón Valderrama, N. Barnea, and U. Van Kolck, Phys. Rev. Lett. 122, 143001 (2019).
  * Note [5] Closing the contour in the lower half of the complex plane would give the same answer.
  * Louck [1960] J. D. Louck, J. Mol. Spectr 4, 298 (1960).
  * Granzow [1963] K. D. Granzow, J. Math. Phys 4, 897 (1963).
  * Efthimiou and Frye [2014] C. Efthimiou and C. Frye,  _Spherical harmonics in p dimensions_ (World Scientific, 2014).
  * Arfken  _et al._ [2013] G. B. Arfken, H. J. Weber, and F. E. Harris, in  _Mathematical Methods for Physicists (Seventh Edition)_ (Academic Press, Boston, 2013) pp. 643–713.

Supplemental Material

##  I Overview

In this Supplemental Material, we present details omitted in the main text and
generalize the results to higher dimensions, certain non-analytic dispersion
relations and delta-function potential scattering. In Sec. II, we derive the
expression for $L(\omega)$ in Eq. (13) in the main text. In Sec. III, we prove
Eq. (11) in the main text for the case of $N\geq 2$ and define bright zero-
energy eigenstates. In Sec. IV, we prove Levinson’s theorem [Eqs. (18) in the
main text].  In Sec. V, we generalize our results to higher dimensions and
dispersion relations $\epsilon(k)=|\bm{k}|^{a}$ with non-integer values of
$a>0$. In Sec. VI, we generalize our results to separable potential
scattering.

##  II Calculation of $L(\omega)$

In this section, we derive the expression for $L(\omega)$ in Eq. (13) in the
main text. We start with the definition of $L(\omega)$ in Eq. (9) in the main
text:

$\displaystyle
L(\omega)=\int_{-\infty}^{+\infty}dk\frac{1}{\omega-\epsilon(k)}.$ (S1)

The dispersion relation is given by $\epsilon(k)=\sigma|d|k^{m}$, where
$\sigma=\pm 1$ and $m\geq 2$ is a positive integer. To compute the integral,
we close the integration contour in the upper half 555Closing the contour in
the lower half of the complex plane would give the same answer  of the complex
plane and apply the residue theorem:

$\displaystyle L(\omega)=-2\pi
i\sum_{\text{Im}[y_{j}]>0}\epsilon^{\prime}(y_{j})^{-1},$ (S2)

where the complex numbers $y_{i}$ satisfy
$\epsilon(y_{j})=\sigma|d|y_{i}^{m}=\omega$ and $\text{Im}[y_{j}]>0$. Given
the parametrization of $\omega$ in polar coordinates as
$\omega=\sigma\exp(i\theta)|d|p^{m}$, we have
$y_{j}=\exp(i\theta/m)p\mu^{2j}$, where $\mu=\exp(i\pi/m)$ and
$j\in\\{0,1,\dots,m-1\\}$. Define $A$ as the set of $j$ for which $y_{j}$ is
above the real line. Equation (S2) can then be expressed as

$\displaystyle L(\omega)$ $\displaystyle=-2\pi i\sum_{j\in
A}\frac{1}{m|d|y_{j}^{m-1}},$ (S3) $\displaystyle=-\pi
i\frac{2}{m|d|p^{m-1}}\exp\left(-i\theta\frac{m-1}{m}\right)\sum_{j\in
A}(-\mu)^{2j},$ (S4)

where the set $A$ and the value of $\kappa_{m}\equiv\sum_{j\in A}(-\mu)^{2j}$
are given in Table 1 for both odd and even $m$. Note that the prefactor
$\frac{2}{m|d|p^{m-1}}$ in Eq. (S4) is equal to the density of states
$\rho(|\omega|)=\frac{2}{m|d|^{1/m}}|\omega|^{-1+1/m}$. Hence, we have proved
that $L(\omega)$ is given by Eq. (13) in the main text.

Table 1: The set $A$ and the value of $\kappa_{m}=\sum_{j\in A}(-\mu)^{2j}$ for both odd and even $m$. |  | $A$ | $\kappa_{m}\equiv\sum_{j\in A}(-\mu)^{2j}$
---|---|---|---
Odd $m$ | $\theta\in(0,\pi)$ | $(0,1,2,\dots\frac{m-1}{2})$ | $-\frac{1}{\mu-1}$
$\theta\in(\pi,2\pi)$ | $(0,1,2,\dots\frac{m-3}{2})$ | $-\frac{1}{(\mu-1)\mu}$
Even $m$ | $\theta\in(0,2\pi)$ | $(0,1,2,\dots\frac{m-2}{2})$ | $\frac{2}{1-\mu^{2}}$

##  III Emitter scattering

In this section, we prove that if there exists no bright zero-energy
eigenstate, Eq. (11) in the main text holds for the class of models where
$\ket{v_{k}}=V(k)\ket{u}$, even when $N\geq 2$. Before diving into the proof,
we give the definition of bright zero-energy eigenstates and give a physical
explanation as to why our universality results require their absence.

Due to the multi-component nature of our emitter scattering problems, we find
it necessary to categorize all eigenstates of the Hamiltonian into bright,
dark, and emitter eigenstates. Bright eigenstates have a nonzero photon and
emitter wavefunction, while dark eigenstates have only a nonzero photonic
amplitude, and emitter eigenstates have only a nonzero emitter amplitude. With
this terminology established, we now give an overview of the properties of the
different types of eigenstates at zero energy. The zero-energy emitter states
correspond to the null vectors of $\bm{K}^{R}$ that are orthogonal to
$\ket{v_{k}}=V(k)\ket{u}$. They are decoupled from the photon channel, hence
their existence has no impact on the universal behavior of the $S$-matrix. For
$V(k)$ with nonzero derivatives at $k=0$, there generally exist uncountably
many zero-energy dark states independent of $\bm{K}^{R}$, which are polynomial
functions with degree less than $m-1$. Bright states at zero-energy are fine-
tuned and have a constant photon wavefunction in space. As we show below,
these states come into existence precisely when the universal scattering
behavior fails.

To give a heuristic explanation for why universal scattering at zero energy
fails at these fine-tuned parameters, we consider the classic model of 1D
potential scattering with quadratic dispersion relation ($m=2$), i.e., a 1D
quantum mechanical problem described by the Schrödinger equation

$-\frac{d^{2}\psi(z)}{dz^{2}}+V(z)\psi(z)=E\psi(z),$ (S5)

where we set the mass equal to $1/2$. A particle being scattered off a
generic, short-range potential $V(z)$ would experience a total reflection in
the limit $E\rightarrow 0$, similarly to what happens in our 1D emitter
scattering models. Another feature of these 1D potential scattering problems
is that there exists a fine-tuned, critical regime when the scattering in the
limit $E\rightarrow 0$ becomes total transmission instead of total reflection.
This occurs when there is a zero-energy eigenstate and there is no energy
scale to compare with when the limit $E\rightarrow 0$ is taken. The zero-
energy eigenstate can be understood as the effective “transition state” when a
new bound state emerges or disappears upon the continuous tuning of
parameters.

Similarly, in our emitter scattering models, the universal scattering behavior
that takes place for generic parameters would fail at certain fine-tuned
parameters. An important difference to note is that, unlike in potential
scattering, not all zero-energy eigenstates in emitter scattering are
associated with the critical regime where the universal scattering behavior
fails. For the particular type of interactions $\ket{v_{k}}$ being considered
in this Letter, we discover that the critical regime can be associated with
the existence of a particular type of eigenstates at zero energy, which we
call bright zero-energy states (defined above).

In order to state our goal more explicitly, we rewrite the Hamiltonian given
by Eqs. (1)-(3) in the main text in the single-excitation manifold:

$\displaystyle H^{(1)}$ $\displaystyle=\int_{-\infty}^{+\infty}dk\
\epsilon(k)C^{\dagger}(k)C(k)+\int_{-\infty}^{+\infty}dk\left[C^{\dagger}(k)V^{*}(k)\bra{u}+C(k)V(k)\ket{u}\right]+\bm{K}^{R},$
(S6)

where we have used the matrix representation $\bm{K}^{R}$ to replace
$\sum_{i,j=1}^{N}K^{R}_{ij}b_{i}^{\dagger}b_{j}$ and the vector
$\ket{v_{k}}=V(k)\ket{u}$ to replace the emitter creation operators
$\sum_{i=1}^{N}V_{i}(k)b^{\dagger}_{i}$.

Our goal in this section is to prove the following theorem:

######  Theorem 1.

Suppose $V(k)$ is a locally square-integrable function continuous at $k=0$ and
$V(k=0)\neq 0$. When $k\rightarrow\pm\infty$, $|V(k)|^{2}=o(k^{m-\gamma})$ for
some $\gamma>1$. Consider the class of emitter interactions
$\ket{v_{k}}=V(k)\ket{u}$, where $\ket{u}$ is a unit vector. The single-
particle $T$-matrix given by Eqs. (6)- (8) in the main text reads:

$\displaystyle T(\omega,k,k^{\prime})$
$\displaystyle=V^{*}(k^{\prime})V(k)\langle
u|\frac{1}{\omega\mathbb{1}_{N}-\bm{K}^{R}-\bm{K}(\omega)}|u\rangle,$ (S7)
$\displaystyle\bm{K}(\omega)$ $\displaystyle=|u\rangle\langle
u|K(\omega),\quad
K(\omega)\equiv\int_{-\infty}^{+\infty}dk\frac{|V(k)|^{2}}{\omega-\epsilon(k)}.$
(S8)

When $H^{(1)}$ in Eq. (S6) has no bright zero-energy eigenstates, Eq. (12) in
the main text holds, namely,

$\displaystyle\lim_{\omega\rightarrow
0}L(\omega)T(\omega,k,k^{\prime})=-\frac{V^{*}(k^{\prime})V(k)}{|V(0)|^{2}}.$
(S9)

Note that $T(\omega,k,k^{\prime})$ and $\bm{K}(\omega)$ are defined for
$\omega$ outside the continuum spectrum, hence the limit $\omega\rightarrow 0$
is taken in any direction except from within the continuum spectrum.

###### Proof.

Our proof consists of two lemmas linked by a condition on $\bm{K}^{R}$. The
idea of the proof is that the absence of bright zero-energy eigenstates can be
translated into a condition on $\bm{K}^{R}$, which turns out to be necessary
for the proof of Eq. (S9).

Choose an orthonormal basis
$\\{|u_{1}\rangle,|u_{2}\rangle,\dots|u_{N}\rangle\\}$ for the single-emitter
Hilbert space, where $\ket{u_{1}}\equiv\ket{u}$ is the first vector in this
new basis. The link between the two lemmas is the submatrix
$\bm{K}^{R}_{\cancel{11}}$ constructed from deleting the first row and first
column of $\bm{K}^{R}$; $\bm{K}^{R}_{\cancel{11}}$ can be considered as an
operator on the emitter-excitation subspace
$\\{|u_{2}\rangle,\dots|u_{N}\rangle\\}$ orthogonal to $\ket{u}$. In Lemma 2,
we prove that Eq. (S9) holds if any null vector of $\bm{K}^{R}_{\cancel{11}}$
also corresponds to the null vector of $\bm{K}^{R}$. In Lemma 3, we prove that
the condition Lemma 2 relies on is guaranteed by the absence of bright zero-
energy eigenstates. Combining the two lemmas completes the proof of Theorem 1.

######  Lemma 2.

If any null vector of $\bm{K}^{R}_{\cancel{11}}$ also corresponds to the null
vector of $\bm{K}^{R}$, Eq. (S9) follows.

###### Proof.

Using Eq. (S7), the l.h.s of Eq. (S9) can be written as

$\displaystyle\lim_{\omega\rightarrow 0}L(\omega)T(\omega,k,k^{\prime})$
$\displaystyle=V^{*}(k^{\prime})V(k)\lim_{\omega\rightarrow
0}L(\omega)\bra{u}\bm{H}(\omega)^{-1}\ket{u},$ (S10)

where $\bm{H}(\omega)\equiv\omega\mathbb{1}_{N}-\bm{K}^{R}-\bm{K}(\omega)$.
Hence, our goal, Eq. (S9), is equivalent to

$\displaystyle\lim_{\omega\rightarrow
0}L(\omega)\bra{u}\bm{H}(\omega)^{-1}\ket{u}=-\frac{1}{|V(0)|^{2}}.$ (S11)

In the new basis where $|u_{1}\rangle=\ket{u}$ is the first basis vector,
$\bra{u}\bm{H}(\omega)^{-1}\ket{u}$ is the $(1,1)$ matrix element of the
inverse of $\bm{H}(\omega)$, and can be computed from the $(N-1)\times(N-1)$
submatrix $\bm{H}_{\cancel{11}}(\omega)$ constructed from deleting the first
row and first column of $\bm{H}(\omega)$:

$\displaystyle\bra{u}\bm{H}(\omega)^{-1}\ket{u}=\frac{\det(\bm{H}_{\cancel{11}}(\omega))}{\det(\bm{H}(\omega))}.$
(S12)

Using $\bm{K}(\omega)=|u\rangle\langle u|K(\omega)$, we have

$\displaystyle\det(\bm{H}_{\cancel{11}}(\omega))$
$\displaystyle=\det(\omega\mathbb{1}_{N-1}-\bm{K}^{R}_{\cancel{11}}),$ (S13)
$\displaystyle\det(\bm{H}(\omega))$
$\displaystyle=-K(\omega)\det(\omega\mathbb{1}_{N-1}-\bm{K}^{R}_{\cancel{11}})+\det(\omega\mathbb{1}_{N}-\bm{K}^{R}).$
(S14)

Combining Eqs. (S12), (S13) and (S14), the l.h.s of Eq. (S11) becomes

$\displaystyle\lim_{\omega\rightarrow
0}L(\omega)\bra{u}\bm{H}(\omega)^{-1}\ket{u}$
$\displaystyle=\lim_{\omega\rightarrow
0}L(\omega)\left(-K(\omega)+\frac{\det(\omega\mathbb{1}_{N}-\bm{K}^{R})}{\det(\omega\mathbb{1}_{N-1}-\bm{K}^{R}_{\cancel{11}})}\right)^{-1}.$
(S15)

Let us label the $N$ roots of the characteristic polynomial of $\bm{K}^{R}$ by
$E_{i}$ for $i=1,\dots N$, and the $N-1$ roots of the characteristic
polynomial of $\bm{K}_{\cancel{11}}^{R}$ by $\bar{E}_{i}$ for $i=1,\dots N-1$.
$E_{i}$ and $\bar{E}_{i}$ correspond to the eigenvalues of $\bm{K}^{R}$ and
$\bm{K}_{\cancel{11}}^{R}$, respectively, where any eigenvalue with
multiplicity $n\geq 2$ is assigned to $n$ different indices. We have

$\displaystyle\lim_{\omega\rightarrow
0}\frac{\det(\omega\mathbb{1}_{N}-\bm{K}^{R})}{\det(\omega\mathbb{1}_{N-1}-\bm{K}^{R}_{\cancel{11}})}=\lim_{\omega\rightarrow
0}\frac{\prod_{i=1}^{N}(\omega-
E_{i})}{\prod_{i=1}^{N-1}(\omega-\bar{E}_{i})}.$ (S16)

Since any null vector of $\bm{K}^{R}_{\cancel{11}}$ corresponds to a null
vector of $\bm{K}^{R}$ by the assumption of the Lemma, if $\bm{K}^{R}$ has
null vectors, its zero-eigenvalue multiplicity must be greater or equal to
that of $\bm{K}_{\cancel{11}}$. Hence, the limit in Eq. (S16) is finite.

In the main text, we have introduced the identity $\lim_{\omega\rightarrow
0}L^{-1}(\omega)\frac{1}{\omega-\epsilon(k)}=\delta(k)$; hence
$\lim_{\omega\rightarrow 0}L^{-1}(\omega)K(\omega)=|V(0)|^{2}\neq 0$ and Eq.
(S15) leads to Eq. (S11). The proof of Lemma 2 is complete.

∎

If we can prove that the absence of bright zero-energy eigenstates of Eq. (S6)
guarantees that any null vector of $\bm{K}^{R}_{\cancel{11}}$ also corresponds
to the null vector of $\bm{K}^{R}$, Eq. (S9) would immediately follow from
Lemma 2. To do this, we prove the contrapositive statement in the following
lemma:

######  Lemma 3.

When there exists a vector $\ket{e_{0}}=\sum_{i=2}^{N}e_{i}\ket{u_{i}}$
orthogonal to $\ket{u}$, such that $\bm{K}^{R}\ket{e_{0}}\neq 0$ and
$\bm{K}_{\cancel{11}}^{R}\ket{e_{0}}=0$, then there exists a bright zero-
energy eigenstate of the Hamiltonian in Eq. (S6).

###### Proof.

We plan to write down an ansatz with a nonzero photon and emitter wavefunction
and verify that it is a zero-energy eigenstate of the Hamiltonian in Eq. (S6).
The ansatz we propose is the following:

$\displaystyle\ket{\psi_{0}}$ $\displaystyle=\int_{-\infty}^{+\infty}dz\
\psi_{0}(z)C^{\dagger}(z)\ket{0,g}+\ket{e_{0}},$ (S17)
$\displaystyle\psi_{0}(z)$
$\displaystyle=-V(0)^{-1}\braket{u}{\bm{K}^{R}}{e_{0}},$ (S18)

where the photon wavefunction $\psi_{0}(z)$ in the coordinate space is a
constant function. By definition, $\ket{e_{0}}$ is orthogonal to $\ket{u}$.
Because $\bm{K}^{R}\ket{e_{0}}\neq 0$ and
$\bm{K}_{\cancel{11}}^{R}\ket{e_{0}}=0$, $\bm{K}^{R}\ket{e_{0}}$ is a nonzero
vector proportional to $\ket{u}$. Hence, $\psi_{0}(z)\neq 0$.

Our goal is to prove that the ansatz given by Eqs. (S17) and (S18) is the
zero-energy eigenstate of the Hamiltonian in Eq. (S6). Applying $H^{(1)}$ in
Eq. (S6) to the Fourier transform of the ansatz in Eq. (S17), we get

$\displaystyle H^{(1)}\ket{\psi_{0}}$
$\displaystyle=\int_{-\infty}^{+\infty}dk\
\left[\epsilon(k)\psi_{0}(k)C^{\dagger}(k)\ket{0,g}+V^{*}(k)\braket{u}{e_{0}}C^{\dagger}(k)\ket{0,g}+\psi_{0}(k)V(k)\ket{u}\right]+\bm{K}^{R}\ket{e_{0}},$
(S19)

where the momentum-space photon wavefunction
$\psi_{0}(k)=-V(0)^{-1}\braket{u}{\bm{K}^{R}}{e_{0}}\delta(k)$ is the Fourier
transform of Eq. (S18). Since the dispersion relation satisfies
$\epsilon(0)=0$, the first term on the r.h.s of Eq. (S19) is zero:
$\int_{-\infty}^{+\infty}dk\ \epsilon(k)\psi_{0}(k)C^{\dagger}(k)\ket{0,g}=0$.
Because $\braket{u}{e_{0}}=0$, the second term on the r.h.s of Eq. (S19) is
also equal to $0$. The third term

$\displaystyle\int_{-\infty}^{+\infty}dk\
\psi_{0}(k)V(k)\ket{u}=-\ket{u}\bra{u}\bm{K}^{R}\ket{e_{0}}$ (S20)

cancels with the fourth term $\bm{K}^{R}\ket{e_{0}}$ on the r.h.s of Eq. (S19)
because $\bm{K}^{R}\ket{e_{0}}$ is proportional to $\ket{u}$. Therefore,
$H^{(1)}\ket{\psi_{0}}=0$, and this is the end of the proof for Lemma 3.

∎

Combining Lemmas 2 and 3, we can obtain Theorem 1.

∎

##  IV Levinson’s theorem

In this section, we prove Levinson’s theorem for the class of emitter
scattering models with $\ket{v_{k}}=V(k)\ket{u}$, i.e. Eq. (18) in the main
text. Let us restate the objective of our proof in the following theorem:

######  Theorem 4.

Denote the continuum spectrum by $\mathcal{R}_{c}$. We assume that there are
no bound states in the continuum or bright zero-energy eigenstates in the
system. For dissipative systems, we assume that $\det[\bm{S}(E)]\neq 0$ for
$E\in\mathcal{R}_{c}$. The winding phase $\Delta\delta$ of $\det[\bm{S}(E)]$
around the origin is defined as

$\displaystyle 2\Delta\delta$
$\displaystyle=-i\int_{\mathcal{R}_{c}}dE\frac{\partial_{E}\det[\bm{S}(E)]}{\det[\bm{S}(E)]}.$
(S21)

Suppose $\ket{v_{k}}=V(k)\ket{u}$ satisfies the properties listed in Theorem 1
and the dispersion relation is given by $\epsilon(k)=\sigma|d|k^{m}$, where
$\sigma=\pm 1$ and $m\geq 2$ is an integer. We have

$\displaystyle\Delta\delta=\pi(N-N_{B})+\pi\frac{m-1}{m},$ (S22)

where $N$ is the number of emitters and $N_{B}$ is the number of bound states.

The main idea of the proof is to define an analytic continuation of
$\det[\bm{S}(E)]$ to the complex plane and observe the fact that the bound
state energies are the poles of this function. The proof is similar to our
previous work [53], where we proved Levinson’s theorem for photon-emitter
models with linear dispersion relations.

In preparation for the proof of Theorem 4, we introduce Theorem 5, where we
propose an analytic function $s(\omega)$ that is equal to the analytic
continuation of $\det[\bm{S}(E)]$ to the complex plane. Though introduced here
as a tool for proving Theorem 4, Theorem 5 provides a quick method to compute
$\det[\bm{S}(E)]$ using $\bm{K}^{R}$ and $\bm{K}(E+i0^{\pm})$ and is an
important theorem itself. We comment that the range of application of Theorem
5 is well beyond the class of photon-emitter models discussed in this letter:
it can be applied to general photon-emitter interactions $\ket{v_{k}}$ and
other dispersion relations beyond $\epsilon(k)=\pm|d|k^{m}$.

######  Theorem 5.

Define $J(\omega)=\det[\omega\mathbb{1}_{N}-\bm{K}^{R}-\bm{K}(\omega)]$ as a
function on the complement of the continuum spectrum $\mathcal{R}_{c}$ in the
complex plane. For the values of $\omega$ s.t. $J(\omega)\neq 0$, we can
define $s(\omega)=\frac{J(\omega^{*})}{J(\omega)}$. When $E$ is not equal to
the energy of a bound state in the continuum,

$\displaystyle s(E+i0^{+})=\det[\bm{S}(E)].$ (S23)

We comment that the bound state energies $E_{B}$ correspond to the poles of
the emitter propagator
$\bm{G}(\omega)=[\omega\mathbb{1}_{N}-\bm{K}^{R}-\bm{K}(\omega)]^{-1}$, hence
they satisfy $J(E_{B})=0$.

###### Proof.

Let $n(E)$ denote the momentum degeneracy at energy $E$ and $k_{1},\dots
k_{n(E)}$ the degenerate momenta at energy $E$. When $\epsilon(k)=\pm|d|k^{m}$
and $E\in\mathcal{R}_{c}$, $n(E)=1$ for odd $m$ and $n(E)=2$ for even $m$.
According to Eqs. (5)-(8) in the main text, the $S$-matrix $\bm{S}(E)$ is a
$n(E)\times n(E)$ matrix, whose matrix elements are given by

$\displaystyle S_{\alpha\beta}(E)=\delta_{\alpha\beta}-\frac{2\pi
i}{\sqrt{|\epsilon^{\prime}(k_{\alpha}(E))\epsilon^{\prime}(k_{\beta}(E))|}}\braket{v_{k_{\beta}(E)}}{\bm{G}(E+i0^{+})}{v_{k_{\alpha}(E)}},$
(S24)

where $\alpha,\beta\in(1,2,\dots,n(E))$.

Note that in writing down Eq. (S24), we have implicitly assumed that the limit
$\bm{G}(E+i0^{+})\equiv\lim_{\eta\rightarrow 0^{+}}\bm{G}(E+i\eta)$ exists.
However, if $E_{c}\mathbb{1}_{N}-\bm{K}^{R}-\bm{K}(E_{c}+i0^{+})$ has a zero
eigenvalue for some energy $E_{c}\in\mathcal{R}_{c}$, $\bm{G}(E_{c}+i\eta)$
does not have a limit when $\eta\rightarrow 0^{+}$ and $E_{c}$ corresponds to
the energy of a bound state in the continuum. This is why the theorem only
applies to $E\neq E_{c}$.

Construct $\bm{A}$ as a $N\times n(E)$ matrix and $\bm{A}^{\dagger}$ its
Hermitian conjugate:

$\displaystyle\bm{A}$
$\displaystyle=\left(\begin{matrix}\frac{1}{\sqrt{{|\epsilon^{\prime}(k_{1})|}}}|v_{k_{1}}\rangle,\dots\frac{1}{\sqrt{{|\epsilon^{\prime}(k_{n(E)})|}}}|v_{k_{n(E)}}\rangle\end{matrix}\right),\quad$
(S25)

then the $n(E)\times n(E)$ matrix $\bm{S}(E)$ for $E\neq E_{c}$ can be written
as

$\displaystyle\bm{S}(E)=\mathbb{1}_{n(E)}-2\pi
i\bm{A}^{\dagger}\bm{G}(E+i0^{+})\bm{A},$ (S26)

where $\mathbb{1}_{n(E)}$ is an identity matrix of dimension $n(E)$.

Using the definitions of $s(\omega),J(\omega)$ and the properties of
determinant, we have,

$\displaystyle s(E+i0^{+})$
$\displaystyle=\det(\mathbb{1}_{N}+(\bm{K}(E+i0^{+})-\bm{K}(E+i0^{-}))\bm{G}(E+i0^{+})),$
(S27)

where $\bm{K}(E+i0^{+})-\bm{K}(E+i0^{-})$ can be re-written as

$\displaystyle\bm{K}(E+i0^{+})-\bm{K}(E+i0^{-})$
$\displaystyle=\int_{-\infty}^{+\infty}dk\ |v_{k}\rangle\langle
v_{k}|\left(\frac{1}{E+i0^{+}-\epsilon(k)}-\frac{1}{E-i0^{-}-\epsilon(k)}\right),$
$\displaystyle=-\int_{-\infty}^{+\infty}dk\ |v_{k}\rangle\langle v_{k}|2\pi
i\delta(E-\epsilon(k)),$ $\displaystyle=-2\pi
i\sum_{\alpha=1}^{n(E)}\frac{1}{|\epsilon^{\prime}(k_{\alpha})|}|v_{k_{\alpha}}\rangle\langle
v_{k_{\alpha}}|=-2\pi i\bm{A}\bm{A}^{\dagger}.$ (S28)

Inserting Eq. (S28) into Eq. (S27), we get

$\displaystyle s(E+i0^{+})=\det\left[\mathbb{1}_{N}-2\pi
i\bm{A}\bm{A}^{\dagger}\bm{G}(E+i0^{+})\right].$ (S29)

According to a standard result in linear algebra known as the extension of the
matrix determinant lemma, given an invertible $N\times N$ matrix $-2\pi
i\bm{G}(E+i0)$ and a $N\times n(E)$ matrix $\bm{A}$,

$\displaystyle\det[\mathbb{1}_{N}-2\pi
i\bm{A}\bm{A}^{\dagger}\bm{G}(E+i0^{+})]=\det[\mathbb{1}_{n(E)}-2\pi
i\bm{A}^{\dagger}\bm{G}(E+i0^{+})\bm{A}].$ (S30)

Using Eqs. (S26) and (S29), we see that the l.h.s and r.h.s of Eq. (S30) are
equal to $s(E+i0^{+})$ and $\det[\bm{S}(E)]$, respectively. This is the end of
the proof for Theorem 5.

∎

We proceed to prove Theorem 4 with the help of Theorem 5.

###### Proof.

(Theorem 4)

First consider $\epsilon(k)=|d|k^{m}$ with odd $m$, in which case the
continuum spectrum is $\mathcal{R}_{c}=(-\infty,0)\cup(0,+\infty)$. Since we
have assumed that there is no bound state in the continuum, using Theorem 5,
we can replace $\det[\bm{S}(E)]$ by $\frac{J(E+i0^{-})}{J(E+i0^{+})}$ for
$E\in\mathcal{R}_{c}$ and rewrite Eq. (S21) in terms of a contour integral in
the complex plane:

$\displaystyle 2\Delta\delta$
$\displaystyle=-i\int_{\mathcal{R}_{c}}dE\bigg{[}\frac{\partial_{E}J(E+i0^{-})}{J(E+i0^{-})}-\frac{\partial_{E}J(E+i0^{+})}{J(E+i0^{+})}\bigg{]}=-i\int_{\mathcal{R}_{1}^{+}+\mathcal{R}_{1}^{-}+\mathcal{R}_{2}^{+}+\mathcal{R}_{2}^{-}}d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)},$
(S31)

where $E$ is a real coordinate, $\omega$ is a complex coordinate, and the
integration contours $\mathcal{R}_{1}^{\pm}$ and $\mathcal{R}_{2}^{\pm}$ are
illustrated by the dashed lines in Fig. S1a.

(a) $\epsilon(k)=|d|k^{m}$, odd $m$

(b) $\epsilon(k)=|d|k^{m}$, even $m$

Figure S1: Illustration of the integration contours for the calculation of the
winding phase of $\det[\bm{S}(E)]$. (a) Contours for a dispersion relation
$\epsilon(k)=|d|k^{m}$ with odd $m$. (b) Contours for a dispersion relation
$\epsilon(k)=|d|k^{m}$ with even $m$. The density of states diverges at the
origin $E=0$, marked by the red dot. The black lines represent the continuum
spectrum, while the yellow stars represent bound-state energies. The dashed
lines with arrows are the integration paths for the evaluation of the winding
number of $\det[\bm{S}(E)]$. The semicircles (circles) are added to form
closed integration contours so that the residue theorem can be invoked. The
red semicircles (circle) go around the origin with an infinitesimal radius,
while the black semicircles (circle) have an infinite radius.

$\mathcal{R}_{2}^{\pm}$ and $\mathcal{R}_{1}^{\pm}$ represent the contours
just above/below the real line for $E<0$ and $E>0$, respectively. We can
obtain two closed integration contours by adding a pair of semicircles
$\mathcal{C}^{\pm}$ with an infinitesimal radius around $0$ and a pair of
semicircles $\mathcal{S}^{\pm}$ with radius $|\omega|\rightarrow\infty$.
Equation (S31) can then be rewritten as

$\displaystyle 2\Delta\delta$ $\displaystyle=-i\oint
d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)}+i\int_{\mathcal{S}^{+}+\mathcal{S}^{-}}d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)}+i\int_{\mathcal{C}^{+}+\mathcal{C}^{-}}d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)},$
(S32)

where $\oint$ represents the sum of integrals over the two closed contours.
For odd $m$, $J(\omega)$ is analytic in the complement of the real line in the
complex plane. The poles of $J^{-1}(\omega)$ correspond to the bound state
energies; they can only be located below the real line given our assumption
that there is no bound state in the continuum. This also implies that when
$\bm{K}^{R}$ is Hermitian, $N_{B}=0$. Applying the residue theorem, the closed
contours in the upper and lower half planes yield $0$ and $-2\pi iN_{B}$,
respectively. Hence,

$\displaystyle-i\oint
d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)}=-2\pi N_{B}.$ (S33)

Next, we evaluate the integrals along the small semicircles. $J(\omega)$ is
equal to $\det[\bm{H}(\omega)]$ in Eq. (S14), which shows that $J(\omega)\sim
K(\omega)\sim L(\omega)$ when $\omega\rightarrow 0$. Intuitively, the winding
phases of $J(\omega)$ along $\mathcal{C}^{\pm}$ are equal to the winding
phases of $L(\omega)$ along $\mathcal{C}^{\pm}$, which contribute to the term
$\pi\frac{m-1}{m}$ in Eq. (S22). To demonstrate it rigorously, we write
$J(\omega)$ as the product of $L(\omega)$ and another function $g(\omega)$:

$\displaystyle J(\omega)$ $\displaystyle=L(\omega)g(\omega),$ $\displaystyle
g(\omega)$
$\displaystyle\equiv-L^{-1}(\omega)K(\omega)\det(\omega\mathbb{1}_{N-1}-\bm{K}^{R}_{\cancel{11}})+L^{-1}(\omega)\det(\omega\mathbb{1}_{N}-\bm{K}^{R}).$
(S34)

This way the winding phases of $J(\omega)$ along $\mathcal{C}^{\pm}$ can be
evaluated as the sum of the winding phases of $g(\omega)$ and $L(\omega)$:

$\displaystyle
i\int_{\mathcal{C}^{\pm}}d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)}$
$\displaystyle=i\int_{\mathcal{C}^{\pm}}d\omega\frac{\partial_{\omega}L(\omega)}{L(\omega)}+i\int_{\mathcal{C}^{\pm}}d\omega\frac{\partial_{\omega}g(\omega)}{g(\omega)}.$
(S35)

Using Eqs. (S4), the winding number of $L(\omega)$ can be evaluated explicitly
in polar coordinates:

$\displaystyle
i\int_{\mathcal{C}^{+}}d\omega\frac{\partial_{\omega}L(\omega)}{L(\omega)}$
$\displaystyle=i\lim_{r\rightarrow 0}\int_{0^{+}}^{\pi^{-}}d\theta\
\frac{\partial_{\theta}L(r,\theta)}{L(r,\theta)},$
$\displaystyle=i\lim_{r\rightarrow 0}\int_{0^{+}}^{\pi^{-}}d\theta\
-i(m-1)/m,$ $\displaystyle=\pi(m-1)/m,$ (S36)

where $\int_{0^{+}}^{\pi^{-}}d\theta\equiv\lim_{\theta_{1}\rightarrow
0^{+},\theta_{2}\rightarrow\pi^{-}}\int_{\theta_{1}}^{\theta_{2}}d\theta$. The
integral along $\mathcal{C}^{-}$ can be evaluated similarly; and it has the
same value as the integral along $\mathcal{C}^{+}$.

Next, we argue that the winding phases of $g(\omega)$ along
$\mathcal{C}^{\pm}$ are equal to $0$. Note that the contour
$\mathcal{C}^{+}/\mathcal{C}^{-}$ is defined through two limiting processes
taken consecutively on an arc centered at the origin of the complex plane. In
the first limit, we fix the radius of the arc and send both endpoints of the
arc to infinitesimal distances above/below the real line, so the arc almost
becomes a semicircle. In the second limit, the radius of the arc is sent to
$0$. Because of this, we need to first examine $g(E+i\eta)$ when
$\eta\rightarrow 0^{\pm}$, and then send $E\rightarrow 0$.

Using Eq. (S34), we see that $g(\omega)$ is an analytic function in the
complement of the real line on the complex plane for odd $m$. Since
$\lim_{\eta\rightarrow 0^{\pm}}L^{-1}(E+i\eta)K(E+i\eta)$ and
$\lim_{\eta\rightarrow 0^{\pm}}L^{-1}(E+i\eta)$ exist for $E$ anywhere on the
real line $\mathcal{R}$, $\lim_{\eta\rightarrow 0^{\pm}}g(E+i\eta)\equiv
g(E+i0^{\pm})$ exist for $E\in\mathcal{R}$. Furthermore, since
$\lim_{\omega\rightarrow 0}L^{-1}(\omega)K(\omega)=|V(0)|^{2}$,
$g(E+i0^{\pm})$ as functions of $E\in\mathcal{R}$ are continuous at $E=0$.

The winding phase of $g(\omega)$ along $\mathcal{C}^{+}$ is equal to the phase
difference between $g(-|E|+i0^{+})$ and $g(|E|+i0^{+})$ in the limit
$E\rightarrow 0$. Similarly, the winding phase of $g(\omega)$ along
$\mathcal{C}^{-}$ is equal to the phase difference between $g(|E|+i0^{-})$ and
$g(-|E|+i0^{-})$ in the limit $E\rightarrow 0$. Because of the continuity of
$g(E+i0^{\pm})$ at $E=0$,
$i\int_{\mathcal{C}^{+}+\mathcal{C}^{-}}d\omega\frac{\partial_{\omega}g(\omega)}{g(\omega)}=0$.
Therefore,

$\displaystyle
i\int_{\mathcal{C}^{+}+\mathcal{C}^{-}}d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)}=i\int_{\mathcal{C}^{+}+\mathcal{C}^{-}}d\omega\frac{\partial_{\omega}L(\omega)}{L(\omega)}=2\pi(m-1)/m.$
(S37)

At last, we evaluate the integral along the large semicircles, which can be
written in polar coordinates as

$\displaystyle
i\int_{\mathcal{S}^{+}+\mathcal{S}^{-}}d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)}=i\lim_{r\rightarrow\infty}\left(\int_{\pi^{-}}^{0^{+}}d\theta\frac{\partial_{\theta}J(r,\theta)}{J(r,\theta)}+\int_{2\pi^{-}}^{\pi^{+}}d\theta\frac{\partial_{\theta}J(r,\theta)}{J(r,\theta)}\right).$
(S38)

Using Eq. (S14), $J(\omega)$ can be written as

$\displaystyle J(\omega)=-K(\omega)P_{N-1}(\omega)+P_{N}(\omega),$ (S39)

where
$P_{N-1}(\omega)\equiv\det(\omega\mathbb{1}_{N-1}-\bm{K}^{R}_{\cancel{11}})$
and $P_{N}(\omega)\equiv\det(\omega\mathbb{1}_{N}-\bm{K}^{R})$ are polynomial
functions of $\omega$ with degrees $N-1$ and $N$, respectively. From the
definition of $K(\omega)$ in Eq. (S8), we can see that
$\lim_{|\omega|\rightarrow\infty}K(\omega)=0$, hence $J(\omega)\sim
P_{N}(\omega)\sim\omega^{N}$ when $|\omega|\rightarrow\infty$; and we expect
that the sum of the winding phases of $J(\omega)$ around the large semicircles
is equal to $2\pi N$. In the following, we provide a careful mathematical
analysis to verify this intuitive result.

Taking the derivative of Eq. (S39) w.r.t $\theta$, we have

$\displaystyle\partial_{\theta}J(r,\theta)=i\omega\partial_{\omega}J(\omega)=-i\omega\partial_{\omega}K(\omega)P_{N-1}(\omega)-i\omega
K(\omega)\partial_{\omega}P_{N-1}(\omega)+i\omega\partial_{\omega}P_{N}(\omega).$
(S40)

We can observe from Eq. (S8) that
$\lim_{|\omega|\rightarrow\infty}\partial_{\omega}K(\omega)=\lim_{|\omega|\rightarrow\infty}K(\omega)=0$.
In addition,
$\lim_{|\omega|\rightarrow\infty}\frac{P_{N-1}(\omega)}{P_{N}(\omega)}=\lim_{|\omega|\rightarrow\infty}\frac{\partial_{\omega}P_{N-1}(\omega)}{P_{N}(\omega)}=0$,
hence

$\displaystyle\lim_{r\rightarrow\infty}\frac{\partial_{\theta}J(r,\theta)}{J(r,\theta)}=\lim_{|\omega|\rightarrow\infty}\frac{i\omega\partial_{\omega}P_{N}(\omega)}{P_{N}(\omega)}=iN$
(S41)

uniformly in $\theta\in(0,\pi)\cup(\pi,2\pi)$. Applying the dominated
convergence theorem, we can evaluate the $r\rightarrow\infty$ limit of the
following definite integral as a function of the integration end points
$\theta_{1},\theta_{2}\in(0,\pi)$ (or $\theta_{1},\theta_{2}\in(\pi,2\pi)$):

$\displaystyle
i\lim_{r\rightarrow\infty}\int_{\theta_{1}}^{\theta_{2}}d\theta\
\frac{\partial_{\theta}J(r,\theta)}{J(r,\theta)}=(\theta_{1}-\theta_{2})N.$
(S42)

The limit in Eq. (S42) is uniform in $\theta_{1},\theta_{2}$ because the limit
in Eq. (S41) is uniform in $\theta$. This implies that, when we evaluate Eq.
(S38), we can exchange the limit in $r$ and the limits in the integration end
points:

$\displaystyle
i\int_{\mathcal{S}^{+}}d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)}$
$\displaystyle\equiv i\lim_{r\rightarrow
0}\lim_{\theta_{1}\rightarrow\pi^{-}}\lim_{\theta_{2}\rightarrow
0^{+}}\int_{\theta_{1}}^{\theta_{2}}d\theta\
\frac{\partial_{\theta}J(r,\theta)}{J(r,\theta)}$
$\displaystyle=i\lim_{\theta_{1}\rightarrow\pi^{-}}\lim_{\theta_{2}\rightarrow
0^{+}}\lim_{r\rightarrow 0}\int_{\theta_{1}}^{\theta_{2}}d\theta\
\frac{\partial_{\theta}J(r,\theta)}{J(r,\theta)}=\pi N,$ (S43)

where we have used Eq. (S42) in evaluating Eq. (S43). The integration along
$\mathcal{S}^{-}$ can be evaluated similarly, and it has the same value as Eq.
(S43); hence we get

$\displaystyle
i\int_{\mathcal{S}^{+}+\mathcal{S}^{-}}d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)}=2\pi
N.$ (S44)

Combining Eqs. (S32), (S33), (S37) and (S44), we obtain Eq. (S22) for the
dispersion relation $\epsilon(k)=|d|k^{m}$ with odd $m$. The case of
$\epsilon(k)=-|d|k^{m}$ with odd $m$ can be proved identically once we replace
$E$ with $-E$.

Next, we discuss the case of $\epsilon(k)=|d|k^{m}$ with even $m$. Similarly
as in the case of odd $m$, the winding phase of $\det[\bm{S}(E)]$ can be
evaluated as

$\displaystyle 2\Delta\delta$
$\displaystyle=\int_{0^{+}}^{\infty}dE\bigg{[}\frac{\partial_{\omega}J(E-i0)}{J(E-i0)}-\frac{\partial_{\omega}J(E+i0)}{J(E+i0)}\bigg{]}=\int_{\mathcal{R}^{+}+\mathcal{R}^{-}}d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)}$
$\displaystyle=\oint
d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)}-\int_{\mathcal{S}}d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)}-\int_{\mathcal{C}}d\omega\frac{\partial_{\omega}J(\omega)}{J(\omega)},$
(S45)

where the integration contours are illustrated in Fig. S1b.
$\mathcal{R}^{\pm}$ represent the contours just above and below the real line
along the continuum spectrum. $\oint$ represents the integration over the
closed contour. Following a procedure similar to the case of odd $m$, it is
easy to show that the result of this integral is also given by Eq. (S22). The
case of $\epsilon(k)=-|d|k^{m}$ with even $m$ can be proved similarly.

This is the end of the proof for Theorem 4. ∎

##  V Generalization to spatial dimension $D$ and non-integer $m$

### Angular momentum eigenstates in $D$ dimension

In the main text and Sec. VI of the Supplement, we have focused on 1D systems
with dispersion relations $\epsilon(k)=\pm|d|k^{m}$, where $m$ is a positive
integer. To demonstrate the generality of the principle that divergent density
of states leads to a nontrivial universal limit of the $S$-matrix, we extend
the discussion to all dimensions $D\geq 1$ and non-integer values of $m$. Let
$\bm{k}$ denote the momentum vector in integer spatial dimension $D\geq 1$.
For simplicity, we assume a dispersion relation with rotational symmetry:
$\epsilon(k)=|\bm{k}|^{a}$, where $a>0$ does not have to be an integer. These
dispersion relations are natural extensions of the even integer $m$ case of
$\epsilon(k)=\sigma|d|k^{m}$ in the one-dimensional models. The odd integer
extensions of this analytic dispersion relation do not have natural analogs
for $D>1$. The density of states is defined as

$\displaystyle\rho(E)$ $\displaystyle=\int d^{D}k\
\delta(E-|\bm{k}|^{a})=b(D)\int_{0}^{+\infty}dk\
k^{D-1}\delta(E-k^{a})=b(D)k^{D-1}\epsilon^{\prime}(k)^{-1},$ (S46)

where the constant $b(D)=\frac{2\pi^{D/2}}{\Gamma(D/2)}$ comes from the
integration over the solid angle of a $(D-1)$-sphere. $\Gamma(z)$ is the gamma
function. Evaluating Eq. (S46), we have

$\displaystyle\rho(E)=\frac{b(D)}{a}E^{-1+\zeta^{-1}}=\frac{b(D)}{a}k^{D-a},\quad\zeta\equiv\frac{a}{D}.$
(S47)

where $k\equiv E^{1/a}$. When $D=1$, we have $b(D)=2$ and $\zeta=a$, so that
Eq. (S47) agrees with the value of $\rho(E)$ in the main text for $a=m$ for
positive even integer $m$. For general values of $D$, $\rho(E)$ diverges when
$m>D$; we will show that the $S$-matrix goes to a nontrivial limit dependent
on $a/D$ at zero energy. $\rho(E)$ has a finite limit at $E=0$ when $a\leq D$;
we will show that the $S$-matrix goes to the identity matrix at zero energy.

Let us first study the $S$-matrix in $D$ dimensions. The momentum-space
representation of the scattering operator is given by

$\displaystyle\mathcal{S}(\bm{k},\bm{k^{\prime}})$
$\displaystyle=\delta(\bm{k}\\!-\\!\bm{k}^{\prime})-2\pi
i\delta[\epsilon(\bm{k})\\!-\\!\epsilon(\bm{k^{\prime}})]T(E\\!+\\!i0^{+},\bm{k},\bm{k^{\prime}}),$
(S48)

where $T(E\\!+\\!i0^{+},\bm{k},\bm{k^{\prime}})$ is the momentum-space
representation of the $T$-operator $\mathcal{T}(\omega)$ which we specify
later. Energy is preserved in the scattering process, hence we can define an
operator $\bm{S}(E)$ that describes the scattering process at energy $E$. In
1D, the momentum degeneracy is $2$ at any energy; $\bm{S}(E)$ is a $2\times 2$
matrix the same as in the case of positive even integer $m$ discussed in the
main text. In 3D and higher dimensions, there are uncountably many momentum
eigenstates at the same energy $E>0$; $\bm{S}(E)$ is an integral operator in
the momentum basis instead of a discrete, finite matrix.

In the familiar cases of quadratic dispersion relation $a=2$ and $D=2,3$, it
is a common practice to choose the common eigenstates of the angular momentum
operator and the kinetic energy operator as the basis states for the
representation of the $S$-matrix. For example, when $D=3$, the angular
momentum eigenbasis can be labelled by two integers, $l$ and $m_{l}$, where
$l=0,1,2,\dots$ is called the angular momentum quantum number and
$m_{l}=-l,-l+1,\dots,l$ the magnetic quantum number. In the angular momentum
eigenbasis, the scattering operator at energy $E$ can be represented as a
matrix, describing the transmission coefficients between different angular
momentum eigenstates at energy $E$. Note that the dispersion relation
$|\bm{k}|^{a}$ shares the same eigenbasis as the quadratic dispersion relation
$|\bm{k}|^{2}$, hence we can use the same angular momentum eigenbasis for the
representation of the $S$-matrix. In the following, we give an overview of the
angular momentum eigenstates in arbitrary dimensions.

In $D\geq 2$ dimensional space with Cartesian coordinates $\\{z_{1},\dots
z_{D}\\}$, we can introduce generalized polar coordinates
$\\{r,\theta_{1},\dots\theta_{D-1}\\}$ such that $r=\sum_{i=1}^{D}z_{i}^{2}$
is the radial distance to the origin of the coordinate frame. The set
$\bm{\theta}=\\{\theta_{1},\dots\theta_{D-1}\\}$ specifies coordinates on the
surface of a $(D-1)$-sphere [60, 61]. The $D$-dimensional total orbital
angular-momentum operator is given by $L^{2}\equiv-\nabla_{\bm{\theta}}^{2}$,
where $\nabla_{\bm{\theta}}^{2}$ is the Laplacian operator on the unit
$(D-1)$-sphere—a partial differential operator defined purely in terms of
$\bm{\theta}$. The eigenvalues and eigenvectors of $L^{2}$ are given by

$\displaystyle
L^{2}Y_{l,q_{l}}(\bm{\theta})=l(l+D-2)Y_{l,q_{l}}(\bm{\theta}),$ (S49)

where $l=0,1,2,\dots$ is the generalization of the angular momentum quantum
number to $D$ dimensions and $q_{l}=1,2,\dots N_{l}$ labels the degenerate
eigenstates. $Y_{l,q_{l}}(\bm{\theta})$ is the generalization of spherical
harmonics to $D$ dimensions [62]. When $l=0$, $N_{l}=1$, i.e. the eigenstate
is non-degenerate. When $l\geq 1$, $N_{l}=\frac{D+2l-2}{l}C^{D+l-3}_{l-1}$.
For example, when $D=2$, $N_{l}=2$ for $l\geq 1$;
$Y_{l,q_{l}=1}(\theta)=(2\pi)^{-1/2}\exp(il\theta)$ and
$Y_{l,q_{l}=2}(\theta)=(2\pi)^{-1/2}\exp(-il\theta)$, where
$\theta=\arctan(z_{2}/z_{1})$. When $D=3$, $N_{l}=2l+1$ for $l\geq 1$; $q_{l}$
has a one-to-one correspondence with the magnetic quantum number
$m_{l}=-l,-l+1,\dots l$.

The orthogonality relations of the spherical harmonics are given by

$\displaystyle\int d\Omega\
Y^{*}_{l,q_{l}}(\bm{\theta})Y_{n,q_{n}}(\bm{\theta})=\delta_{nl}\delta_{q_{l},q_{n}},$
(S50)

where $\int d\Omega$ is the integration over the solid angle of the
$(D-1)$-sphere.

In scattering theory with $D\geq 2$, states with $l=0,1,2\dots$ are often
referred to as s-waves, p-waves, d-waves…, respectively. When $D=1$, the
dispersion relation is symmetric about $k=0$; the s-wave and p-wave refer to
the symmetric and antisymmetric combinations of the degenerate momentum
eigenstates at a given energy, respectively. In the main-text discussion of 1D
systems, we have shown that the scattering of the s-wave is decoupled from the
p-wave when $E\rightarrow 0$; the s-wave transmission coefficient has a
nontrivial limit $\exp(i\pi/a)$, while the p-wave transmission coefficient is
$1$. The goal of this section is to generalize the zero-energy scattering
behavior in 1D to higher dimensions. Specifically, in systems with
nonvanishing interactions at zero energy, s-wave scattering is decoupled from
all other channels in the zero-energy limit in any dimension; the s-wave
transmission coefficient goes to a universal limit $\exp(2\pi iD/a)$ when
$a>D$, while the scattering in other channels $l\geq 1$ goes to a trivial
limit—the identity matrix.

The different zero-energy behaviors for $l=0$ and $l\geq 1$ are due to the
different behaviors of the radial wavefunctions $R_{l}(r)\sim(kr)^{l}$ at
small $r$. The main idea is that, when $E\rightarrow 0$, $R_{l}(r)$ goes to a
constant at any finite $r$ for $l=0$ and vanishes for $l\geq 1$. Therefore,
the s-wave experiences interactions at zero energy while the higher channels
do not see the interactions. To substantiate the argument, let us compute
$R_{l}(r)$ below.

The eigenvalue equation for the kinetic energy operator at positive energy $E$
is given by

$\displaystyle\nabla^{2}\phi(\bm{r})=k^{2}\phi(\bm{r}),$ (S51)

where $k\equiv E^{1/a}$ and $\nabla^{2}$ is the Laplacian operator in $D$
dimension. In the spherical coordinate system $(r,\bm{\theta})$, we have

$\displaystyle\nabla^{2}=r^{1-D}\frac{\partial}{\partial
r}\left(r^{D-1}\frac{\partial}{\partial r}\right)-\frac{L^{2}}{r^{2}}.$ (S52)

Inserting the separable ansatz
$\phi_{l,q_{l},E}(\bm{r})=R_{l,k}(r)Y_{l,q_{l}}(\bm{\theta})$ into Eq. (S51)
and using Eq. (S52), we obtain the radial equation

$\displaystyle\left[\frac{d^{2}}{dr^{2}}+\frac{D-1}{r}\frac{d}{dr}-\frac{l(l+D-2)}{r^{2}}+k^{2}\right]R_{l,k}(r)=0.$
(S53)

Defining $R_{l,k}(r)=r^{-\frac{D-1}{2}}y(r)$, Eq. (S53) can be written as

$\displaystyle\left[\frac{d^{2}}{dr^{2}}-\frac{l^{\prime}(l^{\prime}+1)}{r^{2}}+k^{2}\right]y(r)=0,\quad
l^{\prime}\equiv l+\frac{D-3}{2},$ (S54)

where $l^{\prime}\geq 0$ for $D\geq 2$. When $D=1$, the centrifugal term
$\frac{l^{\prime}(l^{\prime}+1)}{2}$ vanishes and $R_{l}(r)=y(r)$; when $D=3$,
we have $l^{\prime}=l$, $R_{l}(r)=y(r)/r$. When $D\geq 2$, as $r\rightarrow
0$, the centrifugal term $\frac{l^{\prime}(l^{\prime}+1)}{2}$ dominates the
energy term $k^{2}$, and the solutions behave like solutions of the
corresponding equation with $E=0$; namely, like combinations of
$r^{l^{\prime}+1}$ and $r^{-l^{\prime}}$. Thus, the physically acceptable wave
function behaves like $r^{l^{\prime}+1}$— the Riccati-Bessel function
$\hat{j}_{l^{\prime}}(kr)$[63]:

$\displaystyle\hat{j}_{l^{\prime}}(z)$ $\displaystyle\equiv
zj_{l^{\prime}}(z)\equiv\sqrt{\frac{\pi z}{2}}J_{l^{\prime}+\frac{1}{2}}(z)$
(S55)
$\displaystyle=z^{l^{\prime}+1}\sqrt{\frac{\pi}{2}}\sum_{n=0}^{\infty}\frac{(-z^{2}/2)^{n}}{n!2^{n+l^{\prime}+1/2}\Gamma(l^{\prime}+n+3/2)}$
(S56)

where $j_{l^{\prime}}(z)$ is the spherical Bessel function, and
$J_{\lambda}(z)$ the ordinary Bessel function. The Riccati-Bessel functions
satisfy the following orthogonality relations

$\displaystyle\int_{0}^{\infty}dr\hat{j}_{l^{\prime}}(k^{\prime}r)\hat{j}_{l^{\prime}}(kr)=\frac{\pi}{2}\delta(k-k^{\prime}).$
(S57)

Hence we obtain $R_{l,k}(r)$ and $\phi_{l,q_{l},E}(r,\bm{\theta})$:

$\displaystyle R_{l,k}(r)$
$\displaystyle=\sqrt{\frac{2}{\pi}}r^{-\frac{D-1}{2}}\hat{j}_{l^{\prime}}(kr),$
(S58) $\displaystyle\phi_{l,q_{l},E=k^{a}}(r,\bm{\theta})$
$\displaystyle=[\rho(E)\epsilon^{\prime}(k)]^{-1/2}R_{l,k}(r)Y_{l,q_{l}}(\bm{\theta}),$
$\displaystyle=\sqrt{\frac{2}{\pi}}b(D)^{-1/2}(kr)^{-\frac{D-1}{2}}\hat{j}_{l^{\prime}}(kr)Y_{l,q_{l}}(\bm{\theta}).$
(S59)

Here, we have chosen normalization constants such that the following
orthogonality and completeness relations are satisfied:

$\displaystyle\braket{\phi_{l,q_{l},E}}{\phi_{n,q_{n},E^{\prime}}}=\rho(E)^{-1}\delta(E-E^{\prime})\delta_{l,n}\delta_{q_{l},q_{n}},$
(S60) $\displaystyle\int_{0}^{\infty}dE\
\sum_{l,q_{l}}\rho(E)\,\ket{\phi_{l,q_{l},E}}\bra{\phi_{l,q_{l},E}}=\mathbb{1},$
(S61)

where $\mathbb{1}$ is the identity operator in the Hilbert space of a particle
in $D$ dimensions.

Using Eqs. (S56), (S58) and (S59), we have, for $kr\ll 1$,

$\displaystyle\phi_{l,q_{l},E}(r,\bm{\theta})=$ $\displaystyle
b(D)^{-1/2}Y_{l,q_{l}}(\bm{\theta})\frac{(kr)^{l}}{2^{l+D/2-1}\Gamma(l+\frac{D}{2})}\\{1+O[(kr)^{2}]\\}.$
(S62)

Hence, using $Y_{0,1}(\bm{\theta})=b(D)^{-\frac{1}{2}}$, we can derive the
point-wise convergence

$\displaystyle\lim_{E\rightarrow
0^{+}}\phi_{l,q_{l},E}(\bm{r})=\begin{cases}b(D)^{-1}\frac{1}{2^{D/2-1}\Gamma(\frac{D}{2})}=\left(\frac{1}{\sqrt{2\pi}}\right)^{D}&l=0\\\
0&l\geq 1\end{cases},$ (S63)

which confirms our earlier claim that the s-wave has a constant wavefunction
at zero energy.

Eq. (S63) is all we need to know about the angular momentum wavefunctions to
derive the $S$-matrix universal limits. Eq. (S63) implies that the $S$-matrix
universal limit is only nontrivial for $l=0$; the quantum number $q_{l}$ plays
no role in the discussions. For simplicity and uniformity of notation with the
1D case, we will use a single variable $\alpha=1,2,\dots$ to denote the pair
of quantum numbers $(l,q_{l})$. In particular, $\alpha=1$ corresponds to $l=0$
and $\alpha=2,3,\dots$ correspond to states with $l\geq 1$. The orthogonality
relation in Eq. (S60) can be rewritten as

$\displaystyle\braket{\phi_{\alpha,E}}{\phi_{\beta,E^{\prime}}}=\rho(E)^{-1}\delta(E-E^{\prime})\delta_{\alpha,\beta}.$
(S64)

Finally, we are ready to define the $S$-matrix in the angular momentum basis
in arbitrary dimension. In the basis $\\{\ket{\phi_{\alpha,E}}\\}$, the
scattering operator at energy $E$ can be represented by a matrix $\bm{S}(E)$:

$\displaystyle\rho(E)\braket{\phi_{\beta,E}}{\mathcal{S}}{\phi_{\alpha,E^{\prime}}}$
$\displaystyle\equiv\delta(E-E^{\prime})S_{\alpha,\beta}(E),$ $\displaystyle
S_{\alpha,\beta}(E)$ $\displaystyle=\delta_{\alpha,\beta}-2\pi
i\rho(E)T(E\\!+\\!i0^{+},\alpha,E,\beta,E),$ (S65)

where $S_{\alpha,\beta}(E)$ is the matrix element of $\bm{S}(E)$ and

$\displaystyle
T(\omega,\alpha,E,\beta,E)\equiv\bra{\phi_{\beta,E}}\mathcal{T}(\omega)\ket{\phi_{\alpha,E}}.$
(S66)

Eq. (5) can be compared to Eq. (5) in the main text for 1D systems.

### Universal scattering

In this subsection, we consider emitter scattering for arbitrary integer
spatial dimension $D\geq 1$ and dispersion relation
$\epsilon(k)=|\bm{k}|^{a}$, where $a>0$ is not required to be an integer. The
Hamiltonian is given by

$\displaystyle H$ $\displaystyle=H_{0}+V,$ $\displaystyle H_{0}$
$\displaystyle=\int d^{D}k\
\epsilon(\bm{k})C^{\dagger}(\bm{k})C(\bm{k})+\sum_{i,j=1}^{N}K^{R}_{ij}b_{i}^{\dagger}b_{j},$
$\displaystyle V$ $\displaystyle=\int
d^{D}k\left[\sum_{i=1}^{N}V_{i}(\bm{k})C(\bm{k})b^{\dagger}_{i}+\text{h.c}\right],$
(S67)

where we have either commutation or anti-commutation relations:
$[C(\bm{k}),C^{\dagger}(\bm{k}^{\prime})]_{\pm}=\delta(\bm{k}-\bm{k}^{\prime}),[b_{i},b^{\dagger}_{j}]_{\pm}=\delta_{ij}$.
We assume that the emitter-photon interaction has the form
$\ket{v_{\bm{k}}}\equiv(V_{1}(\bm{k}),\dots
V_{N}(\bm{k}))^{T}=V(\bm{k})\ket{u}$, where $\ket{u}$ is a unit vector. Let
$\bm{0}$ be the null vector in dimension $D$. We require that $V(\bm{k})$ is
locally square-integrable and continuous at $\bm{k}=\bm{0}$ and that
$V(\bm{0})$ is nonzero.

Similarly to the main text, we can define a $N\times N$ matrix
$\bm{K}(\omega)$ describing the effective interactions between emitters:

$\displaystyle K_{ij}(\omega)=\int d^{D}k\
\frac{V_{i}(\bm{k})V_{j}^{*}(\bm{k})}{\omega-|\bm{k}|^{a}}.$ (S68)

The momentum-space representation of the $T$-operator $\mathcal{T}(\omega)$ is
given by

$\displaystyle
T(\omega,\bm{k},\bm{k}^{\prime})=\braket{v_{\bm{k}^{\prime}}}{\frac{1}{\omega\mathbb{1}_{N}-\bm{K}(\omega)-\bm{K}^{R}}}{v_{\bm{k}}}.$
(S69)

Since $V(\bm{k})$ is square-integrable, its Fourier transform
$\tilde{V}(\bm{z})=\left(\frac{1}{\sqrt{2\pi}}\right)^{D}\int
d^{D}k\exp(i\bm{k}\bm{z})V(\bm{k})$ exists. To find the representation of the
$T$-operator in the basis $\\{\ket{\phi_{\alpha,E}}\\}$, define vector
$|v_{\alpha,E}\rangle=V_{\alpha,E}\ket{u}$, where

$\displaystyle V_{\alpha,E}$ $\displaystyle\equiv\int d^{D}z\
\tilde{V}(\bm{z})\phi^{*}_{\alpha,E}(\bm{z}).$ (S70)

The vector elements of $|v_{\alpha,E}\rangle$ represent the interaction
coefficients between the emitters and the angular momentum mode $\alpha$ at
energy $E$.

The representation of the $T$-operator in the angular momentum basis is given
by

$\displaystyle
T(\omega,\alpha,E,\beta,E^{\prime})=V^{*}_{\beta,E^{\prime}}V_{\alpha,E}\braket{u}{\frac{1}{\omega\mathbb{1}_{N}-\bm{K}(\omega)-\bm{K}^{R}}}{u}.$
(S71)

The $S$-matrix in the angular momentum basis is related to
$T(\omega,\alpha,E,\beta,E^{\prime})$ by Eq. (5). The generalization of 1D
universal scattering to arbitrary dimension $D$ and to all (including non-
integer) $m>0$ is given in the following theorem:

######  Theorem 6.

Suppose $V(\bm{k})$ is a locally square-integrable function continuous at
$\bm{k}=0$ and suppose $V(\bm{k}=0)\neq 0$. In the absence of bright zero-
energy eigenstates, when $a\leq D$, $\lim_{E\rightarrow
0^{+}}S_{\alpha,\beta}(E)=\delta_{\alpha,\beta}$; when $a>D$,

$\displaystyle\lim_{E\rightarrow
0^{+}}S_{\alpha,\beta}(E)=\begin{cases}\exp(2\pi iD/a)&\alpha=\beta=1,\\\
\delta_{\alpha,\beta}&\mathrm{Otherwise}.\end{cases}$ (S72)

###### Proof.

In the orthonormal basis where $\ket{u}$ is the first basis vector, the only
nonzero matrix element of $\bm{K}(\omega)$ is $K_{11}(\omega)\equiv
K(\omega)$:

$\displaystyle K(\omega)$ $\displaystyle\equiv\int d^{D}k\
\frac{V(\bm{k})V^{*}(\bm{k})}{\omega-|\bm{k}|^{a}},$ (S73)
$\displaystyle\bm{K}(\omega)$ $\displaystyle=\ket{u}\bra{u}K(\omega).$ (S74)

Using Eq. (S71), we have

$\displaystyle\lim_{\omega\rightarrow
0}T(\omega,\alpha,E,\beta,E^{\prime})=\lim_{\omega\rightarrow
0}V^{*}_{\beta,E^{\prime}}V_{\alpha,E}\left(-K(\omega)+\frac{\det(\omega\mathbb{1}_{N}-\bm{K}^{R})}{\det(\omega\mathbb{1}_{N-1}-\bm{K}^{R}_{\cancel{11}})}\right)^{-1}.$
(S75)

It is easy to show that Lemma 3 applies to general dispersion relations and
any dimension. Hence, the absence of bright zero-energy eigenstates implies
that any null vector of $\bm{K}^{R}_{\cancel{11}}$ corresponds to the null
vector of $\bm{K}^{R}$. Using a similar argument as in Eq. (S16),
$\lim_{\omega\rightarrow
0}\frac{\det(\omega\mathbb{1}_{N}-\bm{K}^{R})}{\det(\omega\mathbb{1}_{N-1}-\bm{K}^{R}_{\cancel{11}})}$
exists in the absence of bright zero-energy eigenstates.

We first prove the theorem for cases when the S-matrix has a trivial zero-
energy limit. When $a<D$, $\lim_{E\rightarrow 0^{+}}\rho(E)=0$ from Eq. (S47);
$\lim_{\omega\rightarrow 0}K(\omega)$ is a constant. Hence,
$\lim_{\omega\rightarrow 0}T(\omega,\alpha,E,\beta,E^{\prime})$ exists from
Eq. (S75). Using Eq. (5), we can conclude that $\lim_{E\rightarrow
0^{+}}S_{\alpha,\beta}(E)=\delta_{\alpha,\beta}$ when $a<D$.

When $a=D$, $\lim_{E\rightarrow 0}\rho(E)$ is finite from Eq. (S47);
$K(\omega)$ diverges logarithmically in the limit of $\omega\rightarrow 0$.
Hence, we have $\lim_{\omega\rightarrow
0}T(\omega,\alpha,E,\beta,E^{\prime})=0$ from Eq. (S75). Using Eq. (5), we see
that $\lim_{E\rightarrow 0^{+}}S_{\alpha,\beta}(E)=\delta_{\alpha,\beta}$ when
$a=D$.

We continue with the proof of nontrivial universal limit of $\bm{S}(E)$ when
$a>D$. Similarly, as in the 1D case, define

$\displaystyle L(\omega)=\int
d^{D}k\frac{1}{\omega-|\bm{k}|^{a}}=b(D)\int_{0}^{+\infty}dk\
\frac{k^{D-1}}{\omega-k^{a}}=\frac{b(D)}{D}\int_{0}^{+\infty}dp\frac{1}{\omega-p^{\zeta}},$
(S76)

where, in the last equality, $\zeta=\frac{a}{D}$ and we have changed the
integration variable from $k$ to $p=k^{D}$. The integral converges for
$\zeta>1$, and the value of $L(\omega)$ is given by

$\displaystyle L(\omega)=-\pi i\rho(|\omega|)\frac{2}{1-\exp(2\pi
i/\zeta)}\exp\left(-i\theta\frac{\zeta-1}{\zeta}\right),$ (S77)

which diverges at the same rate as the density of states $\rho(|\omega|)$ when
$|\omega|\rightarrow 0$. Eq. (S77) agrees with Eq. (13) in the main text for
even $m=a$ and $D=1$.

When $\zeta>1$, following a standard relation in functional analysis,
$\lim_{\omega\rightarrow
0}L(\omega)\frac{1}{\omega-|\bm{k}|^{a}}=\delta(\bm{k})$. We have

$\displaystyle\lim_{\omega\rightarrow
0}K(\omega)L^{-1}(\omega)=|V(\bm{0})|^{2}\neq 0.$ (S78)

Using Eqs. (S75) and (S78), we have, in the absence of bright zero-energy
eigenstates,

$\displaystyle\lim_{\omega\rightarrow
0}L(\omega)T(\omega,\alpha,E,\beta,E^{\prime})=-\frac{V^{*}_{\beta,E^{\prime}}V_{\alpha,E}}{|V(\bm{k}=\bm{0})|^{2}},$
(S79)

which can be compared to Eq. (S9) for the case of $D=1$. Using Eqs. (S63) and
(S70), we have

$\displaystyle\lim_{E\rightarrow
0}V_{\alpha,E}=\begin{cases}\left(\frac{1}{\sqrt{2\pi}}\right)^{D}\int d^{D}z\
\tilde{V}(\bm{z})=V(\bm{k}=\bm{0})&\alpha=1,\\\ 0&\alpha\geq 2,\end{cases}$
(S80)

where $\alpha=1$ corresponds to $l=0$ and $\alpha\geq 2$ corresponds to $l\geq
1$. Using Eq. (S80), Eq. (S79) becomes

$\displaystyle\lim_{\omega\rightarrow
0}L(\omega)T(\omega,\alpha,E,\beta,E^{\prime})=\begin{cases}-1&\alpha=\beta=1,\\\
0&\text{ Otherwise.}\end{cases}$ (S81)

Using Eqs. (5), (S77), and (S81), we obtain Eq. (S72). We are done with the
proof of Theorem 6 for all values of $a>0$. ∎

### Levinson’s theorem

Levinson’s theorem can also be generalized to $D\geq 2$. We define the
determinant of the infinite-dimensional matrix $\bm{S}(E)$ through the
$n\rightarrow\infty$ limit of the series $\det[\bm{S}_{n}(E)]$, where
$\bm{S}_{n}(E)$ is the $n\times n$ submatrix of $\bm{S}(E)$ in the subspace of
$\alpha=0,1,\dots n-1$:

$\displaystyle\det[\bm{S}(E)]\equiv\lim_{n\rightarrow\infty}\det[\bm{S}_{n}(E)].$
(S82)

As $l$ increases, $\phi_{l,q_{l},E}(r,\bm{\theta})\sim(rk)^{l}$ vanishes
increasingly fast close to the scattering center because of the centrifugal
barrier. This implies that modes with high angular momentum
($\alpha\rightarrow\infty$) have trivial scattering amplitudes and the limit
in Eq. (S82) exists.

######  Theorem 7.

Define the winding phase $\Delta\delta$ of $\det[\bm{S}(E)]$ similarly to the
1D case in Theorem 4. Suppose $\ket{v_{\bm{k}}}=V(\bm{k})\ket{u}$ satisfies
the properties listed in Theorem 6, and the dispersion relation is given by
$\epsilon(k)=|\bm{k}|^{a}$, where $m>0$. We have, in the absence of bright
zero-energy eigenstates and bound states in the continuum,

$\displaystyle\Delta\delta=\begin{cases}\pi(N-N_{B})+\pi\frac{a-D}{a}&a>D,\\\
\pi(N-N_{B})&a\leq D,\end{cases}$ (S83)

where $N$ is the number of emitters and $N_{B}$ is the number of bound states.

Theorem 7 can be proven through a procedure similar to the one used in Theorem
4 for 1D systems. Below, we provide the extension of Theorem 5 to arbitrary
dimension $D$. The rest of the proof is quite straightforward and we omit it
here.

######  Theorem 8.

Define $J(\omega)=\det[\omega\mathbb{1}_{N}-\bm{K}^{R}-\bm{K}(\omega)]$, where
$\bm{K}(\omega)$ is defined in Eq. (S68). When $E$ is not equal to the energy
of a bound state in the continuum, we have

$\displaystyle\det[\bm{S}(E)]=\frac{J(E-i0)}{J(E+i0)}.$ (S84)

###### Proof.

The proof follows the same procedure as the proof for Theorem 5. Construct
$\bm{A}_{n}$ as a $N\times n$ matrix and $\bm{A}_{n}^{\dagger}$ its Hermitian
conjugate:

$\displaystyle\bm{A}_{n}$
$\displaystyle=\rho^{1/2}(E)\left(\begin{matrix}|v_{\alpha=0,E}\rangle,\dots|v_{\alpha=n-1,E}\rangle\end{matrix}\right).\quad$
(S85)

Then the $n\times n$ matrix $\bm{S}_{n}(E)$ for $E\neq E_{c}$ can be written
as

$\displaystyle\bm{S}_{n}(E)=\mathbb{1}_{n}-2\pi
i\bm{A}_{n}^{\dagger}\bm{G}(E+i0^{+})\bm{A}_{n},$ (S86)

where $\mathbb{1}_{n}$ is the identity matrix of dimension $n$ and
$\bm{G}(\omega)\equiv(\mathbb{1}_{N}-\bm{K}(\omega)-\bm{K}^{R})^{-1}$. Using
the properties of determinant, the definition of $J(\omega)$, and the identity

$\displaystyle\bm{K}(E+i0^{+})-\bm{K}(E+i0^{-})$
$\displaystyle=\sum_{\alpha}\int_{0}^{+\infty}dE^{\prime}\
\rho(E^{\prime})|v_{\alpha,E^{\prime}}\rangle\langle
v_{\alpha,E^{\prime}}|\left(\frac{1}{E+i0^{+}-E^{\prime}}-\frac{1}{E-i0^{-}-E^{\prime}}\right),$
$\displaystyle=-2\pi i\sum_{\alpha}\rho(E)|v_{\alpha,E}\rangle\langle
v_{\alpha,E}|=-2\pi i\lim_{n\rightarrow\infty}\bm{A}_{n}\bm{A}_{n}^{\dagger},$
(S87)

the r.h.s of Eq. (S84) can be written as

$\displaystyle\frac{J(E-i0)}{J(E+i0)}=\det\left[\mathbb{1}_{N}-2\pi
i\lim_{n\rightarrow 0}\bm{A}_{n}\bm{A}_{n}^{\dagger}\bm{G}(E+i0^{+})\right].$
(S88)

According to the matrix determinant lemma, given an invertible $N\times N$
matrix $-2\pi i\bm{G}(E+i0)$ and a $N\times n(E)$ matrix $\bm{A}_{n}$,

$\displaystyle\det[\mathbb{1}_{N}-2\pi
i\bm{A}_{n}\bm{A}_{n}^{\dagger}\bm{G}(E+i0^{+})]=\det[\mathbb{1}_{n}-2\pi
i\bm{A}_{n}^{\dagger}\bm{G}(E+i0^{+})\bm{A}_{n}].$ (S89)

Using Eqs. (S82), (S26) and (S88), we see that the l.h.s and r.h.s of Eq.
(S89) are equal, respectively, to $\det[\bm{S}(E)]$ and
$\frac{J(E-i0)}{J(E+i0)}$ in the limit of $n\rightarrow\infty$. This is the
end of the proof of Theorem 8.

∎

##  VI Separable potential scattering

The purpose of this Letter is to demonstrate the principle that divergent
density of states leads to nontrivial universal behavior of the $S$-matrix. To
demonstrate that this principle is not limited to emitter-photon interactions,
in this section, we show that the $S$-matrix has the same universal limit in a
particular class of potential scattering. To be specific, we study separable
potential scattering. Seperable potentials generalize delta-function potential
scattering and serve as an important effective model to describe the long-
distance behavior in many scattering systems.

Assume any integer spatial dimension $D\geq 1$. The time-independent
Schrodinger equation in momentum space is given by

$\displaystyle\epsilon(\bm{k})\psi(\bm{k})+\,\int d^{D}k^{\prime}\
V(\bm{k^{\prime}},\bm{k})\psi(\bm{k^{\prime}})=E\psi(\bm{k}),$ (S90)

where the dispersion relation $\epsilon(\bm{k})$ is any of those being
considered for emitter scattering in 1D in the main text or for arbitrary $D$
in Sec. V. For local potentials,
$V(\bm{k^{\prime}},\bm{k})=V(\bm{k^{\prime}}-\bm{k})$ depend only on the
momentum difference $\bm{k^{\prime}}-\bm{k}$, where
$\bm{k}~{}(\bm{k^{\prime}})$ is the incoming (outgoing) momentum of the
incident particle. For scattering with a separable potential, the potential is
non-local in real space and takes the form
$\tilde{V}(\bm{z^{\prime}},\bm{z})=g\tilde{v}(\bm{z^{\prime}})\tilde{v}(\bm{z})$,
where $\bm{z}~{}(\bm{z^{\prime}})$ is the incoming (outgoing) position of the
incident particle and $\tilde{v}(\bm{z})$ is normalized such that
$\left(\frac{1}{\sqrt{2\pi}}\right)^{D}\int d^{D}z\,\tilde{v}(\bm{z})=1$. Let
$v(\bm{k})$ be the Fourier transform of $\tilde{v}(\bm{z})$; it is clear that
$v(\bm{k})$ is uniformly continuous and $v(\bm{k}=\bm{0})=1$. The separable
potential in momentum space takes the form
$V(\bm{k^{\prime}},\bm{k})=g\tilde{v}(\bm{k^{\prime}})\tilde{v}(\bm{k})$; the
time-independent Schrodinger equation can then be written as

$\displaystyle\epsilon(\bm{k})\psi(\bm{k})+g\,v(\bm{k})\int
d^{D}k^{\prime}v(\bm{k^{\prime}})\psi(\bm{k^{\prime}})=E\psi(\bm{k}).$ (S91)

In potential scattering, the $T$-matrix can be solved from the Lippmann-
Schwinger equation:

$\displaystyle
T_{\text{sep}}(\omega,\bm{k},\bm{k^{\prime}})=g\,v(\bm{k})v(\bm{k^{\prime}})+g\,v(\bm{k}^{\prime})\int
d^{D}k^{\prime\prime}\frac{v(\bm{k^{\prime\prime}})}{\omega-\epsilon(\bm{k^{\prime\prime}})}T_{\text{sep}}(\omega,\bm{k},\bm{k^{\prime\prime}}).$
(S92)

Solving Eq. (S92), we obtain the solution for the $T$-matrix:

$\displaystyle T_{\text{sep}}(\omega,\bm{k},\bm{k^{\prime}})$
$\displaystyle=\frac{v(\bm{k})v(\bm{k^{\prime}})}{g^{-1}-K(\omega)},$ (S93)
$\displaystyle K_{\text{sep}}(\omega)$ $\displaystyle\equiv\int
d^{D}k\frac{v(\bm{k})^{2}}{\omega-\epsilon(\bm{k})},$ (S94)

where the momentum dependence is simply captured by the momentum dependence of
the potential. The scattering operator is related to the $T$-matrix through
Eq. (S48). Let us compare the $T$-matrix for separable potential scattering to
single-particle emitter scattering with $K^{R}=0$ and
$V_{1}(\bm{k})=v(\bm{k})$:

$\displaystyle T(\omega,\bm{k},\bm{k^{\prime}})$
$\displaystyle=\frac{v(\bm{k})v^{*}(\bm{k^{\prime}})}{\omega-K(\omega)},$
(S95) $\displaystyle K(\omega)$ $\displaystyle=\int
d^{D}k\frac{|v(\bm{k})|^{2}}{\omega-\epsilon(\bm{k})}.$ (S96)

The similarity between $T_{\text{sep}}(\omega,\bm{k},\bm{k}^{\prime})$ and
$T(\omega,\bm{k},\bm{k}^{\prime})$ allows the $S$-matrix universal limits to
be similarly derived for separable potential scattering. To be specific, the
zero-energy limit of the $S$-matrix behaves identically to the single-emitter
scattering for any dispersion relation studied in earlier sections of the
paper.

The Levinson’s theorem can also be generalized to separable potential
scattering. Defining the $S$-matrix $\bm{S}_{\text{sep}}(E)$ similarly to
$\bm{S}(E)$ in the case of emitter scattering, it is easy to derive a
generalization of Theorems 5 and 8 to separable potential scattering. In any
dimension, we have

$\displaystyle\det[\bm{S}_{\text{sep}}(E)]=\frac{g^{-1}-K_{\text{sep}}(E-i0)}{g^{-1}-K_{\text{sep}}(E+i0)}.$
(S97)

The solutions of $g^{-1}-K_{\text{sep}}(\omega)=0$ correspond to bound state
energies. Define $\Delta\delta_{\text{sep}}$ as the difference of the
scattering phase of $\det[\bm{S}_{\text{sep}}(E)]$ between the two ends of the
continuum spectrum similarly to Eq. (S21). Following a proof similar to that
of Theorem 4, we have, for dispersion relation $\epsilon(k)=\pm|d|k^{m}$ in
1D,

$\displaystyle\Delta\delta_{\text{sep}}=-\pi N_{B}+\pi\frac{m-1}{m},$ (S98)

where $N_{B}$ is the number of bound states. For dispersion relation
$\epsilon(\bm{k})=|\bm{k}|^{a}$ in integer dimension $D\geq 1$,

$\displaystyle\Delta\delta_{\text{sep}}=\begin{cases}-\pi
N_{B}+\pi\frac{a-D}{a}&a>D,\\\ -\pi N_{B}&0<a\leq D.\end{cases}$ (S99)