# Schrödinger’s Black Hole Cat

Joshua Foo111Corresponding Author<EMAIL_ADDRESS>Centre for
Quantum Computation & Communication Technology, School of Mathematics &
Physics, The University of Queensland, St. Lucia, Queensland, 4072, Australia
Robert B. Mann<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada  Perimeter Institute
for Theoretical Physics, Waterloo, Ontario N2L 6B9, Canada  Magdalena Zych
<EMAIL_ADDRESS>Centre for Engineered Quantum Systems, School of Mathematics
and Physics, The University of Queensland, St. Lucia, Queensland, 4072,
Australia

###### Abstract

In the absence of a fully-fledged theory of quantum gravity, we propose a
“bottom-up” framework for exploring quantum-gravitational physics by pairing
two of the most fundamental concepts of quantum theory and general relativity,
namely quantum superposition and spacetime. We show how to describe such
“spacetime superpositions” and explore effects they induce upon quantum
matter. Our approach capitalises on standard tools of quantum field theory in
curved space, and allows us to calculate physical observables like transition
probabilities for a particle detector residing in curvature-superposed de
Sitter spacetime, or outside a mass-superposed black hole. Crucially, such
scenarios represent genuine quantum superpositions of spacetimes, in contrast
with superpositions of metrics which only differ by a coordinate
transformation and thus are not different according to general relativity.

Essay written for the Gravity Research Foundation 2022 Awards for Essays on
Gravitation.

Submitted on March 31, 2022

A unified theory of quantum gravity continues to elude us despite numerous
attempts over the last century to construct one. The dynamical character of
the spacetime metric obstructs the implementation of standard approaches to
its quantization. Top-down attempts at unification such as string theory
Gubser  _et al._ (1998); Seiberg and Witten (1999); Polchinski (1998); Witten
(1995) and loop quantum gravity Rovelli (1998); Rovelli and Vidotto (2014);
Thiemann (2003) have yet to succeed, raising questions as to the necessity or
well-posedness of quantum gravity Dyson (2013); Garay (1995); Carlip (2008).

We propose taking a “bottom-up” perspective on this problem, one that allows
exploration of quantum gravity effects that do not require the full dynamical
theory of quantum gravity. Specifically, we focus on the most signature
feature of quantum physics: superposition.

If indeed relativistic gravitation has a quantum description, it must be
meaningful to consider a spacetime metric in a genuine quantum superposition.
Any theory combining spacetime and quantum theory should admit as a special
case superpositions of semiclassical geometries. Recent experimental proposals
attempting to gravitationally induce entanglement between two mesoscopic
particles each in a spatial superposition Bose  _et al._ (2017); Marletto and
Vedral (2017), if successful, have been argued to provide evidence for
superpositions of semiclassical spacetime geometries Christodoulou and Rovelli
(2019). Other studies have argued for inevitable decoherence mechanisms in
systems involving macroscopically superposed gravitational sources (e.g. black
holes and dark matter Arrasmith  _et al._ (2019); Demers and Kiefer (1996);
Allali and Hertzberg (2021)). Rapid advances in tabletop techniques Schut  _et
al._ (2022); Howl  _et al._ (2021); Carney  _et al._ (2021) show that such
questions are not only of fundamental, but also practical interest, especially
when considering analog experiments, for example with Bose-Einstein
condensates Nova  _et al._ (2019); Barceló  _et al._ (2021).

The common feature of these investigations is that they focus on effects
produced by spatial superpositions of massive objects. Such scenarios involve
classical metrics that differ only by a coordinate transformation and are thus
diffeomorphic. One can equivalently describe these “spacetime superpositions”
in terms of a single classical spacetime where quantum systems residing within
are prepared and measured in appropriate quantum states Zych  _et al._ (2018);
Foo  _et al._ (2021a) (we present this argument further in this essay for
completeness).

Our new approach admits superposition of genuinely different spacetimes, i.e.
superpositions of unique solutions of Einstein’s field equations, which are
thus not diffeomorphic. It can of course also describe superpostions of
spacetimes differing by a mere coordinate transformation, such as those
arising from the above mentioned superpositions of locations of a source mass.
It allows us to explicitly show how and why the latter elicit the same effects
as the corresponding quantum states of matter on a fixed background in a
concrete scenario.

We do not propose a full quantum-gravitational theory for the emergence of
quantum spacetimes, but rather develop an operational framework describing
dynamics of quantum systems residing within them. The framework is based on
the standard paradigm of quantum fields in curved spacetime and quantum matter
coupled to these fields. We are predominantly interested in the effects
produced by spacetime superpositions on the quantum matter system(s),
henceforth called detectors or probes, and the new insights they may engender.
These quantum probes are modelled as 2-level systems linearly coupled to
quantum fields, otherwise known as Unruh-deWitt (UdW) detectors Unruh and Wald
(1984). This model has been utilized widely in relativistic and gravitational
settings to investigate properties of quantum fields, vacuum entanglement,
black holes, and causal structure Ng  _et al._ (2014, 2017, 2018); Henderson
_et al._ (2020a); Hodgkinson and Louko (2012a, b); Henderson  _et al._
(2020b).

Before proceeding towards more complex situations involving superpositions of
gravitational sources, it is instructive to first consider flat space
analogues to such scenarios. For example, consider placing a UdW detector on a
classical uniformly accelerated (Rindler) trajectory. Its response to the
field (i.e. its transition probability) is a Planckian distribution with an
associated temperature proportional to the magnitude of its acceleration, the
well-known Unruh effect. This effect has a direct correspondence with the
thermalization of the Hartle-Hawking vacuum outside a Schwarzschild black
hole, which will likewise cause a detector to “thermalize” to the Hawking
temperature, inversely proportional to the black hole’s mass.

Let us now place the detector in a superposition of semiclassical trajectories
with different uniform accelerations Foo  _et al._ (2020); Barbado  _et al._
(2020); Foo  _et al._ (2021b):

$\displaystyle|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|\text{acceleration
1}\rangle+|\text{acceleration 2}\rangle).$ (1)

Due to the aforementioned correspondence between the Rindler and Schwarzschild
spacetimes, we expect that the detector’s response may provide insight into
the quantum-gravitational effects produced by a mass-superposed Schwarzschild
black hole.

Surprisingly, such a detector does not thermalize, not even at a temperature
given by some function of the temperatures associates with the individual
trajectories Foo  _et al._ (2020). Even more surprisingly, the detector does
not thermalize even if the two trajectories have the same acceleration and
thus the same associated Unruh temperature but are simply translated in space
in the direction of motion. This lack of thermalisation is due to quantum
interference between the amplitudes associated with the different paths, which
can be traced back to the fact that the detector interacts with two different
sets of spatially translated field modes.

We now move to the main focus of our framework and consider a detector
residing in a superposition of spacetimes. Consider first a superposition of
spacetime manifolds that are related by a coordinate transformation, e.g.
spatial translation. This is a scenario where “two” manifolds are related by a
diffeomorphism and thus do not actually differ according to classical general
relativity. Moreover, in any theory where the considered transformation is a
symmetry of the dynamics, any probability amplitude involving matter on such
“quantum” spacetime is equivalent to an amplitude arising from a scenario in
which the matter is in an appropriate superposition but on a single, classical
spacetime manifold.

To understand this, note that the total Hilbert space of the relevant systems
is
$\mathcal{H}=\mathcal{H}_{\text{UdW}}\otimes\mathcal{H}_{F}\otimes\mathcal{H}_{C}$
where $\mathcal{H}_{\text{UdW}}$, $\mathcal{H}_{F}$ and $\mathcal{H}_{C}$ are
respectively associated with the internal states of the detector, the field
degrees of freedom (DoF), and a DoF which we call a control. Conceptually, the
control is some ancillary system to which the field, spacetime and/or detector
are coupled.

The interaction between all DoFs in our context very generally can be written
as

$\displaystyle\hat{U}=\sum_{i}\hat{U}_{i}\otimes{{}_{C}|i\rangle}\langle
i|_{C}$ (2)

where $\\{|i\rangle_{C}\\}$ are orthonormal states of the control (e.g.
position of the detector), and $\hat{U}_{i}$ is the interaction between the
remaining DoFs for a given state of the control (e.g. interaction between the
field and internal DoF of the detector for a given position of the latter).
Given an initial superposition state of the control

$\displaystyle|\psi\rangle_{C}$
$\displaystyle=\frac{1}{\sqrt{2}}(|1\rangle_{C}+|2\rangle_{C})$ (3)

and an arbitrary state $|\varphi\rangle_{FD}$ of the remaining DoFs (e.g. the
field and the internal DoF of an UdW detector) the conditional state of the
latter two given the control is measured in $|\psi\rangle_{C}$ reads

$\displaystyle{{}_{C}\langle\psi|}\hat{U}|\psi\rangle_{C}|\varphi\rangle_{FD}$
$\displaystyle=\frac{1}{2}(\hat{U}_{1}+\hat{U}_{2})|\varphi_{FD}\rangle.$ (4)

Note that the state of the control on which we condition is chosen to be the
same as the initial state just for simplicity – it can be arbitrary.

Suppose the control DoF describes the detector’s position. The two control
states can then be related using a possibly time-dependent function $L\equiv
L(\tau)$ relating the coordinates $\xi\equiv\xi(\tau)$ of the superposed
positions of the detector, such that $|1\rangle_{C}\equiv|\xi\rangle_{C}$,
$|2\rangle_{C}\equiv|\xi+L\rangle_{C}$. There thus exists an operator
$\hat{T}_{C}$ implementing spatial translations, such that
$|\xi+L\rangle_{C}=\hat{T}_{C}(L)|\xi\rangle_{C}$. The corresponding change of
coordinates is implemented on the total system by an operator
$\hat{T}_{C}(L)\otimes\hat{T}_{F}(L)$ and since it is a symmetry of the
dynamics we have Zych  _et al._ (2018)

$\displaystyle\Big{[}\hat{U}(\xi)+\hat{U}(\xi+L)\Big{]}|\varphi\rangle_{FD}={{}_{C}\Big{\langle}\xi\Big{|}}\Big{[}\hat{U}+\hat{T}_{F}(L)\hat{U}\hat{T}^{\dagger}_{F}(L)\Big{]}\Big{|}\xi\Big{\rangle}_{C}\Big{|}\varphi\Big{\rangle}_{FD}.$
(5)

The right-hand side describes the scenario where the detector is traversing a
single trajectory $\xi(\tau)$, but undergoes dynamics in which the field
operators are enacted in a “superposition” of translations. The left-hand side
is Eq. (4) for the special case of the control states being two positions
$\xi,\xi+L$ of the detector, and describe a conditional state of the remaining
DoFs given that the detector is prepared and measured in superposition (e.g.
traversing a Mach-Zender-type interferometer). Equation (5) is a formal
statement of the fact that a scenario involving a particle in a spatially
superposed spacetime is fully equivalent to a scenario where the particle
follows superposed trajectories in one spacetime. The equivalence between the
two can be interpreted as describing the same physical situation simply using
two sets of coordinates related by a “superpostion” of classical
transformations Zych  _et al._ (2018), similar to scenarios arising in the
context of quantum reference frames Giacomini  _et al._ (2019), i.e. frames
operationally established using quantum systems. This general equivalence
further implies that a detector linearly coupled to the quantum field responds
identically (i.e. has the same transition probability) in both scenarios.

The crux of the argument is that only relative configurations between the
interacting systems are physically relevant, for example a superposition of
two distances between a pair of particles; a global (joint) translation of all
DoFs is irrelevant.

This argument also applies to the thought experiments by Bose Bose  _et al._
(2017) and Marletto Marletto and Vedral (2017), where two spatially superposed
source masses gravitationally interact with each other, which is argued to
lead to a superposition of spacetime metrics Christodoulou and Rovelli (2019).
However, our argument shows that this is still equivalent to a situation in
which one of the masses is on a fixed trajectory while the other undergoes two
interference experiments in superposition – closer and further from the source
mass – so that all relative distances are kept unchanged (see Fig. 1). All the
amplitudes (here leading to an entangled final state) are then also unchanged,
but the explanation now only involves a superposition of distances between
masses where the one designated as the source is fixed. Hence, one concludes
that the spacetime is fixed as well. This highlights again the inherent
ambiguity in the notion of “spacetime superposition” if the superposed
amplitudes differ by a diffeomorphism (or any symmetry of the dynamics).

Figure 1: (Top) An illustration of the thought experiments of Bose Bose  _et
al._ (2017) and Marletto Marletto and Vedral (2017), where two spatially
superposed source masses (depicted here as 2-level particles) interact via
gravity. (Bottom) However such a system is equivalent to the case where one
mass is on a fixed trajectory with the other undergoing two interference
experiments in superposition. All relative distances remain unchanged.

We can now apply this argument to the conceptually simple example of static de
Sitter spacetime (and a basic model of our expanding universe), with line
element

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-\left(1-\frac{r^{2}}{l^{2}}\right)\mathrm{d}t^{2}+\left(1-\frac{r^{2}}{l^{2}}\right)^{-1}\mathrm{d}r^{2}+r^{2}(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2})$
(6)

where $l$ is the de Sitter length parametrizing the constant positive
curvature of the spacetime, $1/l<r<\infty$, and $\theta,\phi\in[0,2\pi)$.
Placing the detector in a superposition of spatially translated worldlines –
for example, a superposition of $r$’s or $\theta$’s – is equivalent to one in
which the detector resides on a manifold in a quantum superposition of spatial
translations Zych  _et al._ (2018); Foo  _et al._ (2021c), as illustrated in
Figure 2(a,b).

Figure 2: Visualisation of (a) a detector residing in a de Sitter spacetime in
a superposition of rotations, which is diffeomorphic to (b) a detector in a
superposition of rotations on a classical de Sitter spacetime. Diagram (c) is
a visualization of de Sitter spacetime in a superposition of curvatures.

Compare this with a de Sitter universe whose curvature can be quantized and so
is in a superposition of expansion rates (de Sitter lengths) Foo  _et al._
(2021c), see Figure 2(c). We thus associate the control DoF with the
superposition

$\displaystyle|\psi\rangle_{C}$
$\displaystyle=\frac{1}{\sqrt{2}}(|l_{1}\rangle_{C}+|l_{2}\rangle_{C}).$ (7)

Here we have a truly novel situation: since the individual amplitudes of the
superposition (each defined by different de Sitter length $l_{1}$ or $l_{2}$)
are unique solutions to Einstein’s field equations, there is no diffeomorphism
that maps the spacetime superposition to a superposition of matter
configurations on a single metric having a fixed value of $l$. We might
consider such a scenario to more uniquely describe a quantum-gravitational
superposition, rather than a mere superposition of spatial configurations.

We might also expect such a spacetime to elicit unambiguous signatures of a
quantum-gravitational effects picked up by the detector. Interestingly
however, owing to the equivalence between uniformly accelerated motion in flat
Minkowski spacetime and static worldlines in expanding de Sitter spacetime,
the effects produced by curvature-superposed de Sitter spacetime can be shown
to be functionally equivalent to those experienced by an acceleration-
superposed detector in flat Minkowski spacetime Foo  _et al._ (2021c): The
detector’s transition probability has a functional form that is equivalent in
both scenarios. This means that at least in this specific example, the UdW
detector cannot unambiguously discern the effects produced by a genuine
spacetime superposition of different curvatures, and that induced by a
superposition of accelerated motions in flat spacetime. Note that this is not
the same as mapping the two amplitudes arising from different curvatures to a
single curved spacetime, but points out the equivalence of each amplitude to a
flat-spacetime via the equivalence principle.

To determine whether matter coupled to spacetime via quantum fields is
sensitive to genuinely quantum-gravitational effects, we can exploit a model
system in which technical progress can be made: the $(2+1)$ Banados-
Teitelboim-Zanelli (BTZ) dimensional black hole Bañados  _et al._ (1992),
obtained as a quotient of AdS-Rindler spacetime Hodgkinson and Louko (2012c);
Langlois (2006); Smith and Mann (2014), giving the line element

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-\left(\frac{r^{2}}{l^{2}}-M\right)\mathrm{d}t^{2}+\left(\frac{r^{2}}{l^{2}}-M\right)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\phi^{2},$
(8)

where $\sqrt{M}l<r<\infty$, $-\infty<t<\infty$, $\phi\in[0,2\pi)$ and
$r_{H}=\sqrt{M}l$ is the radial coordinate of the event horizon. Our general
approach can be now applied to the BTZ black hole in a superposition of masses
$M_{1}$, $M_{2}$ Foo  _et al._ (2021d). We thus identify the control DoF with
the mass of the black hole and obtain the conditional evolution of the
remaining DoFs in the form of Eq. (4) where $\hat{U}_{i}\equiv\hat{U}_{M_{i}}$
now describe the evolution of the field and detector for BTZ mass $M_{i}$.

By tracing the field DoF we can even obtain the probability for the detector
to register a field particle (Hawking quanta), which takes a general form

$\displaystyle P_{D}$
$\displaystyle=\text{Tr}_{F}\left[\frac{1}{2}(\hat{U}_{M_{1}}+\hat{U}_{M_{1}})|\varphi\rangle_{FD}\right].$
(9)

Plotting the response of the detector as a function of the mass ratio
$\sqrt{M_{2}/M_{1}}$ in Fig. 3, we observe in both cases resonant peaks that
occur at rational values of this ratio.

Figure 3: (a) Schematic diagram of the black hole mass superposition, with a
detector situated at a fixed radial distance from the origin of coordinates.
(b) Normalised probability $P_{E}/\sigma$ of the detector as a function of
$\sqrt{M_{2}/M_{1}}$ ($\sigma$ is the timescale of the interaction). The
measurement basis of the control is indicated in the legend. The dashed lines
correspond to $\sqrt{M_{2}/M_{1}}=1/n$ where $n=\\{1,\ldots 6\\}$.

We ascribe this behaviour to constructive interference between the field modes
associated with topologically closed AdS spacetimes. These ratios are
commensurate with the quantum black hole conjecture of Bekenstein Bekenstein
(1973, 2020); Regge and Wheeler (1957), who posited that a complete theory of
quantum gravity must account for the treatment of black holes as quantum
objects and that the horizon area (and hence its mass) is an adiabatic
invariant with an associated discrete quantization Hod (1998). Specifically,
the allowed mass values of the BTZ black hole, assuming the Bohr-Sommerfeld
quantization scheme for the horizon radius, are given by Kwon and Nam (2010):

$\displaystyle r_{H}$ $\displaystyle=\sqrt{M}l=n,\qquad n=1,2,\ldots$ (10)

While our construction does not require that the superposed masses are
quantized in integer values, we see that the detector responds uniquely to
black hole mass superpositions with mass ratios corresponding to those
predicted by Bekenstein’s quantum gravity conjecture Foo  _et al._ (2021d).
Our results demonstrate how a detector’s transition probability can reveal
genuinely quantum-gravitational properties of a spacetime superposition.

Our approach has much potential and scope for investigating spacetime
superpositions and the quantum-gravitational effects they induce upon quantum
matter Rather than developing a full-fledged quantization scheme for gravity
itself, as numerous “top-down” approaches have attempted, we have shown that
combining the simple but fundamental notion of quantum superposition with the
equally foundational notion of spacetime leads to unique effects that can in-
principle be detected by a first-quantized system. Because we have approached
the problem using the accessible tools of quantum field theory in curved
spacetime, we are also able to ask and hopefully resolve further important
questions of fundamental importance.

One such question is how one may construct a “quantum metric”, describing a
superposed spacetime. Using our UdW detector model, one can in-principle
measure the field correlations between different points in such a spacetime.
Utilizing the fact that the strength of correlations in quantum field theory
is an operational measure of spacetime distance Saravani  _et al._ (2016), we
can write a “conditional quantum metric” (i.e. conditioned on the measurement
of the control system):

$\displaystyle g_{\mu\nu}$ $\displaystyle\propto\lim_{x\to
x^{\prime}}\frac{\partial}{\partial x^{\mu}}\frac{\partial}{\partial
x^{\prime\nu}}\sum_{i,j}\Big{[}\underbrace{{{}_{F}\langle\varphi|}\hat{\Phi}(x_{i})\hat{\Phi}(x_{j})|\varphi\rangle_{F}}_{\text{two-
point correlator}}\Big{]}^{\frac{2}{d-2}}$ (11)

where $|\varphi\rangle_{F}$ is the state of the field, $\hat{\Phi}(x_{i})$ is
the field operator pulled back to the worldline $x_{i}$ of the $i$th amplitude
of the spacetime superposition, and $d$ is the spacetime dimension.
Calculating geodesics in such a spacetime poses an interesting question for
further study.

Another problem that has garnered recent interest has been the study of
universal mechanisms of decoherence, particularly gravitational mechanisms. In
our case, the interaction between the spacetime superposition, field, and
matter is expected to decohere the system. However the extent of such
decoherence remains to be seen. Conceivably one might be able to apply our
approach to incorporate the “self-decoherence” of a black hole due to
interaction with its own Hawking radiation, as suggested in refs. Arrasmith
_et al._ (2019); Demers and Kiefer (1996), and analyze the effects therein.

Finally, spacetime superpositions may find important applications in quantum
foundations and quantum information. For example, they can be considered in
the context of quantum-gravitational causal structures Zych  _et al._ (2019)
or quantum information processing in the presence of quantum gravity Howl  _et
al._ (2022). In particular, a detector situated outside a mass-superposed
black hole will experience a “superposition” of redshifts and proper times –
its interaction with the field being fundamentally indeterminate. This
presents an important stepping stone towards understanding the notion of time
in quantum gravity.

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# Hierarchical band gaps in complex periodic systems

Lucas Dunckley Department of Mathematics, Imperial College London, 180 Queen’s
Gate, London SW7 2AZ, United Kingdom. Bryn Davies11footnotemark: 1

###### Abstract

Complex periodic structures inherit spectral properties from the constituent
parts of their unit cells, chiefly their spectral band gaps. Exploiting this
intuitive principle, which is made precise in this work, means spectral
features of periodic systems with very large unit cells can be predicted
without numerical simulation. We study a class of difference equations with
periodic coefficients and show that they inherit spectral gaps from their
constituent elements. This result shows that if a frequency falls in a band
gap for every constituent element then it must be in a band gap for the
combined complex periodic structure. This theory and its instantaneous utility
is demonstrated in a series of vibro-acoustic and mechanical examples.

##  1 Introduction

A wealth of advanced technologies depend on the varied (and sometimes
surprising) properties exhibited by waves propagating through systems with
periodically varied material parameters. These technologies include photonic
and phononic crystals capable of precise wave control [11] and metamaterials
that display exotic effective properties [24]. Such metamaterials have been
realised in almost every corner of wave physics and can perform impressive
feats such as negative refraction and cloaking [20, 5]. Partly as a result of
this ongoing technological revolution, there is a wealth of literature devoted
to studying the spectra of the associated operators (either differential or
discrete) with periodic coefficients.

As photonic crystals and metamaterials grow in complexity, there is a need to
understand the spectra of periodic systems with increasingly complex unit
cells. There has even been some interest in unit cells with fractal geometries
[3, 27], which presents its own specific theoretical challenges. Under
reasonable assumptions (and excluding fractals), the complexity of the unit
cell typically does not cause any issues for the standard theoretical tools
such as Floquet-Bloch analysis [15] or two-scale homogenisation [1]. The
challenge, however, when faced with a periodic system with a highly complex
unit cell, is to develop some intuition for the main features of its spectrum.

The main spectral feature that it is useful to be able to predict is the
occurrence of band gaps. These are ranges of frequencies at which waves are
unable to propagate through the periodic system. Instead, the amplitude decays
exponentially as a function of position and energy is typically reflected back
in the direction of incidence. There are a variety of approaches for
predicting if a system will support band gaps, without computing the spectrum
in full. These are typically based on examining the symmetries of the material
and can be framed in terms of either point symmetry groups [18], frieze groups
[28] or topological invariants [2].

In this work, we characterise the spectra of complex periodic materials by
decomposing the unit cell into simpler component elements. Equivalently, we
are interested in complex periodic materials that are formed by combining the
unit cells of simpler periodic materials. This is depicted in Figure 1. The
key question is whether the complex material inherits the band gaps of the
simpler materials that make up its unit cell. We consider systems with two
degrees of freedom, which could include a broad variety of mechanical,
acoustic or suitably polarized electromagnetic systems (such that the
Helmholtz equation captures the wave dynamics). In such systems, wave
propagation can be described by a two-by-two transfer matrix [19] and band
gaps are precisely the values at which the trace of the transfer matrix is
larger than 2 in absolute value. As a result, the question at the heart of
this study is whether the set

$\mathcal{M}:=\left\\{M\in\text{SL}_{2}(\mathbb{R}):|\tr(M)|>2\right\\},$
(1.1)

is closed under matrix multiplication. Here, $\text{SL}_{2}(\mathbb{R})$ is
the special linear group of $2\times 2$ matrices with real entries and
determinant 1. It is clear that this is not true in general. To see this, take
two arbitrary matrices $A,B\in\mathcal{M}$ such that there exists constants
$\alpha\neq\pm 1,a,b,c,d\in\mathbb{R}$ and some
$P\in\text{GL}_{2}(\mathbb{R})$ such that

$A\sim P^{-1}AP=\begin{bmatrix}\alpha&0\\\
0&\frac{1}{\alpha}\end{bmatrix}\quad\text{and}\quad B\sim
P^{-1}BP=\begin{bmatrix}a&b\\\ c&d\end{bmatrix},$ (1.2)

where $\sim$ is used to denote matrix similarity. Then, it holds that

$\tr(AB)=\tr(P^{-1}ABP)=\alpha a+\frac{d}{\alpha},$

from which it is clear that there are uncountably many matrices $B$ with
$|a+d|>2$ such that $|\alpha a+d/\alpha|\geq 2.$ Therefore, we must constrain
the problem and consider a subset of $\mathcal{M}$.

Figure 1: This work develops a theory for hierarchical band gaps in complex
periodic systems. These are band gaps in complex periodic systems which are
common with the band gaps of all the constituent simple systems. For example,
the spectra of three materials A, B and C are shown here, with their common
band gaps highlighted and inherited by the combined periodic material with
unit cell ABC. This presents a simple way to predict some, but not all, of the
band gaps in complex periodic systems.

In this work, we prove a partial result about the closure of subsets of
$\mathcal{M}$ that is sufficient for yielding insight on applications. Our
result concerns products of matrices belonging to the subset
$\mathcal{T}\subset\mathcal{M}$, given by

$\mathcal{T}:=\left\\{\begin{bmatrix}0&-1\\\ 1&t\end{bmatrix}:|t|>2\right\\}.$
(1.3)

We show that any finite product of elements of $\mathcal{T}$ will be an
element of $\mathcal{M}$. This result can be used to predict the spectra of a
class of one-dimensional difference equations with on-site modulation. In
particular, it shows that if the intersection of the band gaps of the
constituent simple (1-periodic) materials is non-empty, then that interval
will be within a band gap of the complex periodic material (as depicted in
Figure 1). We will refer to this band gap that has been inherited from the
constituent elements as a _hierarchical band gap_.

This paper will begin by laying out the problem setting in section 2,
including specifying how the set $\mathcal{T}$ is related to the Floquet-Bloch
spectrum of a class of difference equations. Section 3 contains the main
theoretical results. Finally, in section 4, we will consider some specific
examples of mechanical and acoustic systems to demonstrate the value of our
theory for predicting spectral features.

##  2 Problem setting

In this work, we consider a canonical problem given by the difference equation

$g(i+1)+g(i-1)+V(i)g(i)=F(i,\omega)g(i),$ (2.1)

where $V:\mathbb{Z}\to\mathbb{R}$ and
$F:\mathbb{Z}\times\mathbb{R}\to\mathbb{R}$ are given functions and
$\omega\in\mathbb{R}$ is a scalar eigenvalue that describes the frequency of
waves propagating through the system. The spectral problem in question is to
find all the values of $\omega\in\mathbb{R}$ such that (2.1) has a non-zero
solution $g:\mathbb{Z}\to\mathbb{R}$. Difference equations of this type arise
naturally in many different settings, such as tight-binding models for
electron dynamics, one-dimensional wave transmission problems or in systems of
coupled mechanical oscillators [19]. We will explore some examples in more
detail in section 4. The difference equation (2.1) is a typical example of a
discrete Hamiltonian with nearest-neighbour coupling and on-site modulation.

We are interested in the case where $V:\mathbb{Z}\to\mathbb{R}$ is periodic
and $F:\mathbb{Z}\times\mathbb{R}\to\mathbb{R}$ is periodic in its first
variable. That is, there is some $n>0$ such that $V(i+n)=V(i)$ and
$F(i+n,\cdot)=F(i,\cdot)$ for all $n\in\mathbb{Z}$. In this case, the spectrum
is known to consist of a countable number of continuous spectral bands. These
bands can be resolved using so-called Floquet-Bloch analysis, of which we now
summarise the main ideas (for a more detailed overview, see _e.g._ [15, 4]).

The difference equation (2.1) can be rewritten as

$\displaystyle\begin{bmatrix}g(i)\\\ g(i+1)\end{bmatrix}$
$\displaystyle=-\begin{bmatrix}0&-1\\\
1&V(i)-F(i,\omega)\end{bmatrix}\begin{bmatrix}g(i-1)\\\ g(i)\end{bmatrix}.$
(2.2)

Letting $\mathbf{u_{i}}=[g(i-1),g(i)]^{\dagger}$, where † is used to denote
the transpose, we can rewrite (2.2) as

$\mathbf{u}_{i+1}=-T_{i}\mathbf{u}_{i}\quad\text{where}\quad
T_{i}=\begin{bmatrix}0&-1\\\ 1&V(i)-F(i,\omega)\end{bmatrix}.$ (2.3)

Further, across the unit cell of length $n$ it holds that

$\mathbf{u}_{n+1}=(-1)^{n}\left(\prod_{i=1}^{n}T_{i}\right)\mathbf{u}_{1}.$
(2.4)

The key insight of Floquet-Bloch analysis is that the spectrum can be
decomposed into modes satisfying

$\mathbf{u}_{n+1}=e^{\mathrm{i}nk}\mathbf{u}_{1}$ (2.5)

for some $k\in\mathbb{R}$. For each $k$, there exists a discrete set of
countably many eigenvalues and the spectral bands of the periodic problem are
recovered by taking the union over all possible $k$. Clearly, it is sufficient
to take this union over $k\in[-\pi/n,\pi/n)$. Equating (2.4) with (2.5) and
taking the determinant gives

$\det\left((-1)^{n}\tr\left(\prod_{i=1}^{n}T_{i}\right)-e^{\mathrm{i}nk}I\right)=0.$

Using the fact that $\det T_{i}\equiv 1$, this reduces to the well-known
dispersion relation

$2\cos(nk)=(-1)^{n}\tr\left(\prod_{i=1}^{n}T_{i}\right).$ (2.6)

It is clear that (2.6) has a real solution for $k$ if and only if the right-
hand side falls in $[-2,2]$. Hence, the condition for an eigenvalue $\omega$
to belong to a spectral band of the periodic system is that the associated
transfer matrix has trace smaller than 2 in absolute value (with the
associated matrix being defined by (2.4), where it should be noted that the
matrices in the product depend on $\omega$).

The intervals of frequencies between spectral bands are known as _band gaps_
and are significant as waves at these frequencies are unable to propagate
through the material. The aim of this work is to characterise how complex
systems inherit band gaps from their constituent elements. Hence, we will show
that the product of matrices of the form (2.3) with
$|t_{i}|=|V(i)-F(i,\omega)|>2$ must also have trace larger than 2. We will say
that any $\omega$ which satisfies this condition belongs to a _hierarchical
band gap_ of the complex periodic material.

######  Definition 2.1.

Given $n\in\mathbb{N}$ and functions $t_{i}:\mathbb{R}\to\mathbb{R}$ for
$1\leq i\leq n$, a _hierarchical band gap_ is some interval
$(a,b)\subset\mathbb{R}$ such that for $\omega\in(a,b)$ it holds that both

$\left|\tr\begin{bmatrix}0&-1\\\
1&t_{i}\end{bmatrix}(\omega)\right|>2\quad\text{for all }1\leq i\leq n$ (2.7)

and

$\left|\tr\left(\,\prod_{i=1}^{n}\begin{bmatrix}0&-1\\\
1&t_{i}(\omega)\end{bmatrix}\right)\right|>2.$ (2.8)

The theory developed in section 3 will show that, in fact, (2.7) implies (2.8)
(_i.e._ the product of $n$ matrices belonging to $\mathcal{T}$ must belong to
$\mathcal{M}$). This demonstrates how band gaps persist in systems with
increasingly complex arrangements. Some examples of such arrangements will be
explored in section 4.

##  3 Main results

To see why products of matrices belonging to $\mathcal{T}$ must themselves
belong to $\mathcal{M}$, we exploit the patterns that occur when multiplying
matrices of the form (2.3). Before stating any results we introduce some
notation to simplify statements:

$\Pi_{k}^{l}:=\prod_{i=k}^{l}T_{i},$ (3.1)

where $T_{i}$ is some arbitrary matrix of the form

$T_{i}=\begin{bmatrix}0&-1\\\ 1&t_{i}\end{bmatrix}.$ (3.2)

We will use the notation $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ to denote the
standard basis elements in $\mathbb{R}^{2}$:

$\mathbf{e}_{1}=\begin{bmatrix}1\\\
0\end{bmatrix}\quad\text{and}\quad\mathbf{e}_{2}=\begin{bmatrix}0\\\
1\end{bmatrix}.$ (3.3)

It will be useful to begin by establishing that the result holds for edge
cases (_i.e._ $t_{i}\equiv 2$ and $t_{i}\equiv-2$), before moving on to the
general case.

######  Proposition 3.1.

For any $n\in\mathbb{N}$, it holds that

$\begin{bmatrix}0&-1\\\ 1&2\end{bmatrix}^{n}=\begin{bmatrix}0&-1\\\
1&2\end{bmatrix}+(n-1)\begin{bmatrix}-1&-1\\\ 1&1\end{bmatrix},$

the trace of which is always equal to 2.

###### Proof.

$\displaystyle\begin{bmatrix}0&-1\\\ 1&2\end{bmatrix}^{n}$
$\displaystyle=\begin{bmatrix}0&-1\\\
1&2\end{bmatrix}\cdot\left\\{\begin{bmatrix}0&-1\\\
1&2\end{bmatrix}+(n-1)\begin{bmatrix}-1&-1\\\ 1&1\end{bmatrix}\right\\}$
$\displaystyle=\begin{bmatrix}-1&-2\\\
2&3\end{bmatrix}+(n-1)\begin{bmatrix}-1&-1\\\ 1&1\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0&-1\\\
1&2\end{bmatrix}+n\begin{bmatrix}-1&-1\\\ 1&1\end{bmatrix}$

∎

######  Proposition 3.2.

For any $n\in\mathbb{N}$, it holds that

$\begin{bmatrix}0&-1\\\
1&-2\end{bmatrix}^{n}=(-1)^{n}\left\\{(n-1)\begin{bmatrix}-1&1\\\
-1&1\end{bmatrix}-\begin{bmatrix}0&-1\\\ 1&-2\end{bmatrix}\right\\},$

the trace of which is always equal to $\pm 2$.

###### Proof.

$\displaystyle\begin{bmatrix}0&-1\\\ 1&-2\end{bmatrix}^{n}$
$\displaystyle=(-1)^{n}\begin{bmatrix}0&-1\\\
1&-2\end{bmatrix}\left\\{(n-1)\begin{bmatrix}-1&1\\\
-1&1\end{bmatrix}-\begin{bmatrix}0&-1\\\ 1&-2\end{bmatrix}\right\\}$
$\displaystyle=(-1)^{n}n\begin{bmatrix}-1&1\\\
-1&1\end{bmatrix}+(-1)^{n+2}\begin{bmatrix}0&-1\\\ 1&-2\end{bmatrix}$
$\displaystyle=(-1)^{n}\left\\{n\begin{bmatrix}-1&1\\\
-1&1\end{bmatrix}-\begin{bmatrix}0&-1\\\ 1&-2\end{bmatrix}\right\\}$

∎

We are now ready to handle the general case of $|t_{i}|\geq 2$.

######  Lemma 3.1.

Let $t_{1},...,t_{n}\in\mathbb{R}$ be such that $\min_{1\leq i\leq
n}|t_{i}|\geq 2$, then

$\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{2}\right|>\left|\mathbf{e}_{1}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{2}\right|+1\quad\text{and}\quad\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{2}\right|>\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{1}\right|+1.$

###### Proof.

This follows by induction. For $n=1$, it holds that

$\Pi_{1}^{1}\equiv\begin{bmatrix}0&-1\\\ 1&t\end{bmatrix}\quad\text{for some
}t\in(-\infty,-2]\cup[2,\infty)$ (3.4)

so the result is trivially true.

To prove the first of the two statements, consider splitting the product as

$\Pi_{1}^{n}=\begin{bmatrix}0&-1\\\ 1&t_{1}\end{bmatrix}\cdot\Pi_{2}^{n},$
(3.5)

from which it is clear that

$\displaystyle\mathbf{e}_{1}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{2}=-\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2},$
(3.6)
$\displaystyle\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{2}=\mathbf{e}_{1}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}+t_{1}\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}.$
(3.7)

Where $\dagger$ denotes transpose. Taking absolute values gives

$\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{2}\right|\geq\left|\left\\{\left|t_{1}\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|-\left|\mathbf{e}_{1}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|\right\\}\right|,$
(3.8)

which can be simplified to see that

$\displaystyle\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{2}\right|$
$\displaystyle\geq\left|t_{1}\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|-\left|\mathbf{e}_{1}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|$
$\displaystyle=\left(|t_{1}|-1\right)\left|\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|+\left|\mathbf{e}_{2}\Pi_{2}^{n}\mathbf{e}_{2}\right|-\left|\mathbf{e}_{1}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|$
$\displaystyle>\left(|t_{1}|-1\right)\left|\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|+1$
$\displaystyle>\left|\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|+1,$

where the penultimate inequality follows from the inductive hypothesis. Using
(3.6), this becomes

$\displaystyle\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{2}\right|$
$\displaystyle>\left|\mathbf{e}_{1}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{2}\right|+1.$
(3.9)

The second statement can be shown using a similar approach. We use the formula

$\Pi_{1}^{n}=\Pi_{1}^{n-1}\cdot\begin{bmatrix}0&-1\\\ 1&t_{n}\end{bmatrix},$
(3.10)

to see that

$\displaystyle\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{1}=\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n-1}\mathbf{e}_{2},$
(3.11)
$\displaystyle\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{2}=-\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n-1}\mathbf{e}_{1}+t_{n}\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n-1}\mathbf{e}_{2}.$
(3.12)

Therefore, we have that

$\displaystyle\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{2}\right|$
$\displaystyle\geq\left|\left\\{\left|t_{n}\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n-1}\mathbf{e}_{2}\right|-\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n-1}\mathbf{e}_{1}\right|\right\\}\right|$
$\displaystyle=\left|t_{n}\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n-1}\mathbf{e}_{2}\right|-\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n-1}\mathbf{e}_{1}\right|$
$\displaystyle=\left(|t_{n}|-1\right)\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n-1}\mathbf{e}_{2}\right|+\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n-1}\mathbf{e}_{2}\right|-\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n-1}\mathbf{e}_{1}\right|$
$\displaystyle>\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n-1}\mathbf{e}_{2}\right|+1$
$\displaystyle=\left|\mathbf{e}_{2}^{\dagger}\Pi_{1}^{n}\mathbf{e}_{1}\right|+1,$

which proves the second condition. ∎

######  Theorem 3.1.

If for some arbitrary $n\in\mathbb{N}$, for all $i\leq n$, $|t_{i}|>2$, then

$\left|\tr\left\\{\prod_{1}^{n}\begin{bmatrix}0&-1\\\
1&t_{i}\end{bmatrix}\right\\}\right|>2.$

###### Proof.

Re-using the simple formula (3.5), we can see that

$\tr\Pi_{1}^{n}=\tr\left\\{\begin{bmatrix}0&-1\\\
1&t_{1}\end{bmatrix}\Pi_{2}^{n}\right\\}=-\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{1}+\mathbf{e}_{1}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}+t_{1}\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}.$
(3.13)

Taking absolute values and applying Lemma 3.1 gives that

$\displaystyle\left|\tr\Pi_{1}^{n}\right|$
$\displaystyle\geq\left|\left\\{\left|t_{1}\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|-\left|\mathbf{e}_{1}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|-\left|\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|\right\\}\right|$
$\displaystyle=\left|t_{1}\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|-\left|\mathbf{e}_{1}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|-\left|\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|$
$\displaystyle>\left\\{|t_{1}|-2\right\\}\left|\mathbf{e}_{2}^{\dagger}\Pi_{2}^{n}\mathbf{e}_{2}\right|+2$
$\displaystyle>2.\qed$

Theorem 3.1 is the main theoretical result of this work. It shows that the
product of a finite number of matrices in $\mathcal{T}$ will be an element of
$\mathcal{M}$. In other words, given a complex periodic material that has a
unit cell with length $n$, if there is an eigenvalue that is in the band gaps
of all of the $n$ periodic materials associated to the $n$ individual sites,
then that eigenvalue must be in a band gap of the complex periodic material.
Since the intersection of a finite number of open intervals is itself an open
interval, any such eigenvalues must appear in continuous bands. These are the
sought hierarchical band gaps.

It should be emphasised that Theorem 3.1 describes a sufficient but not
necessary condition for the complex periodic material to have a band gap.
Indeed, it is easy to construct examples where the trace of the product has
absolute value greater than two in spite of the fact that there is at least
one $i$ for which $|t_{i}|\leq 2$ (typically, many of the other values would
be $|t_{i}|\gg 2$ to compensate). Predicting the spectrum in full needs more
knowledge of the specific configuration of the system than is required for
Theorem 3.1. Nevertheless, the simple condition derived in Theorem 3.1 is
sufficient for predicting the main band gaps in complex systems where there is
some overlap between the spectra of the individual constituent elements.

##  4 Examples

The difference equation (2.1) is a canonical model for wave propagation in
one-dimensional systems. Some specific examples of appropriate systems are
coupled masses and springs, coupled pendulums and locally resonant phononic
crystals. These examples are surveyed below, to demonstrate the utility of
Theorem 3.1.

###  4.1 Masses and springs

The simplest canonical model for wave propagation in a lattice structure is
that of an array of masses coupled by springs, as depicted in figure 2. These
models have been studied as early as the 17th century (such as by Newton in
his Principia of 1686) and remain one of the primary toy models for exploring
wave dynamics in both quantum and classical settings [4].

Figure 2: The canonical difference equation studied here can be used to model
a system of masses (with modulated masses $m_{i}$) coupled by identical
springs (with spring constant $\kappa$).

We consider the propagation of time-harmonic waves with frequency $\omega$
through a chain of masses with mass $m_{i}$ that are connected by identical
springs with stiffness $\kappa$. Then, the dynamics are described by the
difference equation [17]

$u_{i+1}+u_{i-1}-2u_{i}=-\frac{\omega^{2}m_{i}}{\kappa}u_{i}.$ (4.1)

This is equivalent to (2.1) with $V(i)-E=\omega^{2}m_{i}/\kappa-2$. This
difference equation can be rewritten as

$\begin{bmatrix}u_{i}\\\ u_{i+1}\end{bmatrix}=-\begin{bmatrix}0&-1\\\
1&\frac{\omega^{2}m_{i}}{\kappa}-2\end{bmatrix}\begin{bmatrix}u_{i-1}\\\
u_{i}\end{bmatrix}$ (4.2)

where the matrix is an element of $\mathcal{T}$ if and only if

$\left|\frac{\omega^{2}m_{i}}{\kappa}-2\right|>2.$ (4.3)

It is immediate that for any simple singly periodic system with this matrix
there is one high-frequency band gap that occurs when
$\omega^{2}>\frac{4\kappa}{m_{i}}$. Therefore, it is expected that a complex
periodic arrangement of masses and springs will also inherit a high-frequency
band gap.

An example is shown in Figure 3, where we study a system of 5 masses with
masses $m_{1}=0.6$, $m_{2}=1$, $m_{3}=0.8$, $m_{4}=0.5$ and $m_{5}=1$, coupled
with identical springs with spring constant $\kappa=0.75$. The top five rows
show the pattern of pass and stop bands for each of the five periodic systems
corresponding to the single mass $m_{i}$ repeated periodically. The bottom row
shows the pass and stop bands for the complex system obtained by combining the
five masses. The hierarchical band gap, as predicted by our theory, is
highlighted by the hatched region. The bottom of the (high-frequency)
hierarchical band gap is highlighted by the dashed line, which occurs at
$4\kappa/m_{N}$ where $N=\operatorname*{arg\,inf}_{i\leq n}m_{i}$.

Figure 3: Hierarchical band gaps of a system of coupled masses and springs,
where the complex system inherits the common high-frequency band gap of its
constituent elements. A simple periodic structure is considered, composed of
five constituent elements.

The high-frequency band gap of a modulated system of masses and springs is
widely observed in the physical literature. Modifying the asymptotic arguments
from [7, Theorem 4.1], for example, it is easy to show that any periodic mass-
spring system will exhibit a band gap at sufficiently high frequencies. The
theory developed in the present work extends this result by giving a
quantitative estimate for the minimum of this high-frequency band gap, showing
that it is at most the minimum of the intersection of band gaps of the
constituent elements.

It is important to note that our theory for hierarchical band gaps does not
contain any information about the relative positioning of the masses within
the unit cell of the complex material. As a result, the hierarchical band gap
will exist for any arrangement of the constituent masses (including adding
many repetitions of some of them). This is one reason why we cannot expect, in
general, the hierarchical band gap to provide tight bounds on the band gaps of
the complex periodic system.

###  4.2 Coupled pendulums

The addition of local interactions is an immediate and important development
of the mass-spring model. This can be achieved, for example, by adding
pendulums to the masses, as depicted in Figure 4. This is a canonical model
for wave propagation in locally resonant systems and mimics the key properties
of the tight-binding models used in quantum physics [8].

####  4.2.1 Homogeneous Pendulums

Consider the system presented in Figure 4 with identical springs of stiffness
$\kappa$ and identical pendulums of length $l$ attached to masses, which are
allowed to vary. We will address the case of non-identical pendulums in
section 4.2.2.

Figure 4: A system of coupled pendulums (which are coupled by indentical
springs) can be described by difference equations of the form considered in
this work.

Let $u_{n}$ be the one-dimensional displacement of the sphere with mass
$m_{n}$, then resolving the forces due to both the springs and the pendulum we
arrive at the difference equation [19]

$\left(\omega_{i}^{2}+\frac{1}{m_{i}}(2\kappa-\omega^{2})\right)u_{i}=\frac{1}{m_{i}}(\kappa
u_{i-1}+\kappa u_{i+1}).$ (4.4)

Here, $\kappa$ is the spring constant, $\omega$ is the frequency of the wave
in the system and $\omega_{i}$ is the eigenfrequency of the pendulum with
index $i$. This eigenfrequency is given by $\omega_{i}^{2}=\frac{g}{l}$ where
$l$ is the length of the pendulum and $g$ is the acceleration due to gravity.
For simplicity, in the case of identical pendulums we denote this as
$\omega_{0}$ to remove dependence on $i$. We can rewrite (4.4) as the linear
system

$\begin{bmatrix}u_{i}\\\ u_{i+1}\end{bmatrix}=\begin{bmatrix}0&1\\\
-1&2-\frac{\omega^{2}m_{i}}{\kappa}+\frac{\omega_{0}^{2}m_{i}}{\kappa}\end{bmatrix}\begin{bmatrix}u_{i-1}\\\
u_{i}\end{bmatrix}$ (4.5)

This matrix has trace larger than 2 if

$\left|2-\frac{\omega^{2}m_{i}}{\kappa}+\frac{\omega_{0}^{2}m_{i}}{\kappa}\right|>2,$

which is equivalent to

$\omega^{2}<\omega_{0}^{2}\quad\text{or}\quad\omega^{2}>\frac{4\kappa}{m_{i}}+\omega_{0}^{2}.$
(4.6)

We can see that the introduction of the pendulums has introduced a low-
frequency band gap that was not present in the mass-spring system considered
in section 4.1.

Figure 5: Hierarchical band gaps of a system of coupled pendulums, with a
simple periodic unit cell consisting of five elements. The complex pendulum
periodic structure inherits both high- and low-frequency hierarchical band
gaps from the constituent systems.

An example of the band gaps corresponding to a system of five masses and
pendulums is shown in Figure 5. The masses have been chosen to be such that
$m_{1}=0.6$, $m_{2}=0.4$, $m_{3}=0.8$, $m_{4}=0.7$ and $m_{5}=0$, with
$\kappa=0.5$ and $\omega_{0}^{2}=2$. This gives a range of different band gaps
(with common lower limits due to the identical pendulums). The corresponding
complex periodic system has hierarchical band gaps at either ends of its
spectrum, as the complex system inherits both the low- and high-frequency band
gaps from the constituent elements. Of course, the behaviour of the spectrum
in the frequency range $\omega_{0}^{2}<\omega<w_{0}^{2}+4m_{N}\kappa$ is
undetermined, a reminder of the sufficient but not necessary nature of the
hierarchical band gap condition.

####  4.2.2 Modulated Pendulums

More intricate spectra can be obtained by varying the lengths of the
pendulums. In which case, the eigenfrequency $\omega_{i}$ in (4.4) varies and
we obtain the linear system

$\begin{bmatrix}u_{i}\\\ u_{i+1}\end{bmatrix}=\begin{bmatrix}0&1\\\
-1&2-\frac{\omega^{2}m_{i}}{\kappa}+\frac{\omega_{n}^{2}m_{i}}{\kappa}\end{bmatrix}\begin{bmatrix}u_{i-1}\\\
u_{i}\end{bmatrix}.$ (4.7)

Similar to (4.6), we find the condition for each individual mass-pendulum
system to support a band gap as

$\omega^{2}<\omega_{i}^{2}\quad\text{or}\quad\omega^{2}>\frac{4\kappa}{m_{i}}+\omega_{i}^{2}.$
(4.8)

Then, under the same assumptions as Figure 5, we build a system with varying
lengths. In particular, we suppose that the pendulums have one of two
alternating lengths: $l_{2n}=20$ and $l_{2n+1}=5$. If, for simplicity, we take
$g=10$, then this leads to $\omega_{2n}^{2}=0.5$ and $\omega_{2n+1}^{2}=2$.
Choosing the spring stiffness and masses as $\kappa=0.5$, $m_{1}=1.2$,
$m_{2}=2$, $m_{3}=1$, $m_{4}=2.2$ and $m_{5}=1.2$, we reach the pattern of
spectral bands shown in Figure 6.

Figure 6: Hierarchical band gaps of a modulated array of coupled pendulums,
again with a simple periodic unit cell consisting of five elements. Here, the
modulation means that there is a mid-frequency hierarchical band gap in
addition to the high- and low-frequency hierarchical band gaps (_cf._ Figure
5). The hierarchical band gaps are $0<\omega^{2}<\Omega_{1}$,
$\Omega_{2}<\omega^{2}<\Omega_{3}$ and $\Omega_{4}<\omega^{2}$, as specified
in (4.9).

In Figure 6, we can see that modulating the pendulum length causes the band
gaps of the individual systems to be shifted. As a result, there are
overlapping band gaps in the centre of the frequency range, meaning that the
complex periodic system has a hierarchical band gap here, in addition to those
at low and high frequencies, that were observed in Figure 5. In particular,
the hierarchical band gaps occur at $0<\omega^{2}<\Omega_{1}$,
$\Omega_{2}<\omega^{2}<\Omega_{3}$ and $\Omega_{4}<\omega^{2}$, where

$\begin{split}\Omega_{1}=\min_{1\leq i\leq
n}\omega_{i}^{2},\qquad\Omega_{2}=\min_{1\leq i\leq
n}\frac{4\kappa}{m_{i}}+\omega_{i}^{2},\\\ \Omega_{3}=\max_{1\leq i\leq
n}\omega_{i}^{2},\qquad\Omega_{4}=\max_{1\leq i\leq
n}\frac{4\kappa}{m_{i}}+\omega_{i}^{2}.\end{split}$ (4.9)

That $\Omega_{2}\leq\Omega_{3}$ in this example is the key property that leads
to the hierarchical band gap in the centre of the spectrum.

This model of coupled pendulums is significant due to its applications in
understanding electron dynamics, as outlined in [8]. The arrangement of
alternating pendulums was chosen for simplicity of the example and could
easily be generalised to a finite number of different lengths to accommodate
the variety of problems arising from quantum physics.

###  4.3 Resonant phononic crystals

Lastly we consider periodic systems of coupled acoustic resonators. This could
be either a one-dimensional acoustic waveguide containing resonant material
inclusions, as studied by _e.g._ [30, 16] and sketched in Figure 7(a), or a
coupled array of Helmholtz resonators, as studied by _e.g._ [9, 23, 25, 29]
and sketched in Figure 7(b). The local resonance in both systems can be
modelled using a frequency-dependent effective mass term that is singular at
resonance.

(a) Phononic crystals composed of a soft material (such as air or rubber)
surrounded by a much less compressible material (such as water) experience
subwavelength local resonances. The resonant frequencies can be modulated by
varying the size, shape or material parameters of the inclusions.

(b) Coupled Helmholtz resonators. The resonant frequencies can be modulated by
varying the size and shape of the Helmholtz resonators.

Figure 7: The propagation of acoustic waves in one-dimensional locally
resonant phononic crystals can be modelled by the difference equations
considered in this work. The local resonances could be due to, for example,
Helmholtz resonators or high-contrast material inclusions.

The canonical model for wave propagation in locally resonant phononic systems
is the difference equation

$\left(2\kappa-M_{i}^{\mathrm{eff}}(\omega)\omega^{2}\right)u_{i}=(\kappa
u_{i-1}+\kappa u_{i+1}),$ (4.10)

where the frequency-dependent effective mass $M_{i}^{\mathrm{eff}}(\omega)$ is
given by

$M_{i}^{\mathrm{eff}}(\omega)=M+\frac{m\omega_{i}^{2}}{\omega_{i}^{2}-\omega^{2}}.$
(4.11)

Here, $\omega_{i}$ is the resonant frequency of the locally resonant elements
and $m$ and $M$ are constants. This was derived in the appendices of [30], for
example.

An example of the hierarchical band gaps that exist within the discrete system
(4.10) is given in Figure 8. In this example, the effective mass term
$M_{i}^{\text{eff}}$ has been modulated by varying the resonant frequency so
that $\omega_{1}^{2}=6$, $\omega_{2}^{2}=4$, $\omega_{3}^{2}=8$,
$\omega_{4}^{2}=6$ and $\omega_{5}^{2}=10$, with $m=0.5$ and $M=2$ fixed. In
this case, all the unit cells have low frequency pass bands, so as a result
there is no low frequency band gap. However, the resonances create common band
gaps at intermediate frequencies, leading to a hierarchical band gap in the
interval $\Omega_{1}<\omega^{2}<\Omega_{2}$, as highlighted in Figure 8. The
constituent resonant elements also have high-frequency band gaps, that are
similarly inherited by the complex structure and predicted by Theorem 3.1.

Figure 8: The hierarchical band gaps of a locally resonant phononic crystal
with a complex unit cell inherits some of the band gaps from its constituent
resonators, including a high-frequency hierarchical band gap. The hierarchical
band gaps are $\Omega_{1}<\omega^{2}<\Omega_{2}$ and $\Omega_{3}<\omega^{2}$.

###  4.4 Fibonacci tilings

An example of periodic materials with complex unit cells that have been
studied extensively are those based on Fibonacci tilings. That is, given two
simple materials A and B, a sequence of materials can be formed through the
substitution rule

$\text{A}\mapsto\text{AB},\quad\text{B}\mapsto\text{A}.$ (4.12)

This leads to the sequence of materials A, AB, ABA, ABAAB, ABAABABA, and so
on. This sequence earns the name “Fibonacci” as each element can be obtained
by combining the previous two and the number of elements in each unit cell is
given by the corresponding Fibonacci number.

Systems based on the Fibonacci tiling have exotic properties, particularly in
the limit of the sequence, which converges to a material that is non-periodic
and quasicrystalline [10]. Interesting questions concern the system’s
topological properties [14] and its ability to localise wave energy [12].
Materials based on Fibonacci tilings have been used to create waveguides [6]
and realise negative refraction [21].

An important question for wave systems based on Fibonacci tilings is how the
Floquet-Bloch spectrum behaves in the limit of the sequence. It is known that
the limiting system typically has a pattern of spectral bands that is a Cantor
set [13, 26]. Further, it has been observed that the sequence is such that
there are band gaps which emerge early in the sequence and persist for all
subsequent iterations (sometimes known as _super band gaps_) [7, 22].

The stability of band gaps in the sequence of Fibonacci tilings can be
partially but straightforwardly explained using our theory of hierarchical
band gaps. This is demonstrated in Figure 9, where we consider the system of
modulated coupled pendulums from Section 4.2.2. We take the constituent
materials A and B to be the materials with associated matrices $T_{1}$ and
$T_{2}$ as in Figure 6. Then, we consider unit cells given by successive
Fibonacci tilings:

$F_{1}=T_{1},\,F_{2}=T_{2},\,F_{3}=T_{2}T_{1},\,F_{4}=T_{2}T_{1}T_{2},\,F_{5}=T_{2}T_{1}T_{2}T_{2}T_{1},\,F_{6}=T_{2}T_{1}T_{2}T_{2}T_{1}T_{2}T_{1}T_{2}.$
(4.13)

Figure 9: Coupled pendulums arranged in periodic unit cells based on Fibonacci
tilings exhibit intricate spectra, some features of which can be predicted
using the theory of hierarchical band gaps. The system inherits a high- and
low-frequency band-gaps as well as a mid-frequency band gap.

It can be seen from Figure 9 that the hierarchical band gaps, predicted by
Theorem 3.1, persist throughout the sequence of Fibonacci tilings as the unit
cell grows and the spectrum becomes increasingly intricate. Clearly, however,
Theorem 3.1 cannot hope to describe the exotic behaviour of the intricate
pattern of spectral bands and gaps that emerges outside of these intervals.
This behaviour is due to the geometric self similarity of the system (having
arbitrarily large sections that repeat infinitely often, even in the limiting
non-periodic system) and requires approaches similar to _e.g._ [26, 7] to
handle. Nevertheless, particularly with wave control applications in mind,
being able to predict some of the main spectral gaps without needing to invoke
such sophisticated machinery is valuable.

##  5 Concluding remarks

This work provides a simple approach to predict band gaps of complex periodic
systems by decomposing them into simpler (singly periodic) systems, whose
spectra can often be calculated explicitly. This is achieved by re-framing the
Floquet-Bloch spectral problem as a problem of matrix algebra. We have proved
that, although the set $\mathcal{M}\subset\text{SL}_{2}(\mathbb{R})$ of real
unimodular two-by-two matrices with trace greater than two is not closed under
matrix multiplication, it is possible to write down a subset $\mathcal{T}$ of
matrices whose products must be in $\mathcal{M}$. This result, stated in
Theorem 3.1, exploits the symmetries of the subset $\mathcal{T}$. This implies
that, given a collection of singly periodic systems (each described by a
version of the difference equation (2.1)) that has a common band gap, any
complex periodic system formed by combining their unit cells must have a
(hierarchical) band gap at that frequency range.

Some valuable generalisations of this result would be, firstly, to discrete
systems with modulated couplings in addition to the on-site modulation present
in (2.1). We conjecture that it should similarly be possible to formulate a
subset of matrices in $\mathcal{M}$ whose elements correspond to these systems
and whose products are themselves in $\mathcal{M}$. Further, generalisations
to multi-dimensional systems (either discrete or differential) would be
valuable but would require a modification of the two-by-two matrix framework
used here (either to larger matrix systems or to continuous operators).
Finally, it would be valuable to develop a theory capable of handling cases
where one (or more) of the matrices in a large product has trace less than
two. As is clear from the examples presented in section 4, Theorem 3.1 yields
a sufficient but not necessary condition for a complex periodic system to have
a band gap. While more refined approaches exist for specific examples (_e.g._
Fibonacci tilings [26, 7]), it would be useful to develop approaches in the
spirit of the method developed in this work, that reveal fundamental
principles for guiding the design of complex periodic materials without the
need for expensive computations.

## Declaration of Interests

The authors do not work for, advise, own shares in, or receive funds from any
organisation that could benefit from this article, and have declared no
affiliation other than their research organisations.

## Acknowledgement

The work of B.D. was supported the Engineering and Physical Sciences Research
Council (EPSRC) under grant number EP/X027422/1.

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