Distributing circuits over heterogeneous, modular quantum computing network architectures

1,7]Pablo Andres-Martinez
2,7]Tim Forrer
1,7]Daniel Mills
3]Jun-Yi Wu
1]Luciana Henaut
1]Kentaro Yamamoto
2,5]Mio Murao
1,4,6]Ross Duncan

[1]Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, UK
[2]Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
[3]Department of Physics, Tamkang University, 151 Yingzhuan Rd
[4]Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, UK
[5]Trans-scale Quantum Science Institute, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
[6]Department of Computer and Information Sciences, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK
[7]These authors contributed equally. Corresponding authors: {daniel.mills<EMAIL_ADDRESS>
We consider a heterogeneous network of quantum computing modules, sparsely connected via Bell states. Operations across these connections constitute a computational bottleneck and they are likely to add more noise to the computation than operations performed within a module. We introduce several techniques for transforming a given quantum circuit into one implementable on a network of the aforementioned type, minimising the number of Bell states required to do so.

We extend previous works on circuit distribution over fully connected networks to the case of heterogeneous networks. On the one hand, we extend the hypergraph approach of [Andres-Martinez & Heunen. 2019] to arbitrary network topologies. We additionally make use of Steiner trees to find efficient realisations of the entanglement sharing within the network, reusing already established connections as often as possible. On the other hand, we extend the embedding techniques of [Wu, et al. 2022] to networks with more than two modules. Furthermore, we discuss how these two seemingly incompatible approaches can be made to cooperate. Our proposal is implemented and benchmarked; the results confirming that, when orchestrated, the two approaches complement each other's weaknesses.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Virtual, USA

Association for Computing Machinery

booktitleProceedings of the 26th ACM International Conference on Architectural Support for Programming Languages and Operating Systems
seriesASPLOS '21
titleCutQC: Using Small Quantum Computers for Large Quantum Circuit Evaluations
Hybrid Computing,Quantum Circuit Cutting,Quantum Computing (QC)

titleCircuit knitting with classical communication
Quantum Physics (quant-ph),FOS: Physical sciences,FOS: Physical sciences

American Physical Society

journaltitlePRX Quantum
titleDoubling the Size of Quantum Simulators by Entanglement Forging

World Scientific

journaltitleInt. J. Q. Inf.
titleDistributed quantum computation architecture using semiconductor nanophotonics

American Physical Society (APS)

journaltitlePhysical Review A
titleLarge-scale modular quantum-computer architecture with atomic memory and photonic interconnects

IOP Publishing

journaltitleNew Journal of Physics
titleMinimally complex ion traps as modules for quantum communication and computing

abstractAs indispensable as the internet has become in our daily lives, it still has many shortcomings, not least of which is that communication can be intercepted and information stolen. If, however, the internet attained the capability of transmitting quantum information—qubits—many of these security concerns would be addressed. Wehner et al. review what it will take to achieve this so-called quantum internet and propose stages of development that each correspond to increasingly powerful applications. Although a full-blown quantum internet, with functional quantum computers as nodes connected through quantum communication channels, is still some ways away, the first long-range quantum networks are already being planned. Science, this issue p. eaam9288 The internet—a vast network that enables simultaneous long-range classical communication—has had a revolutionary impact on our world. The vision of a quantum internet is to fundamentally enhance internet technology by enabling quantum communication between any two points on Earth. Such a quantum internet may operate in parallel to the internet that we have today and connect quantum processors in order to achieve capabilities that are provably impossible by using only classical means. Here, we propose stages of development toward a full-blown quantum internet and highlight experimental and theoretical progress needed to attain them.
titleQuantum internet: A vision for the road ahead

New York, NY, USA

Association for Computing Machinery

journaltitleACM Transactions on Quantum Computing
titleOptimized Compiler for Distributed Quantum Computing
Quantum circuit compilation,Integer Linear Programming

American Physical Society (APS)

journaltitlePhysical Review A
titleAutomated distribution of quantum circuits via hypergraph partitioning

Dagstuhl, Germany

Schloss Dagstuhl – Leibniz-Zentrum für Informatik

annotationKeywords: Distributed Quantum Computing, Hypergraph Min-Cut
booktitle35th International Symposium on Distributed Computing (DISC 2021)
seriesLeibniz International Proceedings in Informatics (LIPIcs)
titleEfficient Distribution of Quantum Circuits

titleDistribution of Quantum Circuits Over General Quantum Networks
Emerging Technologies (cs.ET),Quantum Physics (quant-ph),FOS: Computer and information sciences,FOS: Computer and information sciences,FOS: Physical sciences,FOS: Physical sciences

titleEntanglement-efficient bipartite-distributed quantum computing with entanglement-assisted packing processes
Quantum Physics (quant-ph),FOS: Physical sciences,FOS: Physical sciences

Dagstuhl, Germany

Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik

annotationKeywords: Quantum Computing, Qubit routing, Compilation
booktitle14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)
seriesLeibniz International Proceedings in Informatics (LIPIcs)
titleOn the Qubit Routing Problem

Dagstuhl, Germany

Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik

annotationKeywords: quantum circuit, quantum architectures, circuit mapping
booktitle14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019)
seriesLeibniz International Proceedings in Informatics (LIPIcs)
titleCircuit Transformations for Quantum Architectures

IOP Publishing

journaltitleQuantum Science and Technology
titlet$\vert$ket⟩: a retargetable compiler for NISQ devices

American Physical Society (APS)

journaltitlePhysical Review A
titleOptimal local implementation of nonlocal quantum gates

journaltitleACM J. Exp. Algorithmics
titleHigh-Quality Hypergraph Partitioning

Open Publishing Association

journaltitleElectronic Proceedings in Theoretical Computer Science
titlePyZX: Large Scale Automated Diagrammatic Reasoning

American Physical Society

journaltitlePhys. Rev. Lett.
titleEfficient Behavior of Small-World Networks

American Association for the Advancement of Science (AAAS)

titleEmergence of Scaling in Random Networks

abstractNetworks of coupled dynamical systems have been used to model biological oscillators1,2,3,4, Josephson junction arrays5,6, excitable media7, neural networks8,9,10, spatial games11, genetic control networks12 and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks `rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them `small-world' networks, by analogy with the small-world phenomenon13,14 (popularly known as six degrees of separation15). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
titleCollective dynamics of `small-world' networks

American Physical Society (APS)

journaltitlePhysical Review A
titleValidating quantum computers using randomized model circuits

Open Publishing Association

journaltitleElectronic Proceedings in Theoretical Computer Science
titlePhase Gadget Synthesis for Shallow Circuits

Springer Science
Business Media LLC

journaltitleCommunications in Mathematical Physics
titleEfficient Quantum Algorithms for Simulating Sparse Hamiltonians

Springer Science
Business Media LLC

journaltitleNature Communications
titleA variational eigenvalue solver on a photonic quantum processor

American Physical Society

journaltitlePhys. Rev. A
titleQuantum algorithms for electronic structure calculations: Particle-hole Hamiltonian and optimized wave-function expansions

Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften

titleApplication-Motivated, Holistic Benchmarking of a Full Quantum Computing Stack

titleAn Introduction to Cartan's KAK Decomposition for QC Programmers
Quantum Physics (quant-ph),FOS: Physical sciences,FOS: Physical sciences

titleChemically Aware Unitary Coupled Cluster with ab initio Calculations on System Model H1: A Refrigerant Chemicals Application
Quantum Physics (quant-ph),Materials Science (cond-mat.mtrl-sci),FOS: Physical sciences,FOS: Physical sciences


# An intrinsic topological model in the absence of symmetry

Ye Xiong<EMAIL_ADDRESS>Department of Physics and Institute of
Theoretical Physics , Nanjing Normal University, Nanjing 210023, P. R. China

###### Abstract

We study the vibrational spectrum of a constrained classical ring. Due to the
presence of 2-order exceptional points, a topologically trivial band at the
infinity can make the vibrational band topologically nontrivial. The symmetry,
which is believed to be indispensable in topological models, is absent in this
model. The fractional boundary states can be found in such classical system.
Furthermore, the other aspect of the bulk boundary correspondence is revealed:
an extra fractional exceptional point is topologically protected in bulk by
the boundary states.

non-hermitian, exceptional point, fractional state

##  I Introduction

The exceptional point (EP) is playing very important rules in the non-
hermitian Hamiltonian systemsMoiseyev (2011); Berry (2004); Heiss (2012);
Mehri-Dehnavi and Mostafazadeh (2008); Liang and Huang (2013); Malzard  _et
al._ (2015); Cerjan  _et al._ (2016); Lin  _et al._ (2016); Xu  _et al._
(2017a); Bergholtz  _et al._ (2021). It can be obtained by further splitting a
2-fold degenerate weyl point when the Hamiltonian becomes non-hermitian. The
bands in the region between these EPs are connected so that one can reach the
other bands smoothly by varying the wave vector $k$ around the EP
continuouslyBergholtz  _et al._ (2021). Although the exceptional ring, a ring
of EPs in higher dimensional Brillouin zone (BZ), has been discussedXu  _et
al._ (2017b); Cerjan  _et al._ (2019), most of the physicists avoid to perform
the arguments right at the EP because the Hamiltonian at that point is
defective and cannot be diagonalized into the distinct bands completely. As a
result, the project operator on the bands and the metric matrix in the
eigenstate representation are both not well defined at the EP. But this is not
the case when the infinity emerges. Because a meaningful state cannot live in
the subspace with the infinite eigenvalue, one can avoid to define the metric
matrix there. Now if the eigenvalue of the Jordan matrix (the part of the
Hamiltonian that cannot be diagonalized) is approaching to the infinity, it is
possible to redefine the metric matrix to measure the length of the states
only in the remaining subspace with the finite eigenvalues.

Figure 1:  (a) In the upper panel, the two bands in the $k$ space for the
topological SSH model are shown. In the lower panel, the chain in the open
boundary condition has one boundary state at each end, which is composed by
the two half states (colored half circles) contributed from the two bands,
respectively. (b) In the upper panel, the band for the 1D topological
superconducting model is shown. Its duplication in the Nambu representation
below the fermi energy is not plotted. In the lower panel, the fractional
modes (majorana fermions) at the ends will couple nonlocally to form one
majorana zero mode. (c) In the upper panel, the vibrational band of our
classical model is schematically shown. There is only one band at the finite
value. The shape of the line has nothing to do with the actual dispersion of
the band and is only used to indicate the nontrivial zak phase of the band.
The non-hermitian skin effect will destroy the topology of the bulk states in
the open boundary condition. So in the lower panel, we consider a ring with
one part in the topologically trivial phase. Here the green part is the chain
of our model and the black part is a traditional elastic string in the
topologically trivial phase. The fractional modes should appear at the two
interfaces.

In the most cases, the states whose eigenvalues are at the infinity will have
nothing to do with the remaining states whose spectrum are at the finite
values (SAFV). The Hamiltonian can be written as

$H=\begin{pmatrix}H_{1}&V\\\ U&H_{2}\end{pmatrix},$ (1)

where $H_{1}$ is the Hamiltonian in the subspace with the finite spectrum,
$H_{2}$ is that with the infinite spectrum and their couplings are $U$ and
$V$, respectively. After projecting out the subspace of $H_{2}$, the finite
eigenvalue $\omega$ of $H$ can be obtained by diagonalizing the matrix
$H_{1}+V(\omega-H_{2})^{-1}U$. As $H_{2}$ is approaching to the infinite
matrix, $(\omega-H_{2})^{-1}$ is the zero matrix so that the second term in
the above expression is zero. It is the $H_{1}$ itself that provides the SAFV
and the corresponding eigenstates of the Hamiltonian. But when $H_{2}$ is
defective, e.g., $H_{2}=\infty^{2}\begin{pmatrix}0^{1}&1\\\
0&0^{1}\end{pmatrix}$, then $(\omega-H_{2})^{-1}=\begin{pmatrix}-0^{1}&1\\\
0&-0^{1}\end{pmatrix}$, where the superscript indicates the order of the
infinity and the infinitesimals with the rule $0^{1}\infty^{1}=1$. So if the
EP appears in $H_{2}$, the second term is not zero matrix any more and the
states at the infinity must be considered in the diagonalization. As a demo
model, we will consider the vibrational spectrum of a classical chain. At any
$k$ in the BZ, the equations of motion (EOM) have $3$ linear equations for the
$3$ unknown variables. One of the eigen-frequencies is at the finite value and
the other two are at the infinity with only one left (right) eigenstate. So
there is an exceptional BZ: a 2-order EP at every point in the BZ. The single
band at the finite value is topologically nontrivial, a band with $\pi$ zak
phase. We want to emphasize that the zak phase of the infinite band is $0$ so
that even counting the band at the infinity, the total zak phase is also in a
nontrivial value. In this sense, the topology in this model is intrinsic:
symmetry is not needed to balance a pair of bands to protect the nontrivial
topological index.

There is also correspondence between the topology of the bulk states and the
boundary states at the ends. In Fig. 1, we begin to present our arguments from
the other well-known 1-dimensional (1D) models. The spectrum of the 1D Su-
Schrieffer-Heeger (SSH) modelSu  _et al._ (1979, 1980); Kivelson and
Schrieffer (1982); Li  _et al._ (2014); Rhim  _et al._ (2017) are illustrated
in the upper panel in Fig. 1 (a). The two bands are denoted by the red and the
blue lines, respectively. The zak phase of each band is $\pi$ in the
topologically nontrivial phase. The chiral symmetry is needed to balance the
two bands. When the open boundary condition is taken (shown in the lower
panel), each band leaves one half state at the end which is denoted by the
colored half circle in the figure. The topologically protected boundary states
(large circles) are the combination of the two half states at each end. In
Fig. 1 (b), we show the band of a 1D topological superconducting chain. There
is only one band because we have omitted its duplication below the Fermi
energy $E_{f}=0$ in the Nambu representation. The chiral symmetry is present
implicitly between the band and its duplication. When the boundaries are
introduced, the single band leaves one half state at each end. These
fractional states are the majorana states and their nonlocal combination is
the majorana zero mode.

The vibrational band of our classical model is schematically shown in the
upper panel in Fig. 1 (c). The exact vibrational spectrum are on a complex
plane and the shape of the line here has nothing to do with the dispersion of
the spectrum. It is only used to indicate that the zak phase of such band is
$\pi$. The chiral symmetry is absent because there is only one band at the
finite value. Adopting the similar arguments in the previous subfigures, it
seems that we can find isolated fractional states at the ends of the chain.
But, unfortunately, the non-hermitian skin effect emerges first and destroys
the topology of the bulk statesSilveirinha (2019); Lee and Thomale (2019); Lee
_et al._ (2020); Helbig  _et al._ (2020); Zhang  _et al._ (2021). To avoid the
skin effect, we embed a topologically trivial chain (black part of the ring in
the lower panel) in our chain. The interfaces between the regions with
different topological phases are equivalent to the open ends in the
topological sense. But due to the absence of the chiral symmetry, we don’t
find a zero frequency mode. Instead, the whole spectrum of the black part of
the ring are modified. Such phenomenon breaks some of our knowledges on the 1D
vibrational chain. The black part of the ring is in the topologically trivial
phase and is equivalent to a normal elastic string in our traditional life.
The textbooks on phonons tell us that the local decorations, including the
connections to the other chains at the ends, can only introduce a few local
vibrational modes. For what reason the connections to the green chain at the
two far ends can modify the bulk spectrum of an elastic string?

We find that it can be traced back to the fractional states at the interfaces
and the origin of the topology in the green part. First, as that has been
discussed in the previous models, the nontrivial zak phase in the green chain
ensures that there are fractional states at the interfaces. Second, the bulk
boundary correspondence actually declares that the topologically protected
boundary states are caused by a kind of split of the states in the bulk
topological band. While in our model, the bulk topological band is related to
the band at the infinity through the help of EPs. The similar relation should
also exist behind the fractional boundary states: one more state at the
infinity is required to accompany with the boundary states. For a ring with
$L_{1}$ unit cells in the green part, we find $L_{1}-1$ EPs (remember that
there is one 2-order EP at each $k$ which will map to $L_{1}$ EPs in the real
space representation) are still kept in the 2-order. But one of the EPs is
changed from 2 to 3 order. This is topologically equivalent to an appearance
of a half 2-order EP. Such phenomenon is universal, regardless of the details
near the interfaces, the parameters of the model and the disorders in the
bulk. So this model actually reveals a boundary bulk correspondence: an extra
fractional EP in bulk is topologically protected by the boundary states. This
explains the spectrum modification because the whole spectrum must be modified
dramatically when a new EP appearsXiong (2018). The eigenstates of these
modified spectrum are spatially localized at one interface. But they can
respond to the global effective magnetic flux so that the nonlocal correlation
in between the fractional states is still present in such classical system.

##  II  a 1D single band topological lattice

Figure 2:  (a) Our model is based on a 1D classical vibrational chain. The
nearest neighboring MPs are connected by the springs. On the top of each MP,
there is a speedometer that measuring its velocity. The rigid bonds are
connecting each MP with the pointer of the speedometer in its front. This will
restrict the displacement of the $i$th MP, $x_{i}$, with the velocity of the
$i+1$th MP, $\dot{x}_{i+1}$, by the constraint $x_{i}-\dot{x}_{i+1}=0$. Figure
3:  The inverse spectrum $\frac{1}{\omega}$ for the ring shown in the lower
panel in Fig. 1 (c). The lengths of the topologically nontrivial (green) part
and topological trivial (black) part are $L_{1}$ and $L_{2}$, respectively.
(a) $L_{1}=39$ and $L_{2}=1$. (b) $L_{1}=32$ and $L_{2}=8$. (c) $L_{1}=2000$
and $L_{2}=1000$. The inverse spectrum on the real axis outside $[-3:3]$ are
not shown.

The 1D classical lattice is made up by the identical mass points (MPs) with
the mass $m=1$. Each MP can only move in the direction along the chain in the
$x$ direction. The springs, whose coefficients are $K=1$, are connecting the
nearest neighboring MPs. The active constraints (ACs),
$x_{i}-\dot{x}_{i+1}=0$, are restricting the displacement of the $i$th MP with
the velocity of the next one. They are realized by the rigid bonds between the
$i$th MP and the pointer of the speedometer on the top of the $i+1$th MP, as
the Fig. 2 shows. For each MP, there are $3$ unknown variables: its
displacement $x_{i}$, its velocity $\dot{x}_{i}$ and the stress force in the
bond $\lambda_{i}$. The linear first order time derivative EOM for the $i$th
MP are

$\displaystyle\dot{x}_{i}$ $\displaystyle\equiv$ $\displaystyle\dot{x}_{i}$
(2) $\displaystyle m\ddot{x}_{i}\equiv m\dot{\dot{x}}_{i}$ $\displaystyle=$
$\displaystyle K(x_{i+1}+x_{i}-2x_{i})+\lambda_{i}$ (3) $\displaystyle
x_{i}-\dot{x}_{i+1}$ $\displaystyle=$ $\displaystyle 0.$ (4)

We want to emphasize that one will find a wrong EOM by applying the Lagrange
multiplier directly in the Lagrangian. This is because the Newton’s third law
is not obeyed on the chain itself: the reaction force in the bond does not act
back on the $(i+1)$th MP but on the pointer of the speedometer. After the
Fourier transformation on the $3$ variables, we can find the EOM for
$|\psi_{k}\rangle\equiv(x_{k},\dot{x}_{k},\lambda_{k})^{T}$,

$\begin{pmatrix}1&0&0\\\ 0&m&0\\\
0&0&0\end{pmatrix}\dot{|\psi_{k}\rangle}=\begin{pmatrix}0&1&0\\\
-4K\sin^{2}(k/2)&0&1\\\ 1&-e^{ik}&0\end{pmatrix}|\psi_{k}\rangle.$ (5)

After substituting $|\psi_{k}\rangle=e^{-i\omega t}|\psi_{k}(\omega)\rangle$,
the eigen-frequency $\omega$ and the eigenstate $|\psi_{k}(\omega)\rangle$ are
found by solving a general eigen-problem likes, $-i\omega
C|\psi(\omega)\rangle=D|\psi(\omega)\rangle$, where $C$($D$) is the $3\times
3$ matrix on the left(right) side of Eq. 5. Here $C$ is an irreversible matrix
and $D$ is reversible. By multiplying $i\frac{D^{-1}}{\omega}$ from the left,
the above equation is equivalent to the eigen-problem of a non-hermitian
matrix $D^{-1}C$,

$\frac{i}{\omega}|\psi_{k}(\omega)\rangle=D^{-1}C|\psi_{k}(\omega)\rangle.$
(6)

We find that the matrix $D^{-1}C$ is defective and its eigenvalues are,
$e^{ik}$ and $0$. The first one describes the vibration band whose dispersion
is $\omega=ie^{-ik}$. The other is the band at the infinity with a 2-order
exceptional point. The eigenstates for the SAFV are
$|\psi^{1}_{k}\rangle=(1,e^{-ik},me^{-2ik}+4K\sin^{2}(k/2))^{T}$, while that
for the band at the infinity is $(0,0,1)^{T}$. This result is what we expected
in the introduction: there is an EP at every $k$ in the BZ and the
corresponding spectrum are at the infinity.

In the presence of EP, one will fail to decompose an arbitrary state into the
superposition of the eigenstates. Just as the above equations show, the
dimension of the Hilbert space is $3$ at every $k$ while there are only $2$
available (right) eigenstates. But our model is special at the point that the
spectrum corresponding to the EP are at the infinity. All modes carrying the
infinite vibrational frequency are unobservable in the real experiment. The
available state in the Hilbert space is just $c|\psi^{1}_{k}\rangle$, where
$c$ is a real number. So the effective dimension of the Hilbert space is one
and this gives us more freedom to define the metric matrix $\hat{g}$, which
measures the length of a vector in the Hilbert space by
$\langle\psi|\hat{g}|\psi\rangle$. Actually, any hermitian matrix with nonzero
$\langle\psi^{1}_{k}|\hat{g}|\psi^{1}_{k}\rangle$ is a good candidate. But
because we are going to connect the chain to a traditional elastic string, to
avoid further discussions on the domain wall between regions with different
metric matrix, the metric matrix here is taken as

$\hat{g}=\frac{1}{2}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&0\end{pmatrix},$ (7)

will be the same as that in a topologically trivial chain.

The berry phase is defined as
$-i\langle\psi^{1}_{k}|\hat{g}\frac{\partial}{\partial k}|\psi^{1}_{k}\rangle$
and the zak phase is the integration of the berry phase in the BZ:

$\int_{0}^{2\pi}-i\langle\psi^{1}_{k}|\hat{g}\frac{\partial}{\partial
k}|\psi^{1}_{k}\rangle dk.$ (8)

It is easy to verify that this zak phase is $\pi$, at the topologically
nontrivial value.

By changing the representation for $k$ space to the real space, for a ring
with $N$ unit cells, there are $N$ inverse eigen-frequencies on the unit
circle on the complex plane. They are belong to the band at the finite value.
Simultaneously, there are also $2N$ inverse eigen-frequencies at $0$ that
corresponding to the $N$ 2-order EPs with the spectrum at the infinity. When
the open boundary condition is taken, a non-hermitian skin effect appears and
destroys all bulk states of the ring.

To avoid the skin effect, we embed a topologically trivial chain into the
ring. It is realized by erasing a sequence of ACs in our model because in the
absence of AC, the chain is just the simplest model on phonon in many
textbooks. It is absolutely topologically trivial and can be considered as the
simplified model of a traditional elastic string.

We first show the spectrum for the ring with $L_{1}=39$ and $L_{2}=1$ in Fig.
3 (a). It is realized by erasing one AC on the perfect ring. In most of our
beliefs, when a constraint is erased, one more degree of freedom is left free
so that a new vibrational mode with the finite frequency should be observed on
the spectrum. But the result is surprisingly opposite. In the figure, there is
$39$ points near the unit circle. They are the bulk states with the SAFV in
the green part. Besides these, the rest states are at $\frac{1}{\omega}=0$.
This means, when one AC is removed from the ring, one more vibrational mode is
frozen, instead of being released.

When we increase the length of black part to $L_{2}=8$, as the Fig. 3 (b)
shows, the vibrational modes are not frozen exactly. But their frequencies are
still strongly amplified. The inverse spectrum of the green part are on a
circle a little smaller than the unit circle. This is the finite size effect.
They will approach to the unit circle as increasing $L_{1}$.

In Fig. 3 (c), we show the case when $L_{1}=2000$ and $L_{2}=1000$. For a long
isolated black chain, its spectrum are $\omega=\pm
2\sqrt{\frac{K}{m}}\sin(k/2)$ in the $k$ space. So besides the boundary
states, the bulk spectrum should be in the range $[-2,2]$ on the real axis.
But our results show that all the inverse spectrum within the unit circle have
been altered from the real axis. The percentage of these modified states is
kept at about $66\%$ and does not decrease as increasing $L_{2}$.

Now we summarize the common features in the above subfigures. For a ring with
$L_{1}$ topologically nontrivial part and $L_{2}$ topologically trivial part,
there are $L_{1}$ states with the inverse spectrum on the unit circle. This
means that the bulk states in the green part are not modified by the
embedment. It will ensure the validity of the topological arguments in the
following.

As we have discussed in the introduction, the existence of the fractional
states at the interfaces is protected by the nontrivial topology in the green
bulk. These fractional states may couple together nonlocally to form one
state, just like that in the topological superconducting case. Although there
is no chiral symmetry to fix its energy to zero, the above inverse spectrum
still confirm its existence. The trace of the $D^{-1}C$ matrix, $Tr(D^{-1}C)$,
is a real number. This means the summation of the spectrum is an imaginary
number. In most cases, the eigenvalues will appear in pair with $\pm
Re(\frac{1}{\omega})$. If we look at the imaginary axis with
$Re(\frac{1}{\omega})=0$, besides the spectrum at the infinity that
corresponding to the EPs, there are even number of eigenstates on the axis.
But if further eliminating the bulk states in the green part, there is only
one state left on the imaginary axis. This confirms our conclusions on the
nontrivial topology in the green bulk and the fractional states at the
interfaces.

There are also $L_{1}$ EPs accompanied with the above bulk states. Their
spectrum are still at the infinity, $\frac{1}{\omega}=0$. But different from
that in the $k$ space, only $L_{1}-1$ of them are still the $2$-order EPs. The
rest one EP is changed to the $3$-order EPs. This feature is robust against
the disorders in the ring or the details near the interfaces. So mostly, such
feature should also have a topological explanation. When the boundary
condition of a topological model is changes from open to periodic, the extra
hoppings between the boundaries will coupling the boundary states back to the
topologically nontrivial band states. In another word, the boundary states are
a kind of split of the band states, i.e., the edge states in 2D quantum Hall
system can be considered as the split of bulk state with different chirality
or different sign of velocity and the end state of the SSH model is half from
the upper band and half from the lower band. So it seems that the two
fractional boundary states in our model are a split of one bulk state in the
topological band and they must also inherit one 2-order EP at the infinity
because every state in the band is accompanied with one EP. But different from
the case of the SSH model, as Fig. 3 (a) shows, a hopping between the
fractional states, which is realized by the spring here, cannot recombine
them. The bulk state can only be restored when the missing constraint is added
back. But one more variable, the force in the bond, is enrolled in this
process. So the boundary states should only accompany with one dimensional
subspace with infinite eigenvalue instead of two dimensional. To keep the
connection between the states at the infinity and the finite value, such one
dimensional subspace should also be the eigen-space of an EP. So the
fractional boundary states are accompanied with a half $2$-order EP. This is
consistent with our numerical results because a $3$-order EP is equivalent to
$1+\frac{1}{2}$ $2$-order EP topologically. So besides the bulk states that
can protect the existence of the boundary states, the boundary states can also
protect an extra state in the infinite bulk band.

The appearance of a new EP or the modification of the order of an EP must be
accompanied with the deformation of spectrum on the complex planeXiong (2018).
This is why the spectrum of the black part is different from that of a free
chain. By focusing on the eigenstates of those modified spectrum, we find that
they are localized spatially at one interface, instead of extended in the
black bulk or localized at the two interfaces simultaneously. But they still
represent the nonlocal correlation between the fractional states in between
the two interfaces because they can respond to the effective magnetic flux
$e^{ik^{\prime}}$ in the ring. At the first sight, a magnetic flux is
inapplicable in a classical vibrational ring. But if we consider the ring in
Fig. 1(c) as one supercell and build a long lattice with this supercell,
$k^{\prime}$ here is the wave vector for this lattice and the effective
magnetic flux $e^{ik^{\prime}}$ is added naturally during the calculation of
the spectrum in the $k^{\prime}$ space. The inverse spectrum are plotted in
Fig. 4 for $L_{1}=32$ and $L_{2}=8$. As $k^{\prime}$ is varying from $0$ to
$2\pi$, the spectrum of the green part are varying smoothly on the unit
circle. It further confirms that the spectrum on the unit circle are
correspond to the bulk states in the topologically nontrivial region. But the
inverse spectrum inside the unit circle, whose eigenstates are localized at
one interface, are still changing with $k^{\prime}$. Such behavior should not
exist in the absence of extra fractional EP because the effective magnetic
flux can be gauge transfered to a phase jump at the sites far away from the
localized states. The localized states cannot feel this phase jump so that
they cannot respond to the effective magnetic flux. But when the EP is
present, the spectrum can be very sensitive to the parameters so that the
localized states can feel the phase jumping far away from them. This also
explains why these inverse spectrum are varying chaosly with $k^{\prime}$.

Figure 4:  (a) the evolution of inverse spectrum with $k^{\prime}$ for the
ring with $L_{1}=32$ and $L_{2}=8$. Those spectrum on the unit circle are
varying on the circle smoothly and are outside the visitable region of this
figure. The modified spectrum do respond to the effective magnetic flux and
they are varying with $k^{\prime}$ chaosly. Figure 5:  (a) the inverse
spectrum when both the spring constant and the constraint are randomly
distributed along the ring. The length of the ring is the same as that in Fig.
3 (b).

All the above conclusions are robust against disorder. We consider both
constraint disorder and spring disorder: the constraint at the $i$th site is
$x_{i}-w_{i}\dot{x}_{i+1}=0$ and the spring constant between the $i$th and the
$i+1$th sites is $K_{i}=z_{i}K$, where $w_{i}$ and $z_{i}$ are randomly
distributed in $[0,2]$ and $[0.5,1.5]$, respectively. We find the inverse
spectrum in Fig. 5 are qualitatively same as that in Fig. 3 (b). Here all
other parameters are the same in these two figures.

##  III Conclusions and discussions

We find a new class of topological models that are not protected by the
symmetry. In traditional topological models, including non-hermitian models,
one needs to split the bands into two groups. In the fermion system, the bands
are naturally separated into the valence bands and the conduction bands by the
fermi energy. In the boson case, one still need to specify the gap that
separates the two groups of bands. This gap or the fermi energy is never taken
on the top or the bottom of the whole bands because the project operator on
one group of the bands becomes $\hat{P}=1$ or $\hat{P}=0$ in this case. Such
group of the bands must be topologically trivial because its project operator
commutes with the translation operator in the reciprocal space,
$e^{i\hat{r}\cdot\delta k}$, where $\hat{r}$ is the position operator and
$\delta k$ is an infinitesimal displacement along the Wilson loops in the BZYu
_et al._ (2011); Soluyanov and Vanderbilt (2011); Kivelson (1982); Zhao  _et
al._ (2014); Marzari and Vanderbilt (1997); Taherinejad  _et al._ (2014); Li
_et al._ (2016); Gresch  _et al._ (2017); Liu  _et al._ (2017);
Alexandradinata  _et al._ (2016); Lian  _et al._ (2017). So one must split the
bands in the middle gap, and the symmetry is needed to balance these bands
intergrouply or intragrouply. But in our model, there is only one band at the
finite value. The split of bands into two groups is not needed and not even
applicable. As a result, symmetry is not the key ingredient in this class of
topological model.

To be topologically nontrivial, the project operator on this single band
should not be $\hat{P}=1$. This can be realized only when there is an
invisible band at the far infinity and there are EPs to effectively couple
these two bands. In principle, the EPs with SAFV may also play a similar rule.
But one cannot define the length of the eigenstates in the full Hilbert space
in that case.

The model shows many interesting behaviors: it shows fractional states at the
interfaces in a classical chain, it reveals another aspect of the bulk
boundary correspondence that the topologically protected boundary states
further protect the existence of the fractional EP in bulk, the spectrum are
varying with the effective magnetic flux chaosly in such linear system.
Besides these, the model also leaves us with many interesting questions. For
instance, due to the freedom in the definition of metric matrix, it is
possible to find a topologically protected boundary state on an interface in
between two topologically trivial regions. Such boundary state should be
traced back to the metric matrix interface instead of the interface for the
topology of the bulk states. The model can also be extended to the higher
dimension. We have found that the in-gap boundary states are protected to be
extended in bulk on a 2D lattice with nonzero Chern number.

One of the weak points of the present model is the unstableness: there are
spectrum on the upper half plane of the complex plane. So one can only observe
the phenomena in a short time before the chain falls apart in experiment. We
are looking for a way to stabilize the chain. If we succeed in this, such
model will have potential application in engineering: one can obtain a stiff
chain (with higher vibrational eigen-frequencies) from a soft chain by
connecting the two ends to our model.

Acknowledgments.— The work was supported by the National Foundation of Natural
Science in China Grant Nos. 10704040.

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Now, it just remains to prove the two lemmas.

###### Proof of Lemma 5.3.2.2.

On $V\in\mathcal{J}$, the sequence

$(A\vee B)(V)\to(A\circledast B)(V)\to(A\circledast_{\text{red}}B)(V)$

takes the form

${\operatorname{colim}\displaylimits_{U_{1}\oplus U_{2}\to V}A(U_{1})\vee
B(U_{2})}\to{\operatorname{colim}\displaylimits_{U_{1}\oplus U_{2}\to
V}A(U_{1})\times B(U_{2})}\to{\operatorname{colim}\displaylimits_{U_{1}\oplus
U_{2}\to V}A(U_{1})\wedge B(U_{2})}$

after using

${\operatorname{colim}\displaylimits_{U_{1}\oplus U_{2}\to V}A(U_{1})\vee
B(U_{2})}\simeq A(V)\vee B(V)$

Hence it is a cofibre sequence since the sequences

$A(U_{1})\vee B(U_{2})\to A(U_{1})\times B(U_{2})\to A(U_{1})\wedge B(U_{2})$

are cofibre sequences.

On the level of derivative spectra, the required splitting is induced by

${{A\circledast B}}$${{A\times B}}$${{A\circledast*\vee*\circledast
B}}$${{A\vee B}}$$\scriptstyle{\simeq}$

where the map $A\circledast B\to A\times B$ is the product of $A\circledast
B\to A\circledast*$ and $A\circledast B\to*\circledast B$ (and we are using
the stability of $Sp^{\text{BO}(n)}$).

∎

###### Proof of Lemma 5.3.2.3.

Let $A=\Phi M$, $B=\Phi N$ for $M,N\in Sp^{\text{BO}(n)}$. Recall here that we
denote by $\Phi$ the inverse equivalence to
$\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})\to Sp^{\text{BO}(n)}$.

Then, by Yoneda, the following calculation for arbitrary $C\in
Sp^{\text{BO}(n)}$ shows that

$\Theta^{n}(A\circledast_{\text{red}}B)\simeq D_{O(n)}^{\vee}\otimes M\otimes
N$
$\displaystyle\text{Map}_{Sp^{\text{BO}(n)}}(\Theta^{n}(A\circledast_{\text{red}}B),C)$
$\displaystyle\simeq\text{Nat}_{*}(L_{n}(\Phi A\circledast_{\text{red}}\Phi
B),\Phi C)$ $\displaystyle\simeq\text{Nat}_{*}(T_{n}(\Phi
A\circledast_{\text{red}}\Phi B),\Phi C)$
$\displaystyle\simeq\text{Nat}_{*}(\Phi A\circledast_{\text{red}}\Phi B,\Phi
C)$ $\displaystyle\simeq\text{Nat}_{*}(\Phi A,\mathcal{F}_{*}(\Phi B,\Phi C))$
$\displaystyle\simeq\text{Nat}_{*}(\Phi A,L_{n}\mathcal{F}_{*}(\Phi B,\Phi C)$
$\displaystyle\simeq\text{Nat}_{*}(\Phi A,\Phi(\text{Map}_{Sp}(B,C)\otimes
D_{O(n)}))$
$\displaystyle\simeq\text{Map}_{Sp^{\text{BO}(n)}}(M,\text{Map}_{Sp}(N,C)\otimes
D_{O(n)})$
$\displaystyle\simeq\text{Map}_{Sp^{\text{BO}(n)}}(D_{O(n)}^{\vee}\otimes
M,\text{Map}_{Sp}(N,C))$
$\displaystyle\simeq\text{Map}_{Sp^{\text{BO}(n)}}(D_{O(n)}^{\vee}\otimes
M\otimes N,C)$

Almost all of the steps are straightforward applications of the equivalence
$\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})\to Sp^{\text{BO}(n)}$ and the
adjunctions for
$T_{n}\colon\text{Fun}(\mathcal{J},\mathcal{S}_{*})\to\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S}_{*})$ and ${L_{n}}_{|\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S}_{*})}\colon\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S}_{*})\to\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})$.
For the second step, we use that

$T_{n}(\Phi A\circledast_{\text{red}}\Phi B)\simeq L_{n}(\Phi
A\circledast_{\text{red}}\Phi B)$

which holds as $T_{n-1}(\Phi A\circledast_{\text{red}}\Phi B)\simeq*$, which
follows from the pullback square

${{\text{Poly}^{\leq
n}(\mathcal{J},\mathcal{S}_{*})}}$${\text{Homog}^{n}(\mathcal{J},\mathcal{S}_{*})}$${{\text{Poly}^{\leq
n-1}(\mathcal{J},\mathcal{S}_{*})}}$${{*}}$$\scriptstyle{\lrcorner}$

and monoidality of $T_{n-1}$.

∎

###  5.4 Final Result on Derivative Spectra of Monoidal Functors

One of the questions which started this project was

For a monoidal functor $F\colon\mathcal{J}\to\mathcal{S}$, what can be said
about its $n$-th derivative spectra $\Theta^{n}F$?

Using Proposition 5.3.1.1, we can finally give the following answer

######  Corollary 5.4.0.1.

Let $F\colon\mathcal{J}\to\mathcal{S}$ be a monoidal functor. Then, for any
$n\in\mathbb{N}$, there are $O(n)$-equivariant maps

$\Theta^{n}F\otimes\Theta^{n}F\to\Theta^{n}F\otimes D_{O(n)}$

###### Proof.

Fix $n\in\mathbb{N}$. Since $F$ is monoidal, it lifts to pointed spaces by
Proposition 5.1.0.2. By our main theorem (Theorem 4.2.0.1), $T_{n}F$ and
$T_{n-1}F$ are monoidal. By Corollary 5.2.0.1, the $n$-homogeneous layer
$L_{n}F$, which corresponds to $\Theta^{n}F$ under the classification of
homogeneous functors (Theorem 3.3.0.6) is monoidal. Hence there is a structure
map

$L_{n}F\circledast L_{n}F\to L_{n}F$

Using the internal hom - adjunction and Proposition 5.3.0.4, we obtain

$L_{n}F\to\mathcal{F}(L_{n}F,L_{n}F)\to\mathcal{F}_{*}(L_{n}F,L_{n}F)$

Applying $\Theta^{n}$ to this and using Proposition 5.3.1.1, we get a map

$\Theta^{n}F\to\text{Map}_{Sp}(\Theta^{n}F,\Theta^{n}F)\otimes D_{O(n)}$

As we pointed out in Remark 5.3.1.7 (last point), $D_{O(n)}\simeq
S^{\text{Ad}_{n}}$ has an inverse $D_{O(n)}^{\vee}\simeq S^{-\text{Ad}_{n}}$.
Thus this map is equivalent to a map

$\Theta^{n}F\otimes
D_{O(n)}^{\vee}\to\text{Map}_{Sp}(\Theta^{n}F,\Theta^{n}F)$

By the internal hom adjunction in $Sp^{\text{BO}(n)}$, this corresponds to a
map

$\Theta^{n}F\otimes D_{O(n)}^{\vee}\otimes\Theta^{n}F\to\Theta^{n}F$

which corresponds to the desired map

$\Theta^{n}F\otimes\Theta^{n}F\to\Theta^{n}F\otimes D_{O(n)}$

∎

######  Remark 5.4.0.2.

These maps on derivative spectra will satisfy additional consistencies such as
homotopies making

${{\Theta^{n}F\otimes\Theta^{n}F\otimes\Theta^{n}F}}$${{\Theta^{n}F\otimes\Theta^{n}F\otimes
D_{O(n)}}}$${{\Theta^{n}F\otimes
D_{O(n)}\otimes\Theta^{n}F}}$${{\Theta^{n}F\otimes D_{O(n)}^{\otimes 2}}}$

commute, since the map

$m\colon F\circledast F\to F$

comes with homotopies making the square

${{F\circledast F\circledast F}}$${{F\circledast F}}$${{F\circledast
F}}$${F}$$\scriptstyle{m\circledast
id}$$\scriptstyle{m}$$\scriptstyle{m}$$\scriptstyle{id\circledast m}$

commute.

######  Conjecture 5.4.0.3.

For $F\colon\mathcal{J}\to\mathcal{S}_{*}$ lax symmetric monoidal, these maps
make $\widetilde{\Theta^{n}}F:=\Theta^{n}F\otimes D_{O(n)}^{\vee}$ into a
nonunital $E_{\infty}$-ring spectrum.

######  Proposition 5.4.0.4.

There is a (non-equivariant) equivalence

$\widetilde{\Theta^{n}}F\simeq\Omega^{\frac{n(n-1)}{2}}\Theta^{n}F$

###### Proof.

As

$\dim O(n)=\frac{n(n-1)}{2}$

we have (non-equivariantly)

$D_{O(n)}\simeq\Sigma^{\infty}S^{\text{Ad}_{n}}\simeq\mathbb{S}^{\frac{n(n-1)}{2}}$

thus

$D_{O(n)}^{\vee}\simeq\mathbb{S}^{-\frac{n(n-1)}{2}}$

thus

$\widetilde{\Theta^{n}}F\simeq
D_{O(n)}^{\vee}\otimes\Theta^{n}F\simeq\Omega^{-\frac{n(n-1)}{2}}\Theta^{n}F$

∎

5.4.0.3 would then imply

######  Conjecture 5.4.0.5.

For $F\colon\mathcal{J}\to\mathcal{S}$ lax symmetric monoidal, $\Theta^{1}F$
is a nonunital $E_{\infty}$-ring spectrum.

###### Proof.

As $O(1)\cong\mathbb{Z}/2$ is $0$-dimensional, the dualising spectrum
$D_{O(1)}$ is the sphere spectrum, hence
$\widetilde{\Theta^{1}}F\simeq\Theta^{1}F$. ∎

##  Chapter 6 Applications and Open Questions

This short final chapter discusses the following topics:

  * •

In Section 6.1, examples of monoidal functors $\mathcal{J}\to\mathcal{S}_{*}$
which motivated this work and to which the main results, i.e. Theorem 4.2.0.1
and Corollary 5.4.0.1, can be applied, are discussed. However, as the proofs
of the main results do not provide explicit formulas for the resulting
structure on the polynomial approximations and on the derivative spectra,
concrete formulas remain out of reach for now.

  * •

In Section 6.2, we explain a potential alternative description of the
multiplication on $\Theta^{1}F$ of a monoidal functor $F$.

  * •

In Section 6.3, we present some open questions and challenges, as well as some
directions for future work. These include more sophisticated examples of
monoidal structures, the existence of a similar theorem in Goodwillie
Calculus, as well as some more general open questions in Orthogonal Calculus
which we find interesting.

###  6.1 Examples of Monoidal Functors

######  Example 6.1.0.1.

The following functors $\mathcal{J}\to\mathcal{S}$ (already mentioned in
Example 3.1.0.5) are lax symmetric monoidal.

  1. a)

$\mathcal{J}\to\mathcal{S}\\\ V\mapsto S^{V}\simeq V^{c}$

  2. b)

$\mathcal{J}\to\mathcal{S}\\\ V\mapsto O(V)$

  3. c)

$\mathcal{J}\to\mathcal{S}\\\ V\mapsto\text{BO}(V)$

  4. d)

$\mathcal{J}\to\mathcal{S}\\\ V\mapsto\text{B}\text{TOP}(V)$

  5. e)

$\mathcal{J}\to\mathcal{S}\\\ V\mapsto\text{B}G(S^{V})$

(Here, $G(X)$ denotes the grouplike topological monoid of homotopy
automorphisms, i.e. self homotopy-equivalences, of $X$.)

###### Proof.

In all cases, we indicate the required maps

$F(V)\times F(W)\to F(V\oplus W)$

  1. a)

For $V\mapsto S^{V}$, up to identifying $S^{V\oplus W}$ with $S^{V}\wedge
S^{W}$, these are the quotient maps $S^{V}\times S^{W}\to S^{V}\wedge
S^{W}\simeq(S^{V}\times S^{W})/(S^{V}\vee S^{W})$.

  2. b)

For $V\mapsto O(V)$, taking block sums of matrices/automorphisms defines maps

$\displaystyle O(V)\times O(W)$ $\displaystyle\to O(V\oplus W)$
$\displaystyle(A,B)$ $\displaystyle\mapsto\begin{pmatrix}A&0\\\
0&B\end{pmatrix}$

  3. c)

For $V\mapsto\text{B}O(V)$, apply the functor B to the maps for $O(V)$.

  4. d)

The cases $V\mapsto\text{TOP}(V)$ and $V\mapsto\text{B}\text{TOP}(V)$ work
analogously to $O(V)$ and $\text{BO}(V)$.

  5. e)

For $V\mapsto G(S^{V})$, the identification $S^{V}\wedge S^{W}\simeq
S^{V\oplus W}$ gives maps

$\text{Map}(S^{V},S^{V})\times\text{Map}(S^{W},S^{W})\to\text{Map}(S^{V\oplus
W},S^{V\oplus W})$

which restrict to the desired maps. ∎

By Theorem 4.2.0.1 and Corollary 5.4.0.1 the $n$-polynomial approximations of
the functors of Example 6.1.0.1 are lax symmetric monoidal, and their
derivative spectra admit multiplication maps

$\Theta^{n}F\otimes\Theta^{n}F\to\Theta^{n}F\otimes D_{O(n)}$

We would like to be able to describe these maps explicitly, but are so far
unable to do so. There are some guesses in easy cases.

######  Example 6.1.0.2.

Let $F(V)=BO(V)$. By [39], the first few derivative spectra are given by

$\displaystyle\Theta^{1}\text{BO}$ $\displaystyle\simeq\mathbb{S}$
$\displaystyle\Theta^{2}\text{BO}$ $\displaystyle\simeq\Omega\mathbb{S}$
$\displaystyle\Theta^{3}\text{BO}$ $\displaystyle\simeq\Omega^{2}\mathbb{S}/3$

The multiplication maps predicted by Corollary 5.4.0.1 thus are maps of the
form

  * •

($n=1$) $\mathbb{S}\otimes\mathbb{S}\to\mathbb{S}$

  * •

($n=2$) $\mathbb{S}^{-2}\otimes\mathbb{S}^{-2}\to\mathbb{S}^{-2}$

  * •

($n=3$)
$\Omega^{5}\mathbb{S}/3\otimes\Omega^{5}\mathbb{S}/3\to\Omega^{5}\mathbb{S}/3$

We conjecture that the first map is the usual multiplication map on the sphere
spectrum. About the second, we remark that it must be non-equivariantly
nullhomotopic, but we think that this is not true as an $O(2)$-equivariant
map.
For general $n$, there is also the description of $\Theta^{n}\text{BO}$ by
Arone ([2]) which states that

$\Theta^{n}\text{BO}\simeq\text{Map}_{*}(L_{n},D_{O(n)})$

where $L_{n}$ is the unreduced suspension of the geometric realisation of the
category of non-trivial direct-sum decompositions of $\mathbb{R}^{n}$.
A possible guess would be that the multiplication map is in general related to
the diagonal map on $L_{n}$.

######  Example 6.1.0.3.

Let $F(V)=\text{BTOP}(V)$. Then

$\Theta^{1}F\simeq A(*)\simeq
K(\mathbb{S})\simeq\mathbb{S}\vee\text{Wh}^{\text{DIFF}}(*)$

where $A(*)$ denotes Waldhausen ’s A-theory spectrum of a point and
$\text{Wh}^{\text{DIFF}}$ denotes the smooth Whitehead spectrum). (For a
reference, see [39], p.3, which references [36, Theorem 2, Addendum 4] for
this fact.)
By 5.4.0.5 we obtain a nonunital multiplication on $A(*)$. We conjecture that
it is the usual multiplication. A possible explanation for this fact uses work
of Weiss-Williams which is yet to appear111 likely as “Automorphisms of
Manifolds and Algebraic K-Theory IV”. .

###  6.2 A Potential Alternative Description of the Multiplication On
$\Theta^{1}F$

######  Remark 6.2.0.1 (Potential alternative description of a multiplication
on the first derivative spectrum).

There is a potential alternative explicit description of a multiplication on
the first derivative spectrum of a monoidal functor using the explicit
construction of the homotopy fibre of a map $f\colon(X,x_{0})\to(Y,y_{0})$ as

$\mathrm{hofib}f=\\{(a,p)\in X\times PY\mid p(0)=f(a)\text{ and
}p(1)=y_{0}\\}$

Recall that (e.g. as a special case of [39, Prop. 5.3]) there is a fibre
sequence

$F^{(1)}(V)\to F(V)\to F(V\oplus\mathbb{R})$

Applying this we get

$F^{(1)}(V)=\\{(a,p)\in F(V)\times PF(V\oplus\mathbb{R})\mid
p(0)=i_{*}(a),p(1)=*\\}$

The monoidal structure on $F$ provides maps

$m_{V,W}\colon F(V)\times F(W)\to F(V\oplus W)$

Using this one can also write down a multiplication map in the following way.

Define

$\tilde{m}_{V,W}\colon F^{(1)}(V)\times F^{(1)}(W)\to\Omega F^{(1)}(V\oplus
W\oplus\mathbb{R})$

by

$\displaystyle\tilde{m}((a,p),(b,q))=(\eta,\nu)$ $\displaystyle\in\Omega
F^{(1)}(V\oplus W\oplus\mathbb{R})$ $\displaystyle\subset\Omega F(V\oplus
W\oplus\mathbb{R})\times\Omega PF(V\oplus W\oplus\mathbb{R}\oplus\mathbb{R})$

where

$\eta=i_{*}(q)\ast
m_{V\oplus\mathbb{R},W}(p,b)\ast\overline{m_{V,W\oplus\mathbb{R}}(a,q)}\ast
i_{*}(\overline{p})$

Here, $\overline{\gamma}$ for a path $\gamma$ denotes the reverse path.

The construction of $\nu$, which needs to be a path in $\Omega F^{(1)}(V\oplus
W\oplus\mathbb{R}\oplus\mathbb{R})$, proceeds similarly. Concretely, if $s$ is
the path-space variable, $\nu$ can be defined by

$v_{s}(t)=\begin{cases}m(p,q)(s\cdot 4t,1)&\text{if }0\leq t\leq\frac{1}{4}\\\
m(p,q)(s,s\cdot 4(t-\frac{1}{4}))&\text{if }\frac{1}{4}<t\leq\frac{1}{2}\\\
m(p,q)(s\cdot(1-4(t-\frac{1}{2})),s)&\text{if }\frac{1}{2}<t\leq\frac{3}{4}\\\
m(p,q)(s(1-4(t-\frac{3}{4})),s)&\text{if }\frac{3}{4}<t\leq 1\end{cases}$

We have now described maps

$F^{(1)}(V)\times F^{(1)}(W)\to\Omega F^{(1)}(V\oplus W\oplus\mathbb{R})$

Since

$(\Theta^{1}F)_{n}\simeq F^{(1)}(\mathbb{R}^{n})$

these maps together describe a map

$m\colon\Theta^{1}F\otimes\Theta^{1}F\to\Theta^{1}F\simeq\Sigma\Omega\Theta^{1}F$

for a monoidal functor $F\colon\mathcal{J}\to\mathcal{S}_{*}$.
We conjecture that this map models the map

$\Theta^{1}F\otimes\Theta^{1}F\to\Theta^{1}F$

that is predicted by our theory (see 5.4.0.5). If so, this much more concrete
perspective might enable concrete calculations. It seems poorly suited for
establishing more abstract coherence properties though.

###  6.3 Further Open Questions

Here are some questions that we can’t answer as of now but which might be
interesting for further study.

######  Question 6.3.0.1 (Rational $K$-Theory of $\mathbb{Z}$).

We already mentioned in Example 6.1.0.3 that

$\Theta^{1}(V\mapsto\text{BTOP}(V))\simeq A(*)\simeq
K(\mathbb{S})\simeq\mathbb{S}\vee\text{Wh}^{\text{DIFF}}(*)$

Using $S(V)$ to denote the unit sphere in $V$, one can similarly show that

$\Theta^{1}(V\mapsto\text{BTOP}(S(V)))\simeq\mathbb{S}\vee\Omega\text{Wh}^{\text{DIFF}}(*)$

Rationally, the former is identified with $K(\mathbb{Z})$, the second with
$\mathbb{S}\vee\Omega K(\mathbb{Z})$. Borel ([6]) proved that

$\pi_{*}K(\mathbb{Z})\otimes\mathbb{Q}=\left\\{\begin{array}[]{lr}\mathbb{Q},&\text{for
}*=0\\\ \mathbb{Q},&\text{for }*=4k+1\text{ with }k\geq 1\\\
0,&\text{otherwise}\\\ \end{array}\right.$

so one sees that

$\pi_{*}\Theta^{1}(V\mapsto\text{BTOP}(S(V)))\otimes\mathbb{Q}=\left\\{\begin{array}[]{lr}\mathbb{Q},&\text{for
}*=0\\\ \mathbb{Q},&\text{for }*=4k\text{ with }k\geq 1\\\
0,&\text{otherwise}\\\ \end{array}\right.$

By virtue of the monoidality theorem for $\Theta^{1}$, this will carry a
graded ring structure. It would now be very interesting to determine it. In
particular, is this ring generated by the class in degree $4$?

######  Question 6.3.0.2.

Although unexplained in this work, one possible purpose of Orthogonal Calculus
is to generalise the theory of characteristic classes. E.g. for
$F\colon\mathcal{J}\to\mathcal{S}$ one can construct a cohomology class on
$F(\mathbb{R}^{\infty})$ with values in $\Theta^{1}F$ which in the case of
$V\mapsto\text{BO}(V)$ recovers (up to change of coefficients) the classical
total Stiefel-Whitney class ([40]). Can the product structure on $\Theta^{1}F$
in case of a monoidal functor $F$ thus recover the classical product formula
for the total Stiefel-Whitney class as well as prove generalisations of it,
e.g. for Euclidean fibre bundles ($V\mapsto\text{BTOP}(V)$) or spherical
fibrations ($V\mapsto\text{BG}(V)$)?

######  Question 6.3.0.3 (The gluing datum in the orthogonal tower and
chromatic localization).

Even if one understands all the homogeneous layers of the orthogonal tower,
the question how to reconstruct the $n$-polynomial approximations can remain
difficult. In Goodwillie Calculus, the gluing data can be described as maps to
certain Tate constructions, e.g. for functors to spectra, by a result of Kuhn
([23], Prop. 5.13), there is a pullback diagram

${T_{d}F}$${(\Delta_{d}F)^{h\Sigma_{d}}}$${T_{d-1}F}$${(\Delta_{d}F)^{t\Sigma_{d}}}$

In orthogonal calculus, a similar concrete description of the gluing data is
still missing and should be investigated.

A fascinating consequence of the description via Tate constructions in
Goodwillie Calculus is that, after certain chromatic localizations, i.e.
$T(n)$-locally, Goodwillie towers split, as the Tate constructions are trivial
(Kuhn, [23, Thm. 7.6]).

Orthogonal calculus too can be done locally with respect to such localisations
(see e.g. Taggart [32]). An interesting follow-up question would therefore be
if an analogue of Kuhn’s theorem holds in orthogonal calculus too.

######  Question 6.3.0.4.

In Remark 3.1.0.6, we observed that several functors of interest
$\mathcal{J}\to\mathcal{S}$, such as $V\mapsto\text{BO}(V)$, are also defined
on a smaller category which does not know about inner products, only about
injective maps and natural complements.

This motivates the following question, which to our knowledge is open.

Let $K$ be a field, and let $\mathcal{J}_{K}$ be the category of finite-
dimensional $F$-vector spaces, with $\text{mor}_{\mathcal{J}_{K}}(V,W)$ given
by

$\\{f\colon V\to W,U_{V,W}\subseteq W\mid f\text{ linear and
injective},U_{V,W}\text{ a complement to \text{im}(f)}\\}$

with composition maps

$\text{mor}_{\mathcal{J}_{K}}(V,W)\times\text{mor}_{\mathcal{J}_{K}}(W,X)\to\text{mor}_{\mathcal{J}_{K}}(V,X)$
$((f\colon V\to W,U_{V,W}),(g\colon W\to X,U_{W,X})\mapsto(g\circ f\colon V\to
X,U_{W,X}+g(U_{V,W}))$

Then by definition, independent of the choice of $K$, there are functors such
as

  * •

$V\mapsto\text{GL}(V)$

  * •

$V\mapsto\text{SL}(V)$

For $F=\mathbb{R}$ there is a functor

$\mathcal{J}\to\mathcal{J}_{\mathbb{R}}$ $V\mapsto V$
$f\mapsto(f,(\text{im}f)^{\perp})$

inducing (by precomposition) a calculus for functors on
$\mathcal{J}_{\mathbb{R}}$.
The question is if there are such calculi for other fields too. Speculatively,
such calculi might be applied to questions of arithmetic, e.g. about
$\text{SL}_{n}(F)$ or maybe even $\text{SL}_{n}(\mathbb{Z})$.

We do not know the answer. We speculate that for local fields such as
$\mathbb{Q}_{p}$ (on which there exists $p$-adic analysis), such a calculus
might look rather similar to Orthogonal Calculus, while for even more general
fields such as global or finite fields, we assume the proofs would need to use
quite different, rather more algebro-geometric methods222 Could motivic
homotopy theory play a role here?.

######  Question 6.3.0.5.

It seems very likely that Theorem 4.2.0.1 holds equally, with the same proof
strategy, for Goodwillie Calculus for functors $\mathcal{C}\to\mathcal{D}$
when $\mathcal{C}$ and $\mathcal{D}$ are sufficiently well behaved, e.g.
$\mathcal{D}$ closed monoidal. Does this have interesting consequences in
Goodwillie Calculus? What examples of monoidal functors to which Goodwillie
Calculus can be applied are there?

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