# Solitons in Weakly Non-linear Topological Systems: Linearization,
Equivariant Cohomology and K-theory

Daniel Sheinbaum  Facultad de Ciencias, Universidad Nacional Autónoma de
México. Circuito Exterior S/N, Cd. Universitaria, Colonia Copilco el Bajo,
Delegación Coyoacán, 04510, Ciudad de México, México<EMAIL_ADDRESS>
###### Abstract

There is a lack of knowledge about the topological invariants of non-linear
$d$-dimensional systems with a periodic potential. We study these systems
through a classification of the linearized NLS/GP equation around their
soliton solutions. Stability conditions under linearized (mode) adiabatic
evolution can be interpreted topologically and we can use equivariant
$\mathit{K}$-theory and cohomology for their classification. On a lattice with
crystallographic point group $P$, modes around stable, $P$-symmetric solitons
are coarsely classified by the groups
$\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^{d})\oplus H^{2}(BP;\mathbb{Z})$.
Similarly, for $P$-symmetric gap solitons that are oscillatory stable, we have
$\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^{d})\oplus\tilde{R}(P)$ instead. If
we include a boundary, we can replace
$\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^{d})$ with
$\mathit{K}^{-1}_{P}(\mathbb{T}^{d-1})$. Finally, we mention how to use these,
and the spaces of soliton solutions $M_{D}(E_{gap})$ and $M_{O}(E_{gap})$ to
provide global invariants for the system.

Suggested keywords

††preprint: APS/123-QED

##  I Introduction

The field of topological systems in photonics Ozawa  _et al._ (2019), exciton-
polaritons Gulevich  _et al._ (2017) and Bose-Einstein Condensates (BEC)
Pitaevskii and Stringari (2016) has brought to the fore the interplay between
non-linear effects such as solitons Whitham (2011), Boyd (2019) and linear
topological phases of matter Thouless  _et al._ (1982), Kane and Mele (2005),
Avron  _et al._ (1983). An interesting question arises when we couple a non-
linear medium to a periodic potential Sivan  _et al._ (2008), particularly
when the linear system is topologically non-trivial, e.g. quantum Hall effect
(QHE), SQHE and topological insulators (TI) Ozawa  _et al._ (2019). Are
systems which combine both still topological? And if so, which topological
invariants classify these new non-linear systems? Though there are already
many theoretical and experimental results on the stability of solitons, non-
linear Bloch waves and their interactions as well as resulting macroscopic
properties of materials Smirnova  _et al._ (2020), Mukherjee and Rechtsman
(2020), Gulevich  _et al._ (2017), there are not many proposals for
topological invariants classifying these systems (see Liu and Fu (2010),
Bomantara  _et al._ (2017) for some results in this direction). For weakly
non-linear systems, the Non-linear Schrödinger/Gross-Pitaevskii equation
(NLS/GP) Boyd (2019), Smirnova  _et al._ (2020), Weinstein (2006) and
variations of these are good approximations to low-energy behaviour. Here we
shall consider weakly non-linear $d$-dimensional systems with a periodic
potential Sivan  _et al._ (2008) and magnetic field described by

$i\partial_{z}\Psi(\vec{x},z)=[-(i\nabla-\vec{A}(\vec{x}))^{2}+V(\vec{x})-f(\vec{x},|\Psi|^{2})]\Psi(\vec{x},z),$
(1)

where $z$ can be either time (BEC) or the distance along the direction of
propagation (Optics, here $z\geq 0$). We also assume that
$\vec{A}(\vec{x}+\vec{a})=\vec{A}(\vec{x});\,\,V(\vec{x}+\vec{a})=V(\vec{x})$
for all $\vec{a}\in\Lambda$, where $\Lambda$ is a $d$-dimensional lattice.
Note that in general the assumption is not true for the magnetic potential
$\vec{A}$ and we should include disorder Prodan and Schulz-Baldes (2016). We
denote the linear part by $\mathcal{H}_{l}=-(i\nabla-\vec{A})^{2}+V$. We can
simultaneously include systems with a boundary in our discussion by splitting
$\vec{x}=(x_{\perp},\vec{x}_{||})$ and making $V(\vec{x})=A(\vec{x})=0$ for
$\,x_{\perp}\leq 0$ and periodic only in the $\vec{x}_{||}$-direction with
respect to $\Lambda_{||}$, a $d-1$-dimensional lattice, parallel to the
boundary. We would like $\mathcal{H}_{l}$ to correspond to a linear
topological system (phase) with a gap Avron  _et al._ (1983) or gapped bulk
condition (systems with boundary) Adem  _et al._ (2016) under adiabatic
evolution Avron  _et al._ (1987) at a fixed energy scale $E_{gap}$, as there
is no analogue of the Fermi energy $E_{F}$ for our non-linear systems, since
they are generally bosonic Ozawa  _et al._ (2019), De Nittis and Lein (2017).

$\mathit{K}$-theory Hatcher (2003) and cohomology Hatcher (2002) have by now
been used in different fields of physics, for example $\mathit{K}$-theory was
used in Kane and Mele (2005), Hořava (2005) and cohomology in Wang  _et al._
(2015), and Kapustin and Thorngren (2014), among many others. For those
readers who are not familiar with algebraic topology, we present a brief
description of what it entails. Given a topological space $X$, cohomology and
$\mathit{K}$-theory assign abelian groups $H^{n}(X;\mathbb{Z})$ and
$\tilde{\mathit{K}}^{0}(X)$ to it, hence the name algebraic topology. A
homotopy between two maps from a topological space $X$ to $Y$ is simply a
continuous family of maps parametrized by $s$ in $[0,1]$ such that at 0 and 1
it coincides with the original ones. Homotopies are the backbone of algebraic
topology. Said differently, everything in algebraic topology (and topological
phases) has an expression in terms of homotopy theory, e.g. cohomology,
$\mathit{K}$-theory and their equivariant versions Freed and Moore (2013) are
defined using homotopy theory, the all-encompassing framework. In particular,
the groups assigned to $X$, $H^{n}(X;\mathbb{Z})$ and
$\tilde{\mathit{K}}^{0}(X)$ can be constructed out of maps (up to homotopy)
from $X$ to another space called the classifying space, which is independent
of $X$, but does depend on whether it is $H^{n}(X;\mathbb{Z})$ or
$\tilde{\mathit{K}}^{0}(X)$. Here, as it is done in the physics community, we
interpret adiabatic evolution as a homotopy Avron  _et al._ (1987) and show
how these classifying spaces arise from physical and stability conditions. For
our purposes, equivariant topology can be thought of in the following way: If
we have symmetries represented by a group $P$, we only want to consider maps
that preserve them and see which groups we get with this restriction. We
represent these as $H^{n}_{P}(X;\mathbb{Z})$ and $\mathit{K}_{P}^{0}(X)$
respectively. With this in mind let us proceed.

For fully periodic systems with a gap, there is a coarse classification
111Coarse means we do not care about adding trivial valence bands to our
systems. where a phase
$[\mathcal{H}_{l}-E_{gap}I]\in\tilde{\mathit{K}}^{0}(\mathbb{T}^{d})$, the
$\mathit{K}$-theory group constructed out of vector bundles over the Brillouin
torus $\mathbb{T}^{d}$ arising from the Bloch bands below the Fermi energy
Kitaev (2009), Freed and Moore (2013). Meanwhile, for systems with a boundary,
$\mathit{K}$-theory arises naturally and
$[\mathcal{H}_{l}-E_{gap}I]\in\mathit{K}^{-1}(\mathbb{T}^{d-1})$, where
$\mathbb{T}^{d-1}$ is the surface Brillouin torus Adem  _et al._ (2016).
Ideally we would imitate the linear classification for systems with
$f(\vec{x},|\Psi|^{2})$ by defining a non-linear gap condition and a notion of
non-linear adiabatic evolution Liu  _et al._ (2003). The problem is that so
far there is no analogue of a gap condition for non-linear systems Smirnova
_et al._ (2020). We will consider the simplified problem of classifying the
topological behaviour of modes around soliton solutions, stationary solutions
of the form $\Psi(\vec{x},z)=e^{-i\lambda z}\Phi^{\lambda}(\vec{x})$ to eq
(1), which decay exponentially as we go to spatial infinity.

But, can all solitons have topological modes? And even if some do, could these
be destroyed by the non-linearity? If a soliton is unstable it will eventually
disappear and its modes together with it. Positive (ground state) solitons
have two types of instabilities, one is a focusing instability Sivan  _et al._
(2008), where without increasing, its energy focuses towards a single point,
yielding an arbitrary high density which blows up. The other is a drift
instability, where via asymmetric distortions, infinitesimal displacements of
its original position make it drift towards infinity Sivan  _et al._ (2008).
What happens for non-positive solitons, so-called gap solitons, i.e. those
which live between the spectral bands of the linear problem? These suffer from
a different set of instabilities Pelinovsky  _et al._ (2004). One of these is
an oscillatory instability, where among other mechanisms, a vibration mode
appears and resonates with radiation, causing the soliton to oscillate at
higher and higher frequencies Barashenkov  _et al._ (1998). We would like our
solitons to avoid such instabilities; therefore, we shall further impose some
stability conditions Sivan  _et al._ (2008), Pelinovsky  _et al._ (2004).
However, how compatible are these conditions with those necessary for
topological modes? Do these conditions have a topological interpretation? We
shall elucidate their topological character and their relation to the other
topological restrictions in what follows.

##  II Linearization and stability conditions

We linearize eq (1) around $\Phi^{\lambda}$ Zhou and Weinstein (2008), Sivan
_et al._ (2008) which yields

$\partial_{z}\vec{\chi}=\mathcal{L}(\lambda)\vec{\chi},$ (2)

where

$\mathcal{L}(\lambda)=\begin{pmatrix}0&L_{-}(\lambda)\\\
-L_{+}(\lambda)&0\end{pmatrix},$ (3)

with self-adjoint operators

$\displaystyle L_{-}(\lambda)$ $\displaystyle=$
$\displaystyle\mathcal{H}_{l}-\lambda I-f(x,|\Phi^{\lambda}|^{2}),$ (4)
$\displaystyle L_{+}(\lambda)$ $\displaystyle=$
$\displaystyle\mathcal{H}_{l}-\lambda
I-f(x,|\Phi^{\lambda}|^{2})-2df|_{\phi^{\lambda}}.$

Our linearized problem has a clear analogue of a gap condition for the mode
operators $L_{\pm}(\lambda)$. We note that $\mathcal{H}_{l}-\lambda I$ has its
spectrum shifted and hence the scale at which the soliton satisfies a gapped
modes condition is at

$\displaystyle E_{modes}(\lambda)$ $\displaystyle=$ $\displaystyle
E_{gap}-\lambda,$ (5) $\displaystyle 0$ $\displaystyle<$
$\displaystyle\lambda<E_{gap}.$ (6)

Thus, the first constraint we put on solitons to have topological modes is
$\lambda<E_{gap}$. We also have extra potentials determined by
$f(x,|\Phi^{\lambda}|^{2})$ and
$f(x,|\Phi^{\lambda}|^{2})-2df|_{\phi^{\lambda}}$, which we name the soliton
potentials. For the soliton potentials not to destroy the topological
character of $\mathcal{H}_{l}-\lambda I-E_{modes}(\lambda)$, we need them to
behave as a perturbation/impurity/defect that does not break the gapped modes
condition

$\displaystyle|f(x,|\Phi^{\lambda}|^{2})|$ $\displaystyle<<$ $\displaystyle
E_{modes}(\lambda),$ (7)
$\displaystyle|f(x,|\Phi^{\lambda}|^{2})-2df|_{\phi^{\lambda}}$
$\displaystyle<<$ $\displaystyle E_{modes}(\lambda).$ (8)

These constrain both the non-linearities and the width of solitons with
topological modes. Just as an example, for power non-linearities
$f(|\Psi|)=|\Psi|^{p-1},\,\,p>1$, solitons which have a peak of the order
$E_{modes}(\lambda)^{\frac{1}{p-1}}$ will destroy the gapped modes condition
and this will be easier as $\lambda$ increases. For systems with a boundary
the constraint is only necessary for solitons that are surface-localized. Here
arises our first connection with instabilities. Positive (ground-state)
solitons that are focusing unstable Sivan  _et al._ (2008), that is, solitons
whose modulus blows up will quickly break the gapped modes condition. This
means that our solitons should satisfy the Vakhitov-Kolokolov stability
condition

$\frac{dP}{d\lambda}<0,$ (9)

where $P=\int|\Psi|^{2}d\vec{x}$ is the particle number (BEC) or optical
power, which is conserved. Let us now consider solitons that are drift stable,
i.e. those which stay put under small displacements of their initial position.
These have to satisfy the spectral condition

$n_{-}(L_{+}(\lambda))=1,$ (10)

where $n_{-}(L_{+}(\lambda))$ is the number of negative eigenvalues. Note that
for positive solitons, these conditions are necessary and sufficient for full
stability Sivan  _et al._ (2008). The latter condition can be interpreted as
topological if we further note that from the positivity of $\mathcal{H}_{l}$,
there is a restriction on the continuous spectrum $\sigma_{c}(L_{+}(\lambda))$
(scattering modes) to be positive. Modes associated to $L_{+}(\lambda)$ live
in a Hilbert space $\mathfrak{H}^{2}(\mathbb{R}^{d},\mathbb{C})$ and the above
means there is a natural split

$\mathfrak{H}^{2}(\mathbb{R}^{d},\mathbb{C})=\mathfrak{H}_{-1}(\lambda)\oplus\mathfrak{H}_{\geq
0}(\lambda).$ (11)

The set of all such $1$-dimensional subspaces $\mathfrak{H}_{-1}(\lambda)$ of
$\mathfrak{H}^{2}(\mathbb{R}^{d},\mathbb{C})$ forms a topological space known
as the infinite dimensional Grassmannian
$Gr_{1}(\mathfrak{H}^{2}(\mathbb{R}^{d},\mathbb{C}))$ Abbondandolo and Majer
(2009), which we denote $Gr_{1}$ for shortness. The space $Gr_{1}$ is a
classifying space for the second cohomology group $H^{2}$ Abbondandolo and
Majer (2009), Hatcher (2002). This means that (up to homotopy) maps to
$Gr_{1}$ are used to construct the cohomology group $H^{2}$. Note that the
gapped modes condition never entered into our discussion of drift stability.
These two conditions are independent as the drift stability is about what
happens below $\sigma_{c}(L_{+}(\lambda))$, while the gapped modes condition
is about what happens in between (same for gapped Bulk-modes). Thus, we can
view linearization around the soliton as a map

$\Phi^{\lambda}\mapsto\bigg{\\{}\begin{pmatrix}0&P_{\geq
0}(\lambda)L_{-}(\lambda)\\\ -P_{\geq
0}(\lambda)L_{+}(\lambda)&0\end{pmatrix},P_{-1}(\lambda)\bigg{\\}},$ (12)

where $P_{-1}(\lambda)$ and $P_{\geq 0}(\lambda$) are projections to
$\mathfrak{H}_{-1}(\lambda)$ and $\mathfrak{H}_{\geq 0}(\lambda)$. Because of
conditions (7), (8), the first component can be seen to be equivalent, up to
mode adiabatic evolution (for definition see sec (III)), to a Hamiltonian
operator in $Gap(L^{2}(\mathbb{R}^{d}))$, the space of gapped $d$-dimensional
single-particle periodic Hamiltonians. In Freed and Moore (2013), using
Bloch’s theorem, it is shown using the periodicity that
$Gap(L^{2}(\mathbb{R}^{d}))$ is coarsely equivalent (by adding trivial bands)
to $Map(\mathbb{T}^{d},BGL_{\infty})$, the space of continuous maps from the
$d$-dimensional Brillouin torus to the classifying space $BGL_{\infty}$, which
can be thought of as an ever increasing sequence of Grassmannians. Maps (up to
homotopy) to $BGL_{\infty}$ give rise to the group $\tilde{\mathit{K}}^{0}$
Hatcher (2003). Analogously, for systems with a boundary, we instead have
$Gap_{Bulk}(L^{2}(\mathbb{R}^{d}))$, itself being equivalent to
$Map(\mathbb{T}^{d-1},\mathcal{F}^{sa}_{*}(\mathfrak{H}))$ Adem  _et al._
(2016), where $\mathbb{T}^{d-1}$ is now the surface Brillouin torus and
$\mathcal{F}^{sa}_{*}(\mathfrak{H})$ is a subspace of self-adjoint Fredholm
operators Atiyah and Singer (1969). Maps to
$\mathcal{F}^{sa}_{*}(\mathfrak{H})$ now give rise to the group
$\mathit{K}^{-1}$ instead of $\tilde{\mathit{K}}^{0}$. The bulk-boundary
correspondence can be seen by noticing that the $-1$ in the degree of the
$\mathit{K}$-group compensates the $-1$ in the dimension of the surface
Brillouin torus for $d=2$ Prodan and Schulz-Baldes (2016). The projection
$P_{-1}(\lambda)$ in the second component represents a point in $Gr_{1}$, as
discussed previously.

Regarding the scattering of an unstable soliton with a stable soliton, we can
speculate that if the unstable soliton implodes, then as the two solitons get
closer to each other, it will behave as a strong potential that breaks the
gapped modes condition. If the soliton does not implode, however, the
expectation is that the other instabilities do not affect these modes, as
generally solitons go right through each other under collision. Further non-
linear analysis is required. Nevertheless, for solitons sufficiently isolated,
these modes should be topologically stable to other perturbations such as
defects Gulevich  _et al._ (2017), Smirnova  _et al._ (2020).

##  III Mode adiabatic evolution

Consider now eq (1) with potential, gap energy and non-linearity, which are
also $z$-dependent but in such a way that the linearized evolution of the
modes around the soliton is adiabatic Avron  _et al._ (1987). We set
$1/E_{gap}$ as the adiabatic scale and set $s=E_{gap}z$ to be the
dimensionless variable that replaces $z$. We name this key concept mode
adiabatic evolution. We now use the homotopy interpretation Avron  _et al._
(1983), Freed and Moore (2013) for our mode adiabatic evolution. Two solitons
$\Phi^{\lambda_{0}}(V_{0},f_{0}),\,\Phi^{\lambda_{1}}(V_{1},f_{0})$ are in the
same topological class if there exists an $s$-dependent family of soliton
solutions $\Phi^{\lambda(s)}(V(s),f(s))$ such that

$\displaystyle[0,1]$ $\displaystyle\longrightarrow$ $\displaystyle
Gap(L^{2}(\mathbb{R}^{d}))\times Gr_{1}$ $\displaystyle s$
$\displaystyle\mapsto$ $\displaystyle\bigg{\\{}\begin{pmatrix}0&P_{\geq
0}(\lambda(s))L_{-}(\lambda(s))\\\ -P_{\geq
0}(\lambda(s))L_{+}(\lambda(s))&0\end{pmatrix},P_{-1}(\lambda(s))\bigg{\\}},$
$\displaystyle\Phi^{\lambda(0)}(V(0),f(0))$ $\displaystyle=$
$\displaystyle\Phi^{\lambda_{0}}(V_{0},f_{0}),$
$\displaystyle\Phi^{\lambda(1)}(V(1),f(1))$ $\displaystyle=$
$\displaystyle\Phi^{\lambda_{1}}(V_{1},f_{1}).$ (13)

Thus, using this homotopy interpretation of mode adiabatic evolution, we can
separate modes into topological classes. Employing the homotopy type of the
spaces discussed above, we have that for periodic systems the set of distinct
classes of topological modes around solitons is equivalent to the groups:

$\tilde{\mathit{K}}^{0}(\mathbb{T}^{d})\oplus H^{2}(*;\mathbb{Z}),$ (14)

where $*$ denotes (from here on) a point, viewed as a topological space. For
systems with a boundary we replace $Gap(L^{2}(\mathbb{R}^{d}))$ with
$Gap_{Bulk}(L^{2}(\mathbb{R}^{d}))$ and using the results of Adem  _et al._
(2016), we obtain $\mathit{K}^{-1}(\mathbb{T}^{d-1})\oplus
H^{2}(*;\mathbb{Z})$ instead. We remark that many solitons of interest such as
those that are surface-localized are often gap solitons Gulevich  _et al._
(2017) and do not satisfy the spectral condition mentioned above. The
topological interpretation of the drift stability condition might seem
irrelevant since the group $H^{2}(*;\mathbb{Z})$ is trivial, but we shall see
it yields new classes for systems with more symmetry. Let us reflect on what
this result implies. If our solitons satisfy (7) and (8), and we ignore issues
of soliton stability (momentarily), we can approximate (up to mode adiabatic
evolution) eq (2) Zhou and Weinstein (2008) for

$\partial_{z}\chi\approx-i[\mathcal{H}_{l}-\lambda
I]\chi\,;\,\,\chi=\chi_{1}+i\chi_{2},\,\,\begin{pmatrix}\chi_{1}\\\
\chi_{2}\end{pmatrix}=\vec{\chi},$ (15)

which is a linear Schrödinger equation for the mode $\chi$. This means that
all the well-known invariants, such as the Chern number Thouless  _et al._
(1982), the spectral flow (for systems with boundary) Adem  _et al._ (2016),
the $\mathbb{Z}_{2}$ index (if we add time-reversal symmetry) or mirror Chern
number (when we add crystallographic symmetries), etc., have a physical
analogue for the mode $\chi$, as discussed for photonic systems in Ozawa  _et
al._ (2019), if they exist for the original linear system $\mathcal{H}_{l}$.
These invariants can be seen as generating the different $\mathit{K}$-groups
which arise. Thus, $\mathit{K}$-groups handle in a single swoop all the
different topological mode invariants, instead of discussing each one of them
separately.

Let us now switch gears a bit and consider solitons which lie between the gaps
of the linear periodic system Pelinovsky  _et al._ (2004), Mukherjee and
Rechtsman (2020), Ilan and Weinstein (2010). These are not generally positive
and the above spectral condition does not apply. Gap solitons may suffer an
oscillatory instability Barashenkov  _et al._ (1998), Pelinovsky  _et al._
(2004), and the condition for gap solitons to be oscillatory stable can also
be stated in topological terms. Let $L(\lambda)=\mathcal{H}_{l}-\lambda I$,
the neutral (sometimes called internal) modes (eigenvectors with positive
eigenvalues below $\sigma_{c}(L_{+}(\lambda))$) of $L_{+}(\lambda)$ must lie
between the band gaps of the inverted operator $-L(\lambda)$. If they embed in
the bands of $-L(\lambda)$, the eigenvalues of $\mathcal{L}(\lambda)$ start
bifurcating into complex pairs Pelinovsky  _et al._ (2004). If there are $m$
neutral modes between the inverted bands, then there is an $m$-dimensional
vector space $\mathfrak{H}_{N}$ associated to these, which will not change
dimension under mode adiabatic evolution, as long as it remains oscillatory
stable. Thus, again, we have a natural splitting

$\mathfrak{H}^{2}(\mathbb{R}^{d},\mathbb{C})=\mathfrak{H}_{N}(\lambda)\oplus\mathfrak{H}_{rest}(\lambda).$
(16)

Once again, the space of all possible $m$-dimensional subspaces
$\mathfrak{H}_{N}(\lambda)$ of $\mathfrak{H}^{2}(\mathbb{R}^{d},\mathbb{C})$
is the $m$-dimensional Grassmannian
$Gr_{m}(\mathfrak{H}^{2}(\mathbb{R}^{d},\mathbb{C}))$. We can again simplify
matters by replacing the finite-dimensional Grassmannian with $BGL_{\infty}$
Hatcher (2003). Repeating the same analysis as before we obtain the
topological classes for oscillatory stable gap solitons, which are given by

$\tilde{\mathit{K}}^{0}(\mathbb{T}^{d})\oplus\tilde{\mathit{K}}^{0}(*).$ (17)

We have not seen if this condition generalizes to systems with a boundary;
however, considering the appearance of topological surface gap solitons
Gulevich  _et al._ (2017) and further assuming there is no qualitative
difference in their oscillatory behaviour (relative to their bulk analogues),
we could suggest that for systems with a boundary, their gap soliton
topological classes are given by
$\mathit{K}^{-1}(\mathbb{T}^{d-1})\oplus\tilde{\mathit{K}}^{0}(*)$. Once
again, the reformulation of oscillatory stability seems irrelevant but let us
see what happens when we add symmetries.

On a last note, surface gap solitons can suffer from other types of
instabilities such as decaying to small amplitude linear waves. It was found
in Leykam and Chong (2016) that solitons with topological edge modes are
stable and propagate unidirectionally as in the linear case. It would be
interesting to express these results in terms of our mode invariants.

##  IV Crystallographic and Time-reversal symmetries

Consider systems which further have a crystallographic symmetry with point
group $P\subset O(d)$ Fu (2011), Freed and Moore (2013). If we restrict to
$P$-symmetric soliton solutions, their corresponding $L_{\pm}(\lambda)$ will
be $P$-invariant. Further, if they satisfy all of the conditions discussed
above and the mode adiabatic evolution respects this $P$-invariance, then the
crystalline topological classes of modes around $P$-symmetric positive
solitons are given by:

$\bar{\mathit{K}}^{0,\tau}_{P}(\mathbb{T}^{d})\oplus H^{2}_{P}(*;\mathbb{Z}).$
(18)

The groups $\bar{\mathit{K}}^{0}_{P}$ and $H^{2}_{P}$ denote a twisted
equivariant version of $\mathit{K}$-theory 222By
$\bar{\mathit{K}}^{0}_{P}(\mathbb{T}^{d})$ we mean the kernel
$\mathit{K}^{0}_{P}(\mathbb{T}^{d})\rightarrow\mathit{K}^{0}(*)$. This is
because the coarser classification does not care about adding trivial bands
and changing the dimension of our bundles., Freed and Moore (2013) and
equivariant cohomology Gomi (2017), Adem and Milgram (2013), respectively. The
interesting thing here is that $H^{2}_{P}(*;\mathbb{Z})$ is no longer trivial!
Instead it is equivalent to $H^{2}(BP;\mathbb{Z})$, where $BP$ is an infinite
dimensional space known as the classifying space of $P$ Hatcher (2002). To
have an example in mind note that for $P=\mathbb{Z}_{2}$,
$B\mathbb{Z}_{2}\simeq\mathbb{R}P^{\infty}$, the infinite dimensional real
projective space. Hence, the spectral condition becomes topologically non-
trivial when we include more symmetries. For systems with a boundary we
replace $\bar{\mathit{K}}^{0,\tau}_{P}(\mathbb{T}^{d})$ with
$\mathit{K}^{-1,\tau}_{P}(\mathbb{T}^{d-1})$, where $P$ now denotes surface
crystallographic symmetry and $d\geq 2$ Sheinbaum Frank (2019). This easily
extends to $P$-symmetric oscillatory stable solitons and yields the groups
$\bar{\mathit{K}}^{0,\tau}_{P}(\mathbb{T}^{d})\oplus\bar{\mathit{K}}^{0}_{P}(*)$
and
$\mathit{K}_{P}^{\tau,-1}(\mathbb{T}^{d-1})\oplus\bar{\mathit{K}}^{0}_{P}(*)$.
Once again, $\bar{\mathit{K}}^{0}_{P}(*)$ is not trivial, it is equivalent to
the well-known representation ring of $P$, $\bar{R}(P)$ Bröcker and Tom Dieck
(2013).

What is the physical interpretation of these classes? On the one hand for
positive solitons, $\mathfrak{H}_{-1}$ is a direction of instability which has
to be controlled Weinstein (2006). For the perturbed solution to remain at
most $\epsilon$-distance from $\Phi^{\lambda}$ at any value of $z$, the
initial perturbation needs to be at a distance
$\delta(\epsilon,\mathfrak{H}_{-1})$ from $\Phi^{\lambda}$. As we
adiabatically evolve $\mathcal{L}(\lambda)(s)$, we would expect an
$s$-dependence $\delta(\epsilon,s)$; however, the topological character of
$\mathfrak{H}_{-1}$ and $P$-symmetry will mean that $\delta$ is
$s$-independent. Thus, how close the initial perturbation has to be to our
soliton depends only on the topological action of $P$ on $\mathfrak{H}_{-1}$
and the $\epsilon$ chosen. On the other hand for oscillatory stability, the
neutral mode subspace $\mathfrak{H}_{N}(\lambda)$ is a $P$-representation. As
discussed in Soffer and Weinstein (2005), neutral modes are relevant for
scattering. In particular, there is a mechanism in which non-linear excited
bound states dissipate their energy towards radiation modes and the ground
state. The dissipation coefficient $\mathit{\Gamma}$ is a function of the
neutral modes ($\mathit{\Gamma}\neq 0$ is the non-linear analogue of Fermi’s
golden rule). As we mode adiabatically evolve our system, so will
$\mathit{\Gamma}(s)$ vary; however, we should always be able to split it
$\mathit{\Gamma}(s)=\gamma(\mathfrak{H}_{N}(\lambda))\mathit{\Gamma}_{*}(s)$,
with $\gamma(\mathfrak{H}_{N}(\lambda))$ only depending on
$\mathfrak{H}_{N}(\lambda)$ as a $P$-representation. Furthermore, we speculate
that since these modes must be linearly topologically robust,
$\gamma(\mathfrak{H}_{N})$ should dominate and imply a qualitatively slower
decay of the modes. We leave the interesting task of explicitly determining
$\gamma$ to future work. We remark that both of these new invariants do not
arise in linear systems.

We now briefly discuss the inclusion of time-reversal symmetry $\Theta$. Since
our systems are bosonic, we only have the so called class AI ($\Theta^{2}=I$)
in the AZ classification Altland and Zirnbauer (1997). We thus have to replace
the $\mathit{K}$-groups with $\mathit{KR}$-groups in the sense of Atiyah
Atiyah (1966), Kitaev (2009). Let us focus on the invariants arising from
soliton stability. For drift stability, instead of $H^{2}(*;\mathbb{Z})$, we
have the group $H^{2}_{\mathbb{Z}_{2}}(*;\mathbb{Z}(1))$ De Nittis and Gomi
(2014), which is trivial. Similarly for oscillatory stability, instead of
$\tilde{\mathit{K}}^{0}(*)$ we have the group $\tilde{\mathit{KR}}^{-6}(*)$
Kitaev (2009), which using Bott periodicity (a fundamental result in
$\mathit{K}$-theory) turns out to be, again, trivial. Hence, we can conclude
that for individual solitons in bosonic systems, time-reversal symmetry
$\Theta^{2}=I$ does not distinguish between the linear classification Kitaev
(2009), Adem  _et al._ (2016).

$d$ |  Boundary (y/n) | $P$ | $\Theta$ |  L+DS |  L+OS
---|---|---|---|---|---
2 | n | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}$
2 | y | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}$
2 | n | pm | 0 | $\mathbb{Z}^{2}\oplus\mathbb{Z}_{2}$ | $\mathbb{Z}^{2}\oplus\mathbb{Z}$
3 | y | pm | 0 | $\mathbb{Z}^{3}\oplus\mathbb{Z}_{2}$ | $\mathbb{Z}^{3}\oplus\mathbb{Z}$
2 | n | 0 | $\Theta^{2}=I$ | 0 | 0
2 | y | 0 | $\Theta^{2}=I$ | 0 | 0
Table 1: Some examples of topological classes for modes around solitons in
$d$-dimensional systems, where $P$ is the crystallographic point group,
$\Theta$ is the time-reversal operator, L+DS stands for linear plus drift
stable topological classes and L+OS for linear plus oscillatory stable
topological classes.

We present a few examples in dimension $d=2,\,3$ with either of these
symmetries in Table 1.

##  V Spaces of soliton solutions and Global classes

So far our analysis tells us the different topological character of individual
solitons, but does a single soliton define the character of eq (1)? Given
$\vec{A},V$ and $f$ there will generally be many solitons which satisfy
conditions (5, 7, 8, 9). The set of soliton solutions can be given a topology.
Let $M_{D}(E_{gap},\vec{A},V,f),\,M_{O}(E_{gap},\vec{A},V,f)$ be subspaces of
soliton solutions, which further satisfy either eq (9, 10) or eq (16)
respectively. $M_{D}(E_{gap})$ actually forms a manifold Zhou and Weinstein
(2008), but to our knowledge not much is known about $M_{O}(E_{gap})$. Let us
further identify two solitons in $M_{D}(E_{gap})$ (or $M_{O}(E_{gap})$) as
equivalent if they only differ by a translation ($\mathbb{Z}^{d}$), a Galilean
boost ($\mathbb{R}^{d}$) or a phase ($\mathit{S}^{1}$) when these are
symmetries of $(\vec{A},V,f)$. Abusing notation we will employ the same
symbols for the spaces resulting from the identification. Hence a triple
$(\vec{A},V,f)$ induces the maps

$G^{D}_{\vec{A},V,f}:M_{D}(E_{gap})\longrightarrow
Map(\mathbb{T}^{d},BGL_{\infty})\times Gr_{1},$ (19)

and

$G^{O}_{\vec{A},V,f}:M_{O}(E_{gap})\longrightarrow
Map(\mathbb{T}^{d},BGL_{\infty})\times BGL_{\infty}.$ (20)

The analogues for systems with boundaries are obtained using
$Map(\mathbb{T}^{d-1},\mathcal{F}^{sa}_{*}(\mathfrak{H}))$ instead. Then, if
we mode adiabatically evolve the system $(\vec{A}(s),V(s),f(s))$, these spaces
$M_{D}(E_{gap},s),\,M_{O}(E_{gap},s)$ will also change and not necessarily in
a continuous fashion. However, as long as they can be deformed into one
another via a homotopy, we can use $M_{D}(E_{gap}),\,M_{O}(E_{gap})$ as a
characteristic of the entire system and study the maps
$G^{D}_{\vec{A},V,f},\,G^{O}_{\vec{A},V,f}$ up to homotopy to define global
topological classes. We can also easily extend this to include symmetries. Let
us consider the easiest example for $f(s,\Psi)=f(s,|\Psi|^{2})$ where
$M_{D}(E_{gap})$ is the same as an interval Zhou and Weinstein (2008) once we
have made the proper identifications. This interval is simply the $\lambda$’s
which satisfy eqs (5), (7) and (8). Then $M_{D}(E_{gap})$ can be contracted to
a point and the global drift topological classes for these systems are the
same (eq (14)) as those for individual solitons. However, suppose that we
allow our systems to have $f$’s whose $\Psi$-dependence is not only on its
modulus. Then the phase symmetry $\Phi^{\lambda}\mapsto
e^{i\theta}\Phi^{\lambda}$ Zhou and Weinstein (2008) is lost, then we have
$M_{D}(E_{gap})=\mathit{S}^{1}\times(\lambda_{min},\lambda_{max})$ Tao (2008)
and the topological classes are equivalent to

$\bar{\mathit{K}}^{0,\tau}_{P}(\mathit{S}^{1}\times\mathbb{T}^{d})\oplus
H^{2}_{P}(\mathit{S}^{1};\mathbb{Z}),$ (21)

or the analogue for systems with boundary. Just as an appetizer we note that
for $P=0$, the trivial group, we obtain for fully periodic systems in $d=2$,
the group $\tilde{\mathit{K}}^{0}(\mathbb{T}^{3})=\mathbb{Z}^{3}$ and with
boundary $\mathit{K}^{-1}(\mathbb{T}^{2})=\mathbb{Z}^{2}$, so the presence of
a topologically non-trivial space $M_{D}(E_{gap})$ can add more non-trivial
topological classes than there are for individual solitons. We leave
equivariant computations for future work.

##  VI Summary and Conclusions

We have attacked, for the first time (to our knowledge), the problem of
assigning topological invariants to non-linear topological systems in
photonics, exciton-polaritons and BECs simultaneously Smirnova  _et al._
(2020), Gulevich  _et al._ (2017). Our analysis provides under which
conditions the modes around solitons have the same topological character as
linear phases do and describes how soliton stability conditions become
topologically non-trivial when including crystallographic symmetries,
providing new invariants for modes, that have no analogue in linear systems.
The same does not happen if we instead have bosonic time-reversal symmetry.
Using these constructions together with the space of soliton solutions, we
build novel, global invariants for the entire system, providing a partial
answer to the problem of classifying non-linear topological systems. General
optical systems are non-hermitian as they have gains and losses. Thus, a
natural extension of this work is to consider the $\mathit{K}$-theory arising
from the point and line gap generalizations for these systems Kawabata  _et
al._ (2019). It would also be very interesting to see if other stability
conditions for gap solitons have a topological character and to combine them
with the non-linear topological nature of vortices, skyrmions and other
standard topological solitons Manton and Sutcliffe (2004).

We thank O. Antolín-Camarena, K. Ramos-Musalem and J. Sheinbaum for useful
comments.

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