aainstitutetext: Facultad de Química, Universidad Nacional Autónoma de México,
04510 Mexico City, Mexico.bbinstitutetext: Instituto de Matematicas,
Universidad Nacional Autónoma de México, 04510 Mexico City, Mexico.

# Crystallographic Interacting Topological Phases and Equivariant Cohomology:
To assume or not to assume

Daniel Sheinbaum,111Corresponding author. b,2 Omar Antolín Camarena
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

For symmorphic crystalline interacting gapped systems we derive a
classification under adiabatic evolution. This classification is complete for
non-degenerate ground states. For the degenerate case we discuss some
invariants given by equivariant characteristic classes. We do not assume an
emergent relativistic field theory nor that phases form a topological
spectrum. We also do not restrict to systems with short-range entanglement,
stability against stacking with trivial systems nor assume the existence of
quasi-particles as is done in SPT and SET classifications respectively. Using
a slightly generalized Bloch decomposition and Grassmanians made out of ground
state spaces, we show that the $P$-equivariant cohomology of a $d$-dimensional
torus gives rise to different interacting phases, where $P$ denotes the point
group of the crystalline structure. We compare our results to bosonic
symmorphic crystallographic SPT phases and to non-interacting fermionic
crystallographic phases in class A. Finally we discuss the relation of our
assumptions to those made for crystallographic SPT and SET phases.

###### Keywords:

Topological States of Matter, Discrete Symmetries, Topological Field Theories

##  1 Introduction

The role of algebraic topology in the classification of topological phases of
matter began with the pioneering work of Thouless et al. Thouless  _et al._
(1982) and Avron et al. Avron  _et al._ (1983) on the integer quantum Hall
effect in the 80’s. There was a new surge of interest in topological phases
due to the discovery of topological insulators Kane and Mele (2005) in the mid
2000’s, culminating in the classification for non-interacting fermion systems
using characteristic classes De Nittis and Gomi (2014), De Nittis and Gomi
(2015) and equivariant $K$-theory Freed and Moore (2013),Kitaev (2009) which
still appears even in the presence of disorder Prodan and Schulz-Baldes
(2016), Thiang (2016).

Interacting topological phases, on the other hand, have been much harder to
analyse, beginning with the work on the fractional quantum Hall effect
Laughlin (1983), Arovas  _et al._ (2002) and the concept of topological order
Wen (1990). For symmetry protected (SPT) topological phases there are many
constructive classifications Chen  _et al._ (2013), Kapustin  _et al._ (2015),
Freed (2014), Debray (2021), Debray (2019), Gaiotto and Johnson-Freyd (2019),
Thorngren and Else (2018) to mention the trendy ones. We refer to these as
constructive because they do not derive their classification from first
principles, but rather construct their classification either by assuming that
there is an effective relativistic extended topological field theory emerging
at low energy, or that the set of short-range entangled invertible phases in
all dimensions conspire to form a spectrum in the sense of algebraic topology,
and there are different possible choices of spectra competing out there in the
literature (see Xiong (2018) for a discussion). We will instead derive a
classification for symmorphic crystallographic interacting systems under
adiabatic evolution Avron  _et al._ (1987), Bachmann  _et al._ (2017) without
these added assumptions: neither restricting to short-range entangled systems,
nor declaring that phases are stable under stacking with a system in a trivial
phase. Using a modest generalization of the Bloch decomposition to the many-
body case, we show that different equivariant characteristic classes Bott and
Tu (2001) in the cohomology ring $H^{*}_{P}(\mathbb{T}^{d};\mathbb{Z})$ give
rise to different interacting topological phases for degenerate and non-
degenerate ground states. For the non-degenerate case we provide a full
classification for symmorphic crystallographic groups given by
$H^{2}_{P}(\mathbb{T}^{d};\mathbb{Z})$ and compare our results in $d=2$ with
those of Thorngren and Else Thorngren and Else (2018) for the analogue SPT
systems. Here $P$ denotes the point group of the crystalline structure. We
find that every representative of these non-degenerate phases can be
adiabatically connected to a non-interacting system with a single valence
band. We also compare, in the simplest cases, our non-degenerate fermionic
classification to the non-interacting fermionic gapped phases of Shiozaki et
al. Shiozaki  _et al._ (2017) classified by equivariant $K$-theory. Let us
mention from the outset that our results do not imply that the published SPT
classifications are incorrect as we do not have all of the same starting
assumptions. Our calculations can be seen as measuring the effect of removing
the assumptions mentioned previously. We discuss in section 8 some plausible
scenarios. Symmetry enriched topological phases (SET) Mesaros and Ran (2013),
Cheng  _et al._ (2016), Lu and Vishwanath (2016) are traditionally classified
by their fractionalized quasi-particle excitations (sometimes arising from an
emergent relativistic field theory). The classification method for these is
completely different from the ones for SPTs. Our derivation applies to both
the degenerate and non-degenerate ground state of symmorphic crystalline
systems in a unifying manner (but it should be noted that in the degenerate
case, the cohomological invariants which are well understood in the algebraic
topology literature only allow a partial classification of these phases).
However we also do not have the same assumptions as the SET classifications
since we do not assume (and hence do not use) the existence of quasi-particle
excitations nor an algebraic structure associated to these.

##  2 Translation and Fock space decomposition

Common physical lore states that there is no Brillouin zone and no generalized
single particle picture for interacting systems. We now explain that the same
mathematical construction of the Brillouin zone used for non-interacting
systems can also be used for interacting systems. Consider for continuous
systems the one-particle Hilbert space $L^{2}(\mathbb{R}^{d};W)$, with $W$ an
$n$-dimensional representation of $\mathfrak{su}(2)$ corresponding to the spin
of the particle. We can construct a Fock space
$\mathfrak{F}(L^{2}(\mathbb{R}^{d};W))=\bigoplus_{n\geq
0}\,L^{2}(\mathbb{R}^{d};W)^{\otimes n}$ and there are bosonic and fermionic
Fock spaces $\mathfrak{F}_{+}(L^{2}(\mathbb{R}^{d};W))=\bigoplus_{n\geq
0}\mathrm{Sym}^{n}(L^{2}(\mathbb{R}^{d};W))$ and
$\mathfrak{F}_{-}(L^{2}(\mathbb{R}^{d};W))=\bigoplus_{n\geq
0}\Lambda^{n}(L^{2}(\mathbb{R}^{d};W))$ made out of symmetric and
antisymmetric wave functions. Let $\mathfrak{H}^{n}$ denote either
$\mathrm{Sym}^{n}(L^{2}(\mathbb{R}^{d};W))$ or
$\Lambda^{n}(L^{2}(\mathbb{R}^{d};W))$. Since $\mathbb{Z}^{d}$ acts on
$\mathbb{R}^{d}$ by translations, it also acts on $L^{2}(\mathbb{R}^{d};W)$,
and therefore on $\mathfrak{H}^{n}$ diagonally by acting on each factor. Just
as in the single particle case ($n=1$) we can perform a Bloch decomposition
Gomi (2017) using the dual group
$\widehat{\mathbb{Z}}^{d}=\mathrm{Hom}(\mathbb{Z}^{d},U(1))$ and write it as a
direct integral decomposition:

$\mathfrak{H}^{n}\simeq\int^{\oplus}_{\widehat{\mathbb{Z}}^{d}}\mathfrak{H}^{n}(\vec{k})d\vec{k}\,.$
(1)

This decomposition can be intuitively thought of as a direct sum of Hilbert
spaces, one for each point of $\widehat{\mathbb{Z}}^{d}\cong\mathbb{T}^{d}$, a
$d$-dimensional torus. We call this torus a generalized Brillouin zone, since,
though mathematically it is obtained in the same way, its physical
interpretation is rather different as $\vec{k}$ is a many-body crystal
momentum. The spaces $\mathfrak{H}^{n}(\vec{k})$ are easily defined:

$\mathfrak{H}^{n}(\vec{k})=\\{\psi\in\mathfrak{H}^{n}|\,T_{\vec{a}}\psi=e^{i\vec{k}\cdot\vec{a}}\psi\,\,\forall\vec{a}\in\mathbb{Z}^{d}\\}$
(2)

Obtaining this decomposition is as simple as for the single particle case, the
only differences being in a few details presented in Appendix A. Note that
this is simply a rewriting of the Hilbert space and is completely independent
of the Hamiltonian. Now we can repeat the same process for any $n$ and hence
decompose the entire Fock space as

$\mathfrak{F}_{\pm}(L^{2}(\mathbb{R}^{d};W))\simeq\int^{\oplus}_{\mathbb{T}^{d}}\bigoplus_{n\geq
0}\mathfrak{H}^{n}(\vec{k})d\vec{k}\,.$ (3)

where, just as one can swap integrals with sums, one can do the same for
direct integrals and direct sums. This rewriting of Fock space by itself is
not useful; it becomes useful when a many-body Hamiltonian operator
$\mathcal{H}$ has interaction terms which are discrete translation invariant
(e.g.
$\mathcal{H}=\sum_{j}(i\nabla_{j}-e\vec{A}(x_{j}))^{2}+V(x_{j})+\sum_{i\neq
j}e^{2}/|x_{i}-x_{j}|$), with $\vec{A}(x_{j}),\,V(x_{j})$ the periodic and
magnetic potentials associated to the ions and hence it commutes with the
unitary representation $U$ of $\mathbb{Z}^{d}$ on
$\mathfrak{F}_{\pm}(L^{2}(\mathbb{R}^{d};W))$, that is
$[\mathcal{H},U(\gamma)]=0$ for all $\gamma\in\mathbb{Z}^{d}$. In that case
$\mathcal{H}$ can be decomposed (“diagonalized”) into a family of operators
parametrized by $\mathbb{T}^{d}$ and written as:

$\mathcal{H}\simeq\int^{\oplus}_{\mathbb{T}^{d}}\mathcal{H}(\vec{k})d\vec{k}.$
(4)

The spectrum $\sigma(\mathcal{H}(\vec{k}))$ varies continuously with
$\vec{k}$. From the analytical point of view we have not made much progress at
solving the many-body problem as the spectral subspaces of
$\mathcal{H}(\vec{k})$ are still many-body wave functions and $\vec{k}$ is a
quantum number behaving like a total momentum of all particles combined.
Perhaps this is why this decomposition had been completely dismissed. However,
from the topological point of view it is a big gain as we now have the
topological space $\mathbb{T}^{d}$ to work with.

##  3 Gapped phases and Grassmanians

We now assume that the spectrum of the Hamiltonian is gapped: that for some
$\delta>0$, $\sigma(\mathcal{H})\cap(0,\delta)=\emptyset$. Note that there are
many examples of Hamiltonians which are not gapped so this is a strong
restriction already. Also note that the gapped condition is physically
meaningful only in the thermodynamic limit. General theorems Reed and Simon
(1980) for direct integral decompositions imply that if a translation
invariant $\mathcal{H}$ is gapped then $\mathcal{H}(\vec{k})$ is also gapped
except at most for a measure zero set in $\mathbb{T}^{d}$, and such cases are
non-generic so without loss of generality we can further restrict to
Hamiltonians $\mathcal{H}$ such that
$\sigma(\mathcal{H}(\vec{k}))\cap(0,\delta)=\emptyset$ for all $\vec{k}$. Thus
for each $\vec{k}$, $\mathcal{H}(\vec{k})$ singles out a subspace
$\mathfrak{G}(\vec{k})$ of $\mathfrak{F}_{\pm}(\vec{k}):=\bigoplus_{n\geq
0}\mathfrak{H}^{n}(\vec{k})$ corresponding to the ground state space. Let $m$
be the ground state degeneracy, so $\dim\mathfrak{G}(\vec{k})=m$. The set of
all $m$-dimensional subspaces of $\mathfrak{F}_{\pm}(\vec{k})$ can be given a
topology making it into a space called a Grassmanian Abbondandolo and Majer
(2009), denoted by $\mathcal{G}_{m}(\mathfrak{F}_{\pm}(\vec{k}))$ and
$\mathfrak{G}(\vec{k})$ is a point in this space.

By Kuiper’s theorem Kuiper (1965), any bundle of Hilbert spaces is trivial, so
the bundle $\mathfrak{F}_{\pm}(\vec{k})$ is isomorphic to one with no
dependence on $\vec{k}$, and thus the $\mathfrak{F}_{\pm}(\vec{k})$ can be
replaced by a fixed Hilbert space $\mathfrak{H}^{{}^{\prime}}$. We then obtain
a continuous function
$\mathbb{T}^{d}\rightarrow\mathcal{G}_{m}(\mathfrak{H}^{{}^{\prime}})$ given
by $\vec{k}\mapsto\mathfrak{G}(\vec{k})$. Here we have, in a sense, a proto-
band theory: a single many-body band corresponding to $\mathfrak{G}(\vec{k})$;
however we believe that here is where the resemblance to semi-classical
conduction theory Ashcroft and Mermin (1976) ends.

Two systems are said to be in the same phase if one can adiabatically evolve
one into the other Avron  _et al._ (1987), Bachmann  _et al._ (2017). Here,
following Avron  _et al._ (1983), we shall model adiabatic evolution as a
continuous path $\mathcal{H}(s),\,s\in[0,1]$ of gapped Hamiltonians such that
$\mathcal{H}(0)=\mathcal{H}_{0}$ and $\mathcal{H}(1)=\mathcal{H}_{1}$. This
homotopy model has limitations which have been discussed in Bachmann  _et al._
(2017), Bachmann and Ogata (2015). Also note that published SPT
classifications only look at adiabatic paths which come from a local unitary
evolution defined by Chen, Gu and Wen Chen  _et al._ (2010). We do not
restrict ourselves to such paths (see Appendix D for a discussion). Thus, we
are interested in homotopy classes of ground state space maps
$\mathbb{T}^{d}\to\mathcal{G}_{m}(\mathfrak{H}^{\prime})$:

$[\mathbb{T}^{d},\mathcal{G}_{m}(\mathfrak{H}^{\prime})]\cong\mathsf{Vect}^{m}(\mathbb{T}^{d}),$
(5)

where $\mathsf{Vect}^{m}(\mathbb{T}^{d})$ is the set of isomorphism classes of
$m$-dimensional vector bundles on the torus. This last equivalence comes from
the fact that the homotopy type of $\mathcal{G}_{m}(\mathfrak{H}^{\prime})$ is
that of the classifying space for vector bundles, $BGL(m)$ Abbondandolo and
Majer (2009). This means that we can associate to each Hamiltonian an
$m$-dimensional vector bundle made out of the ground state space over the
generalized Brillouin zone. Thus far we have only enforced that our
Hamiltonians be gapped and $\mathbb{Z}^{d}$-invariant, is there any difference
at this level between bosons and fermions? No. The homotopy type of
$\mathcal{G}_{m}(\mathfrak{H}^{\prime})$ by definition does not care if the
Hilbert space inside the parenthesis is fermionic, bosonic, one particle or
many particle, so long as its complex and has a countable orthonormal basis
Abbondandolo and Majer (2009) which all of the above fulfil. So, is there any
way to differentiate between bosons and fermions? The distinction is captured
by further imposing on physical grounds that Hamiltonians of fermionic systems
must have fermion parity symmetry $(-1)^{F}$ i.e. $[\mathcal{H},(-1)^{F}]=0$,
and we have

$[\mathbb{T}^{d},\mathcal{G}_{m}(\mathfrak{H}^{\prime})]_{\mathbb{Z}^{F}_{2}}\cong\mathsf{Vect}^{m}_{\mathbb{Z}^{F}_{2}}(\mathbb{T}^{d}),$
(6)

where $\mathbb{Z}^{F}_{2}$ is generated by $(-1)^{F}$, which acts trivially on
$\mathbb{T}^{d}$ but not on $\mathcal{G}_{m}(\mathfrak{H}^{\prime})$; and
$\mathsf{Vect}^{m}_{\mathbb{Z}^{F}_{2}}(\mathbb{T}^{d})$ denotes isomorphism
classes of $\mathbb{Z}^{F}_{2}$-equivariant vector bundles on the torus.
Fermion parity plays a more prominent role when combined with time-reversal
and charge conjugation symmetry but we shall leave such combinations for
future work.

##  4 Crystallographic Interacting Phases

Let us now assume that our Hamiltonian commutes with a crystallographic
symmetry group $G$, containing $\mathbb{Z}^{d}$ and also the point group $P$,
of allowed reflections and rotations. Here there are two important types of
$G$, symmorphic and non-symmorphic Freed and Moore (2013), Gomi (2017). For
now we shall only handle symmorphic groups, which means that $G$ is a semi-
direct product $G=\mathbb{Z}^{d}\rtimes P$. In this case, there is a canonical
$P$-action on the ground state bundle Gomi (2017). We are interested in
adiabatic evolutions which preserve the extra symmetry given by $P$, so our
phases will be described by

$[\mathbb{T}^{d},\mathcal{G}_{m}(\mathfrak{H}^{\prime})]_{P}\cong\mathsf{Vect}^{m}_{P}(\mathbb{T}^{d}).$
(7)

For fermions we instead have to consider a generalized symmetry group $G_{F}$
defined via a group extension Adem and Milgram (2013), Thorngren and Else
(2018)

$1\rightarrow\mathbb{Z}_{2}^{F}\rightarrow G_{F}\rightarrow G\rightarrow 1,$
(8)

as these symmetries may combine non-trivially with fermion parity. Again, we
only handle the symmorphic case in which $G_{F}=\mathbb{Z}^{d}\rtimes P_{F}$,
where $P_{F}$ is given by an extension,

$1\rightarrow\mathbb{Z}_{2}^{F}\rightarrow P_{F}\rightarrow P\rightarrow 1.$
(9)

Under these assumptions our phases will be given by:

$[\mathbb{T}^{d},\mathcal{G}_{m}(\mathfrak{H}^{\prime})]_{P_{F}}\cong\mathsf{Vect}^{m}_{P_{F}}(\mathbb{T}^{d}).$
(10)

##  5 $m=1$, the non-degenerate case

For $m=1$, the non-degenerate case, our Grassmanian is homotopy equivalent to
$BU(1)\simeq\mathbb{C}P^{\infty}$, the infinite dimensional complex projective
space which is an Eilenberg-Maclane space $K(\mathbb{Z},2)$, and is the
classifying space of the ordinary cohomology group $H^{2}$ Hatcher (2002).
Hence for non-degenerate symmorphic crystallographic interacting phases we
have (see Appendix B for details)

$[\mathbb{T}^{d},\mathcal{G}_{1}(\mathfrak{H}^{\prime})]_{P}\cong
H^{2}_{P}(\mathbb{T}^{d};\mathbb{Z})$ (11)

for bosons and for fermions it is $H^{2}_{P_{F}}(\mathbb{T}^{d};\mathbb{Z})$.
We remark that this isomorphism is only as a set, as the group addition does
not correspond to the so called stacking operation. Nonetheless this outcome
is sufficient to know all possible non-degenerate crystalline phases and to
distinguish among them, and is, in that sense, complete. Ideally we would take
as an example some many-body Hamiltonian with interaction terms and find the
equivariant Chern classes of its ground state bundle. Unfortunately this would
likely involve solving the interacting many-body problem and we were unable to
find exact solutions in the literature to compute to which phase they belong
to. We also note that the only method we are aware of for constructing
interacting toy models, the so called Haldane’s pseudo-potentials Seiringer
and Yngvason (2020) requires one to place the particles on a cylinder or
sphere and it seems non-trivial to adapt our generalized Bloch decomposition
to such cases. Instead here we start at the opposite end and present all
possible bosonic phases for $m=1,d=2$ symmorphic wallpaper groups.
To gain some perspective on the results presented below we note that this is
analogous to employing the fundamental theorem of algebra to conclude how many
roots a given polynomial has. The theorem does not tell one which numbers are
roots of a given polynomial, it only tells us that there are $n$ for a degree
$n$ polynomial. Similarly, our algebraic methods yield all symmorphic
crystalline phases for $m=1$ stable under adiabatic evolution with
interections that do not break symmetry, gap nor non-degeneracy but it does
not say anything about which Hamiltonians are representatives of these phases.
Nevertheless we show there are physical representatives of all of these phases
in section 6.
These groups were previously computed by Gomi Gomi (2017). To compare to the
SPT literature, we picked the work of Thorngren and Else Thorngren and Else
(2018), where they discuss how their approach considers both the group
cohomology and cobordism crystallographic SPTs. They obtain in general the
group $H^{2+d}(G;\mathbb{Z})$, corresponding to group cohomology Adem and
Milgram (2013) and $G=\mathbb{Z}^{d}\rtimes P$ is again the full
crystallographic group. In Table 1 we compare to their $d=2$ case.

$G$ | $P$ | $H^{2}_{P}(\mathbb{T}^{2};\mathbb{Z})$ | $H^{4}(G;\mathbb{Z})$
---|---|---|---
$\mathsf{p2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}\oplus\mathbb{Z}_{2}^{3}$ | $\mathbb{Z}_{2}^{4}$
$\mathsf{p3}$ | $\mathbb{Z}_{3}$ | $\mathbb{Z}\oplus\mathbb{Z}_{3}^{2}$ | $\mathbb{Z}_{3}^{3}$
$\mathsf{p4}$ | $\mathbb{Z}_{4}$ | $\mathbb{Z}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{4}$ | $\mathbb{Z}_{2}\oplus\mathbb{Z}_{4}^{2}$
$\mathsf{p6}$ | $\mathbb{Z}_{6}$ | $\mathbb{Z}\oplus\mathbb{Z}_{6}$ | $\mathbb{Z}_{2}^{2}\oplus\mathbb{Z}_{3}^{2}$
$\mathsf{pm}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}^{2}$ | $\mathbb{Z}_{2}^{2}$
$\mathsf{cm}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$
$\mathsf{pmm}$ | $D_{2}$ | $\mathbb{Z}_{2}^{4}$ | $\mathbb{Z}_{2}^{8}$
$\mathsf{cmm}$ | $D_{2}$ | $\mathbb{Z}_{2}^{3}$ | $\mathbb{Z}_{2}^{5}$
$\mathsf{p31m}$ | $D_{3}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}\oplus\mathbb{Z}_{3}$
$\mathsf{p3m1}$ | $D_{3}$ | $\mathbb{Z}_{2}\oplus\mathbb{Z}_{3}$ | $\mathbb{Z}_{2}$
$\mathsf{p4m}$ | $D_{4}$ | $\mathbb{Z}_{2}^{3}$ | $\mathbb{Z}_{2}^{6}$
$\mathsf{p6m}$ | $D_{6}$ | $\mathbb{Z}_{2}^{2}$ | $\mathbb{Z}_{2}^{4}$
Table 1: $m=1,d=2$ bosonic symmorphic crystallographic phases
($H^{2}_{P}(\mathbb{T}^{2};\mathbb{Z})$) vs $d=2$ bosonic symmorphic
crystallographic SPT phases ($H^{4}(G;\mathbb{Z})$). Input taken from Gomi
(2017) and Thorngren and Else (2018).

It turns out that $H^{2}_{P}(\mathbb{T}^{2};\mathbb{Z})$ is isomorphic to the
second cohomology group of the _reciprocal group_ of the total symmetry group
$G$, i.e., the group $G^{*}$ generated by the point group together with
translations in the reciprocal lattice (see Appendix C) and, because of the
difference in degree and the difference between $G$ and $G^{*}$, in general
these two should yield different results. However for $\mathsf{pm}$ and
$\mathsf{cm}$ the number of phases is the same. For $\mathsf{p2}$,
$\mathsf{p3}$, $\mathsf{p4}$, $\mathsf{p6}$ and $\mathsf{p3m1}$ our $m=1,d=2$
has more phases than the SPTs found by Thorngren and Else whereas for
$\mathsf{pmm}$ and $\mathsf{cmm}$, $\mathsf{p4m}$, $\mathsf{p6m}$ and
$\mathsf{p31m}$ there are more SPT phases. Where there is a difference between
our results and the SPT results it is clearly due to us having made fewer
assumptions: but whether it is solely due to not having imposed short-range
entanglement or stability under stacking with the trivial phase, or whether it
is due to a true physical difference between emergent relativistic topological
field theories and gapped topological phases, remains an interesting open
question.

Notice that neither having more nor fewer phases than in the SPT
classification should be surprising. For example, if the difference in results
were due to us not assuming stability under the so-called stacking operation
with a trivial phase, then we may see more phases consisting of fragile
crystalline topological phases Po  _et al._ (2018), which are robust under
adiabatic evolution to interacting systems Else  _et al._ (2019). On the other
hand, if the difference also lies in the short-range entanglement condition,
it may also happen that we see fewer phases: two distinct SPT phases, one of
which only arises in interacting systems, which by virtue of being distinct
are _not_ connected by adiabatic evolution through short-range entangled
states, could nevertheless be connected by unconstrained adiabatic evolution
—such SPT phases would merge into a single phase in our classification.

We now compare our result with non-interacting phases. It is well known that
there are no non-interacting topological phases for bosonic systems. As we
shall show in section 6, every non-degenerate interacting fermionic system can
be adiabatically evolved to a free fermion system. However, as we show below,
not every free fermion topological phase is stable under adiabatically
evolving through interactions. On the other hand non-interacting fermionic
phases stabilized under stacking are classified by twisted equivariant
$K$-theory Kitaev (2009), Freed and Moore (2013), Shiozaki  _et al._ (2017).
Therefore, our interacting phases are not necessarily a subset of these since
we can also capture fragile phases Po  _et al._ (2018) which are stable under
such adiabatic evolution. We consider the simplest case where
$P_{F}=P\times\mathbb{Z}_{2}^{F}$, which leaves out some interesting purely
fermionic phases, and compare
$H^{2}_{P\times\mathbb{Z}_{2}^{F}}(\mathbb{T}^{2};\mathbb{Z})$ with those of
Shiozaki et al. Shiozaki  _et al._ (2017), which use the $K$-group
$\bar{K}^{0}_{P}(\mathbb{T}^{2})$, those being the symmorphic ones in class
$A$ 222By $\bar{K}^{0}_{P}(\mathbb{T}^{d})$ we mean the kernel of the
homomorphism $K^{0}_{P}(\mathbb{T}^{d})\rightarrow K^{0}(\cdot)$. This is
because Shiozaki  _et al._ (2017) includes the virtual dimension of the
bundle, which has no physical interpretation. This is tantamount to removing a
$\mathbb{Z}$ factor.. As reflections may interact non-trivially with
$\mathbb{Z}^{F}_{2}$, we only consider groups without reflections, i.e.
$\mathsf{p2}$, $\mathsf{p3}$, $\mathsf{p4}$ and $\mathsf{p6}$ as a more
sensible comparison. We compare them in Table 2:

$G$ | $P$ | $H^{2}_{P}(\mathbb{T}^{2};\mathbb{Z})$ | $\bar{K}^{0}_{P}(\mathbb{T}^{2})$
---|---|---|---
$\mathsf{p2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}\oplus\mathbb{Z}_{2}^{3}$ | $\mathbb{Z}^{5}$
$\mathsf{p3}$ | $\mathbb{Z}_{3}$ | $\mathbb{Z}\oplus\mathbb{Z}_{3}^{2}$ | $\mathbb{Z}^{7}$
$\mathsf{p4}$ | $\mathbb{Z}_{4}$ | $\mathbb{Z}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{4}$ | $\mathbb{Z}^{8}$
$\mathsf{p6}$ | $\mathbb{Z}_{6}$ | $\mathbb{Z}\oplus\mathbb{Z}_{6}$ | $\mathbb{Z}^{9}$
Table 2: $m=1,d=2$ fermionic symmorphic crystallographic phases
($H^{2}_{P}(\mathbb{T}^{2};\mathbb{Z})$) without the $\mathbb{Z}_{2}^{F}$
parity index vs $d=2$ symmorphic crystallographic non-interacting fermionic
phases ($\bar{K}^{0}_{P}(\mathbb{T}^{2})$). Symmetry groups are those without
reflections i.e. $\mathsf{p2}$, $\mathsf{p3}$, $\mathsf{p4}$ and
$\mathsf{p6}$. We have removed the non-physical virtual dimension
$\mathbb{Z}$-factor included in Shiozaki  _et al._ (2017). For these groups
there are many more non-interacting phases than interacting ones. Input taken
from Gomi (2017) and Shiozaki  _et al._ (2017).

We note that the case $P_{F}=P\times\mathbb{Z}_{2}^{F}$ for $m=1$ yields

$H^{2}_{P_{F}}(\mathbb{T}^{2};\mathbb{Z})\approx
H^{2}_{P}(\mathbb{T}^{2};\mathbb{Z})\oplus\mathbb{Z}_{2}^{F}$ (12)

which essentially gives the same classes as bosons except for the parity
factor, which we do not include in Table 2. We find that there are far more
non-interacting phases than interacting ones for these groups, indicating that
many non-interacting phases can be connected by adiabatic evolution through
interacting states.

##  6 Explicit models

The purpose of the present section is to provide physical examples of systems
which have non-zero invariants over the generalized Brillouin zone i.e. which
correspond to a non-trivial topological phase in
$H^{2}_{P}(\mathbb{T}^{d};\mathbb{Z})$.

###  6.1 Warm up, only translations, $d=2$

Consider a system of $n$ non-interacting particles (let us ignore fermionic
indistinguishability for simplicity) in $d=2$ i.e. the Hamiltonian is of the
form:

$\mathcal{H}=\sum_{i=1}^{n}\mathcal{L}_{i}\otimes\bigotimes_{j\neq i}I_{j},$
(13)

with $\mathcal{L}_{i}=\mathcal{L}$ the same single particle Hamiltonian.
Furthermore let us consider a single particle Hamiltonian $\mathcal{L}$ which,
by itself, has a non-trivial Chern number over the single particle Brillouin
zone, which we denote as $\mathbb{T}^{2}_{i}$, where the $i$ indicates the
single particle Brillouin zone corresponding to the $ith$-particle. We shall
prove that this non-interacting many-body system has a non-trivial Chern
number over the generalized Brillouin zone, namely, it will be the same as the
Chern number of the single particle wave function. The ground state of
$\mathcal{H}$ is a tensor product of states $\Psi=\bigotimes_{i}^{n}\psi_{i}$.
Let us assume for simplicity that the single particle Hamiltonian
$\mathcal{L}$ has only one energy band $E_{1}(\vec{k})$ below the Fermi energy
$E_{F}$ with corresponding eigenfunction $\psi_{1}(\vec{k})$ and that our
system is gapped. Then we know from the single particle picture that each
$\psi_{i}$ will be a distribution over the $\psi_{1}$:

$\displaystyle\Psi$ $\displaystyle\simeq$
$\displaystyle\bigotimes_{i=1}^{n}\left[\int^{\oplus}_{\mathbb{T}^{2}_{i}}\psi_{1,i}(\vec{k}_{i})d\vec{k}_{i}\right]$
(14) $\displaystyle\simeq$
$\displaystyle\int^{\oplus}_{\mathbb{T}^{2n}}\left(\bigotimes_{i=1}^{n}\psi_{1,i}(\vec{k}_{i})\right)\sqcap_{i}d\vec{k}_{i},$

where we have used the distributive property to rewrite the tensor product as
a direct integral over the product of the single-particle Brillouin zones.

We now wish to rewrite $\Psi$ in terms of the generalized Bloch decomposition,
where each $\vec{k}$ in the generalized Brillouin zone is, in this case, a sum
over the $\vec{k}_{i}$’s i.e. $\vec{k}=\sum_{i=1}^{n}\vec{k}_{i}$. So, by
rewriting, say, $\vec{k}_{n}$ in terms of $\vec{k}$ we can perform the
following decomposition:

$\displaystyle\Psi$ $\displaystyle\simeq$
$\displaystyle\int^{\oplus}_{\mathbb{T}^{2}}\Psi(\vec{k})d\vec{k}\,,$ (15)
$\displaystyle\Psi(\vec{k})$ $\displaystyle=$
$\displaystyle\int^{\oplus}_{\mathbb{T}^{2(n-1)}}\bigotimes_{i=1}^{n-1}\psi_{1,i}(\vec{k}_{i})\otimes\psi_{1,n}\Bigl{(}\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}\Bigr{)}\sqcap_{i}d\vec{k}_{i}.$
(16)

We can therefore compute the many-body Chern number of $\Psi(\vec{k})$ using
the Berry connection:

$\begin{split}A_{\rho}(\vec{k})&=-i\int_{\mathbb{B}^{n}}\Psi^{*}(\vec{k})\frac{\partial}{\partial
k_{\rho}}\Psi(\vec{k})\sqcap_{j}d\vec{x}_{j}\\\
&=-i\int_{\mathbb{B}^{n}}\int_{\mathbb{T}^{2(n-1)}}\left(\prod_{i=1}^{n-1}|\psi_{1,i}(\vec{k}_{i})|^{2}\right)\psi^{*}_{1,n}\Bigl{(}\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}\Bigr{)}\frac{\partial}{\partial
k_{\rho}}\psi_{1,n}\Bigl{(}\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}\Bigr{)}\sqcap_{i}d\vec{k}_{i}\sqcap_{j}d\vec{x}_{j}\end{split}$

and curvature

$\displaystyle F_{\sigma\rho}(\vec{k})$ $\displaystyle=\frac{\partial
A_{\sigma}(\vec{k})}{\partial k_{\rho}}-\frac{\partial
A_{\rho}(\vec{k})}{\partial k_{\sigma}}$
$\displaystyle=i\int_{\mathbb{B}^{n}}\frac{\partial\Psi^{*}(\vec{k})}{\partial
k_{\sigma}}\frac{\partial\Psi(\vec{k})}{\partial
k_{\rho}}-\frac{\partial\Psi^{*}(\vec{k})}{\partial
k_{\rho}}\frac{\partial\Psi(\vec{k})}{\partial
k_{\sigma}}\sqcap_{j}d\vec{x}_{j}$
$\displaystyle\begin{split}=i\int_{\mathbb{B}^{n}}\int_{\mathbb{T}^{2(n-1)}}&\left(\prod_{i=1}^{n-1}|\psi_{1,i}(\vec{k}_{i})|^{2}\right)\\\
&\left(\frac{\partial}{\partial
k_{\sigma}}\psi^{*}_{1,n}\Bigl{(}\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}\Bigr{)}\frac{\partial}{\partial
k_{\rho}}\psi_{1,n}\Bigl{(}\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}\Bigr{)}\right.\\\
&\quad-\left.\frac{\partial}{\partial
k_{\rho}}\psi^{*}_{1,n}\Bigl{(}\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}\Bigr{)}\frac{\partial}{\partial
k_{\sigma}}\psi_{1,n}\Bigl{(}\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}\Bigr{)}\right)\sqcap_{i}d\vec{k}_{i}\sqcap_{j}d\vec{x}_{j},\end{split}$

with $\mathbb{B}^{n}$ the product of $n$ fundamental regions $\mathbb{B}$.
Thus the many-body Chern number is simply:

$\sigma_{xy}=\frac{-i}{2\pi}\int_{\mathbb{T}^{2}}F_{xy}d\vec{k}$ (17)

Putting all the integrals in the $\vec{k}$’s together, we have an integral
over $\mathbb{T}^{2(n-1)}\times\mathbb{T}^{2}$. The main difficulty in
performing the integral are the arguments of the form
$\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}$. We can perform a trick, using the
automorphism $g$ of $\mathbb{T}^{2(n-1)}\times\mathbb{T}^{2}$ which maps

$\bigl{(}\vec{k}_{1},...,\vec{k}_{n-1},\vec{k}\bigr{)}\mapsto\Bigl{(}\vec{k}_{1},...,\vec{k}_{n-1},\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}\Bigr{)},$
(18)

whose Jacobian $Dg$ has $\det Dg=1$. Thus, our integral is equivalent to one
under the image of $g^{-1}$, which transforms the arguments
$\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}$ into $\vec{k}$. Now we can perform the
integral first with respect to the $\vec{k}_{i}$’s and then with respect to
$\vec{k}$. The first integral is equal to the product of the norms of the wave
functions $\psi_{1,i}$, which are normalized to $1$. The second one is simply
equal to the single-particle Chern number $\sigma^{s}_{xy}$, hence

$\sigma_{xy}=\sigma^{s}_{xy}.$ (19)

We have thus constructed a model (a non-interacting one) where the many-body
Chern number over the generalized Brillouin zone has a physical interpretation
and can be non-zero in $H^{2}(\mathbb{T}^{2};\mathbb{Z})$, namely the same one
as the Chern number constructed from the single particle Brillouin zone, i.e.
the conductance $\sigma^{s}_{xy}$ of the system and its class in
$H^{2}(\mathbb{T}_{i}^{2};\mathbb{Z})$. However, we can conclude something
stronger. The derivation in the previous sections implies that the many-body
Chern number will be the same even if we adiabatically evolve the system via
including interaction terms, so long as these do not break the gap nor ground
state degeneracy. We conclude something which is already well known: the
quantization of the conductance of non-interacting gapped systems is stable
under the inclusion of certain types of particle-particle interactions. Which
particle-particle interactions do not break the gap nor degeneracy? This has
been already addressed in quantitative terms in Hastings and Michalakis
(2015), Bachmann  _et al._ (2018).

###  6.2 $P=\mathbb{Z}_{2}\times\mathbb{Z}_{2},\,d=3$

Let us consider now a system with a Hamiltonian of the form (13) in $d=3$,
where each single particle Hamiltonian $\mathcal{L}_{i}$ has crystalline
symmetry group $F222$. The group $F222$ has as point group
$P=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, where each factor is generated by
rotations of order 2 around the $x$ and $y$ axes. The corresponding lattice is
face-centered, and has as reciprocal lattice vectors (in the appropriate
units) $\vec{b}_{1}=(-1,1,1),\,\vec{b}_{2}=(1,-1,1),\,\vec{b}_{3}=(1,1,-1)$.
The model, first presented (to our knowledge) in Shiozaki  _et al._ (2018a)
and Shiozaki  _et al._ (2018b). The wave-function associated to $\mathcal{L}$
produces a Berry connection $\vec{A}_{s}(\vec{k}_{i})$ and curvature
$F_{s}(\vec{k}_{i})$ respectively. The fundamental region for the entire
crystallographic group is a quarter cube of the single particle Brillouin zone
(again see Shiozaki  _et al._ (2018a) for a complete sketch). The quotient of
this boundary by the $P$-action yields a real projective plane
$\mathbb{R}P^{2}$, and we can see that $\mathcal{L}$ represents a nontrivial
phase because we generate the non-zero cohomology class in
$H^{2}(\mathbb{R}P^{2};\mathbb{Z})=\mathbb{Z}_{2}$ using
$\vec{A}_{s}(\vec{k}_{i})$ and $F_{s}(\vec{k}_{i})$ in the formula Freed
(1986)

$c_{1}=\frac{i}{2\pi}\ln\mathsf{Hol}_{l}(\vec{A}_{s})+\frac{1}{2}\frac{i}{2\pi}\int_{X}F_{s}$
(20)

where $\mathsf{Hol}_{l}(\vec{A})$ denotes the holonomy of $\vec{A}$ around the
loop $l$, which is the generator of the first homology group
$H_{1}(\mathbb{R}P^{2};\mathbb{Z})$, and $X$ is the 2-dimensional cycle
attached to it. In terms of the quarter Brillouin zone, $l$ is the cycle made
out of 3 of edges after taking the quotient, and similarly, $X$ is made out of
the corresponding 3 faces and taking the quotient. We now come back to the
many-body system of $n$ non-interacting particles 13 with a single-particle
Hamiltonian as above. Once more, the ground-state will be made of a tensor
product of single-particle wave-functions $\Psi=\bigotimes_{i=1}^{n}\psi_{i}$.
We shall repeat a similar procedure to that of our previous example, but now
at the level of direct integrals. We can first decompose $\Psi$ into a direct
integral over the product of the single-particle Brillouin zones
$\mathbb{T}^{3n}$ rewriting the tensor product of direct integral
decompositions as a direct integral over $\mathbb{T}^{3n}$. We can then pick
the $n$-th wave function and its crystal momentum $\vec{k}_{n}$ and rewrite it
as $\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}$, where
$\vec{k}=\sum_{i=1}^{n}\vec{k}_{i}$ is, once more, a point on the generalized
Brillouin zone $\mathbb{T}^{3}$. Now, we rewrite the direct integral over
$\mathbb{T}^{3(n-1)}\times\mathbb{T}^{3}$ as a direct integral over
$\mathbb{T}^{3}$ only, where the integrand itself is the direct integral

$\Psi(\vec{k})=\int_{\mathbb{T}^{3(n-1)}}^{\oplus}\bigotimes_{i}^{n-1}\psi_{i}(\vec{k}_{i})\otimes\psi_{n}\Bigl{(}\vec{k}-\sum_{i=0}^{n-1}\vec{k}_{i}\Bigr{)}\sqcap_{i}^{n-1}d\vec{k}_{i}.$
(21)

There is a line bundle over $\mathbb{T}^{3}$ associated to this direct
integral which we also denote as $\Psi$ and whose fiber at each $\vec{k}$ is
essentially the span of $\Psi(\vec{k})$. We will now show there is a
$P$-equivariant vector bundle isomorphism between $\Psi$ and the line bundle
we shall call $\Psi^{g}$, defined to have fibers given by the the span of:

$\Psi^{g}(\vec{k})=\int_{\mathbb{T}^{3(n-1)}}^{\oplus}\bigotimes_{i=1}^{n-1}\psi_{i}(\vec{k}_{i})\otimes\psi_{n}(\vec{k})\sqcap_{i}^{n-1}d\vec{k}_{i}$
(22)

Indeed, such an isomorphism $\Psi^{g}\cong\Psi$ is provided by
$\int_{\mathbb{T}^{3}}^{\oplus}\bigotimes_{i=1}^{n}\mathrm{Id}\otimes
Md\vec{k}$, where $M$ is the operator of multiplication by the function
$e^{-i\sum_{i=1}\vec{k}_{i}\cdot\vec{x}_{n}}$, because by Bloch’s theorem,
$\psi_{n}$ satisfies:

$\psi_{n}\Bigl{(}\vec{k}-\sum_{i=1}^{n-1}\vec{k}_{i}\Bigr{)}(\vec{x}_{n})=e^{-i\sum_{i=1}\vec{k}_{i}\cdot\vec{x}_{n}}\psi_{n}(\vec{k})(\vec{x}_{n}).$
(23)

The difference between $\Psi^{g}$ and $\Psi$ is simply that the last factor in
the tensor product is now independent of the $\vec{k}_{i}$’s and thus the
Berry connection and curvature $\vec{A}_{g}$ and $F_{g}$ of $\Psi^{g}$ over
the generalized Brillouin zone $\mathbb{T}^{3}$ are equivalent to the single-
particle ones $\vec{A}_{s}$ and $F_{s}$ associated to our single particle
Hamiltonian $\mathcal{L}$ over the single particle Brillouin zone
$\mathbb{T}^{3}_{i}$ since the integrals over the $\vec{k}_{i}$’s equal 1 in
the definitions of $\vec{A}_{g}$ and $F_{g}$, as in the previous subsection.
Using $\vec{A}_{g}$ and $F_{g}$ in the formula (20) and looking at the same
fundamental region but for the generalized Brillouin zone, will yield a non-
trivial class and since $\Psi^{g}$ and $\Psi$ are isomorphic, we conclude that
$\Psi$ represents a non-trivial phase in
$H^{2}_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}(\mathbb{T}^{3};\mathbb{Z})$. Which
class in
$H^{2}_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}(\mathbb{T}^{3};\mathbb{Z})$ is it?
The calculation for the whole group yields:

$H^{2}_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}(\mathbb{T}^{3};\mathbb{Z})=\mathbb{Z}_{4}\oplus\mathbb{Z}_{2}^{2}$
(24)

Indeed, a rational cohomology calculation shows that the group is torsion, so
the universal coefficients theorem says it is $H_{1}((\mathbb{T}^{3}\times
EP)/P;\mathbb{Z})$, which is the abelianization of
$\pi_{1}((\mathbb{T}^{3}\times
EP)/P)\cong\mathbb{Z}^{3}\rtimes(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$. Thus
the $\mathbb{Z}_{4}$ summand actually comes from the action of
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ on the fundamental group of
$\mathbb{T}^{3}$, as did our invariant. To see what this $\mathbb{Z}_{2}$
invariant corresponds to in
$H^{2}_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}(\mathbb{T}^{3};\mathbb{Z})$
consider that the copy of $\mathbb{RP}^{2}$ used to define the invariant sits
inside of $\mathbb{T}^{3}/P\cong\mathbb{RP}^{3}$. The inclusion of
$\mathbb{RP}^{2}$ in $\mathbb{RP}^{3}$ induces an isomorphism in $H^{2}$, and
the canonical map $(\mathbb{T}^{3}\times EP)/P\to\mathbb{T}^{3}/P$ can be
shown to induce on $H^{2}$ the inclusion of $\mathbb{Z}_{2}$ as the order 2
subgroup of the $\mathbb{Z}_{4}$ summand.

It was argued in Shiozaki  _et al._ (2018b) that this non-interacting phase
extended to a $d=3$ fermionic SPT phase. Hence, this example shows that
systems representing a non-trivial phase in our derivation can also represent
a non-trivial SPT phase.

###  6.3 Stability of free fermion phases with a single valence band under
interacting adiabatic evolution

The above examples are evidence that as long as one has a fermionic non-
interacting system with a single band below the Fermi energy, one can use our
direct integral decomposition over the $dn$-dimensional torus and our
automorphism $g$ defined above to construct an invariant which is stable under
adiabatic evolution to systems with interaction terms, so long as the
evolution does not break the symmetry, gap nor non-degeneracy. Moreover, there
is a one-to-one correspondence between phases of single-particle fermionic
systems with a unique valence band and phases of many-body non-degenerate
fermionic systems, simply exchanging the single-particle Brillouin zone for
its many-body analogue: the generalized Brillouin zone. Thus, if one agrees
with the mathematical interpretations of our physical assumptions (and as we
have explained we are indeed using the same interpretations underlying all
available notions of phases in the physics literature) it implies that the
topological invariants of these non-interacting fermionic systems and their
corresponding physical interpretations are robust to adiabatic evolution
through any types of interactions that respect the above mentioned
constraints. Which interactions do not break the symmetry, gap nor non-
degeneracy? We hope to have a quantitative answer in the spirit of Hastings
and Michalakis (2015), Bachmann  _et al._ (2018) in future work. We further
emphasize that, as mentioned above, these are not all of the non-interacting
phases, but only those which arise when the associated single particle vector
bundle is a line bundle. This means that all other non-interacting fermionic
phases can be connected by an interacting adiabatic path to one of these non-
interacting phases that arise from single valence bands. Note that we did not
assume stability of these phases under stacking with the trivial phase, hence
we are also including the fragile ones Po  _et al._ (2018), some of which have
been shown to be stable under interactions through field theory methods Else
_et al._ (2019) and here we are deducing that only those arising from
$P$-equivariant single particle line bundles are stable under adiabatic
evolution through all interactions that preserve symmetry, gap and non-
degeneracy. Finally, let us remark that nothing analogous can happen for
degenerate systems nor for bosons as all of these must be, strictly speaking,
made from interacting systems.

##  7 $m>1$, the degenerate case

We now have $\mathsf{Vect}^{m}_{P}(\mathbb{T}^{d})$ and
$\mathsf{Vect}^{m}_{P_{F}}(\mathbb{T}^{d})$ as the full set of $m$-degenerate
symmorphic topological phases. Note that this case can only arise in systems
with interactions. Mathematically these objects are not as well understood as
the $m=1$ case since maps to $BGL(m)$ are no longer given by some generalized
cohomology group. One could, analogously to the non-interacting case, consider
stabilizing the dimension of the bundles and obtaining $P$-equivariant
$K$-theory, however it comes at a non-trivial price for we would be
stabilizing the ground state degeneracy and further analysis is required to
see how much physics remains after such an operation. But we could still
obtain a partial classification using equivariant Chern characteristic classes
Bott and Tu (2001) $c_{i}(\mathcal{H})\in
H^{2i}_{P}(\mathbb{T}^{d};\mathbb{Z})$. Phases with different equivariant
Chern classes are different, whereas phases with the same Chern classes may
still be different, but perhaps can only be distinguished through a finer
invariant. Similarly to the non-degenerate case, we do not know what stacking
corresponds to in terms of operations with these cohomology classes. One might
be tempted to think that stacking is a tensor product of the associated vector
bundles, however the direct integral decomposition of a tensor product is not
the fiberwise tensor product but rather what is known as an external tensor
product Hatcher (2003), and this is without taking into account the bosonic
and fermionic nature of the wave-functions, which most likely modifies the
external tensor product operation in a non-trivial manner.

###  7.1 $C_{2}$ Bosonic Examples

The simplest examples are those for bosons where $d=2$ and $P=C_{2}$ , the
group with 2 elements. There are 3 symmorphic crystallographic groups, of
which for brevity we will only discuss two, $\mathsf{pm}$ (reflections over a
fixed parallel axis) and $\mathsf{cm}$ (glide reflections) Gomi (2017). A way
to synthesize all cohomology groups $H^{i}_{P}(\mathbb{T}^{d};\mathbb{Z})$ is
by giving the full cohomology ring $H^{*}_{P}(\mathbb{T}^{d};\mathbb{Z})$. For
$\mathsf{pm}$, the action on the generalized Brillouin zone is
$(k_{1},k_{2})\mapsto(k_{1},-k_{2})$ and we can compute (see Appendix C):

$H^{*}_{C_{2}}(\mathbb{T}^{2};\mathbb{Z})\cong\mathbb{Z}[x,y,t]/(t^{2},2x,2y,xy),$
(25)

where $t$ has degree 1 and $x$ and $y$ have degree 2. On the other hand for
$\mathsf{cm}$, the action is $(k_{1},k_{2})\mapsto(k_{2},k_{1})$ and yields

$H^{*}_{C_{2}}(\mathbb{T}^{2};\mathbb{Z})\cong\mathbb{Z}[t,u,v]/(2t,2v,u^{2},ut,uv,v^{2}),$
(26)

where $u$, $t$ and $v$, have degrees $1$, $2$ and $3$, respectively. We note
that equivariant characteristic classes live in these cohomology rings,
however, which cohomology classes are characteristic classes remains an open
problem.

##  8 Differences with SPT and SET

Besides our lack of relativistic fields and spectra, let us discuss some of
the differences between the assumptions made in the constructive approaches
and ours. SPT phases are generally taken to have a unique ground state, just
as our non-degenerate phases, but said ground state should be short-range
entangled Chen  _et al._ (2013). We found that all systems representing our
non-degenerate interacting fermionic phases can be connected adiabatically to
a non-interacting free fermion system. Does it not contradict our finding that
we can have more of these phases than SPTs according to the results of
Thorngren-Else Thorngren and Else (2018)? At first glance it could seem
contradictory since our derivation should include all SPT paths and free
fermions are systems in a given SPT phase, so should there not be a path
connecting two apparently distinct interacting phases adiabatically connected
to non-interacting ones but which seem to be the in the same SPT? SPT
classifications also assume stability of phases under stacking with the
trivial phase. We have not put in any such criteria and, by the results of
section 6, we know that for fermions any non-trivial fragile phase arising
from single band (line bundle) in non-interacting systems will be stable under
adiabatic evolution through interactions which respect the non-degeneracy,
symmetry and gap. Thus our phases also include ones which are not stable under
the stacking operation, implicitly unaccounted in all SPT constructions and
hence we are seeing all non-interacting fragile phases which are stable to
interactions Else  _et al._ (2019). In our approach systems can transition
adiabatically from short-range entangled states (in fact, for fermions, from
product states) to long-range entangled states and back, so long as the ground
state remains unique. Thus, our non-degenerate phases are in principle more
robust than crystallographic SPTs. What happens in cases where there are fewer
crystalline phases than SPTs? In the case where the difference is due to
short-range entanglement it means that some of these SPT are not stable under
the inclusion of long-range interactions. However, the $F222$ example in
subsection 6.2 constitutes a fermionic crystalline SPT phase which is stable
under including long-range interactions adiabatically.

There are often further restrictions on the allowed actions of $P$ as it is
assumed that certain subspaces should be left invariant Else and Nayak (2014).
Here we did not put any restriction on the type of $P$-actions, except that
they arise from crystallographic groups. Another difference is that some of
the constructive approaches build their invariants through boundary modes,
whereas we assumed our systems had no boundary. As we mentioned in the
introduction these results do not imply that published SPT classifications are
incorrect since we start with a subset of their assumptions. SET phases on the
other hand construct their classification based on the fractionalization of
quasi-particle excitations Cheng  _et al._ (2016) such as anyons. We only
employed the ground state bundle and never addressed excitations. In fact we
did not even assume the existence of quasi-particles, hence our $m$-degenerate
topological phases could in principle occur in strongly interacting systems
that have no quasi-particles. However, in spin systems, Lieb-Schultz-Mattis
type theorems Hastings (2004) for gapped systems satisfying a Lieb-Robinson
bound forces many of them to have quasi-particle excitations and thus, they
may be generally forced upon us.

##  9 Conclusions

We have derived from first principles and a generalized Bloch transform a
classification for crystalline symmorphic interacting gapped phases without
assuming stability under stacking nor short-range entanglement. We have seen
that for non-degenerate fermionic ground states the classification is
complete, is given by $H^{2}_{P}(\mathbb{T}^{d};\mathbb{Z})$ and each phase
has a non-interacting representative, i.e. systems in each non-degenerate
phase can be connected adiabatically to a non-interacting system. Not all non-
trivial non-interacting fermionic phases represent non-trivial interacting
ones. We also showed an example of such a phase in $d=3$ with symmetry group
$F222$ which is both one of our non-degenerate interacting phases and an SPT
phase. We further show how, for the degenerate case, one can employ
characteristic classes and cohomology to achieve a partial classification and
computed some examples. In future work we hope to be able to model the short-
range entanglement condition mathematically (but not the emergent conditions),
so that we can obtain a classification of phases for short-range entangled
states and adiabatic evolution through them. This would help to see if the
discrepancies observed between our classification and existing ones are solely
due to our inclusion of crystalline phases which either are not stable under
stacking and are destroyed when allowing long-range entangled states or
whether the other assumptions (emergent relativistic field theory/spectrum)
are responsible. Independently of either scenario, our results stand on their
own as these new phases can be viewed as crystalline topological phases that
do not require these extra conditions to be satisfied, and are hence more
robust in that sense.

Finally we remark that lattice systems with on-site symmetry given by a
compact Lie group $G$, such as an $SU(2)$ spin chain, the Toric code or Cubic
Haah’s code, are also translation invariant and we expect to have something
similar to $H^{2}_{G}(\mathbb{T}^{d};\mathbb{Z})$. However they are examples
that have projective representations and time-reversal symmetry Cheng  _et
al._ (2016), which adds a real structure De Nittis and Gomi (2014) to the
ground state bundle and are outside our current scope. We will explore this in
future work.

###### Acknowledgements.

We would like to thank the referee for providing critiques which greatly
improved the overall quality of our manuscript. We also thank Ken Shiozaki for
providing us with reference Shiozaki  _et al._ (2018b) and Arun Debray for
useful comments. D. Sheinbaum acknowledges funding from CONACYT Frontera
42821.

##  Appendix A Translation and Fock space decomposition

In this section we follow very closely the work of Gomi Gomi (2017), with
slight modifications. Let $L^{2}(\mathbb{R}^{d};W)$ be the complex separable
Hilbert space of square integrable functions on $\mathbb{R}^{d}$ with values
in an $SU(2)$-representation $W$. $L^{2}(\mathbb{R}^{d};W)$ is the Hilbert
space of a single particle in $\mathbb{R}^{d}$ and the dimension of $W$
determines whether the particle is a boson or a fermion. The $n$-th tensor
product of our Hilbert space $L^{2}(\mathbb{R}^{d};W)^{\otimes n}$ is
isomorphic to $L^{2}(\mathbb{R}^{nd};W^{\otimes n})$. There is a canonical
action of the group $\mathbb{Z}^{d}$ on $\mathbb{R}^{d}$ by discrete
translations and hence there is an induced action on the $n$ copies of
$\mathbb{R}^{d}$ in $\mathbb{R}^{nd}$. This is simply translating each element
of $\mathbb{R}^{d}$ by a fixed vector in $\mathbb{Z}^{d}$. We denote this
action on $\mathbb{R}^{nd}$ by $\vec{a}\cdot
x,\,\vec{a}\in\mathbb{Z}^{d},\,x\in\mathbb{R}^{nd}$. Hence there is a
canonical action on $L^{2}(\mathbb{R}^{nd};W^{\otimes n})$, where the new
function is obtained by evaluating the old function at the translated point.
Here we shall slightly generalize the presentation in Gomi (2017) of the Bloch
transform using this action on $\mathbb{R}^{nd}$. Let
$\widehat{\mathbb{Z}}^{d}=Hom(\mathbb{Z}^{d},U(1))$ be the Pontryagin dual of
$\mathbb{Z}^{d}$. Let us define an intermediate Hilbert space just as in Gomi
(2017)

$\begin{split}L^{2}_{\mathbb{Z}^{d}}(\widehat{\mathbb{Z}}^{d}\times\mathbb{R}^{nd};W^{\otimes
n})=\\{\psi\in L^{2}(\widehat{\mathbb{Z}}^{d}\times\mathbb{R}^{nd};W^{\otimes
n})|\\\ \psi(\vec{k},\vec{a}\cdot
x)=e^{i\vec{k}\cdot\vec{a}}\psi(\vec{k},x)\\}\end{split}$ (27)

We now define $\mathfrak{B}$ and its inverse $\mathfrak{B}^{*}$ as

$\begin{split}\mathfrak{B}:L^{2}_{\mathbb{Z}^{d}}(\widehat{\mathbb{Z}}^{d}\times\mathbb{R}^{nd};W^{\otimes
n})\rightarrow L^{2}(\mathbb{R}^{nd};W^{\otimes n})\\\
(\mathfrak{B}\psi^{*})(x)=\int_{\widehat{\mathbb{Z}^{d}}}\psi^{*}(\vec{k},x)d\vec{k}\end{split}$
(28)

$\begin{split}\mathfrak{B}^{*}:L^{2}(\mathbb{R}^{nd};W^{\otimes n})\rightarrow
L^{2}_{\mathbb{Z}^{d}}(\widehat{\mathbb{Z}}^{d}\times\mathbb{R}^{nd};W^{\otimes
n})\\\
(\mathfrak{B}^{*}\psi)(\vec{k},x)=\sum_{\vec{a}\in\mathbb{Z}^{d}}e^{-i\vec{k}\cdot\vec{a}}\psi(\vec{a}\cdot
x)\end{split}$ (29)

$L^{2}_{\mathbb{Z}^{d}}(\widehat{\mathbb{Z}}^{d}\times\mathbb{R}^{nd};W^{\otimes
n})$ is isomorphic to $L^{2}$-sections of the bundle $E$ over
$\widehat{\mathbb{Z}}^{d}$ defined by

$E=\bigcup_{\vec{k}\in\widehat{\mathbb{Z}}^{d}}L^{2}(\mathbb{R}^{nd}/\mathbb{Z}^{d};\mathcal{L}|_{\vec{k}\times\mathbb{R}^{nd}/\mathbb{Z}^{d}})\otimes
W^{\otimes n}$ (30)

where $\mathcal{L}$ is the Poincaré line bundle
$\mathcal{L}\rightarrow\widehat{\mathbb{Z}}^{d}\times\mathbb{R}^{nd}/\mathbb{Z}^{d}$
built from the quotient identification

$\begin{split}\mathbb{Z}^{d}\times(\widehat{\mathbb{Z}}^{d}\times\mathbb{R}^{nd}\times\mathbb{C})\rightarrow\widehat{\mathbb{Z}}^{d}\times\mathbb{R}^{nd}\times\mathbb{C}\\\
(\vec{a},\vec{k},x,z)\mapsto(\vec{k},\vec{a}\cdot
x,e^{i\vec{k}\cdot\vec{a}}z)\end{split}$ (31)

If we rename
$L^{2}(\mathbb{R}^{nd}/\mathbb{Z}^{d};\mathcal{L}|_{\vec{k}\times\mathbb{R}^{nd}/\mathbb{Z}^{d}})\otimes
W^{\otimes n}$ as $\mathfrak{H}^{n}(\vec{k})$ we have the equivalence to the
direct integral decomposition

$L^{2}(\mathbb{R}^{dn};W^{\otimes
n})\cong\int^{\oplus}_{\widehat{\mathbb{Z}}^{d}}\mathfrak{H}^{n}(\vec{k})d\vec{k}\,,$
(32)

Fock space is defined as the subspace of square integrable functions of
$\bigoplus_{n\geq 0}L^{2}(\mathbb{R}^{dn};W^{\otimes n})$. So rewriting each
$L^{2}(\mathbb{R}^{dn};W^{\otimes n})$ we have

$\mathcal{F}(L^{2}(\mathbb{R}^{d};W))\cong\int^{\oplus}_{\mathbb{T}^{d}}\bigoplus_{n\geq
0}\mathfrak{H}^{n}(\vec{k})d\vec{k}\,.$ (33)

This is simply a rewriting of Fock space. Note that we can restrict to
symmetric and antisymmetric Fock spaces and repeat the split, so it works the
same for bosons and fermions.

##  Appendix B The non-degenerate case leads to Borel cohomology

We explained in the main text that phases of interacting bosonic systems with
a symmorphic crystallographic symmetry group with point group $P$ are
classified by $P$-equivariant rank $m$ vector bundles on $\mathbb{T}^{d}$ or
equivalently by $P$-equivariant homotopy classes of maps
$\mathbb{T}^{d}\to\mathcal{G}_{m}(\mathfrak{H}^{\prime})$ from the torus to
the Grasmmanian of $m$-dimensional subspaces of the integrand
$\mathfrak{H}^{\prime}$ in the Bloch decomposition of the appropriate Fock
space.

In the non-degenerate case, $m=1$, this set of equivariant homotopy classes
simplifies to the Borel cohomology group
$H_{P}^{2}(\mathbb{T}^{d};\mathbb{Z})$. We mentioned the intuition for this in
the main text: that non-equivariantly the Grassmanian is a $K(\mathbb{Z},2)$
space which classifies $H^{2}$. But there is a subtlety in relation to the $P$
action as we now explain. The Grassmanian
$\mathcal{G}_{1}(\mathfrak{H}^{\prime})$ with its $P$-action is a classifying
space for $P$-equivariant rank 1 vector bundles. There is well-known
equivalence between rank 1 vector bundles, i.e., line bundles, and principal
$U(1)$-bundles. Because of this equivalence,
$\mathcal{G}_{1}(\mathfrak{H}^{\prime})$ can also be thought of as a
classifying space for $P$-equivariant principle $U(1)$-bundles. Because $U(1)$
is an Abelian compact Lie group, this means that
$\mathcal{G}_{1}(\mathfrak{H}^{\prime})$ is $P$-equivariantly homotopy
equivalent to the mapping space $\mathrm{Map}(EP,BU(1))$ Lashof  _et al._
(1983) and therefore,

$\displaystyle[\mathbb{T}^{d},\mathcal{G}_{1}(\mathfrak{H}^{\prime})]_{P}$
$\displaystyle\cong[\mathbb{T}^{d},\mathrm{Map}(EP,BU(1))]_{P}$
$\displaystyle\cong[\mathbb{T}^{d}\times EP,BU(1)]_{P}$
$\displaystyle\cong[(\mathbb{T}^{d}\times EP)/P,BU(1)]$ $\displaystyle\cong
H^{2}((\mathbb{T}^{d}\times EP)/P;\mathbb{Z}).$

The last isomorphism is because $BU(1)$ is a $K(\mathbb{Z},2)$ and that last
group is by definition the Borel cohomology group
$H_{P}^{2}(\mathbb{T}^{d};\mathbb{Z})$.

This argument relies heavily on the theorem of Lashof–May–Segal Lashof  _et
al._ (1983) for compact abelian Lie groups like $U(1)$. In particular, that
theorem is not available for $U(m)$ when the degeneracy satisfies $m>1$.

Everything in this section also goes for the fermionic setting (again only in
the symmorphic case) simply replacing $P$ with $P_{F}$ and $G$ with $G_{F}$.

##  Appendix C Equivariant cohomology rings of the 2-torus with $C_{2}$ point
group

To adiabatic evolution classes of non-degenerate bosonic systems with a
symmorphic crystallographic symmetry group $G$ with point group $P$ we
assigned an element of $H_{P}^{2}(\mathbb{T}^{d};\mathbb{Z})$, and explained
that it can help to know the entire cohomology ring
$H_{P}^{*}(\mathbb{T}^{d};\mathbb{Z})$. Here we include sample calculations
for the simplest cases of interest, namely $d=2$ and $P=C_{2}$. There are
three symmorphic plane crystallographic groups with point group of order 2,
but we present only two: $\mathsf{pm}$ which contains reflections in lines
parallel to one of the translation axes and $\mathsf{cm}$ which contains
reflections in line parallel to the bisector of the translation axes.

In both cases the symmetry group is a semi-direct product
$(\mathbb{Z}\times\mathbb{Z})\rtimes C_{2}$; what differs in each case is the
action of $C_{2}$ on $\mathbb{Z}\times\mathbb{Z}$. The generator $\sigma$ of
$C_{2}$ acts as follows: for $\mathsf{pm}$, $\sigma(x,y)=(x,-y)$ and for
$\mathsf{cm}$, $\sigma(x,y)=(y,x)$. From those actions we can compute the
action of $P=C_{2}$ on the Brillouin zone
$\mathbb{T}^{2}=\mathrm{Hom}(\mathbb{Z}^{2},U(1))$. Identifying
$\mathrm{Hom}(\mathbb{Z}^{2},U(1))$ with $U(1)\times U(1)$ via the map
$f\in\mathrm{Hom}(\mathbb{Z}^{2},U(1))\mapsto(f(1,0),f(0,1))$, we can compute
the actions on $\mathbb{T}^{2}$: for $\mathsf{pm}$, $\sigma(w,z)=(w,z^{-1})$
and for $\mathsf{cm}$, $\sigma(w,z)=(z,w)$.

By definition Borel $P$-equivariant cohomology of $\mathbb{T}^{2}$ is given by
ordinary cohomology of the Borel construction, $(\mathbb{T}^{2}\times EP)/P$,
where $EP$ is the total space of the universal principal $P$-bundle, a
contractible space with a free $P$ action. In the case of a torus there is an
important simplification: $\mathbb{T}^{2}$ is the classifying space of the
group $\pi_{1}(\mathbb{T}^{2})\cong\mathbb{Z}\times\mathbb{Z}$, and in that
case the Borel construction is the classifying space of the group
$\pi_{1}(\mathbb{T}^{2})\rtimes P$. With the action of $P=C_{2}$ on
$\mathbb{T}^{2}$ we can compute the induced action on
$\mathbb{Z}\times\mathbb{Z}$ and it turns out to be the same action that
defined $G$ as a semi-direct product. Thus $(\mathbb{T}^{2}\times EP)/P\simeq
BG$.

It is tempting to think that this argument goes through in every symmorphic
case, showing that $(\mathbb{T}^{d}\times EP)/P\simeq BG$, but there is a
subtlety in cases when the Bravais lattice is not just $\mathbb{Z}^{d}$,
namely that, while the symmetry group is generated by $P$ and translations by
lattice vectors, the Brillouin zone is the quotient of $\mathbb{R}^{d}$ by the
_reciprocal lattice_. The argument does go through as far as showing that the
Borel construction is the classifying space of the semi-direct product
$\pi_{1}(\mathbb{T}^{d})\rtimes P$, but $\pi_{1}(\mathbb{T}^{d})$ is really
the reciprocal lattice, and it is on this lattice that we must consider the
action of $P$. If we define the _reciprocal symmetry group_ $G^{*}$ to be the
semi-direct product of the reciprocal lattice with $P$, then we obtain that
$(\mathbb{T}^{d}\times EP)/P\simeq BG^{*}$. In the example above the lattice
was $\mathbb{Z}^{2}$, which is its own reciprocal, so $G\cong G^{*}$, but in
general one must be aware of the distinction. For example, by drawing the
Bravais lattice and its reciprocal it is straighforward to see that the
symmetry groups $\mathsf{p3m1}$ and $\mathsf{p31m}$ are reciprocal to each
other in this sense. But perhaps exactly which group is involved is not as
important as the fact that the Borel construction is the classifying of _some_
readily computed group, which puts the required equivariant cohomology
calculations in the realm of group cohomology for which there is a vast
literature.

_Case $G=\mathsf{pm}$_. In this case the group splits as a direct product:
$G\cong D_{\infty}\times\mathbb{Z}$ where the $D_{\infty}$ factor is generated
by the reflection and a translation in a perpendicular direction, and the
other factor generated by translation parallel to the axis of reflection. The
group $D_{\infty}$ is isomorphic to the free product $C_{2}*C_{2}$, so we get

$\displaystyle BG\simeq B((C_{2}*C_{2})\times\mathbb{Z})$
$\displaystyle\simeq(BC_{2}\vee BC_{2})\times B\mathbb{Z}$
$\displaystyle\simeq(\mathbb{RP}^{\infty}\vee\mathbb{RP}^{\infty})\times
S^{1}.$

Thus $H^{*}(BG;\mathbb{Z})\cong\mathbb{Z}[x,y,t]/(t^{2},2x,2y,xy)$ where $t$
has degree 1 and is the generator of $H^{*}(S^{1};\mathbb{Z})$ and $x$ and $y$
have degree 2 and each generate one copy of
$H^{*}(\mathbb{RP}^{\infty};\mathbb{Z})$.

_Case $G=\mathsf{cm}$_. For semi-direct products one can always attempt to use
of the Lyndon–Hochschild–Serre spectral sequence, which in this case has
$E_{2}^{p,q}=H^{p}(BC_{2};\mathcal{H}^{q}(\mathbb{T}^{2};\mathbb{Z}))$ and
converges to $H^{p+q}(BG;\mathbb{Z})$. Here
$\mathcal{H}^{q}(\mathbb{T}^{2};\mathbb{Z})$ denotes the $C_{2}$-module given
by $H^{q}(\mathbb{T}^{2};\mathbb{Z})$ with the action of $C_{2}$ induced by
its action on $\mathbb{T}^{2}$. These are the following $C_{2}$-modules: for
$q=0$, the trivial module $\mathbb{Z}$; for $q=1$, the module
$\mathbb{Z}\oplus\mathbb{Z}$ with the $C_{2}$ action that swaps the summands;
and for $q=2$, the module $\mathbb{Z}$ with the sign action. The cohomology of
those modules can be readily computed by standard techniques in group
cohomology Adem and Milgram (2013). For $q=0$ it is the integral cohomology
ring of $\mathbb{RP}^{\infty}$, namely, $\mathbb{Z}[t]/(2t)$ where $t$ has
bidegree $(2,0)$. For $q=1$, it is concentrated in degree $p=0$ where it is
$\mathbb{Z}$ generated by, say, $u$. Finally, for $q=2$, as a module over
$H^{*}(\mathbb{RP}^{\infty})$, the cohomology is freely generated by a single
element $v$ with bidegree $(1,2)$.

Assembling those results, we see that the $E_{2}$-page of the spectral
sequence is given by $\mathbb{Z}[t,u,v]/(2t,2v,u^{2},ut,uv,v^{2})$. For a
semi-direct product the spectral sequence always collapses at the
$E_{2}$-page, so this is also the $E_{\infty}$-page. Luckily in each diagonal
$p+q=n$ there is exactly one non-zero entry, so there are no additive
extension problems, and we have found the cohomology groups of $G$. In fact,
the ring structure is also the one given above, but this does not follow
solely from the spectral sequence. Simple algebraic considerations on the
$E_{\infty}$-page do show that $ut$, $u^{2}$, $uv$ are $0$, but as far as the
spectral sequence can tell $v^{2}$ might be either $0$ or $t^{3}$.

To show that in fact $v^{2}=0$, we can appeal to a different space where the
computation is easier. Consider the map $q:\mathbb{T}^{2}\to S^{2}$ which
collapses $\mathbb{T}^{2}$ to a point. This can be made $C_{2}$-equivariant by
equipping $S^{2}$ with the $C_{2}$ action given by reflection in the great
circle which is the image under $q$ of the diagonal of
$\mathbb{T}^{2}=S^{1}\times S^{1}$. Applying the Borel construction to $q$
produces a map $\bar{q}:BG\to(S^{2}\times EP)/P$. The $C_{2}$-action on
$S^{2}$ fixes a great circle pointwise and swaps the two hemispheres it
delimits. From this it is straightforward to compute that $(S^{2}\times
EP)/P\simeq(S^{1}\times\mathbb{RP}^{\infty})/(S^{1}\times\\{x_{0}\\})\cong
S^{1}_{+}\wedge\mathbb{RP}^{\infty}$. The cohomology ring of the latter space
is easy to compute with Künneth’s theorem and in particular there is a class
in degree $3$ squaring to $0$ that
$\bar{q}^{*}:H^{*}(S^{1}_{+}\wedge\mathbb{RP}^{\infty};\mathbb{Z})\to
H^{*}(BG;\mathbb{Z})$ maps to $v$, as required.

##  Appendix D Adiabatic evolution vs Chen, Gu and Wen’s Local Unitary
Evolution and LRE-SRE crossover

We have argued that, in principle, non-degenerate systems in the same phase
can transition between short-range and long-range entanglement as we never
imposed any condition on the type of entanglement systems could have and this
could, again in principle, explain the difference between the traditional SPTs
and our classification. Another possibility is the relativistic or topological
spectrum assumptions. However, this transition between short-range
entanglement and long-range entanglement seems forbidden in the notion of
phase given by Chen, Gu and Wen Chen  _et al._ (2010). The reason seems to be
that the authors substitute the general notion of adiabatic evolution for what
they coined local unitary evolution. The main distinction between the the two
notions is that in a local unitary evolution a Hamiltonian must be of the form

$\mathcal{H}=\sum_{i}\mathcal{O}_{i}$ (34)

with $\mathcal{O}_{i}$ being local. The notion of local here is not entirely
precise but intuitively it is that the spatial decay of the norm of these
operators will go to zero as the distance is comparable to the entire system’s
size. With such a definition it indeed seems impossible to go from a short-
range entangled system to a long-range one. Chen, Gu and Wen Chen  _et al._
(2010) state the reason they prefer this definition is because of practical
purposes, as it is much simpler to check whether two systems can be connected
by a local unitary evolution than an adiabatic one. We argue there are
physically interesting cases which do not fit Chen, Gu and Wen’s notion of
phase and provide the following as an example of interest. The original
explanation of the integer quantum Hall effect is for the non-interacting case
(and hence the Hamiltonian is of the form (34)). Including Coulomb
interactions (non-local terms) opens up new phases (fractional plateaus),
nevertheless, for a suitable range of parameters the interactions yield the
same value for the Hall conductance as the non-interacting case. This means
that we can include non-local interaction terms without breaking the gap,
having no critical phenomena and the same value of the Hall conductance. In
the local unitary evolution picture these two systems would have to be in a
different phase as in one of them the Hamiltonian has these non-local terms.
Hence, we choose to include such cases to our notion of phase. This is also an
example of a system with short-range entanglement being in the same phase as
one with some long-range entanglement. Thus an instantiation of the phenomena
which may explain the discrepancy between our results and those of standard
SPT constructions.

##  Appendix E Dimension and Disorder

For non-degenerate ground states we obtained the cohomology group
$H^{2}_{P}(\mathbb{T}^{d};\mathbb{Z})$ as a full classification. The way that
the dimension $d$ of a system enters is only through the generalized Brillouin
torus $\mathbb{T}^{d}$. Let us remember that the Brillouin zone arises as the
Pontryagin dual of the group of translations, meaning that without translation
symmetry there is no distinction in dimension. Thus, if we remove translation
symmetry completely but keep a symmetry $G^{\prime}$ which is independent of
the dimension (say time-reversal symmetry) we would have something of the form

$[*,\mathcal{G}_{m}(\mathfrak{F}(L^{2}(\mathbb{R};W)))]_{G^{\prime}}.$ (35)

For the non-degenerate case we get $H^{2}_{G^{\prime}}(*;\mathbb{Z})$.
Equation (35) is in stark contrast with results on SPT phases, where the
dimensionality of the system is usually reflected in the dimension of the
cohomology group $H^{d+1}_{G^{\prime}}(*;U(1))$ in Chen  _et al._ (2013) for
example. This discrepancy could be surmounted (perhaps) by the the fact that
SPT phases are often further assumed to have a restricted set of actions that
leave fixed some chosen subspaces Else and Nayak (2014). To include disorder
it would be more reasonable to turn the generalized Brilluoin zone into a
noncommutative $C^{*}$-algebra as in the non-interacting case Prodan and
Schulz-Baldes (2016), Thiang (2016) instead of completely removing the
Brillouin zone from the picture.

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