[labelstyle=]

# A groupoid approach to interacting fermions

Bram Mesland Mathematisch Instituut
Universiteit Leiden
Niels Bohrweg 1
2333CA Leiden, Netherlands
<EMAIL_ADDRESS>and  Emil Prodan Department of Physics and
Department of Mathematical Sciences
Yeshiva University
New York, NY 10016, USA
<EMAIL_ADDRESS>
###### Abstract.

We consider the algebra $\dot{\Sigma}({\mathcal{L}})$ generated by the inner-
limit derivations over the ${\rm GICAR}$ algebra of a fermion gas populating
an aperiodic Delone set ${\mathcal{L}}$. Under standard physical assumptions
such as finite interaction range, Galilean invariance of the theories and
continuity with respect to the deformations of the aperiodic lattices, we
demonstrate that the image of $\dot{\Sigma}({\mathcal{L}})$ through the Fock
representation can be completed to a groupoid-solvable pro-$C^{\ast}$-algebra.
Our result is the first step towards unlocking the $K$-theoretic tools
available for separable $C^{\ast}$-algebras for applications in the context of
interacting fermions.

Emil Prodan was supported by the U.S. National Science Foundation through the
grants DMR-1823800 and CMMI-2131760.

###### Contents

  1. 1 Introduction and Main Statements
  2. 2 Bi-Equivariant Groupoid $C^{\ast}$-Algebras
    1. 2.1 Background: Étale groupoids
    2. 2.2 Background: Reduced $C^{\ast}$-algebra of an étale groupoid
    3. 2.3 Bi-equivariant groupoid $C^{*}$-algebras
    4. 2.4 Left regular representations
  3. 3 Single Fermion Dynamics by groupoid Methods
    1. 3.1 The algebra of local observables
    2. 3.2 Galilean invariant theories
    3. 3.3 Spaces of patterns
    4. 3.4 The groupoid $C^{\ast}$-algebra of single fermion dynamics
  4. 4 Interacting Fermions: The Algebra of Local Observables
    1. 4.1 The CAR and ${\rm GICAR}$ algebras over a lattice
    2. 4.2 Symmetric presentation
    3. 4.3 Algebraic relations and notation
    4. 4.4 Fock representation
    5. 4.5 ${\rm GICAR}$ is a solvable $C^{\ast}$-algebra
  5. 5 Interacting Fermions: Dynamics
    1. 5.1 Hamiltonians with finite interaction range
    2. 5.2 Galilean invariant theories
    3. 5.3 The many-body covers of the Delone space
    4. 5.4 The algebra of finite-range Galilean invariant derivations
    5. 5.5 Fock representation of derivations
  6. 6 Resolving the Algebra of Physical Hamiltonians
    1. 6.1 Filtration by ideals
    2. 6.2 Completing the algebra of the physical Hamiltonians
    3. 6.3 Topological groupoid ${\mathcal{G}}_{N}$ associated to the dynamics of $N$-fermions
    4. 6.4 The 2-action of the symmetric group and the bi-equivariant groupoid algebra
    5. 6.5 Left regular representations and the dynamics of fermions
    6. 6.6 ${\mathcal{G}}_{N}$ as a blow-up groupoid
  7. 7 Analysis of special cases with disconnected order covers
    1. 7.1 Canonical ordering for perturbed periodic patterns
    2. 7.2 Reduction of the $C^{\ast}$-algebra and $K$-theoretic statements
  8. 8 Discussion and Outlook
    1. 8.1 Self-binding versus scattered dynamics
    2. 8.2 Thermodynamic limit

##  1\. Introduction and Main Statements

The physical observables of a self-interacting Fermi gas populating a
uniformly discrete lattice ${\mathcal{L}}\subset{\mathbb{R}}^{d}$ generate the
$C^{\ast}$-algebra of canonical anti-commutation relations (CAR) over the
Hilbert space $\ell^{2}({\mathcal{L}})$ [18], denoted here by ${\rm
CAR}({\mathcal{L}})$. Our interest is in the dynamics of these observables,
formalized as a point-wise continuous one-parameter group

$\bm{\alpha}:{\mathbb{R}}\to{\rm Aut}\big{(}{\rm CAR}({\mathcal{L}})\big{)}.$

We will refer to $\bm{\alpha}$ as the time evolution. Let
${\mathcal{D}}_{\bm{\alpha}}$ be the set of elements $A\in{\rm
CAR}({\mathcal{L}})$ for which $\bm{\alpha}_{t}(A)$ is first order
differentiable w.r.t. $t\in{\mathbb{R}}$. Then ${\mathcal{D}}_{\bm{\alpha}}$
is stable under addition and multiplication, hence it is actually a sub-
algebra of ${\rm CAR}({\mathcal{L}})$. Furthermore,

$\delta_{\bm{\alpha}}(A):=\lim_{t\rightarrow
0}\tfrac{1}{t}\big{(}\bm{\alpha}_{t}(A)-A\big{)}$

defines a ${\rm CAR}({\mathcal{L}})$-valued linear map over
${\mathcal{D}}_{\bm{\alpha}}$, which obeys Leibniz’s rule

$\delta_{\bm{\alpha}}(AA^{\prime})=\delta_{\bm{\alpha}}(A)A^{\prime}+A\delta_{\bm{\alpha}}(A^{\prime}),\quad
A,A^{\prime}\in{\mathcal{D}}_{\bm{\alpha}},$

hence $\delta_{\bm{\alpha}}$ is a (possibly unbounded) derivation [17]. We
will refer to $\delta_{\bm{\alpha}}$ as the generator of ${\bm{\alpha}}$.
Consider now a family of time evolutions whose ${\mathcal{D}}$’s contain a
fixed dense $\ast$-subalgebra ${\mathcal{D}}({\mathcal{L}})\subset{\rm
CAR}({\mathcal{L}})$, which is invariant under the action of all the
generators of the family. Under such conditions, these generators can be
composed and they form a sub-algebra of the algebra of linear maps over
${\mathcal{D}}({\mathcal{L}})$, which we refer to as the core algebra
associated with the family of time evolutions. In this work, we investigate
the core algebra associated with a large family of time evolutions, which we
claim it contains all physically reasonable time evolutions over a Delone set
${\mathcal{L}}$. The goal is to complete this and an associated core algebra
of physical Hamiltonians to pro-$C^{\ast}$-algebras (in the sense of [40]),
and to supply fine characterizations of these completions in a manner that
unlocks the $K$-theoretic tools available for separable $C^{\ast}$-algebras.
To our knowledge, such tools are not yet available in the context of
interacting many-fermion systems.

Before elaborating any further, let us acknowledge that, for the 1-fermion
sector, the program outlined above has been in place for almost three decades.
Indeed, a fundamental result by Bellissard and Kellendonk [8, 9, 10, 11, 30]
states that, for Galilean invariant theories, the generators of the 1-fermion
dynamics over a uniformly discrete pattern form a separable groupoid
$C^{\ast}$-algebra, canonically associated to the given pattern. By
identifying this separable $C^{\ast}$-algebra, Bellissard and Kellendonk
essentially completed the stably-homotopic classification of all the available
Galilean invariant gapped 1-fermion Hamiltonians over a given pattern, which
now can be enumerated via the $K$-theories of the Bellissard-Kellendonk
groupoid $C^{\ast}$-algebra. In [13, 42] and [14], the reader can find models
for implementing this program in the context of disordered and, respectively,
generic lattices in arbitrary dimensions. At the practical level, the program
pioneered by Bellissard and Kellendonk spurred new directions in materials
science, resulting in a high-throughput of novel topological materials and
meta-materials [4, 7, 20, 21, 22, 35, 38, 45, 46, 56], based on the sampling
of different classes of Delone sets. The results in the present can facilitate
similar developments for arbitrary $N$-fermion sectors.

We will be dealing exclusively with time evolutions that conserve the number
of fermions, hence with the ones that drop to time evolutions over the ${\rm
GICAR}$-subalgebra of gauge invariant physical observables. We will enforce
two natural physical conditions: 1) the generators of the time evolutions
depend continuously on the pattern ${\mathcal{L}}$ when the latter is deformed
inside a fixed class of Delone sets and 2) the theories are Galilean invariant
(see section 5.2 and below). A large portion of our work is dedicated to
formulating these two properties in a rigorous and natural framework and to
identifying the most general expression of the generators that deliver such
time evolutions. Prototypical examples are supplied by Hamiltonians generated
from many-body potentials, such as

$\displaystyle
H_{\mathcal{L}}=\sum_{k\in{\mathbb{N}}^{\times}}\sum_{\\{x_{i},x^{\prime}_{i}\\}\in{\mathcal{L}}}\
w_{k}(x_{1},\ldots,x_{k};x^{\prime}_{1},\ldots,x^{\prime}_{k})\,a^{\ast}_{x_{1}}\cdots
a^{\ast}_{x_{k}}\,a_{x^{\prime}_{k}}\cdots a_{x^{\prime}_{1}},$ (1.1)

where the $k$-body potentials
$w_{k}:({\mathbb{R}}^{d})^{k}\times({\mathbb{R}}^{d})^{k}\to{\mathbb{C}}$ obey
several constraints, such as continuity, anti-symmetry against permutations
and invariance under global shifts

$(x_{1},\ldots,x_{k};x^{\prime}_{1},\ldots,x^{\prime}_{k})\mapsto(x_{1}-x,\ldots,x_{k}-x;x^{\prime}_{1}-x,\ldots,x^{\prime}_{k}-x).$

The $a$’s in Eq. (1.1) are the generators of ${\rm CAR}({\mathcal{L}})$. If
the $k$-body potentials have finite interaction range, i.e. $w_{k}$’s vanish
whenever the diameter of the sets $\\{x_{i}\\}\cup\\{x^{\prime}_{i}\\}$ exceed
a fixed value, then $H_{\mathcal{L}}$ defines a generator for a dynamics,
which comes in the form of a closable unbounded derivation

${\rm ad}_{H_{\mathcal{L}}}(A)=\imath[A,H_{\mathcal{L}}],\quad
A\in{\mathcal{D}}({\mathcal{L}})\subset{\rm CAR}({\mathcal{L}}),$

with a fixed dense and invariant domain (see section 5.1). Note that a many-
body potential defines an entire correspondence ${\mathcal{L}}\mapsto
H_{\mathcal{L}}$, which is tacitly assumed to be continuous w.r.t.
${\mathcal{L}}$. Now, if ${\mathcal{L}}$ and ${\mathcal{L}}^{\prime}$ happen
to enter a relation like ${\mathcal{L}}^{\prime}={\mathcal{L}}-x$ for some
$x\in{\mathbb{R}}^{d}$, then

${\rm Ad}_{H_{{\mathcal{L}}^{\prime}}}=S_{x}\circ{\rm
Ad}_{H_{\mathcal{L}}}\circ S_{x}^{-1},$ (1.2)

where $S_{x}:{\rm CAR}({\mathcal{L}})\to{\rm CAR}({\mathcal{L}}-x)$ are the
obvious $C^{*}$-algebra isomorphisms (which properly map the domains). Eq.
(1.2) reflects the Galilean invariance of the theory, which was already
encoded in the many-body potentials.

At first sight, formalizing such Hamiltonians seem straightforward, but a
fundamental difficulty presents itself. As we do not have a topology on the
set of our models, the only way to define the continuity w.r.t.
${\mathcal{L}}$ is through the coefficients of the Hamiltonians, such as the
$w_{k}$’s in Eq. (1.1). Therefore, from both the physical and mathematical
points of view, identifying the correct domain of these coefficients as a
topological space is paramount. Note that, besides the lattice
${\mathcal{L}}$, the orderings of the sets $\\{x_{i}\\}$ and
$\\{x^{\prime}_{i}\\}$ in Eq. (1.1) are part of the data defining the
coefficients and, for a generic Delone set, there is no canonical choice for
these orderings. Furthermore, under continuous cyclic deformations of the
pattern ${\mathcal{L}}$, the orderings of the sets $\\{x_{i}\\}$ and
$\\{x^{\prime}_{i}\\}$ can change and this can happen even when the
deformations occur inside the transversal of a single pattern (see Definition
3.10), as it is the case for topological lattice defects [43] or for patterns
like in our Example 3.13. Such phenomena present a challenge because, while
the Hamiltonian (1.1) returns to itself under such cyclic deformations, the
coefficients do not.

In Section 5, we construct the natural domains of the Hamiltonian coefficients
as covering spaces of the topological space of all Delone sets. First, we
introduce the set of pairs $({\mathcal{L}},V)$, where ${\mathcal{L}}$ is a
Delone set and $V$ is a subset of ${\mathcal{L}}$ of cardinal $|V|=k$, and we
topologize this set such that it becomes an (infinite) cover of the space of
Delone sets. Then we introduce a second, finite cover over the topological
space of pairs $({\mathcal{L}},V)$, which we call the order cover, such that
the group of deck transformations of this covering space is the full
permutation group ${\mathcal{S}}_{k}$ on $k$ elements. A point in this cover
is a triple $\xi=({\mathcal{L}},V_{\xi},\chi_{\xi})$, where
$\chi_{\xi}:\\{1,\ldots,k\\}\to V_{\xi}$ is a bijection, hence an ordering of
the subset $V_{\xi}\subset{\mathcal{L}}$ (see section 5.3). There is one such
space for each $k>1$ and we call $\mathfrak{a}_{k}(\xi):={\mathcal{L}}$ the
$k$-body covering map of the space of Delone sets.

A generic finite interaction range Hamiltonian evaluated at ${\mathcal{L}}$
then takes the form

$H_{\mathcal{L}}=\sum_{k\in{\mathbb{N}}^{\times}}\tfrac{1}{k!}\sum_{\xi,\zeta\in\mathfrak{a}^{-1}_{k}({\mathcal{L}})}h_{k}(\xi,\zeta)\,\bm{a}^{\ast}(\xi)\bm{a}(\zeta),$
(1.3)

where

$\bm{a}(\xi):=a_{\chi_{\xi}(k)}\cdots a_{\chi_{\xi}(1)}\in{\rm
CAR}({\mathcal{L}}),$

and the $h_{k}$ are bi-equivariant coefficients w.r.t. to the deck
transformations of the order cover, invariant under the simultaneous
translation of the arguments, continuous in the arguments and supported on
pairs $(\xi,\zeta)$ for which the diameter of $V_{\xi}\cup V_{\zeta}$ is
smaller than a fixed positive number (see subsection 5.4 for complete
details). The derivations ${\rm ad}_{H_{\mathcal{L}}}$ corresponding to
Hamiltonians of the type (1.3) generate the core algebra
$\dot{\Sigma}({\mathcal{L}})$ we mentioned in the first paragraph.

Of course, the Hamiltonian coefficients can be also defined over a twofold
orientation cover of the space of Delone sets. However, it turns out that the
full order covers over the space of Delone sets not only supply natural
domains for the Hamiltonian coefficients, but also hold the key to our
generalization of the Bellissard-Kellendonk groupoid algebras to the
$N$-fermion sectors. As a topological space, for each $N$, the groupoid
$\mathcal{G}_{N}$ appears naturally as a transversal to the translation action
of $\mathbb{R}^{d}$ on the $N$-order cover. The ordering $\chi_{\xi}$ give us
a handle to equip these transversals with the algebraic structure of a
groupoid. Indeed, viewing each subset $V_{\xi}$ as anchored at $\chi_{\xi}(1)$
allows us to pull $V_{\xi}$ to the origin of the physical space and also gives
us the means to define the set of composable pairs (see section 6.3 for
details). The permutation group $\mathcal{S}_{N}$ acts on $\mathcal{G}_{N}$ by
bisections and this actions are non-trivial as they involve permutations and
translations of the points (see sections 2.3 and 6.4).

We now describe the process that enables us to characterize the algebra
$\dot{\Sigma}({\mathcal{L}})$. The key observation is that these are inner-
limit derivations [17] and such derivations have a special relation with the
double sided ideals: They leave them invariant and, as such, the inner-limit
derivations descend to derivations on quotients of double sided ideals. The
lattice of double sided ideals of ${\rm GICAR}$ algebra was worked out by
Bratteli, who uncover the following structure (see section 4.5):

######  Theorem 1.1 ([16]).

The ${\rm GICAR}({\mathcal{L}})$ algebra is a solvable $C^{\ast}$-algebra (of
infinite length), in the sense of Dynin [25]. Specifically, it accepts a
filtration by primitive ideals

$\cdots\rightarrowtail{\rm
GI}_{N}({\mathcal{L}})\rightarrowtail\cdots\rightarrowtail{\rm
GI}_{1}({\mathcal{L}})\rightarrowtail{\rm GI}_{0}({\mathcal{L}})={\rm
GICAR}({\mathcal{L}})$ (1.4)

and

${\rm GI}_{N}({\mathcal{L}})/{\rm GI}_{N+1}({\mathcal{L}})\simeq{\mathbb{K}},$
(1.5)

for all $N\in{\mathbb{N}}$, where ${\mathbb{K}}$ is the algebra of compact
operators.

Since the derivations descend on the quotients (1.5), we obtain a sequence of
representations of $\dot{\Sigma}({\mathcal{L}})$ by (bounded) derivations on
the algebra of compact operators. The bounded derivations over the algebra of
compact operators have been already completely characterized, cf. [17, Example
1.6.4], specifically, they always take the form of commutators with bounded
operators over the underlying Hilbert space, which in the present context
happens to be the $N$-fermion sector ${\mathcal{F}}_{N}^{(-)}({\mathcal{L}})$
of the Fock space. As a result, there are sub-algebras ${\mathfrak{H}}_{N}$ of
the algebras of bounded operators over
${\mathcal{F}}_{N}^{(-)}({\mathcal{L}})$ as well as a tower of representations
of $\dot{\Sigma}({\mathcal{L}})$ inside
${\mathfrak{H}}_{N}\otimes{\mathfrak{H}}_{N}^{\rm op}$. Furthermore, if $\eta$
is the vacuum state and $\pi_{\eta}$ is the associated Fock representation of
${\rm CAR}({\mathcal{L}})$, then the latter induces a representation
$\dot{\pi}_{\eta}$ of $\dot{\Sigma}_{\mathcal{L}}$ on a dense linear sub-space
of ${\mathcal{F}}_{N}^{(-)}({\mathcal{L}})$ and
$\dot{\Sigma}({\mathcal{L}})/{\rm
ker}\,\dot{\pi}_{\eta}\simeq\varprojlim\,\dot{\Gamma}_{N}({\mathcal{L}})$,
where $\dot{\Gamma}_{N}({\mathcal{L}})=\bigoplus_{n=0}^{N}\mathfrak{H}_{N}$.
Note that $\dot{\Sigma}({\mathcal{L}})/{\rm ker}\,\dot{\pi}_{\eta}$ represents
the core algebra of physical Hamiltonians (see section 5.5 for details and
justification).

In a followup step, we compute the $C^{\ast}$-envelopes of
$\mathfrak{H}_{N}$’s and promote the tower
$\\{\dot{\Gamma}_{N}({\mathcal{L}})\\}$ to a projective tower of
$C^{\ast}$-algebras $\\{\Gamma_{N}({\mathcal{L}})\\}$, whose inverse limit
$\mathfrak{H}({\mathcal{L}})$ is the pro-$C^{\ast}$-algebra mentioned in the
first paragraph of the section. This brings us to our first major result,
stated in Theorem 6.24, saying that the core algebra of physical Hamiltonians
embeds into the pro-$C^{\ast}$-algebra $\mathfrak{H}({\mathcal{L}})$. A
similar statement holds for the algebra $\dot{\Sigma}({\mathcal{L}})$ (see
Theorem 6.27). Our second major result states that
$\mathfrak{H}({\mathcal{L}})$ affords the following explicit characterization:

######  Theorem 1.2.

Let
$\mathfrak{p}_{N}:{\mathfrak{H}}({\mathcal{L}})\twoheadrightarrow\Gamma_{N}({\mathcal{L}})$
be the epimorphisms associated with the inverse limit and define the closed
double sided ideals ${\mathcal{J}}_{N}({\mathcal{L}}):={\rm
ker}\,\mathfrak{p}_{N}$. Then $\\{{\mathcal{J}}_{N}({\mathcal{L}})\\}_{N}$
supplies a filtration of $\mathfrak{H}({\mathcal{L}})$ and

${\mathcal{J}}_{N-1}({\mathcal{L}})/{\mathcal{J}}_{N}({\mathcal{L}})=\pi_{\mathcal{L}}\Big{(}{\mathcal{M}}\big{(}C^{\ast}_{r,{\mathcal{S}}_{N}}({\mathcal{G}}_{N},{\mathbb{C}})\big{)}\Big{)}.$
(1.6)

Here, ${\mathcal{G}}_{N}$ is the groupoid canonically associated to
${\mathcal{L}}$ generalizing the Bellissard-Kellendonk groupoid from the
single to the $N$-fermion sector,
$C^{\ast}_{r,\mathcal{S}_{N}}({\mathcal{G}}_{N},{\mathbb{C}})$ is the
${\mathcal{S}}_{N}$-bi-equivariant groupoid $C^{\ast}$-algebra,
${\mathcal{M}}$ indicates the essential extension to the multiplier algebra
and $\pi_{\mathcal{L}}$ is a left regular representation of the groupoid
$C^{*}$-algebra.

We will show that
${\mathcal{J}}_{N-1}({\mathcal{L}})/{\mathcal{J}}_{N}({\mathcal{L}})$ are
isomorphic to the images of $\mathfrak{H}({\mathcal{L}})$ through its Fock
representations on the $N$-fermion sectors. As such, the algebra of Galilean
invariant Hamiltonians of $N$ fermions over the lattice ${\mathcal{L}}$
coincides with multiplier algebra
${\mathcal{M}}\big{(}C^{\ast}_{r,{\mathcal{S}}_{N}}({\mathcal{G}}_{N},{\mathbb{C}})\big{)}$.
This reduces to the (unital) Bellissard-Kellendonk groupoid algebra for $N=1$.
Our third major result states that the groupoid ${\mathcal{G}}_{N}$ continue
to be second countable, Hausdorff and étale for $N>1$, hence its groupoid
$C^{\ast}$-algebra is separable (see section 6.3). Furthermore, as stated
already, this groupoid accepts a (highly non-trivial) 2-action of the
permutation group $\mathcal{S}_{N}$ (see Definition 2.21 and section 6.4).

In section 8.1, we argue that the $C^{\ast}$-algebra of the groupoid
${\mathcal{G}}_{N}$ generates self-binding dynamics, in the sense that under
the evolution generated by a Hamiltonian from
$C^{\ast}_{r,{\mathcal{S}}_{N}}({\mathcal{G}}_{N},{\mathbb{C}})$, the $N$
fermions evolve in one single cluster. If the Hamiltonian comes from the
corona of the multiplier extension, then the dynamics is scattered, in the
sense that the $N$ fermions evolve in two or more separate clusters. When the
many-body covers are disconnected, as it is the case for periodic and quasi-
periodic lattices, we show in section 7 that
$C^{\ast}_{r,\mathcal{S}_{N}}({\mathcal{G}}_{N},{\mathbb{C}})$ is Morita
equivalent to the 1-fermion Bellissard-Kellendonk $C^{\ast}$-algebra
$C^{\ast}_{r}({\mathcal{G}}_{1},{\mathbb{C}})$. Thus, in these special cases,
the stably-homotopic classification of the self-binding states of $N$ fermions
is no more complicated than that of single fermion states. We point out that
this has been already observed in numerical experiments (see [34]).

We follow up with pointed remarks in order to guide the reader through the
significance of the above results, its relation to other works and to
highlight important aspects that were left to future investigations:

######  Remark 1.3.

A good portion of our program can be repeated for AF-algebras other than the
${\rm GICAR}$ algebra, provided they have a rich lattice of ideals and the
descended derivations on the quotients are by commutators with elements from
their multiplier algebras. It is important to note that the
pro-$C^{\ast}$-algebra completions generated by the mechanism explained above
are not universal, in the sense that they depend in an essential way on the
chosen filtration. In the present context, the filtration mentioned in Theorem
1.1 makes the most physical sense and the associated pro-$C^{\ast}$-algebra
gives a satisfactory framework for the dynamics of fermions. Nevertheless, it
will be very interesting to investigate the range of
pro-$C^{\ast}$-completions that can be generated from other filtrations or
even from other AF-algebras, and to study the relations among these different
completions. Lastly, it will be interesting to see how all this fits into or
benefits from the existing frameworks on representations of $\ast$-algebras by
unbounded operators [36, 48]. $\Diamond$

######  Remark 1.4.

In [3], the algebras ${\rm CAR}({\mathcal{L}})$ of local observables were
formalized as a continuous field of $C^{\ast}$-algebras over the space of
Delone sets. Here, we skip entirely this step and we deal directly with the
algebra of derivations. $\Diamond$

######  Remark 1.5.

The spaces of units of the groupoids ${\mathcal{G}}_{N}$ are only locally
compact, except for $N=1$. Hence, the groupoid algebras for $N>1$ do not have
a unit and the extensions to the multiplier algebras are non-trivial. Since
$C^{\ast}_{r}({\mathcal{G}}_{N})$ does not contain the compacts, neither does
its multiplier extension. As such, this extension is not the whole algebra of
bounded operators over the $N$-sector of the Fock space and we expect
${\mathcal{M}}\big{(}C^{\ast}_{r}({\mathcal{G}}_{N})\big{)}$ to have a very
interesting structure. A glimpse into this structure is supplied by
Proposition 6.39, where a continuous groupoid morphism
${\mathcal{G}}_{N+M}\to{\mathcal{G}}_{N}\times{\mathcal{G}}_{M}$ is defined.
This groupoid morphism is neither continuous nor does it preserve the Haar
system. Nonetheless, its pullback map supplies a ${\mathbb{C}}$-module map
$C^{\ast}_{r}({\mathcal{G}}_{N})\otimes
C^{\ast}_{r}({\mathcal{G}}_{M})\to{\mathcal{M}}\big{(}C^{\ast}_{r}({\mathcal{G}}_{N+M})\big{)}$,
which actually lands in the corona (see Remark 6.41). We conjecture that the
multiplier extensions can be fully characterized by such maps. $\Diamond$

######  Remark 1.6.

In Example 2.37 of [55], Williams describes how to generate new groupoids by
blowing up the unit space of an existing groupoid. As we shall see in Sec.
6.6, the groupoids ${\mathcal{G}}_{N}$, $N>1$, are all blow-ups of
${\mathcal{G}}_{1}$, which automatically implies that the associated
$C^{\ast}$-algebras are Morita equivalent (see Th. 2.52 in [55]). The
${\mathcal{S}}_{N}$-bi-equivariant sub-algebras are, however, not expected to
be Morita equivalent in general. In fact, the 2-action of the permutation
group does not admit a straightforward interpretation when the groupoids
${\mathcal{G}}_{N}$ are presented as blow-ups of ${\mathcal{G}}_{1}$.
Nevertheless, as we already announced, there are special yet important cases
when the ${\mathcal{S}}_{N}$-bi-equivariant sub-algebra is in fact Morita
equivalent to $C^{\ast}_{r}({\mathcal{G}}_{1})$. As a result, for these
special cases, we have precise $K$-theoretic statements.

######  Remark 1.7.

In the light of the previous two remarks, we believe that our formalism
supplies new tools and a fresh framework for the problem of the thermodynamic
limit ($N\to\infty$). These aspects are put in a perspective in Section 8. Let
us specify here that the formalism can be straightforwardly adapted to quasi-
free states (hence situations with $N=\infty$), by using the Fock and anti-
Fock primitive ideals of the ${\rm GICAR}$ algebra (see section 4.5). The
finite $N$-sectors are equally interesting, from both a theoretical and a
practical point of view. For example, [34] supplied numerical evidence that
the $K$-theories of the finite $N$-sectors can become extremely complex even
for $N=3$. As explained in [34], this can be exploited to generate new
topological dynamics in meta-materials and new manifestations of the bulk-
boundary principle. $\Diamond$

######  Remark 1.8.

Any groupoid-invariant open subset of the unit space generates a two-sided
ideal inside the groupoid $C^{\ast}$-algebra [55, Sec. 5.1]. In [43], the
bulk-defect correspondence principle for the 1-fermion sector was re-
formulated in terms of the exact sequences of $C^{\ast}$-algebras
corresponding to such ideals. In the light of this re-formulation, the bulk-
boundary correspondence principle can now be straightforwardly studied for any
$N$-fermion sector. See also [14] for equivalent formulations of the bulk-
boundary correspondence using groupoids and their $C^{\ast}$-algebras.
$\Diamond$

######  Remark 1.9.

In [51], Upmeier introduced the notion of a groupoid-solvable
$C^{\ast}$-algebra (see also [52]), which generalizes the notion of solvable
$C^{\ast}$-algebras of Dynin. In the light of Theorem 1.2, we can say that
$\mathfrak{H}$ is essentially a groupoid-solvable pro-$C^{\ast}$-algebra.
Furthermore, the reader can find in [51] (see also [25]) models of $K$\- and
index-theoretic analyses in the context of groupoid-solvable
$C^{\ast}$-algebras, specifically, for multivariable Toeplitz operators on
domains of ${\mathbb{C}}^{n}$. We, however, will have to leave the
$K$-theoretic aspects to future investigations, although some important
remarks will be made in section 7. $\Diamond$

######  Remark 1.10.

To our knowledge, the notion of bi-equivariant groupoid $C^{\ast}$-algebra has
not been used before. We were naturally led to it once the analysis was lifted
to the order cover and the Hamiltonians were expressed via bi-equivariant
coefficients as in Eq. (1.3). The concept is developed in section 2.3 and is
based on the notion of 2-actions of a group on groupoids and
$C^{\ast}$-algebras. By viewing groupoids as categories, a 2-action is the
natural 2-category analogue of the action of a group on a space. Historically,
such structures arose under the nomer of crossed module of a pair of groups,
originally introduced by Whitehead [54] in the context of homotopy 2-types.
The actions of crossed modules on groupoids and $C^{*}$-algebras are described
in detail in [19, Sections 2 and 3]. We believe that the bi-equivariant
groupoid $C^{\ast}$-algebra for spin-less fermions introduced here can be
generalized, for example, to generate dynamical models for particles with
various statistics, including non-abelian anyons. $\Diamond$

######  Remark 1.11.

The CAR algebra itself accepts a presentation as a groupoid $C^{\ast}$-algebra
(see Example 1.10 in [44], Example 8.3.5 in [49], or our Remark 4.9). This, in
fact, represented the starting point for our investigation but we soon
realized that this presentation is too rigid because it assumes a fixed pre-
ordering of the lattice points. That, unfortunately, is incompatible with a
context where the lattice ${\mathcal{L}}$ can be deformed and the points can
be exchanged. $\Diamond$

######  Remark 1.12.

The completion of the algebra of physical Hamiltonians puts a topology on the
physical models. Indeed, as for any pro-$C^{\ast}$-algebra, the topology is
given by the semi-norms supplied by the $C^{\ast}$-norms of the projective
tower. Let us acknowledge that the completion includes physical Hamiltonians
with infinite interaction range. As we shall see in Proposition 6.46, the
coefficients $h_{k}$ in Eq (1.3) define compactly supported functions over
${\mathcal{G}}_{k}$. The passage from finite to infinite interaction range
happens when the algebra of functions with compact support on
${\mathcal{G}}_{k}$ is completed to the groupoid $C^{\ast}$-algebra
$C^{\ast}_{r}({\mathcal{G}}_{k})$. $\Diamond$

The paper is organized as follows. Section 2 introduces the key mathematical
concept developed by our work, namely, the bi-equivariant groupoid
$C^{\ast}$-algebras. This section starts with a brisk introduction to étale
groupoids and their associated $C^{\ast}$-algebras and continues with a
discussion of the group of bisections of a groupoid. Then it supplies our
definition of a 2-action of a group on groupoids and defines the bi-
equivariant groupoid $C^{\ast}$-algebras relative to such 2-actions of groups.
The use of groupoids and their associated $C^{\ast}$-algebras is showcased in
section 3, where the Bellissard-Kellendonk formalism [8, 9, 30] for the
single-fermion setting is briefly reviewed. This gives us the opportunity to
introduce fundamental concepts related to the space of patterns, such as the
hulls and transversals. In addition, our presentation in this section adopts a
fresh point of view in which the Bellissard-Kellendonk groupoid
$C^{\ast}$-algebra is seen as formalizing the inner-limit derivations over a
local algebra of observables, which in the 1-fermion setting is just the
algebra of compact operators. Then the goal of our work can be understood as
generalizing the accomplishments of Bellissard and Kellendonk to the setting
where the algebra of compact operators is replaced by the GICAR algebra.
Section 4 focuses on the ${\rm CAR}$ and ${\rm GICAR}$ algebras over Delone
sets, specifically, on a symmetric presentation that avoids the use of any
pre-ordering of the points and on the ideal structure, as well as on the
symmetric Fock representations. Section 5 analyzes the dynamics of the local
degrees of freedom. The first part focuses on the physical aspects of the
problem, making sure that our subsequent mathematical analysis covers all
physically sound Hamiltonians. After the investigation of specific model
Hamiltonians generated with many-body potentials, the section concludes that
the Hamiltonian coefficients are naturally defined over many-body covers of
the space of Delone sets. The deck transformations for this covers supply the
2-actions that will eventually lead us to a bi-equivariant theory as presented
in section 2. Section 6 develops the elements needed in Theorem 1.2 and
supplies the proof. In the process, the étale groupoid ${\mathcal{G}}_{N}$
associated with the dynamics of $N$-fermions is introduced and its left
regular representations are shown to reproduce an essential ideal of
${\mathcal{J}}_{N-1}({\mathcal{L}})/{\mathcal{J}}_{N}({\mathcal{L}})$. Section
7 analyses special lattices for which the many-body order covers over the
transversals are disconnected. Section 8 contains concluding remarks and a
list of tools that we see being developed in the future within our formalism.

##  2\. Bi-Equivariant Groupoid $C^{\ast}$-Algebras

This section develops the notion of group 2-actions on groupoids and a bi-
equivariant version of the $C^{\ast}$-algebra associated to a groupoid in the
presence of such a group 2-actions. There will be several specialized
groupoids appearing in this work and, as we already stated in the
introduction, our main results point to a certain groupoid structure in the
dynamics of interacting fermions. As such, the reader needs a minimal
familiarity with these structures and the first two subsections are intended
to supply just that. In parallel, these subsections introduce notation and
examples that will be used throughout.

###  2.1. Background: Étale groupoids

We follow here and throughout the notations and the conventions from [49, Ch.
8]. We start with the algebraic structure of a groupoid. Abstractly, a
groupoid is a small category in which all morphisms are invertible.
Concretely, we have the following:

######  Definition 2.1.

A groupoid consists of

  * •

A set ${\mathcal{G}}$ equipped with an inverse map $\alpha\mapsto\alpha^{-1}$;

  * •

A subset ${\mathcal{G}}^{(2)}\subset{\mathcal{G}}\times{\mathcal{G}}$ equipped
with a multiplication map $(\alpha,\beta)\mapsto\alpha\beta\in{\mathcal{G}}$,

such that

  1. 1)

$(\alpha^{-1})^{-1}=\alpha$ for all $\alpha\in{\mathcal{G}}$;

  2. 2)

If $(\alpha,\beta)$, $(\beta,\gamma)\in{\mathcal{G}}^{(2)}$, then
$(\alpha\beta,\gamma)$, $(\alpha,\beta\gamma)\in{\mathcal{G}}^{(2)}$;

  3. 3)

$(\alpha,\alpha^{-1})\in{\mathcal{G}}^{(2)}$ for all $\alpha\in{\mathcal{G}}$;

  4. 4)

For all $(\alpha,\beta)\in{\mathcal{G}}^{(2)}$,
$(\alpha\beta)\beta^{-1}=\alpha$ and $\alpha^{-1}(\alpha\beta)=\beta$.

The set ${\mathcal{G}}^{(2)}$ is said to contain the pairs of composable
elements and the set

${\mathcal{G}}^{(0)}=\\{\alpha^{-1}\alpha,\
\alpha\in{\mathcal{G}}\\}=\\{\alpha\alpha^{-1},\ \alpha\in{\mathcal{G}}\\}$

is referred to as the space of units. The maps

$\mathfrak{r},\
\mathfrak{s}:{\mathcal{G}}\rightarrow{\mathcal{G}}^{(0)},\quad\mathfrak{r}(\alpha):=\alpha\alpha^{-1},\quad\mathfrak{s}(\alpha):=\alpha^{-1}\alpha$
(2.1)

are called the range and the source maps, respectively. Note that
${\mathcal{G}}^{(2)}$ can be equivalently defined as

${\mathcal{G}}^{(2)}=\big{\\{}(\alpha,\beta)\in{\mathcal{G}}\times{\mathcal{G}},\
\mathfrak{s}(\alpha)=\mathfrak{r}(\beta)\big{\\}}.$

######  Remark 2.2.

It follows that

$\mathfrak{r}(\alpha^{-1})=\mathfrak{s}(\alpha),\quad\mathfrak{s}(\alpha^{-1})=\mathfrak{r}(\alpha).$

Hence, $(\alpha,\beta^{-1})\in{\mathcal{G}}^{(2)}$ iff
$\mathfrak{s}(\alpha)=\mathfrak{s}(\beta)$. This is relevant for the product
(2.2) and for the left regular representations (2.11). Many other useful
properties and identities can be found in [49, Ch. 8]. $\Diamond$

Below, we introduce several standard groupoids, which will appear in our
discussion of aperiodic physical systems.

######  Example 2.3.

For an arbitrary set $X$, one defines the matrix groupoid over $X$ to be the
set ${\mathcal{M}}_{X}=X\times X$. The set
${\mathcal{M}}_{X}^{(2)}\subset{\mathcal{M}}_{X}\times{\mathcal{M}}_{X}$ of
composable elements consists of pairs $((x,y),(z,w))$ such that $y=z$ and the
composition of two composable elements is $(x,y)\cdot(y,z)=(x,z)$. The
inversion map acts as $(x,y)^{-1}=(y,x)$. The space of units
${\mathcal{M}}_{X}^{(0)}$ consists of $(x,x)$, $x\in X$, hence it can be
canonically identified with $X$. As we shall see below, if
$X=\\{1,2,\ldots,N\\}$, this groupoid is related to the standard algebra of
$N\times N$ matrices and, if $X={\mathbb{N}}$ or any other discrete and
infinite topological space, then the groupoid relates to the algebra of
compact operators over the separable Hilbert space $\ell^{2}(X)$. $\Diamond$

######  Example 2.4.

Let $R$ be an equivalence relation over an arbitrary set $X$. The groupoid
associated to $R$ consists of its graph ${\mathcal{R}}=\\{(x,y)\in X\times X,\
x\sim_{R}y\\}$ and the set of composable elements ${\mathcal{R}}^{(2)}$
consists of pairs $((x,y),(z,w))$ such that $y=z$. The composition of two
composable elements is $(x,y)\cdot(y,z)=(x,z)$. The inversion map acts as
$(x,y)^{-1}=(y,x)$ and the space of units ${\mathcal{R}}^{(0)}$ can be
identified again with $X$. The CAR-algebra admits a presentation as the
$C^{\ast}$-algebra associated to a groupoid of this type (see Remark 4.9).
$\Diamond$

######  Example 2.5.

The Bellissard-Kellendonk groupoid associated to the single fermion dynamics
is introduced in Definition 3.17. The groupoids associated with the
$N$-fermion dynamics are introduced in Definition 6.32. $\Diamond$

The groupoids we will consider carry extra structure in the form of a
topology.

######  Definition 2.6.

A groupoid is called a topological groupoid if ${\mathcal{G}}$ is equipped
with a locally compact Hausdorff topology such that

  1. 1)

The inversion, source and range maps are all continuous;

  2. 2)

The composition is also continuous when
${\mathcal{G}}^{(2)}\subset{\mathcal{G}}\times{\mathcal{G}}$ is equipped with
the relative topology.

Although the Hausdorff condition in this definition can be weakened (see [49,
Def. 8.3.1]), the groupoids in this paper will always satisfy it.

######  Example 2.7.

[49, p. 73] Every groupoid is a topological groupoid in the discrete topology.
In particular, if the set $X$ is a discrete topological space, then the
groupoids constructed in examples 2.3 and 2.4 are topological. $\Diamond$

######  Example 2.8.

[49, p. 73] If $X$ in example 2.4 is a second-countable Hausdorff space, then
${\mathcal{R}}$ is a topological groupoid in the relative topology inherited
from $X\times X$. The topology of ${\mathcal{R}}$, however, can be finer than
the relative one, but never coarser. This is a very important observation
because, as we shall see, the compact subsets of the groupoid enter in an
essential way in the definition of the groupoid algebra. As such, by adjusting
the groupoid’s topology, one can drastically alter the character of this
algebra (see Remark 2.11). This issue will be highlighted at several key
points of our narrative (see for example Remark 3.18). $\Diamond$

In this work, we will only consider étale groupoids:

######  Definition 2.9 ([49], Def. 8.4.1).

A topological groupoid ${\mathcal{G}}$ is called étale if the range map
$\mathfrak{r}:{\mathcal{G}}\rightarrow{\mathcal{G}}^{(0)}$ is a local
homeomorphism.

######  Remark 2.10.

Among all topological groupoids, the étale groupoids are the equivalent of the
discrete groups within the topological groups. Indeed, an important
characteristic of étale groupoids is that $\mathfrak{r}^{-1}(\alpha)$ and
$\mathfrak{s}^{-1}(\alpha)$ are discrete topological spaces for any
$\alpha\in{\mathcal{G}}^{(0)}$. Furthermore, any étale groupoid admits a Haar
system [44, Def. 2.2], which is supplied by the system of counting measures on
each fiber [44, Def. 2.6, Lemma 2.7]. As we will be working solely with étale
groupoids, we will not need the general theory of Haar systems and we will
therefore omit a discussion of them. $\Diamond$

###  2.2. Background: Reduced $C^{\ast}$-algebra of an étale groupoid

We develop this section in a generality that is beyond what is needed for our
present work. We do so to facilitate the new concept of bi-equivariant
groupoid $C^{\ast}$-algebras, which will play a central role in our present
and possibly future program (per Remark 1.10).

Given an étale groupoid ${\mathcal{G}}$ and a $C^{*}$-algebra ${\mathcal{A}}$,
one defines the space $C_{c}({\mathcal{G}},{\mathcal{A}})$ of compactly
supported ${\mathcal{A}}$-valued continuous functions on ${\mathcal{G}}$ and
endows it with the multiplication

$(f_{1}\ast
f_{2})(\alpha)=\sum_{\beta\in\mathfrak{s}^{-1}(\mathfrak{s}(\alpha))}f_{1}(\alpha\beta^{-1})f_{2}(\beta)=\sum_{\beta\in\mathfrak{r}^{-1}(\mathfrak{r}(\alpha))}f_{1}(\beta)f_{2}(\beta^{-1}\alpha),$
(2.2)

and the $\ast$-operation

$f^{\ast}(\alpha)=f(\alpha^{-1})^{*}.$ (2.3)

Then $C_{c}({\mathcal{G}},{\mathcal{A}})$ becomes a $\ast$-algebra, which can
be equipped with a norm such that its completion supplies a
$C^{\ast}$-algebra. The physical implication of this process was already
addressed in Remark 1.12.

######  Remark 2.11.

Note that the topology of ${\mathcal{G}}$, specifically the compact subsets of
${\mathcal{G}}$ in this topology, determines which functions are contained in
this $C^{\ast}$-algebra. This somewhat technical point has important physical
implications and will become a central point of discussion in the next
sections. $\Diamond$

We now return to the task of defining a $C^{\ast}$-norm and for this we follow
the procedure from [32], where $C_{c}({\mathcal{G}},{\mathcal{A}})$ is
embedded and then completed in the $C^{\ast}$-algebra of adjointable operators
over a natural Hilbert $C^{\ast}$-module associated with the pair
$({\mathcal{G}},{\mathcal{A}})$. We first recall:

######  Definition 2.12 ([12, 28, 53]).

A right Hilbert C∗-module over the $C^{\ast}$-algebra $\mathcal{B}$ is a right
$\mathcal{B}$-module equipped with an inner product

$\langle\,\cdot\,,\,\cdot\,\rangle\;:\;E_{\mathcal{B}}\times
E_{\mathcal{B}}\rightarrow\mathcal{B}\;,$

which is linear in the second variable and satisfies the following relations
for all $\psi,\psi^{\prime}\in E_{\mathcal{B}}$ and $b\in{\mathcal{B}}$:

  1. (i)

$\langle\psi,\psi^{\prime}b\rangle=\langle\psi,\psi^{\prime}\rangle b$,

  2. (ii)

$\langle\psi,\psi^{\prime}\rangle^{\ast}=\langle\psi^{\prime},\psi\rangle$,

  3. (iii)

$\langle\psi,\psi\rangle\geq 0$,

  4. (iv)

$\psi\neq 0$ implies $\langle\psi,\psi\rangle\neq 0$.

In addition, $E_{\mathcal{B}}$ must be complete in the norm induced by the
inner product

$\|\psi\|\;:=\;\|\langle\psi,\psi\rangle\|^{\frac{1}{2}}_{\mathcal{B}}\;,\qquad\psi\in
E_{\mathcal{B}}\;,$

where $\|\cdot\|_{\mathcal{B}}$ is the $C^{\ast}$-norm of ${\mathcal{B}}$.

######  Example 2.13.

The standard Hilbert $\mathcal{B}$-module is defined to be the set

$\ell^{2}({\mathbb{N}},\mathcal{B})\;=\;\Big{\\{}\\{b_{n}\\}_{n\in{\mathbb{N}}}\
:\ b_{n}\in\mathcal{B},\ \sum_{n=1}^{N}b_{n}^{\ast}b_{n}\ \mbox{converges in}\
\mathcal{B}\ {\rm as}\ N\rightarrow\infty\Big{\\}}\;,$

endowed with the obvious addition and the ${\mathcal{B}}$-valued inner product

$\big{\langle}\\{b_{n}\\}_{n\in{\mathbb{N}}},\\{b^{\prime}_{n}\\}_{n\in{\mathbb{N}}}\big{\rangle}\;=\;\sum_{n\in\mathbb{N}}b_{n}^{\ast}b^{\prime}_{n}.$

The set ${\mathbb{N}}$ can be replaced by any other countable set $X$, in
which case we write $\ell^{2}(X,\mathcal{B})$. $\Diamond$

######  Definition 2.14 ([12, 28, 53]).

The space of adjointable operators ${\rm End}^{\ast}(E_{\mathcal{B}})$ over a
Hilbert $\mathcal{B}$-module consists of the maps
$T:E_{\mathcal{B}}\rightarrow E_{\mathcal{B}}$ for which there exists a linear
map $T^{\ast}:E_{\mathcal{B}}\rightarrow E_{\mathcal{B}}$ such that

$\langle
T\psi,\psi^{\prime}\rangle\;=\;\langle\psi,T^{\ast}\psi^{\prime}\rangle\;,\qquad\psi,\psi^{\prime}\in
E_{\mathcal{B}}\;.$

######  Remark 2.15.

An adjointable operator is automatically a bounded $\mathcal{B}$-module map,
but the reversed implication is not always true. Hence, the attribute
“adjointable” is significant. If the operator $T^{\ast}$ exists, it is
unique.$\Diamond$

######  Proposition 2.16 ([28] p. 4, and [12, 53]).

When endowed with the operator norm,

$\|T\|\;=\;\sup\big{\\{}\|T\psi\|\ :\ \psi\in E_{\mathcal{B}},\ \|\psi\|\leq
1\big{\\}}\;,$

the space ${\rm End}^{\ast}(E_{\mathcal{B}})$ becomes a unital C∗-algebra.

We now describe the Hilbert $C^{\ast}$-module associated to the pair
$({\mathcal{G}},{\mathcal{A}})$. First, note that the space
$C_{c}({\mathcal{G}},{\mathcal{A}})$ is already a right module over
$C_{0}({\mathcal{G}}^{(0)},{\mathcal{A}})$, the $C^{\ast}$-algebra of
${\mathcal{A}}$-valued continuous and compactly supported functions over
${\mathcal{G}}^{(0)}$, endowed with the sup norm. Indeed, for $f\in
C_{c}({\mathcal{G}},{\mathcal{A}})$ and $\chi\in
C_{0}({\mathcal{G}}^{(0)},{\mathcal{A}})$, one defines

$(f\cdot\chi)(\alpha)=f(\alpha)\chi(\mathfrak{s}(\alpha)),\quad\alpha\in{\mathcal{G}}.$

Since ${\mathcal{G}}^{(0)}\subset{\mathcal{G}}$ is closed, one can also
consider the restriction map

$\rho:C_{c}({\mathcal{G}},{\mathcal{A}})\rightarrow
C_{0}({\mathcal{G}}^{(0)},{\mathcal{A}}),$

and define a $C_{0}({\mathcal{G}}^{(0)},{\mathcal{A}})$-valued inner product
on the right $C_{0}({\mathcal{G}}^{(0)},{\mathcal{A}})$-module
$C_{c}({\mathcal{G}},{\mathcal{A}})$ via

$\langle
f_{1}|f_{2}\rangle_{C_{0}({\mathcal{G}}^{(0)},{\mathcal{A}})}(x):=\rho(f_{1}^{\ast}\ast
f_{2})(x)=\sum_{\alpha\in\mathfrak{s}^{-1}(x)}f_{1}(\alpha)^{*}f_{2}(\alpha).$
(2.4)

We denote by $E_{C_{0}({\mathcal{G}}^{(0)},{\mathcal{A}})}$ the Hilbert
$C^{\ast}$-module completion of $C_{c}({\mathcal{G}},{\mathcal{A}})$ in this
inner product. The convolution from the left supplies a left action of the
$\ast$-algebra $C_{c}({\mathcal{G}},{\mathcal{A}})$ on
$E_{C_{0}({\mathcal{G}}^{(0)},{\mathcal{A}})}$ by bounded adjointable
endomorphisms, extending the action of $C_{c}({\mathcal{G}},{\mathcal{A}})$ on
itself.

######  Definition 2.17 ([32]).

The reduced groupoid $C^{\ast}$-algebra of $\mathcal{G}$ over the
$C^{\ast}$-algebra ${\mathcal{A}}$ is the completion of
$C_{c}({\mathcal{G}},{\mathcal{A}})$ in the norm inherited from the embedding

$C_{c}({\mathcal{G}},{\mathcal{A}})\rightarrowtail{\rm
End}^{\ast}(E_{C_{0}({\mathcal{G}}^{(0)},{\mathcal{A}})})$

and is denoted $C^{\ast}_{r}({\mathcal{G}},{\mathcal{A}})$.

In case ${\mathcal{A}}=\mathbb{C}$ it is usually suppressed in the notation.
We refer to $C^{*}_{r}(\mathcal{G}):=C^{*}_{r}(\mathcal{G},\mathbb{C})$ as the
reduced $C^{*}$-algebra of $\mathcal{G}$.

######  Remark 2.18.

For a second countable, Hausdorff, étale groupoid ${\mathcal{G}}$, the reduced
$C^{\ast}$-algebra $C^{\ast}_{r}({\mathcal{G}})$ is separable. In turn, this
implies that its $K$-groups are countable [12] and this is of fundamental
importance for various classifications programs of topological phases of
matter, as we already stressed in the introduction. $\Diamond$

###  2.3. Bi-equivariant groupoid $C^{*}$-algebras

Let $\mathcal{G}$ be a topological groupoid. The group of bisections
$\mathcal{S}(\mathcal{G})$ of $\mathcal{G}$ is the space of continuous maps

$\mathcal{S}(\mathcal{G}):=\left\\{b:\mathcal{G}^{(0)}\to\mathcal{G}:\mathfrak{s}\circ
b=\textnormal{Id},\quad\mathfrak{r}\circ b\textnormal{ is a
homeomorphism}\right\\}.$

The group structure on $\mathcal{S}(\mathcal{G})$ is given by

$b_{1}\cdot b_{2}(\alpha):=b_{1}\big{(}\mathfrak{r}\circ
b_{2}(\alpha)\big{)}b_{2}(\alpha),\quad
b^{-1}(\alpha):=b\big{(}(\mathfrak{r}\circ
b)^{-1}(\alpha)\big{)}^{-1},\quad\alpha\in\mathcal{G}^{(0)}.$ (2.5)

Here, $b^{-1}$ is the inverse of $b$ in ${\mathcal{S}}({\mathcal{G}})$,
$(\mathfrak{r}\circ b)^{-1}$ denotes the inverse homeomorphism to
$\mathfrak{r}\circ b$, whereas $b(\alpha)^{-1}$ denotes the inverse of
$b(\alpha)$ in $\mathcal{G}$. The identity element of
$\mathcal{S}(\mathcal{G})$ is the inclusion
$i:\mathcal{G}^{(0)}\to\mathcal{G}$. In fact, $\mathcal{S}(\mathcal{G})$ is a
locally compact group in the compact open topology [19, Example 13]. It admits
continuous commuting left and right actions on $\mathcal{G}$ via

$b_{1}\cdot\alpha\cdot
b_{2}:=b_{1}\big{(}\mathfrak{r}(\alpha)\big{)}\,\alpha\,b_{2}^{-1}\big{(}\mathfrak{s}(\alpha)\big{)}^{-1},\quad
b\in\mathcal{S}(\mathcal{G}),\quad\alpha\in\mathcal{G}.$

######  Lemma 2.19.

The actions satisfy

$\displaystyle(b_{1}\cdot\alpha\cdot b_{2})^{-1}$
$\displaystyle=b_{2}^{-1}\cdot\alpha^{-1}\cdot b_{1}^{-1},$ (2.6)
$\displaystyle\mathfrak{r}(b_{1}\cdot\alpha\cdot b_{2})$
$\displaystyle=b_{1}\cdot\mathfrak{r}(\alpha)\cdot b_{1}^{-1},$ (2.7)
$\displaystyle\mathfrak{s}(b_{1}\cdot\alpha\cdot b_{2})$
$\displaystyle=b_{2}^{-1}\cdot\mathfrak{s}(\alpha)\cdot b_{2}.$ (2.8)

###### Proof.

Indeed:

$\displaystyle(b_{1}\cdot\alpha\cdot b_{2})^{-1}$
$\displaystyle=b_{2}^{-1}(\mathfrak{s}(\alpha))\,\alpha^{-1}\,b_{1}(\mathfrak{r}(\alpha))^{-1}=b_{2}^{-1}\cdot\alpha^{-1}\cdot
b_{1}^{-1}$ $\displaystyle\mathfrak{r}(b_{1}\cdot\alpha\cdot b_{2})$
$\displaystyle=\mathfrak{r}\circ
b_{1}(\mathfrak{r}(\alpha))=b_{1}(\mathfrak{r}(\alpha))\,\mathfrak{r}(\alpha)\,b_{1}(\mathfrak{r}(\alpha))^{-1}=b_{1}\cdot\mathfrak{r}(\alpha)\cdot
b_{1}^{-1}$ $\displaystyle\mathfrak{s}(b_{1}\cdot\alpha\cdot b_{2})$
$\displaystyle=\mathfrak{s}\circ
b_{2}^{-1}(\mathfrak{s}(\alpha))^{-1}=b_{2}^{-1}(\mathfrak{s}(\alpha))\,\mathfrak{s}(\alpha)\,b_{2}^{-1}(\mathfrak{s}(\alpha))^{-1}=b_{2}^{-1}\cdot\mathfrak{s}(\alpha)\cdot
b_{2},$

which prove the statements.∎

######  Corollary 2.20.

If $(\alpha,\beta)\in\mathcal{G}^{(2)}$, then $(b_{1}\cdot\alpha\cdot
b_{2}^{-1},b_{2}\cdot\beta\cdot b_{3})\in\mathcal{G}^{(2)}$ and

$(b_{1}\cdot\alpha\cdot b_{2}^{-1})(b_{2}\cdot\beta\cdot
b_{3})=b_{1}\cdot\alpha\beta\cdot b_{3}.$

######  Definition 2.21.

Let $\Gamma$ be a discrete group and $\mathcal{G}$ a locally compact Hausdorff
groupoid. A 2-action of $\Gamma$ on $\mathcal{G}$ is a group homomorphism
$\tau:\Gamma\to\mathcal{S}(\mathcal{G})$.

Given a 2-action of $\Gamma$ on $\mathcal{G}$ we obtain commuting left and
right actions by setting

$\gamma_{1}\cdot\alpha\cdot\gamma_{2}:=\tau_{\gamma_{1}}\cdot\alpha\cdot\tau_{\gamma_{2}}=\tau_{\gamma_{1}}\big{(}\mathfrak{r}(\alpha)\big{)}\cdot\alpha\cdot\tau_{\gamma_{2}^{-1}}\big{(}\mathfrak{s}(\alpha)\big{)}^{-1}.$
(2.9)

It should be noted that these actions do not determine the full 2-action of
the group.

######  Remark 2.22.

For the specialized groupoids appearing in our work, a 2-action is introduced
in Proposition 6.42 and the corresponding left and right actions are computed
in Proposition 6.43. $\Diamond$

For a $C^{*}$-algebra ${\mathcal{A}}$, we denote by $UM({\mathcal{A}})$ the
unitary group of the multiplier algebra of ${\mathcal{A}}$. A similar notion
of 2-action of a group on $C^{*}$-algebras is given in the following
definition.

######  Definition 2.23.

Let $\Gamma$ be a discrete group and ${\mathcal{A}}$ a $C^{*}$-algebra. A
2-action of $\Gamma$ on ${\mathcal{A}}$ is a group homomorphism $u:\Gamma\to
UM({\mathcal{A}})$ from $\Gamma$ to the unitary multipliers on
${\mathcal{A}}$.

Given a 2-action of $\Gamma$ on ${\mathcal{A}}$ we obtain left and right
actions on ${\mathcal{A}}$ by setting

$\gamma_{1}\cdot a\cdot\gamma_{2}:=u_{\gamma_{1}}au_{\gamma_{2}}^{*}.$

This in turn induces an action

$\bm{\alpha}:\Gamma\to\textnormal{Aut}({\mathcal{A}}),\quad\bm{\alpha}_{\gamma}(a):=u_{\gamma}au_{\gamma}^{*},$

of $\Gamma$ on ${\mathcal{A}}$ by $*$-automorphisms, making $A$ into a
$\Gamma$-$C^{*}$-algebra. Note that in case ${\mathcal{A}}$ is commutative,
the latter action is trivial but the 2-action is not.

######  Remark 2.24.

In our concrete application, $\Gamma$ is the permutation group
$\mathcal{S}_{N}$ and the $C^{\ast}$-algebra ${\mathcal{A}}$ will be simply
${\mathbb{C}}$. The relevant 2-action is given by the sign homomorphism
$\tau:\mathcal{S}_{N}\to\\{\pm 1\\}\subset\mathbb{C}=UM(\mathbb{C})$. We often
write this action as ${\mathcal{S}}_{N}\ni s\mapsto(-1)^{s}\in
UM({\mathbb{C}})$. $\Diamond$

######  Remark 2.25.

We chose the terminology of 2-actions because, given a group $\Gamma$, the
pair groupoid $\Gamma\times\Gamma$ is an example of a $2$-groupoid (for a
comprehensive definition, see [2, Section 2.2]). A 2-groupoid is a 2-category
in which all 1- and 2-morphisms are invertible. Somewhat informally, a
2-groupoid is a groupoid such that the unit space carries the structure of a
groupoid itself. In case the unit space is a group, we speak of a 2-group.
$\Diamond$

######  Remark 2.26.

For us, the relevant example arises from a discrete group $\Gamma$. The pair
groupoid $\Gamma\times\Gamma$ associated to the full equivalence relation on
$\Gamma$ as in Example 2.4, carries a natural 2-group structure, since
$(\Gamma\times\Gamma)^{(0)}=\Gamma$ is a group. The notion of 2-action in
Definition 2.21 is equivalent to an action of the 2-groupoid
$\Gamma\times\Gamma$ on the 1-groupoid $\mathcal{G}$, and the data in
Definition 2.23 is equivalent to an action of the 2-groupoid
$\Gamma\times\Gamma$ on the $C^{*}$-algebra ${\mathcal{A}}$. $\Diamond$

######  Remark 2.27.

Alternatively, a 2-group can be associated to a so called crossed module of a
pair of groups, originally introduced by Whitehead [54] in the context of
homotopy 2-types. In our case, the relevant crossed module is given by the
group $\Gamma$ acting on itself by conjugation. The equivalence of 2-groups
and crossed modules is described in [39, Section 3]. The actions of crossed
modules on groupoids and $C^{*}$-algebras are described in detail in [19,
Sections 2 and 3]. Applying those definitions to the special case of the
2-group $\Gamma\times\Gamma$ gives our definitions. $\Diamond$

Given 2-actions of $\Gamma$ on a $C^{*}$-algebra ${\mathcal{A}}$ and on a
groupoid $\mathcal{G}$, we write

$C_{c,\Gamma}(\mathcal{G},{\mathcal{A}}):=\Big{\\{}f\in
C_{c}(\mathcal{G},{\mathcal{A}}):f(\gamma_{1}\cdot\alpha\cdot\gamma_{2})=\gamma_{1}\cdot
f(\alpha)\cdot\gamma_{2}\Big{\\}}\subset C_{c}(\mathcal{G},{\mathcal{A}}),$

for the space of $\Gamma$-bi-equivariant maps from $\mathcal{G}$ to
${\mathcal{A}}$. The space $C_{c,\Gamma}(\mathcal{G},{\mathcal{A}})$ carries
the structure of a $*$-algebra:

######  Lemma 2.28.

The subspace $C_{c,\Gamma}(\mathcal{G},{\mathcal{A}})$ is a $*$-subalgebra of
$C_{c}(\mathcal{G},{\mathcal{A}})$.

###### Proof.

The identities (2.6), (2.7) and (2.8) allow us to show that convolution and
involution are compatible with the $\Gamma$-equivariance:

$\displaystyle f*g(\gamma_{1}\cdot\beta\cdot\gamma_{2})$
$\displaystyle=\sum_{\alpha\in\mathfrak{r}^{-1}(\mathfrak{r}(\gamma_{1}\cdot\beta\cdot\gamma_{2}))}f(\alpha)g\big{(}\alpha^{-1}(\gamma_{1}\cdot\beta\cdot\gamma_{2})\big{)}$
$\displaystyle=\sum_{\alpha\in\mathfrak{r}^{-1}(\gamma_{1}\mathfrak{r}(\beta)\gamma_{1}^{-1})}f(\alpha)g\Big{(}\gamma_{1}\cdot\big{(}(\gamma_{1}^{-1}\cdot\alpha^{-1}\cdot\gamma_{1})\beta\big{)}\cdot\gamma_{2}\Big{)}$
$\displaystyle=\sum_{\alpha\in\mathfrak{r}^{-1}(\mathfrak{r}(\beta))}f(\gamma_{1}\cdot\alpha\cdot\gamma_{1}^{-1})g(\gamma_{1}\cdot(\alpha^{-1}\beta)\cdot\gamma_{2})$
$\displaystyle=\sum_{\alpha\in\mathfrak{r}^{-1}(\mathfrak{r}(\beta))}\gamma_{1}\cdot
f(\alpha)\cdot\gamma_{1}^{-1}\cdot\gamma_{1}\cdot
g(\alpha^{-1}\beta)\cdot\gamma_{2}$
$\displaystyle=\sum_{\alpha\in\mathfrak{r}^{-1}(\mathfrak{r}(\beta))}\gamma_{1}\cdot
f(\alpha)g(\alpha^{-1}\beta)\cdot\gamma_{2}=\gamma_{1}\cdot(f*g)(\beta)\cdot\gamma_{2}$
$\displaystyle
f^{*}(\gamma_{1}\cdot\alpha\cdot\gamma_{2})=f((\gamma_{1}\cdot\alpha\cdot\gamma_{2})^{-1})^{*}=f(\gamma_{2}^{-1}\cdot\alpha^{-1}\cdot\gamma_{1}^{-1})^{*}=\gamma_{1}\cdot
f^{*}(\alpha)\cdot\gamma_{2}.$

This proves the lemma. ∎

######  Definition 2.29.

Let $\Gamma$ be a discrete group. Suppose that $\mathcal{G}$ is a locally
compact Hausdorff étale groupoid and ${\mathcal{A}}$ is a $C^{*}$-algebra,
both of which carry a 2-action by $\Gamma$. The $\Gamma$-bi-equivariant
reduced ${\mathcal{A}}$-$C^{*}$-algebra
$C^{*}_{r,\Gamma}(\mathcal{G},{\mathcal{A}})$ of $\mathcal{G}$ is the closure
of the image of $C_{c,\Gamma}(\mathcal{G},{\mathcal{A}})$ inside
$C^{*}_{r}(\mathcal{G},{\mathcal{A}})$.

The following statement supplies a computational tool for projecting onto a
$\Gamma$-bi-equivariant sub-algebra when the group is finite. It can be used,
for example, to project the range of the ${\mathbb{C}}$-module morphisms
mentioned in Remark 1.5.

######  Proposition 2.30.

Let $\Gamma$ be a finite group and
$C^{*}_{r,\Gamma}(\mathcal{G},{\mathcal{A}})$ a $\Gamma$-bi-equivariant
reduced ${\mathcal{A}}$-$C^{*}$-algebra of $\mathcal{G}$. Then the map
$E:C^{*}_{r}(\mathcal{G},{\mathcal{A}})\to
C^{*}_{r}(\mathcal{G},{\mathcal{A}})$,

$\big{(}E(f)\big{)}(\alpha)=\frac{1}{|\Gamma|^{2}}\sum_{\gamma_{1},\gamma_{2}\in\Gamma}\gamma_{1}^{-1}\cdot
f(\gamma_{1}\cdot\alpha\cdot\gamma_{2})\cdot\gamma_{2}^{-1}$ (2.10)

is a conditonal expectation onto
$C^{*}_{r,\Gamma}(\mathcal{G},{\mathcal{A}})$.

###### Proof.

The map $E$ is a projection. Indeed, for $\theta_{1},\theta_{2}\in\Gamma$,

$\displaystyle\big{(}E(f)\big{)}(\theta_{1}\cdot\alpha\cdot\theta_{2})=\frac{1}{|\Gamma|^{2}}\sum_{\gamma_{1},\gamma_{2}\in\Gamma}\theta_{1}(\gamma_{1}\theta_{1})^{-1}\cdot
f(\gamma_{1}\theta_{1}\cdot\alpha\cdot\theta_{2}\gamma_{2})\cdot(\theta_{2}\gamma_{2})^{-1}\theta_{2},$

hence $E(f)$ is bi-equivariant. As such, the range of $E$ is contained in
$C^{*}_{r,\Gamma}(\mathcal{G},{\mathcal{A}})$. Furthermore, for $f\in
C^{*}_{r,\Gamma}(\mathcal{G},{\mathcal{A}})$, we can see directly from Eq.
(2.10) that $E(f)=f$. Lastly, $E$ is a contractive projection, hence an
expectation [50].∎

###  2.4. Left regular representations

After completing the algebra $C_{c}({\mathcal{G}},{\mathcal{A}})$, the
restriction map $\rho:C_{c}({\mathcal{G}},{\mathcal{A}})\rightarrow
C_{0}({\mathcal{G}}^{(0)},{\mathcal{A}})$ extends by continuity to a positive
map of $C^{\ast}$-algebras. By composing $\rho$ with the evaluation maps on
$C_{0}({\mathcal{G}}^{(0)},{\mathcal{A}})$, $j_{x}(\chi)=\chi(x)$ for some
$x\in{\mathcal{G}}^{(0)}$, one obtains a family of ${\mathcal{A}}$-valued
positive maps $\rho_{x}=j_{x}\circ\rho$ on
$C^{\ast}_{r}({\mathcal{G}},{\mathcal{A}})$, indexed by the space of units
${\mathcal{G}}^{(0)}$. Then, for each $x\in{\mathcal{G}}^{(0)}$,
$C^{\ast}_{r}({\mathcal{G}},{\mathcal{A}})$ can be endowed with an
${\mathcal{A}}$-valued inner product $\langle
f_{1},f_{2}\rangle_{x}:=\rho_{x}(f_{1}^{\ast}f_{2})$, which makes
$C^{\ast}_{r}({\mathcal{G}},{\mathcal{A}})$ into a right pre-Hilbert
${\mathcal{A}}$-module. The completion of the latter can be canonically
identified with the standard Hilbert module
$\ell^{2}\big{(}\mathfrak{s}^{-1}(x),{\mathcal{A}}\big{)}$. The action from
the left of the algebra $C^{\ast}_{r}({\mathcal{G}},{\mathcal{A}})$ on itself
can be extended to an action by adjointable operators on
$\ell^{2}(\mathfrak{s}^{-1}(x),{\mathcal{A}})$. Explicitly,

$[\pi_{x}(f)\psi](\alpha)=\sum_{\beta\in\mathfrak{s}^{-1}(x)}f(\alpha\beta^{-1})\psi(\beta),\quad\alpha\in\mathfrak{s}^{-1}(x),\quad\psi\in\ell^{2}\big{(}\mathfrak{s}^{-1}(x),{\mathcal{A}}\big{)}.$
(2.11)

The representations $\pi_{x}$, $x\in{\mathcal{G}}^{(0)}$, are called the left
regular representations of the reduced groupoid $C^{*}$-algebra
$C^{\ast}_{r}({\mathcal{G}},{\mathcal{A}})$.

######  Example 2.31.

For the matrix groupoid from example 2.3,

$\mathfrak{s}^{-1}(x)=\\{(y,x),\ y\in X\\},$

and these sets can all be canonically identified with $X$. If $X$ is a
discrete topological space, then ${\mathcal{M}}_{X}$ is étale and the regular
representations of $C^{\ast}({\mathcal{M}}_{x})$ are on $\ell^{2}(X)$ and
given by

$[\pi_{x}(f)\psi](y)=\sum_{z\in X}f(y,z)\psi(z).$

These are all compact operators over $\ell^{2}(X)$. $\Diamond$

######  Example 2.32.

For the groupoid from example 2.4,

$\mathfrak{s}^{-1}(x)=\\{(y,x)\in X\times X,\ y\sim_{R}x\\}.$

If ${\mathcal{R}}$ is étale, then the regular representations of
$C^{\ast}({\mathcal{R}})$ are on $\ell^{2}(\mathfrak{s}^{-1}(x))$ and given by

$[\pi_{x}(f)\psi](y)=\sum_{z\sim_{R}x}f(y,z)\psi(z).$

Depending on the topology of ${\mathcal{R}}$, these may or may not be compact
operators over $\ell^{2}(\mathfrak{s}^{-1}(x))$. An example of the latter is
the CAR groupoid, for which all the left regular representations are carried
by non-compact operators. $\Diamond$

##  3\. Single Fermion Dynamics by groupoid Methods

Our main goals for this section are to highlight the important role played by
groupoid $C^{*}$-algebras in the analysis of the dynamics of a single fermion
hopping over an aperiodic lattice and to formulate the problem from a
perspective that affords, at the conceptual level, a straightforward
generalization to the case of interacting fermions. For this, we start from an
algebra of local physical observables available to an experimenter. As we
shall see, the time evolutions of these observables are implemented by groups
of outer authomorphisms. Then the specialized groupoid algebras constructed by
Bellissard [8, 9] and Kellendonk [30] can be seen as formalizing the
generators of these groups of outer automorphisms.

###  3.1. The algebra of local observables

The setting is that of a network ${\mathcal{L}}$ of identical quantum
resonators, such as atoms or quantum circuits, rendered in the physical space
${\mathbb{R}}^{d}$ and whose identical internal structures have been fixed.
When placed next to each other, the resonators couple via potentials that are
entirely determined by their internal structure and space arrangement. As
such, under the stated constraints, the dynamics of the collective degrees of
freedom is fully determined by the pattern
${\mathcal{L}}\subset{\mathbb{R}}^{d}$. In Fig. 3.1, we show such a network of
quantum resonators, together with an experimenter who excites the resonators,
perhaps with a laser beam, and then observes the dynamics of the coupled
degrees of freedom. The experimenter can only excite resonators and take
measurements on resonators near its position. In other words, the experimenter
can only probe the local degrees of freedom. Of course, we are referring to
the idealized case of an infinite sample.

Figure 3.1. Schematic of a local observation.

In the 1-fermion setting, the algebra of observables available to the observer
is that of finite-rank operators over the Hilbert space
$\ell^{2}({\mathcal{L}})$. If the resonators have more than one internal
degree of freedom, this Hilbert space is tensored by ${\mathbb{C}}^{N}$ but
this detail can be dealt with later. By a limiting procedure, one can close
this algebra to the $C^{\ast}$-algebra
${\mathbb{K}}\big{(}\ell^{2}({\mathcal{L}})\big{)}$ of compact operators over
$\ell^{2}({\mathcal{L}})$, which represents the algebra of local observables
in the 1-fermion setting. The main interest, however, is not in this algebra
but in the time evolution of the observables, which is formalized by a one
parameter group of $*$-automorphisms

$\bm{\alpha}:{\mathbb{R}}\to{\rm
Aut}\Big{(}{\mathbb{K}}\big{(}\ell^{2}({\mathcal{L}})\big{)}\Big{)}.$ (3.1)

We assume from the beginning that the time evolution is point-wise continuous,
i.e. the map
$t\mapsto\bm{\alpha}_{t}(K)\in{\mathbb{K}}\big{(}\ell^{2}({\mathcal{L}})\big{)}$
is continuous for all $K\in{\mathbb{K}}\big{(}\ell^{2}({\mathcal{L}})\big{)}$.
The generator $\delta_{\bm{\alpha}}$ of the dynamics is defined on the dense
$*$-sub-algebra of elements
$K\in{\mathbb{K}}\big{(}\ell^{2}({\mathcal{L}})\big{)}$ for which
$\bm{\alpha}_{t}(K)$ is first order differentiable in $t$. The derivations
over the algebra of compact operators have been fully characterized in [17,
Example 1.6.4]. In particular, all bounded derivations come in the form

${\rm ad}_{H}(K)=\imath[K,H],\quad(\imath=\sqrt{-1}),$

for some $H$ from the algebra
${\mathbb{B}}\big{(}\ell^{2}({\mathcal{L}})\big{)}$ of bounded operators over
$\ell^{2}({\mathcal{L}})$, which will be referred to as the Hamiltonian.

###  3.2. Galilean invariant theories

As a bounded operator over
${\mathbb{B}}\big{(}\ell^{2}({\mathcal{L}})\big{)}$, the Hamiltonian always
affords a presentation of the form

$H_{\mathcal{L}}=\sum_{x,x^{\prime}\in{\mathcal{L}}}w_{x,x^{\prime}}({\mathcal{L}})\,|x\rangle\langle
x^{\prime}|,\quad w_{x,x^{\prime}}({\mathcal{L}})\in{\mathbb{C}},$ (3.2)

with the sum converging in the strong operator topology. The notation in Eq.
3.2 emphasizes that the Hamiltonian coefficients
$w_{x,x^{\prime}}({\mathcal{L}})$ are fully determined by the pattern itself.
The experimenter can modify the pattern and map again the coefficients, hence,
at least in principle, the entire map
${\mathcal{L}}\mapsto\\{w_{x,x^{\prime}}({\mathcal{L}})\\}$ can be accessed
experimentally. Physical considerations enable us to assume that the
coefficients vary continuously with ${\mathcal{L}}$, in a sense that will be
made precise later.

The Hamiltonians display additional structure steaming from the Galilean
invariance of the non-relativistic physical laws. Indeed, in the absence of
any background field or potential, suppose that two patterns ${\mathcal{L}}$
and ${\mathcal{L}}^{\prime}$ enter the relation
${\mathcal{L}}^{\prime}={\mathcal{L}}-y$. Then necessarily

$w_{x,x^{\prime}}({\mathcal{L}})=w_{x-y,x^{\prime}-y}({\mathcal{L}}-y),\quad
x,x^{\prime}\in{\mathcal{L}},$ (3.3)

and, certainly, this is what the laboratory measurements will return. Now, we
fix a pattern ${\mathcal{L}}_{0}$ and apply these facts to the corresponding
Hamiltonian. We find

$H_{{\mathcal{L}}_{0}}=\sum_{x,x^{\prime}\in{\mathcal{L}}_{0}}w_{0,x^{\prime}-x}({\mathcal{L}}_{0}-x)\,|x\rangle\langle
x^{\prime}|,$

and, if we introduce $y=x^{\prime}-x\in{\mathcal{L}}_{0}-x$ and drop the
trivial index, then

$H_{{\mathcal{L}}_{0}}=\sum_{x\in{\mathcal{L}}_{0}}\sum_{y\in{\mathcal{L}}_{0}-x}w_{y}({\mathcal{L}}_{0}-x)\,|x\rangle\langle
x+y|,$

or

$(H_{{\mathcal{L}}_{0}}\psi)(x)=\sum_{y\in{\mathcal{L}}_{0}-x}w_{y}({\mathcal{L}}_{0}-x)\,\psi(x+y),\quad\psi\in\ell^{2}({\mathcal{L}}_{0}).$
(3.4)

######  Remark 3.1.

We want to stress that the equivariance relation (3.3) is not a statement
about the pattern but about the physical processes involved in the coupling of
the quantum resonators. It is for this reason that we speak of Galilean
invariance as opposed to translational invariance, as the latter is often an
attribute of a pattern rather than of a physical law. $\Diamond$

As it was already pointed out in [15], and will be exemplified again in
subsection 3.4, the expression in Eq. (3.4) is directly related to the left
regular representations of the groupoid algebra canonically attached to the
pattern [8, 9, 30]. Let us state explicitly that operators with the type of
action as in Eq. 3.4 are never compact operators, hence they do not belong to
the algebra of local observations available to an experimenter.

###  3.3. Spaces of patterns

The main goal of this section is to supply the definition of the continuous
hull and of the transversal of a point pattern. The latter will serve as the
space of units for the Bellissard-Kellendonk groupoid.

Since we are dealing with patterns in ${\mathbb{R}}^{d}$, we recall the metric
space $({\mathcal{K}}({\mathbb{R}}^{d}),{\rm d}_{\rm H})$ of compact subsets
equipped with the Hausdorff metric, as well as the larger space
${\mathcal{C}}({\mathbb{R}}^{d})$ of closed subsets of ${\mathbb{R}}^{d}$,
topologized as below. Throughout, if $(X,{\rm d})$ is a metric space, $B(x,r)$
and $\mathring{B}(x,r)$ denote the closed, respectively, open balls centered
at $x$ and of radius $r$. Also, ${\mathcal{K}}(X)$ stands for the set of
compact subsets of $X$.

######  Definition 3.2 ([26, 33]).

Given a closed subset $\Lambda\subset{\mathbb{R}}^{d}$, define

$\Lambda[r]=\big{(}\Lambda\cap B(0,r)\big{)}\cup\partial B(0,r).$

Then

$D(\Lambda,\Lambda^{\prime})=\inf\big{\\{}1/(1+r)\,|\,{\rm d}_{\rm
H}(\Lambda[r],\Lambda^{\prime}[r])<1/r\big{\\}}$ (3.5)

defines a metric on ${\mathcal{C}}({\mathbb{R}}^{d})$.

######  Definition 3.3.

We call the metric space $\big{(}{\mathcal{C}}({\mathbb{R}}^{d}),D\big{)}$ the
space of patterns in ${\mathbb{R}}^{d}$.

The space of patterns is bounded, compact and complete. Furthermore, there is
a continuous action of ${\mathbb{R}}^{d}$ by translations, that is, a
homomorphism $\mathfrak{t}$ between topological groups,

$\mathfrak{t}:{\mathbb{R}}^{d}\rightarrow{\rm
Homeo}\big{(}{\mathcal{C}}({\mathbb{R}}^{d}),D\big{)},\quad\mathfrak{t}_{x}(\Lambda)=\Lambda-x.$
(3.6)

######  Remark 3.4.

The topology of $\big{(}{\mathcal{C}}({\mathbb{R}}^{d}),D\big{)}$ can be
conveniently described in the following way. The space ${\mathbb{R}}^{d}$ can
be canonically embedded in its one-point compactification, the $d$-dimensional
sphere ${\mathbb{S}}^{d}$. As such, there exists a canonical embedding
$\mathfrak{j}:{\mathcal{C}}({\mathbb{R}}^{d})\to{\mathcal{K}}({\mathbb{S}}^{d})$
and the latter can be topologized using the Hausdorff distance. The pullback
topology through $\mathfrak{j}$ defines a topology on
${\mathcal{C}}({\mathbb{R}}^{d})$ that is equivalent with the topology of
$\big{(}{\mathcal{C}}({\mathbb{R}}^{d}),D\big{)}$ [26, p. 17]. $\Diamond$

######  Definition 3.5.

Let ${\mathcal{L}}\in{\mathbb{R}}^{d}$ be discrete and infinite and fix
$0<r<R$.

  1. (1)

${\mathcal{L}}$ is $r$-uniformly discrete if $|B(x,r)\cap{\mathcal{L}}|\leq 1$
for all $x\in{\mathbb{R}}^{d}$.

  2. (2)

${\mathcal{L}}$ is $R$-relatively dense if $|B(x,R)\cap{\mathcal{L}}|\geq 1$
for all $x\in{\mathbb{R}}^{d}$.

An $r$-uniform discrete and $R$-relatively dense set ${\mathcal{L}}$ is called
an $(r,R)$-Delone set.

######  Remark 3.6.

Throughout, if $S$ is a set then $|S|$ denotes its cardinal. $\Diamond$

######  Proposition 3.7.

The set of $(r,R)$-Delone sets in ${\mathbb{R}}^{d}$, ${\rm
Del}_{(r,R)}({\mathbb{R}}^{d})$, is a compact subset of
$({\mathcal{C}}({\mathbb{R}}^{d}),D)$. A basis for its topology is supplied by
the family of sub-sets

$U_{M}^{\epsilon}({\mathcal{L}})=\big{\\{}\mathcal{L}^{\prime}\in{\rm
Del}_{(r,R)}({\mathbb{R}}^{d}),\quad\mathrm{d}_{H}(\mathcal{L}[M],\mathcal{L}^{\prime}[M])<\epsilon\big{\\}},\
M,\epsilon>0.$ (3.7)

Let ${\mathcal{L}}_{0}\subset{\mathbb{R}}^{d}$ be a fixed Delone point set.

######  Definition 3.8.

The continuous hull of ${\mathcal{L}}_{0}$ is the topological dynamical system
$(\Omega_{{\mathcal{L}}_{0}},\mathfrak{t},{\mathbb{R}}^{d})$, where

$\Omega_{{\mathcal{L}}_{0}}=\overline{\\{\mathfrak{t}_{a}({\mathcal{L}}_{0})={\mathcal{L}}_{0}-a,\
a\in{\mathbb{R}}^{d}\\}}\subset{\mathcal{C}}({\mathbb{R}}^{d}),$

with the closure in the metric topology induced by (3.5).

Since $\Omega_{{\mathcal{L}}_{0}}$ is a closed subset of the space of
patterns, every point ${\mathcal{L}}\in\Omega_{{\mathcal{L}}_{0}}$ defines a
closed subset of ${\mathbb{R}}^{d}$. Theorem 2.8 of [10] assures us that all
these closed subsets are in fact Delone point sets.

######  Remark 3.9.

The above dynamical system is always transitive but may fail to be minimal.
Indeed, if ${\mathcal{L}}\in\Omega_{{\mathcal{L}}_{0}}$ belongs to the closure
and not to the orbit of ${\mathcal{L}}_{0}$, then the orbit of ${\mathcal{L}}$
belongs to $\Omega_{{\mathcal{L}}_{0}}$ but might not be dense in
$\Omega_{{\mathcal{L}}_{0}}$, as it is indeed the case for e.g. periodic
patterns in ${\mathbb{R}}$ with one defect [47][p. 9] or the disordered
pattern (see Example 3.12). In fact, the character of the patterns contained
inside the hull of a fixed pattern ${\mathcal{L}}_{0}$ can vary drastically
and $(\Omega_{{\mathcal{L}}_{0}},\mathfrak{t},{\mathbb{R}}^{d})$ can have many
minimal components (see e.g. [43]). It is important to acknowledge that, while
$\Omega_{\mathcal{L}}$ may not coincide with $\Omega_{{\mathcal{L}}_{0}}$ for
${\mathcal{L}}\in\Omega_{{\mathcal{L}}_{0}}$, it is always true that
$\Omega_{\mathcal{L}}\subseteq\Omega_{{\mathcal{L}}_{0}}$. $\Diamond$

######  Definition 3.10.

The canonical transversal of a continuous hull
$(\Omega_{{\mathcal{L}}_{0}},\mathfrak{t},{\mathbb{R}}^{d})$ of a Delone set
${\mathcal{L}}_{0}$ is defined as

$\Xi_{{\mathcal{L}}_{0}}=\\{{\mathcal{L}}\in\Omega_{{\mathcal{L}}_{0}},\
0\in{\mathcal{L}}\\}.$

The transversal is a compact subspace of ${\mathcal{C}}({\mathbb{R}}^{d})$.

######  Example 3.11.

For a periodic pattern such as
${\mathcal{L}}_{0}={\mathbb{Z}}^{d}\subset{\mathbb{R}}^{d}$, the transversal
consists of just one point, the lattice ${\mathcal{L}}_{0}$ itself. $\Diamond$

######  Example 3.12.

For the pattern $\\{n+\lambda(\xi_{n}-\frac{1}{2})\\}_{n\in{\mathbb{Z}}}$,
with the $\xi_{n}$ entries drawn randomly and distinctly from the interval
$[0,1]$ and $\lambda<1$, the transversal is the Hilbert cube
$[0,1]^{\mathbb{Z}}$. $\Diamond$

Figure 3.2. (a) The geometric algorithm for the pattern from Example 3.13
consists of a colinear triplet of points separated by a distance $r$, that are
iteratively translated horizontally by $D>2r$ and rotated by an angle
$\theta\in{\mathbb{R}}\setminus{\mathbb{Q}}$. (b) The practical process of
mapping the transversal of the pattern (see Example 3.13 for details).

######  Example 3.13.

Consider the pattern generated by the geometric algorithm described in Fig.
3.2(a). To map its transversal, an experimenter rigidly moves the pattern so
that each point sits at the origin of the laboratory frame and makes the
observations depicted in Fig. 3.2(b). Specifically, if one of the outer points
of the triplets sits at the origin, such as 0, 2, etc., the experimenter
centers the green circle at the origin and marks its intersection with the
dotted line joining the triplets. If one of the inner points of the triplets
sits at the origin, such as 1, 4, etc., the experimenter centers the red
circle at the origin and marks its intersections with the dotted line joining
the triplets. The experimenter then realizes that these markings are all the
information needed to reproduce the entire sequence of translated patterns.
Indeed, moving a distance $r$ in the direction indicated by a marked point on
the green circle, the experimenter puts down one point and then moves by
another distance $r$ and puts down a second point, hence completing the
triplet (the origin is also counted in). Then the experimenter translates and
rotates the triplet according to the algorithm. In this way, the experimenter
reproduces the translated pattern which produced the marked point on the green
circle in the first place. On the red circle, the experimenter chooses a pair
of opposite markings and walks distances $r$ from the origin in the directions
indicated by those markings and puts down two points, hence completing again
the triplet. Then the experimenter translates and rotates the triplet
according to the algorithm. In this way, the experimenter reproduces the
translated pattern which produced the pair of marked points on the red circle
in the first place. The conclusion is that the translated patterns
${\mathcal{L}}_{0}-x$, $x\in{\mathcal{L}}_{0}$, are in one to one
correspondence with the markings on the green circle and with the pairs of
markings on the red circle. If $\theta/2\pi$ is an irrational number, these
markings densely fill the two circles. Lastly, if the seeding triplet
$\\{0,1,2\\}$ is rotated by an angle $\varphi$ and the algorithm is applied
again, the so obtained patterns vary continuously with $\varphi$ inside
${\mathcal{C}}({\mathbb{R}}^{2})$. This assures us that the closure of the
discrete set of patterns $\\{{\mathcal{L}}_{0}-x,\ x\in{\mathcal{L}}_{0}\\}$
fills entirely the two circles. We can conclude that $\Xi_{{\mathcal{L}}_{0}}$
consists of the green circle and of the red circle with the opposite points
identified (topologically, this is again a circle). Note that moving
continuously along the red connected component of the transversal results in
the exchange of the outer points of each triplet. $\Diamond$

######  Example 3.14.

The reader can find in [22] a general algorithm that generates patterns for
which $\Xi_{{\mathcal{L}}_{0}}$ is a torus of arbitrary dimension. $\Diamond$

######  Example 3.15.

The transversal of the Fibonacci quasicrystal is a cantorized circle (see e.g.
[31]). $\Diamond$

######  Remark 3.16.

In the physics literature, the transversal of a pattern appears under the name
of phason space and a point of this space is called a phason. The patterns
with transversals $\Xi_{{\mathcal{L}}_{0}}$ that have non-trivial topology are
particularly interesting for practical applications and certainly all the
works in materials science cited in our introduction involve such patterns.
This is so because the phason can be physically driven inside
$\Xi_{{\mathcal{L}}_{0}}$ to achieve various exquisite effects. This is one of
the main reasons why, to be relevant for applications, the framework of
interacting fermions must be developed for generic aperiodic patterns.
$\Diamond$

###  3.4. The groupoid $C^{\ast}$-algebra of single fermion dynamics

In this subsection, we review the Bellissard-Kellendonk groupoid associated to
the dynamics of a single fermion over a Delone point set. We then show how the
left regular representations of the groupoid $C^{\ast}$-algebra generates all
Galilean invariant Hamiltonians over the Delone set.

######  Definition 3.17.

The Bellissard-Kellendonk topological groupoid associated to a Delone set
${\mathcal{L}}_{0}$ consists of:

  1. 1.

The set

${\mathcal{G}}_{1}:=\\{({\mathcal{L}},x)\in\Xi_{{\mathcal{L}}_{0}}\times{\mathbb{R}}^{d},\
x\in{\mathcal{L}}\\}$

equipped with the inversion map

$({\mathcal{L}},x)^{-1}=({\mathcal{L}}-x,-x).$

  2. 2.

The subset of ${\mathcal{G}}_{1}\times{\mathcal{G}}_{1}$

${\mathcal{G}}_{1}^{(2)}=\\{(({\mathcal{L}},x),({\mathcal{L}}^{\prime},y))\in{\mathcal{G}}_{1}\times{\mathcal{G}}_{1},\
{\mathcal{L}}^{\prime}={\mathcal{L}}-x\\}$

equipped with the composition

$({\mathcal{L}},x)\cdot({\mathcal{L}}-x,y)=({\mathcal{L}},x+y).$

The topology on ${\mathcal{G}}_{1}$ is the relative topology inherited from
$\Xi_{{\mathcal{L}}_{0}}\times{\mathbb{R}}^{d}$.

######  Remark 3.18.

The algebraic structure of ${\mathcal{G}}_{1}$ can be also described as coming
from the equivalence relation on $\Xi_{{\mathcal{L}}_{0}}$ (see Example 2.4):

$R_{\Xi_{{\mathcal{L}}_{0}}}=\big{\\{}({\mathcal{L}},{\mathcal{L}}^{\prime})\in\Xi_{{\mathcal{L}}_{0}}\times\Xi_{{\mathcal{L}}_{0}},\
{\mathcal{L}}^{\prime}={\mathcal{L}}-a\ \mbox{for some}\
a\in{\mathbb{R}}^{d}\big{\\}}.$

Note, however, that the topology of ${\mathcal{G}}_{1}$ is not the one
inherited from $\Xi_{{\mathcal{L}}_{0}}\times\Xi_{{\mathcal{L}}_{0}}$.
$\Diamond$

######  Proposition 3.19 ([14], Prop. 2.16).

The set ${\mathcal{G}}_{1}$ is a second-countable, Hausdorff, étale groupoid.

######  Remark 3.20.

The above statement remains true if ${\mathcal{L}}_{0}$ is just a uniformly
discrete pattern. This becomes relevant when lattices with defects are
investigated. $\Diamond$

######  Remark 3.21.

It follows from their very definition (2.1) that the range and the source maps
are

$\mathfrak{r}({\mathcal{L}},x)=({\mathcal{L}},0),\quad\mathfrak{s}({\mathcal{L}},x)=({\mathcal{L}}-x,0),$

hence the space of units ${\mathcal{G}}_{1}^{(0)}$ can be canonically
identified with $\Xi_{{\mathcal{L}}_{0}}$ via
$({\mathcal{L}},0)\mapsto{\mathcal{L}}$. Furthermore, we have

$\mathfrak{r}^{-1}\big{(}\mathfrak{r}({\mathcal{L}},x)\big{)}=\\{({\mathcal{L}},y),\,y\in{\mathcal{L}}\\},\quad\mathfrak{s}^{-1}\big{(}\mathfrak{s}({\mathcal{L}},x)\big{)}=\\{({\mathcal{L}}-x,y-x)^{-1},\
y\in{\mathcal{L}}\\},$ (3.8)

hence both spaces
$\mathfrak{r}^{-1}\big{(}\mathfrak{r}({\mathcal{L}},x)\big{)}$ and
$\mathfrak{s}^{-1}\big{(}\mathfrak{s}({\mathcal{L}},x)\big{)}$ can be
canonically identified with the lattice ${\mathcal{L}}$ itself. $\Diamond$

######  Remark 3.22.

We recall that all ${\mathcal{L}}\in\Xi_{{\mathcal{L}}_{0}}$ are Delone point
sets, hence infinite and discrete topological spaces. As such, the Bellissard-
Kellendonk groupoid has a compact space of units but infinite discrete fibers.
$\Diamond$

We now turn our attention to the $C^{\ast}$-algebra
$C_{r}^{\ast}({\mathcal{G}}_{1})$ canonically associated to the groupoid
${\mathcal{G}}_{1}$ and, hence, to the pattern ${\mathcal{L}}_{0}$. Eqs. (2.2)
and (3.8) give

$(f_{1}\ast
f_{2})({\mathcal{L}},x)=\sum_{y\in{\mathcal{L}}}f_{1}({\mathcal{L}},y)f_{2}({\mathcal{L}}-y,x-y),\quad({\mathcal{L}},x)\in{\mathcal{G}}_{1}.$

Also, the involution takes the form
$f^{\ast}({\mathcal{L}},x)=\overline{f({\mathcal{L}}-x,-x)}$.

######  Remark 3.23.

We stress again that $C_{r}^{\ast}({\mathcal{G}}_{1})$ is separable for a
Delone and, in fact, for any uniformly discrete pattern. This in turn assures
us that its $K$-theory is countable, hence we have a sensible and meaningful
classification of the Hamiltonians by stable homotopy. Without the remarkable
insight from Bellissard and Kellendonk, the only available option would have
been to place the Hamiltonians in the large, non-separable algebra
${\mathbb{B}}\big{(}\ell^{2}({\mathcal{L}})\big{)}$, which has trivial
$K$-theory. $\Diamond$

As discussed in section 2.4, the left regular representations of
$C^{\ast}_{r}({\mathcal{G}}_{1})$ are indexed by the points
${\mathcal{L}}\in\Xi_{{\mathcal{L}}_{0}}$ and are carried by the Hilbert
spaces $\ell^{2}\big{(}\mathfrak{s}^{-1}({\mathcal{L}})\big{)}$. In
particular, for ${\mathcal{L}}_{0}$ and
$\varphi\in\ell^{2}\big{(}\mathfrak{s}^{-1}({\mathcal{L}}_{0})\big{)}$,

$\displaystyle\big{(}({\mathcal{L}}_{0},x)^{-1}\big{)}=\sum_{y\in{\mathcal{L}}_{0}-x}f({\mathcal{L}}_{0}-x,y)\varphi\big{(}({\mathcal{L}}_{0},x+y)^{-1}\big{)}.$

We can canonically map $\varphi$ to a function $\psi$ on ${\mathcal{L}}_{0}$,
via $\psi(x)=\varphi\big{(}({\mathcal{L}}_{0},x)^{-1}\big{)}$, and define the
left-regular representation over $\ell^{2}({\mathcal{L}}_{0})$,

$[\pi_{{\mathcal{L}}_{0}}(f)\psi](x)=\sum_{y\in{\mathcal{L}}_{0}-x}f({\mathcal{L}}_{0}-x,y)\psi(x+y).$
(3.9)

Furthermore, for any $a\in{\mathcal{L}}_{0}$, we can define the Hilbert space
isomorphisms

$T_{a}:\ell^{2}({\mathcal{L}}_{0}-a)\rightarrow\ell^{2}({\mathcal{L}}_{0}),\quad
T_{a}\psi=\psi\circ\mathfrak{t}_{a}.$

Then the left regular representations enjoy the following covariant property

$T_{a}^{\ast}\pi_{{\mathcal{L}}_{0}}(f)T_{a}=\pi_{{\mathcal{L}}_{0}-a}(f),\quad
f\in C_{r}^{\ast}({\mathcal{G}}_{1}).$ (3.10)

We arrive now to the main conclusion of the section. If we compare the
expression (3.9) with the action of Galilean invariant Hamiltonians (3.4), we
see that they are identical once we identify $f({\mathcal{L}}_{0}-x,y)$ and
$w_{y}({\mathcal{L}}_{0}-x)$. The outstanding conclusion is that all Galilean
invariant Hamiltonians over ${\mathcal{L}}_{0}$ can be generated from the left
regular representations of $C_{r}^{\ast}({\mathcal{G}}_{1})$, which is the
smallest $C^{\ast}$-algebra with this property.

##  4\. Interacting Fermions: The Algebra of Local Observables

In this section, we first describe the $C^{\ast}$-algebra ${\rm
CAR}({\mathcal{L}})$ of local observables for a system of many-fermions
hopping on a discrete lattice ${\mathcal{L}}$. As in the previous section,
this local algebra plays a similar role as a Hilbert space does when we define
operators: It will help us define an algebra of derivations associated with
the dynamics of the local observables. While our final goal is to characterize
this algebra of derivations, the success of that program rests on a fine
characterization of the structure of ${\rm CAR}({\mathcal{L}})$, which is
supplied in this section. Of particular importance is our choice of presenting
the elements in a symmetric manner that treats all possible orderings of the
anti-commuting generators on equal footing. This leads to the notion of bi-
equivariant coefficients w.r.t. permutation groups, which later will be
connected with the material introduced in section 2.3. The Fock representation
of ${\rm CAR}({\mathcal{L}})$ is treated in the same spirit using frames that
are stable against the actions of the permutation groups. One of our important
observations is that the product of operators continues to manifest itself as
a certain convolution of the “symmetrized” matrix elements. The reader will
also find here the characterization [16] of the lattice of ideals of ${\rm
GICAR}({\mathcal{L}})$, the sub-algebra of gauge invariant elements. Of
essential importance for our program is the observation that ${\rm
GICAR}({\mathcal{L}})$ admits a filtration by primitive ideals and that the
quotient of two consecutive such ideals is isomorphic to the algebra of
compact operators. In other words, ${\rm GICAR}({\mathcal{L}})$ is a solvable
$C^{\ast}$-algebra in the sense of [25].

###  4.1. The CAR and ${\rm GICAR}$ algebras over a lattice

The setting is that of a Delone pattern ${\mathcal{L}}$ whose points are
populated by spin-less fermions. The algebra of local observables is supplied
by the algebra of canonical anti-commutation relations over ${\mathcal{L}}$,
denoted here by ${\rm CAR}({\mathcal{L}})$ [18]. It is constructed from a net
$\\{{\mathcal{L}}_{k}\\}$ of finite subsets of ${\mathcal{L}}$, with
${\mathcal{L}}_{k}\subset{\mathcal{L}}_{k+1}$ and
$\bigcup_{k\in{\mathbb{Z}}}{\mathcal{L}}_{k}={\mathcal{L}}$. For each of these
finite subsets, one defines ${\rm CAR}({\mathcal{L}}_{k})$ as the finite
$\ast$-algebra generated by $a_{x}$, $x\in{\mathcal{L}}_{k}$, and the
relations

$a_{x}a_{x^{\prime}}+a_{x^{\prime}}a_{x}=0,\quad
a_{x}^{\ast}a_{x^{\prime}}+a_{x^{\prime}}a_{x}^{\ast}=\delta_{x,x^{\prime}}.$
(4.1)

######  Definition 4.1.

The algebra ${\rm CAR}({\mathcal{L}})$ is the limit of the inductive tower of
finite algebras ${\rm CAR}({\mathcal{L}}_{k})$, supplied by the canonical
embeddings ${\rm CAR}({\mathcal{L}}_{k})\rightarrowtail{\rm
CAR}({\mathcal{L}}_{k+1})$.

######  Remark 4.2.

We recall that the universal ${\rm CAR}$-algebra admits a unique faithful
tracial state, which will be denoted by ${\mathcal{T}}$. It will play an
essential role in section 5.4 (see Proposition 5.39). $\Diamond$

The local algebra of observables constructed this way formalizes the local
measurements available to an experimenter dealing with a system of fermions
over ${\mathcal{L}}$. As in subsection 3.1, this experimenter analyses the
dynamics of the available local observables and the main task is to figure out
the group of automorphisms $\\{\bm{\alpha}_{t}\\}_{t\in{\mathbb{R}}}$ that
implements the time evolution, which is usually done by experimenting with
many such local observables. As in subsection 3.1, the experimenter will find
that these automorphisms are outer, hence the Hamiltonians generating the
dynamics do not belong to the local algebra. This central aspect will be
addressed in section 5.

###  4.2. Symmetric presentation

Given the anti-commutation relations (4.1), a word constructed from the
generators has many equivalent presentations. The formalism developed in
sections 5 and 6 relies on our specific choice to treat all equivalent word
presentations on equal footing. This leads to a specific presentation of ${\rm
CAR}({\mathcal{L}})$, which we call the symmetric presentation. We describe it
here.

For a finite subset $J\subset{\mathbb{R}}^{d}$ of cardinality $|J|=N$, we
consider the set $\mathcal{S}({\mathcal{I}}_{N},J)$ of bijections from
${\mathcal{I}}_{N}:=\\{1,\ldots,N\\}$ to $J$. Any such bijection supplies a
particular enumeration of the elements of $J$, hence, an order. The group
$\mathcal{S}_{N}$ of ordinary permutations of $N$ objects can and will be
identified with $\mathcal{S}({\mathcal{I}}_{N},{\mathcal{I}}_{N})$. It has
natural left and right actions on $\mathcal{S}({\mathcal{I}}_{N},J)$ via

$s\cdot\chi_{J}:=\chi_{J}\circ s^{-1},\quad\chi_{J}\cdot s:=\chi_{J}\circ
s,\quad s\in\mathcal{S}_{N},\quad\chi_{J}\in\mathcal{S}({\mathcal{I}}_{N},J),$

respectively. Consider now a Delone set ${\mathcal{L}}$. We introduce the
following relations

$(J,\chi_{J})\mapsto a_{J}(\chi_{J}):=a_{\chi_{J}(|J|)}\cdots
a_{\chi_{J}(|1|)}\in{\rm CAR}({\mathcal{L}}),$ (4.2)

for all $J\in{\mathcal{K}}({\mathcal{L}})$ and
$\chi_{J}\in\mathcal{S}({\mathcal{I}}_{|J|},J)$. Obviously,

$a_{J}\big{(}s\cdot\chi_{J}\big{)}=a_{J}\big{(}\chi_{J}\cdot
s\big{)}=(-1)^{s}a_{J}\big{(}\chi_{J}),\quad\forall\ s\in\mathcal{S}_{|J|},$

and

$a_{J}(\chi_{J})^{\ast}=a^{\ast}_{\chi_{J}(1)}\cdots
a^{\ast}_{\chi_{J}(|J|)}.$ (4.3)

It is also useful to introduce the special elements

$n_{J}=a^{\ast}_{J}(\chi_{J})a_{J}(\chi_{J})=\tfrac{1}{|J|!}\sum_{\chi_{J}}a^{\ast}_{J}(\chi_{J})a_{J}(\chi_{J}).$

Throughout, we will use the convention $a_{\emptyset}=a^{\ast}_{\emptyset}=1$,
the unit of ${\rm CAR}({\mathcal{L}})$.

######  Remark 4.3.

Since ${\mathcal{L}}$ is a discrete set, the compact subsets of
${\mathcal{L}}$ are exactly the finite subsets. Throughout, we will only
involve compact subsets, hence the cardinality of the subsets is finite,
everywhere in our discussion. $\Diamond$

One could argue that a choice for the enumeration of the subsets of
${\mathcal{L}}$ can be made once and for all. This is certainly possible if
the pattern ${\mathcal{L}}$ is fixed. However, we want and, in fact, are
forced to allow for deformations of the pattern, e.g. at least those induced
by the simple translations of ${\mathcal{L}}$, already encountered in section
3. Under such deformations, two points of the pattern can be exchanged, as it
was the case in Example 3.13, and the enumeration of the points of a subset
containing those two points becomes ambiguous. This example shows that the
only acceptable option is to use all the available enumerations on equal
footing and this is indeed what we will do when presenting the elements of the
${\rm CAR}$ algebra. However, in doing so, we need to impose restrictions on
the coefficients. It is at this point where the material from section 2.3
makes its entrance.

######  Definition 4.4.

For a pair $(J,J^{\prime})$ of compact subsets of ${\mathcal{L}}$, a bi-
equivariant coefficient is a map
$c_{J,J^{\prime}}:\mathcal{S}({\mathcal{I}}_{|J|},J)\times\mathcal{S}({\mathcal{I}}_{|J^{\prime}|},J^{\prime})\to\mathbb{C}$
such that

$c_{J,J^{\prime}}(s_{1}\cdot\chi_{J},\chi_{J^{\prime}}\cdot
s_{2})=(-1)^{s_{1}}\ c_{J,J^{\prime}}(\chi_{J},\chi_{J^{\prime}})\
(-1)^{s_{2}}.$

######  Remark 4.5.

Given the particular representations of the permutation groups entering in the
above definition, one could argue that just an orientation of the subsets $J$
and $J^{\prime}$ will suffice. This is indeed the case for the presentation of
the CAR algebra. However, the full orders on the sub-sets supply special
points, namely $\chi_{J}(1)$, which will be used in an essential way in our
construction of the groupoid associated with the dynamics of the fermions (see
section 6.3). $\Diamond$

######  Proposition 4.6.

Every element of ${\rm CAR}({\mathcal{L}})$ can be uniquely presented as a
norm convergent sum

$A=\sum_{J,J^{\prime}\in{\mathcal{K}}({\mathcal{L}})}\tfrac{1}{\sqrt{|J|!|J^{\prime}|!}}\sum_{\chi_{J},\chi_{J}^{\prime}}c_{J,J^{\prime}}\big{(}\chi_{J},\chi_{J^{\prime}}\big{)}a_{J}(\chi_{J})^{\ast}a_{J^{\prime}}(\chi_{J^{\prime}}),$
(4.4)

where it is understood that the coefficients are bi-equivariant and that the
second sum runs over the whole set
$\mathcal{S}({\mathcal{I}}_{|J|},J)\times\mathcal{S}({\mathcal{I}}_{|J^{\prime}|},J^{\prime})$.

We will refer to Eq. (4.4) as the symmetric presentation of $A$. The
consideration of the combinatorial factors in front of the second sum will be
justified in Propositions 4.19 and 4.29.

######  Remark 4.7.

The first sum in (4.4) can include $J=\emptyset$, $J^{\prime}=\emptyset$ or
$J=J^{\prime}=\emptyset$. In these cases, we use our convention
$a_{\emptyset}=1$. $\Diamond$

######  Remark 4.8.

It is important to acknowledge that the coefficients $c_{J,J^{\prime}}$ in Eq.
4.4 must display a certain decay as the sets $J$ or $J^{\prime}$ are pushed to
infinity. For example, a formal series with coefficients $c_{J,J}$ that do not
decay to zero as the diameter of the ${\mathbb{R}}^{d}$ sub-set $\\{0\\}\cup
J\cup J^{\prime}$ goes to infinity is not part of ${\rm CAR}({\mathcal{L}})$.
$\Diamond$

The gauge invariant (GI) elements are, by definition, elements that can be
written as in Eq. (4.4), but with the first sum restricted to pairs of subsets
with $|J|=|J^{\prime}|$. Such elements are invariant against the circle action
$a_{x}\mapsto\lambda a_{x}$, $|\lambda|=1$, hence the name “gauge invariant”.
They form a sub-algebra denoted here by ${\rm GICAR}({\mathcal{L}})$. The
symmetric presentation of such elements can be organized as

$A=\sum_{n\in{\mathbb{N}}}\tfrac{1}{n!}\sum_{J,J^{\prime}\in{\mathcal{K}}_{n}({\mathcal{L}})}\sum_{\chi_{J},\chi_{J}^{\prime}}c_{J,J^{\prime}}\big{(}\chi_{J},\chi_{J^{\prime}}\big{)}a_{J}(\chi_{J})^{\ast}a_{J^{\prime}}(\chi_{J^{\prime}}),$
(4.5)

where we introduced the new notation ${\mathcal{K}}_{n}({\mathcal{L}})$ for
the finite un-ordered sub-sets of ${\mathcal{L}}$ of cardinal $n$.

######  Remark 4.9.

A groupoid presentation of the CAR algebra was supplied in Example 1.10 of
[44]. Although not used in our work, we provide it here for completeness. The
construction requires an ordering of the lattice points and it starts from the
configuration space $S=\prod_{x\in{\mathcal{L}}}S_{x}$, where
$S_{x}=\\{0,1\\}$ and $S$ is equipped with the product topology. A
configuration $s=(s_{x})\in S$ singles out the sites $x$ with $s_{x}=1$, which
can be thought of as the sites of ${\mathcal{L}}$ populated by fermions. Two
configurations are declared equivalent iff they differ at at most a finite
number of places. Algebraically, the CAR groupoid is the groupoid
${\mathcal{C}}$ corresponding to this equivalence relation (see Example 2.4).
The étale topology on ${\mathcal{C}}$ is strictly finer than the relative
topology inherited from $S\times S$. To describe this topology, we follow
Example 8.3.5 in [49]. First, one identifies ${\mathcal{L}}$ with
${\mathbb{N}}$ by using the predefined order. Then, for $n\in{\mathbb{N}}$ and
any two finite words $v$ and $w$ from $\\{0,1\\}^{n}$, one defines the subset

$Z_{n}(v,w)=\\{(vs,ws)|s\in S\\}\subseteq{\mathcal{C}},$

where $vs$ means concatenation of the words $v$ and $s$. The collection

$\\{Z_{n}(v,w),\,n\in{\mathbb{N}},\,v,w\in\\{0,1\\}^{n}\\},$

is the basis of a topology that makes ${\mathcal{C}}$ into an étale groupoid.
The associated groupoid algebra $C^{\ast}_{r}({\mathcal{C}})$ is isomorphic to
the CAR algebra. In [44], the isomorphism established by showing that the two
algebras display common Elliot invariants. A direct proof of the isomorphism
can be found in [49, Example 9.2.7 ]. $\Diamond$

###  4.3. Algebraic relations and notation

The commutation relations (4.1) lead to somewhat complex relations between the
monomials. This section lists several key algebraic relations with proofs
omitted. They can be stated more effectively once some straightforward but
essential notation is introduced.

######  Definition 4.10.

For $J,J^{\prime}\in{\mathcal{K}}({\mathcal{L}})$ disjoint subsets and
$\chi_{J}\in\mathcal{S}({\mathcal{I}}_{|J|},J)$,
$\chi_{J^{\prime}}\in\mathcal{S}({\mathcal{I}}_{|J^{\prime}|},J^{\prime})$, we
define $\chi_{J}\vee\chi_{J^{\prime}}\in\mathcal{S}({\mathcal{I}}_{|J\cup
J^{\prime}|},J\cup J^{\prime})$ by

$\chi_{J}\vee\chi_{J^{\prime}}(n)=\left\\{\begin{array}[]{ll}\chi_{J}(n)&{\rm
if}\ n\leq|J|,\\\ \chi_{J^{\prime}}(n-|J|)&{\rm if}\ n>|J|.\end{array}\right.$

In words, $\chi_{J}\vee\chi_{J^{\prime}}$ enumerates first the elements of $J$
and then continue with the enumeration of the elements of $J^{\prime}$, in the
order set by $\chi_{J}$ and $\chi_{J^{\prime}}$.

With this notation, for example, we have:

$a_{J}(\chi_{J})^{\ast}\,a_{J^{\prime}}(\chi_{J^{\prime}})^{\ast}=a_{J\cup
J^{\prime}}(\chi_{J}\vee\chi_{J^{\prime}})^{\ast},$ (4.6)

whenever $J\cap J^{\prime}=\emptyset$. The following relations are useful for
reducing products of generic ${\rm CAR}$ elements to the symmetric form and
for mapping to the Fock representations.

######  Proposition 4.11.

For $J,J^{\prime}\in{\mathcal{K}}({\mathcal{L}})$, the following identity
holds:

$\displaystyle
a_{J}(\chi_{J})a^{\ast}_{J^{\prime}}(\chi_{J^{\prime}})=(-1)^{\sigma}\sum_{K\subseteq
J\cap J^{\prime}}(-1)^{|K|}a^{\ast}_{J^{\prime}\setminus
J}(\chi_{J^{\prime}\setminus J})\,n_{K}\,a_{J\setminus
J^{\prime}}(\chi_{J\setminus J^{\prime}}),$ (4.7)

where the orderings $\chi_{J}$, $\chi_{J^{\prime}}$, $\chi_{J\setminus
J^{\prime}}$ and $\chi_{J^{\prime}\setminus J}$ are independent and the sign
factor in front is determined by the choice of these orderings (its exact form
is not needed and is omitted).

We also mention the following identities, which will be instrumental for
simplifying and manipulating the Fock representations of the ${\rm CAR}$
elements:

######  Proposition 4.12.

For $J,J^{\prime},U\in{\mathcal{K}}({\mathcal{L}})$, the following identities
hold:

$\displaystyle
n_{U}\,a_{J}(\chi_{J})^{\ast}\,a_{J^{\prime}}(\chi_{J^{\prime}})$
$\displaystyle=a_{J}(\chi_{J})^{\ast}\,n_{U\setminus
J}\,a_{J^{\prime}}(\chi_{J^{\prime}}),$ $\displaystyle
a_{J}(\chi_{J})^{\ast}\,a_{J^{\prime}}(\chi_{J^{\prime}})\,n_{U}$
$\displaystyle=a_{J}(\chi_{J})^{\ast}\,n_{U\setminus
J^{\prime}}\,a_{J^{\prime}}(\chi_{J^{\prime}}).$

###  4.4. Fock representation

The Fock representation is associated with the canonical vacuum state:

######  Definition 4.13.

The vacuum state $\eta:{\rm CAR}({\mathcal{L}})\to\mathbb{C}$ is defined by
the rule

$\eta(A)=c_{\emptyset,\emptyset},\quad A\in{\rm CAR}({\mathcal{L}}),$ (4.8)

where $A$ is assumed to be in its symmetric presentation and
$c_{\emptyset,\emptyset}$ is the coefficient corresponding to the unit.

######  Proposition 4.14.

The following relations hold:

$\eta\big{(}a_{J}(\chi_{J})a^{\ast}_{J^{\prime}}(\chi_{J^{\prime}})\big{)}=(-1)^{\chi_{J}^{-1}\circ\chi_{J^{\prime}}}\,\delta_{J,J^{\prime}},\quad
J,J^{\prime}\in{\mathcal{K}}({\mathcal{L}}).$ (4.9)

###### Proof.

They are direct consequences of Eq. (4.7).∎

######  Proposition 4.15.

Let

${\mathcal{N}}_{\eta}=\big{\\{}A\in{\rm CAR}({\mathcal{L}}),\
\eta(A^{\ast}A)=0\big{\\}},$

be the closed left ideal of ${\rm CAR}({\mathcal{L}})$ associated to $\eta$.
Then ${\mathcal{N}}_{\eta}$ is spanned by the monomials
$a^{\ast}_{J}a_{J^{\prime}}$ with $J^{\prime}\neq\emptyset$.

###### Proof.

Clearly, any linear combination of the mentioned monomials belongs to
${\mathcal{N}}_{\eta}$. Then $\eta(Ba^{\ast}_{J}a_{J^{\prime}})=0$ and
$\eta(a^{\ast}_{J^{\prime}}a_{J}B)=0$ for any $B\in{\rm CAR}({\mathcal{L}})$
and $J^{\prime}\neq\emptyset$. Now, let $A\in{\rm CAR}({\mathcal{L}})$ and
assume $A$ is in its symmetric presentation. Then, using the simple facts we
just mentioned, $\eta(A^{\ast}A)$ simplifies to

$\displaystyle\eta(A^{\ast}A)=\sum_{J,J^{\prime}\in{\mathcal{K}}({\mathcal{L}})}\tfrac{1}{\sqrt{|J|!|J^{\prime}|!}}\sum_{\chi_{J},\chi_{J^{\prime}}}\overline{c_{J,\emptyset}(\chi_{J})}c_{J^{\prime},\emptyset}(\chi_{J^{\prime}})\eta\Big{(}a_{J}(\chi_{J})a^{\ast}_{J^{\prime}}(\chi_{J^{\prime}})\Big{)},$
(4.10)

and Eq. (4.9) gives

$\eta(A^{\ast}A)=\sum_{J\in{\mathcal{K}}({\mathcal{L}})}\tfrac{1}{|J|!}\sum_{\chi_{J}}|c_{J,\emptyset}(\chi_{J})|^{2}.$
(4.11)

As such, if $A\in{\mathcal{N}}_{\eta}$, then necessarily all $c_{J,\emptyset}$
are identically zero.∎

######  Remark 4.16.

It follows from the above that the linear space
${\mathcal{N}}_{\eta}+{\mathcal{N}}_{\eta}^{\ast}$ spans the entire ${\rm
CAR}({\mathcal{L}})$, except the identity. From the very definition of $\eta$
in (4.8), we can conclude that
${\mathcal{N}}_{\eta}+{\mathcal{N}}_{\eta}^{\ast}={\rm ker}\,\eta$, which
shows that $\eta$ is a pure state [37, Th. 5.3.4]. Other arguments arriving at
the same conclusion can be found in [27]. $\Diamond$

######  Proposition 4.17.

Let $|U,\chi_{U}\rangle$ be the class of $a^{\ast}_{U}(\chi_{U})$ in the GNS
representation corresponding to $\eta$,

$\big{|}U,\chi_{U}\big{\rangle}:=a^{\ast}_{U}(\chi_{U})+{\mathcal{N}}_{\eta},$
(4.12)

for $U\in{\mathcal{K}}({\mathcal{L}})$ and
$\chi_{U}\in\mathcal{S}({\mathcal{I}}_{|U|},U)$. Then the vectors (4.12) span
the Hilbert space of the GNS representation corresponding to $\eta$.
Furthermore, the scalar product between two such vectors is

$\big{\langle}U,\chi_{U}\big{|}V,\chi_{V}\big{\rangle}=(-1)^{\chi_{U}^{-1}\circ\chi_{V}}\,\delta_{U,V}.$
(4.13)

###### Proof.

The statement is a direct consequence of Proposition 4.15 and Eq. (4.9).∎

######  Remark 4.18.

Proposition 4.17 gives the explicit connection between the GNS representation
induced by $\eta$ and the well known representation on the anti-symmetric Fock
space ${\mathcal{F}}^{(-)}({\mathcal{L}})$, spanned by (4.12). Let us recall
that, since $\eta$ is a pure state, the GNS space coincides with ${\rm
CAR}({\mathcal{L}})/{\mathcal{N}}_{\eta}$, i.e. no Cauchy completion is needed
[37, Th. 5.2.4]. Furthermore, the Fock representation is irreducible.
$\Diamond$

Since we want to avoid making any particular choice between the possible
orderings of the sub-sets, we will work with the frame of
${\mathcal{F}}^{(-)}({\mathcal{L}})$ supplied by the vectors listed in Eq.
(4.12). This frame is invariant against the action of the permutation group.
As already mentioned in our introductory remarks, this is in fact a key point
of our strategy. The following statement describes how operators can be
uniquely presented using such a frame.

######  Proposition 4.19.

If ${\mathbb{B}}\big{(}{\mathcal{F}}^{(-)}({\mathcal{L}})\big{)}$ is the
algebra of bounded operators over the anti-symmetric Fock space, then any of
its elements can be uniquely presented in the form

$F=\sum_{U,U^{\prime}\in{\mathcal{K}}({\mathcal{L}})}\tfrac{1}{\sqrt{|U|!|U^{\prime}|!}}\sum_{\chi_{U},\chi_{U^{\prime}}}F_{U,U^{\prime}}(\chi_{U},\chi_{U^{\prime}})\,\big{|}U,\chi_{U}\big{\rangle}\big{\langle}U^{\prime},\chi_{U^{\prime}}\big{|},$
(4.14)

where the coefficients are bi-equivariant and given by

$F_{U,U^{\prime}}(\chi_{U},\chi_{U^{\prime}})=\tfrac{1}{\sqrt{|U|!|U^{\prime}|!}}\big{\langle}U,\chi_{U}\big{|}F\big{|}U^{\prime},\chi_{U^{\prime}}\big{\rangle}.$
(4.15)

The sum in Eq. (4.14) converges in the strong operator topology of
${\mathbb{B}}\big{(}{\mathcal{F}}^{(-)}({\mathcal{L}})\big{)}$.

We will refer to Eq. (4.14) as the symmetric presentation of the operator. The
following statement describes the rule for the composition of operators in
this symmetric presentation.

######  Proposition 4.20.

Let
$F,F^{\prime}\in{\mathbb{B}}\big{(}{\mathcal{F}}^{(-)}({\mathcal{L}})\big{)}$
and consider their symmetric presentation as in Eq. (4.14). Then the
coefficients of the product
$FF^{\prime}\in{\mathbb{B}}\big{(}{\mathcal{F}}^{(-)}({\mathcal{L}})\big{)}$
in the symmetric presentation are supplied by the convolution

$(FF^{\prime})_{U,U^{\prime}}(\chi_{U},\chi_{U^{\prime}})=\sum_{V,\chi_{V}}F_{U,V}(\chi_{U},\chi_{V})F^{\prime}_{V,U^{\prime}}(\chi_{V},\chi_{U^{\prime}}).$

Note that $(FF^{\prime})_{U,U^{\prime}}$ are indeed bi-equivariant
coefficients.

###### Proof.

From Eq. (4.13) and the symmetric presentation of the operators,

$\displaystyle
FF^{\prime}=\sum_{U,U^{\prime}}\tfrac{1}{\sqrt{|U|!|U^{\prime}|!}}\sum_{V,V^{\prime}}\tfrac{1}{\sqrt{|V|!|V^{\prime}|!}}\sum_{\chi^{\prime}s}$
$\displaystyle(-1)^{\chi_{V}^{-1}\circ\chi_{V^{\prime}}}\delta_{V,V^{\prime}}F_{U,V}(\chi_{U},\chi_{V})$
(4.16) $\displaystyle
F^{\prime}_{V^{\prime},U^{\prime}}\big{(}\chi_{V^{\prime}},\chi_{U^{\prime}})\,\big{|}U,\chi_{U}\big{\rangle}\big{\langle}U^{\prime},\chi_{U^{\prime}}\big{|},$

where the last sum is over all orderings $\chi$ appearing inside this sum. The
sets $V$ and $V^{\prime}$ need to coincide in the second sum, but their
orderings do not. Nevertheless, using the equivariance of the coefficients, we
have

$(-1)^{\chi_{V}^{-1}\circ\chi^{\prime}_{V}}F^{\prime}_{V,U^{\prime}}\big{(}\chi^{\prime}_{V},\chi_{U^{\prime}}\big{)}=F^{\prime}_{V,U^{\prime}}\big{(}\chi_{V},\chi_{U^{\prime}}\big{)}.$

Then the inner summand is independent of $\chi_{V^{\prime}}$ and the sum over
this ordering supplies the factor $|V|!$, bringing Eq. (4.16) to the form
(4.14), with the coefficients given in the statement.∎

######  Remark 4.21.

It is at this point where the need for the cumbersome factorial factors is
being explicitly displayed. $\Diamond$

We now turn our attention to the Fock representation. The representation of
the monomials can be derived directly from the identity (4.6):

######  Proposition 4.22.

The Fock representation of the monomials takes the form,

$\displaystyle\pi_{\eta}\big{(}a_{J}^{\ast}(\chi_{J})\,a_{J^{\prime}}(\chi_{J^{\prime}})\big{)}=\sum_{\begin{subarray}{c}\Gamma\in{\mathcal{K}}({\mathcal{L}})\\\
\Gamma\cap(J\cup J^{\prime})=\emptyset\end{subarray}}\
|J\cup\Gamma,\chi_{J}\vee\chi_{\Gamma}\rangle\langle
J^{\prime}\cup\Gamma,\chi_{J^{\prime}}\vee\chi_{\Gamma}|,$ (4.17)

where the ordering $\chi_{\Gamma}$ can be any choice.

Let ${\mathcal{F}}^{(-)}_{N}$ be the closed linear sub-space of
${\mathcal{F}}^{(-)}$ spanned by the vectors $\big{|}U,\chi_{U}\big{\rangle}$
with $|U|=N$, usually called the $N$-fermion sector.

######  Proposition 4.23 ([16]).

The Fock representation $\pi_{\eta}$ restricted to the ${\rm
GICAR}({\mathcal{L}})$ subalgebra decomposes into a direct sum

$\pi_{\eta}=\bigoplus_{N\in{\mathbb{N}}}\pi_{\eta}^{N},\quad\pi_{\eta}^{N}=\pi_{\eta}\downharpoonright_{{\mathcal{F}}^{(-)}_{N}},$

of irreducible representations.

###### Proof.

Proposition 4.22 assures us that the representations of the elements from the
${\rm GICAR}$ sub-algebra leave the subspaces ${{\mathcal{F}}^{(-)}_{N}}$
invariant. These subspaces are mutually orthogonal and
${\mathcal{F}}^{(-)}=\bigoplus_{N\in{\mathbb{N}}}{{\mathcal{F}}^{(-)}_{N}}$.
Using again Proposition 4.22, one can easily see that the image of ${\rm
GICAR}({\mathcal{L}})$ through $\pi_{\eta}^{N}$ contains the algebra
${\mathbb{K}}\big{(}{\mathcal{F}}_{N}^{(-)}({\mathcal{L}})\big{)}$ of compact
operators over ${\mathcal{F}}_{N}^{(-)}({\mathcal{L}})$. As such, the
commutant of this image is ${\mathbb{C}}\cdot I$, hence the representation is
irreducible.∎

######  Proposition 4.24.

Let $J,J^{\prime}\in{\mathcal{K}}_{n}({\mathcal{L}})$. Then:

  1. (i)

If $n>N$, then $a_{J}(\chi_{J})^{\ast}\,a_{J^{\prime}}(\chi_{J^{\prime}})$ is
sent to zero by $\pi_{\eta}^{N}$.

  2. (ii)

If $n=N$, then

$\pi_{\eta}^{N}\big{(}a_{J}(\chi_{J})^{\ast}\,a_{J^{\prime}}(\chi_{J^{\prime}})\big{)}=|J,\chi_{J}\rangle\langle
J^{\prime},\chi_{J^{\prime}}|.$ (4.18)

  3. (iii)

If $n<N$, then

$\pi_{\eta}^{N}\big{(}a_{J}(\chi_{J})^{\ast}\,a_{J^{\prime}}(\chi_{J^{\prime}})\big{)}=\sum_{\Gamma\in{\mathcal{K}}_{N-n}({\mathcal{L}})}\pi_{\eta}^{N}\big{(}a_{J}(\chi_{J})^{\ast}\,n_{\Gamma}\,a_{J^{\prime}}(\chi_{J^{\prime}})\big{)}.$
(4.19)

The sum can be restricted to those $\Gamma$ with $\Gamma\cap(J\cup
J^{\prime})=\emptyset$ and the rule (4.18) can be applied to the expression
(4.19), if desired.

######  Remark 4.25.

The above expression can be put in its symmetric presentation (4.14), which we
omitted to write out because it is not particularly illuminating. Note that
the sums in (4.17) and (4.19) contain an infinite number of terms, with
coefficients that do not decay to zero. As such, the Fock representations of
the monomials are, in general, not by compact operators. This is a major
difference between the CAR algebra and the algebra of local observables
studied in section 3. $\Diamond$

Figure 4.1. (a) The Bratteli diagram of the ${\rm GICAR}$ is the Pascal
triangle. Its directed and hereditary subsets are triangles like the shaded
one, corresponding to the ideal ${}_{2}{\rm GI}_{1}$. (b) The triangles shaded
in red and blue corresponds to the primitive ideals ${}_{0}{\rm GI}_{1}$ and
${}_{2}{\rm GI}_{0}$, respectively. Their intersection give ${}_{2}{\rm
GI}_{1}$, as it is apparent from the figure.

###  4.5. ${\rm GICAR}$ is a solvable $C^{\ast}$-algebra

We start with a brief review of the work by Bratteli [16] on the global
algebraic structure of the ${\rm GICAR}$ $C^{*}$-algebra. The latter was
completely solved from the Bratteli diagram, which for ${\rm GICAR}$ is the
Pascal triangle ${\mathcal{P}}$, reproduced in Fig. 4.1. We recall that the
double-sided ideals of ${\rm GICAR}$ are in one-to-one correspondence with the
directed hereditary subsets of ${\mathcal{P}}$ [23, Th. III.4.2] and, for the
Pascal triangle, these sub-sets are the triangles described in Fig. 4.1. From
this, the lattice of ideals can be easily derived: The ideal immediately below
two ideals is given by the intersection of their corresponding triangles,
while the ideal immediately above is given by the smallest triangle that
covers both initial triangles. The particular triangles with the tip on the
very left edge of ${\mathcal{P}}$, such as the one highlighted in red in Fig.
4.1(b), form a tower of primitive ideals ${}_{0}{\rm GI}_{N}$,
$N\in{\mathbb{N}}$, associated with the Fock representations $\pi^{N}_{\eta}$.
For orientation, ${}_{0}{\rm GI}_{0}$ coincides with ${\rm GICAR}$ itself. The
triangles with the tip on the very right edge of ${\mathcal{P}}$, such as the
one highlighted in blue in Fig. 4.1(b), form another tower of primitive ideals
${}_{N}{\rm GI}_{0}$, $N\in{\mathbb{N}}$, associated with the anti-Fock
representations. According to [16, Prop. 5.6], ${}_{N}{\rm GI}_{0}$ and
${}_{0}{\rm GI}_{N}$ are the only primitive ideals of the ${\rm GICAR}$
algebra. Furthermore, any other ideal is the intersection of two such
primitive ideals, ${}_{M}{\rm GI}_{N}={}_{M}{\rm GI}_{0}\cap{}_{0}{\rm
GI}_{N}$, as exemplified in Fig. 4.1(b).

Our focus will be exclusively on the filtration supplied by the tower of
ideals ${}_{0}{\rm GI}_{N}$, which will be denoted by ${\rm GI}_{N}$ from now
on. Using Bratteli’s diagram again, one can immediately see that ${\rm
GI}_{N}/{\rm GI}_{N+1}$, whose diagram is given by the subtraction of the
corresponding triangles [23, Th. III.4.4], is isomorphic to the algebra of
compact operators [16, Prop. 5.6]. This tells us that ${\rm GICAR}$ is a
solvable $C^{\ast}$-algebra in the sense of [25] and this particular algebraic
structure will play an essential role in our program. In the rest of the
section, we supply a more detailed account of the statements made above and
supply alternative proofs that rely mostly on the commutation relations, hence
accessible to a readership un-familiar with Bratteli diagrams.

######  Proposition 4.26 ([16]).

The linear sub-spaces ${\rm GI}_{N}({\mathcal{L}})\subset{\rm
GICAR}({\mathcal{L}})$ spanned by the elements

$A=\sum_{n\geq
N}\tfrac{1}{n!}\sum_{J,J^{\prime}\in{\mathcal{K}}_{n}({\mathcal{L}})}\sum_{\chi_{J},\chi_{J}^{\prime}}c_{J,J^{\prime}}\big{(}\chi_{J},\chi_{J^{\prime}}\big{)}a_{J}(\chi_{J})^{\ast}a_{J^{\prime}}(\chi_{J^{\prime}}),$
(4.20)

are two-sided ideals of ${\rm GICAR}({\mathcal{L}})$.

###### Proof.

Focusing on monomials from ${\rm GICAR}$, we have from Eq. (4.7) that

$\displaystyle
a^{\ast}_{J_{1}}(\chi_{J_{1}})a_{J^{\prime}_{1}}(\chi_{J^{\prime}_{1}})a^{\ast}_{J_{2}}(\chi_{J_{2}})a_{J^{\prime}_{2}}(\chi_{J^{\prime}_{2}})=(-1)^{\sigma}\sum_{K\subseteq
J^{\prime}_{1}\cap J_{2}}(-1)^{|K|}$ (4.21) $\displaystyle\qquad\qquad
a^{\ast}_{J_{1}}(\chi_{J_{1}})\big{(}a^{\ast}_{J_{2}\setminus
J^{\prime}_{1}}\,n_{K}\,a_{J^{\prime}_{1}\setminus
J_{2}}\big{)}a_{J^{\prime}_{2}}(\chi_{J^{\prime}_{2}}).$

The above product is zero unless $J_{1}\cap(J_{2}\setminus
J^{\prime}_{1})=\emptyset$, $J^{\prime}_{2}\cap(J^{\prime}_{1}\setminus
J_{2})=\emptyset$ and $|J_{1}|+|J_{2}\setminus
J^{\prime}_{1}|=|J^{\prime}_{2}|+|J^{\prime}_{1}\setminus J_{2}|$. It is then
clear that the symmetric presentation of the product (4.21) involves only
terms with $J$ and $J^{\prime}$ satisfying the constraints

$\max\\{|J_{1}|,|J_{2}|\\}\leq|J|=|J^{\prime}|\leq|J_{1}|+|J_{2}|.$

The statement follows.∎

######  Proposition 4.27 ([16]).

We have ${\rm GI}_{N+1}({\mathcal{L}})\simeq{\rm Ker}\,\pi_{\eta}^{N}$, for
all $N\in{\mathbb{N}}$.

###### Proof.

Assume that $A\in{\rm GICAR}({\mathcal{L}})$ but $A\notin{\rm
GI}_{N+1}({\mathcal{L}})$ and that $A$ is a finite sum of monomials. In this
situation, we can always find a set $\Gamma\in{\mathcal{K}}({\mathcal{L}})$,
$|\Gamma|\leq N$, such that $\Gamma\cap(J\cup J^{\prime})=\emptyset$, for all
pairs $(J,J^{\prime})$ appearing in the standard presentation of $A$, and
$n_{\Gamma}A\in{\rm GI}_{N}({\mathcal{L}})$ but $n_{\Gamma}A\notin{\rm
GI}_{N+1}({\mathcal{L}})$. Then the product $n_{\Gamma}A$ can be brought to
its symmetric presentation using Proposition 4.12 and, given our previous
conclusion, necessarily

$0\neq\pi_{\eta}^{N}(n_{\Gamma}A)=\pi_{\eta}^{N}(n_{\Gamma})\pi_{\eta}^{N}(A),$

hence such element $A$ cannot belong to ${\rm Ker}\,\pi_{\eta}^{N}$.∎

######  Remark 4.28.

According to the above result, the ideals identified in Proposition 4.26 are
the primitive ideals corresponding to the representations $\pi_{\eta}^{N}$.
$\Diamond$

######  Proposition 4.29 ([16]).

We have

${\rm GI}_{N}({\mathcal{L}})/{\rm
GI}_{N+1}({\mathcal{L}})\simeq{\mathbb{K}}\big{(}{\mathcal{F}}^{(-)}_{N}\big{)},$
(4.22)

with the isomorphism supplied by the descent of $\pi_{\eta}^{N}$ onto the
quotient algebra.

###### Proof.

Let $[\cdot]_{N}$ denote the classes in the quotient space. Obviously, the
quotient space is spanned by
$\big{[}a^{\ast}_{J}(\chi_{J})a_{J^{\prime}}(\chi_{J^{\prime}})\big{]}_{N}$,
$J,J^{\prime}\in{\mathcal{K}}_{N}({\mathcal{L}})$, and the class of an element
$A\in{\rm GI}_{N}({\mathcal{L}})$, presented as in Eq (4.20), can be always
represented as

$[A]_{N}=\tfrac{1}{N!}\sum_{J,J^{\prime}\in{\mathcal{K}}_{N}({\mathcal{L}})}\sum_{\chi_{J},\chi_{J}^{\prime}}c_{J,J^{\prime}}\big{(}\chi_{J},\chi_{J^{\prime}}\big{)}\big{[}a_{J}(\chi_{J})^{\ast}a_{J^{\prime}}(\chi_{J^{\prime}})\big{]}_{N}.$

We claim that the map

$\displaystyle\tfrac{1}{N!}\sum_{J,J^{\prime}\in{\mathcal{K}}_{N}({\mathcal{L}})}\sum_{\chi_{J},\chi_{J^{\prime}}}$
$\displaystyle
c_{J,J^{\prime}}(\chi_{J},\chi_{J^{\prime}})[a^{\ast}_{J}(\chi_{J})a_{J^{\prime}}(\chi_{J^{\prime}})]_{N}$
(4.23)
$\displaystyle\quad\mapsto\tfrac{1}{N!}\sum_{J,J^{\prime}\in{\mathcal{K}}_{N}({\mathcal{L}})}\sum_{\chi_{J},\chi_{J^{\prime}}}c_{J,J^{\prime}}(\chi_{J},\chi_{J^{\prime}})\,|J,\chi_{J}\rangle\langle
J^{\prime},\chi_{J^{\prime}}|$

supplies the isomorphism in Eq. (4.22). Indeed, for two monomials,

$\displaystyle\big{[}a^{\ast}_{J}(\chi_{J})a_{J^{\prime}}(\chi_{J^{\prime}})\big{]}_{N}$
$\displaystyle\,[a^{\ast}_{\tilde{J}}(\chi_{\tilde{J}})a_{\tilde{J}^{\prime}}(\chi_{\tilde{J}^{\prime}})\big{]}_{N}$
$\displaystyle\qquad\qquad=(-1)^{\chi_{J^{\prime}}^{-1}\circ\chi_{\tilde{J}}}\delta_{J^{\prime},\tilde{J}}[a^{\ast}_{J}(\chi_{J})a_{\tilde{J}^{\prime}}(\chi_{\tilde{J}^{\prime}})]_{N},$

which follows from the same type of calculations as in the proof of
Proposition 4.26. Using the same arguments as in the proof of Proposition
4.20, we conclude that

$[A]_{N}\,[\tilde{A}]_{N}=\sum_{J,J^{\prime}\in{\mathcal{K}}_{N}({\mathcal{L}})}\tfrac{1}{N!}\sum_{\chi_{J},\chi_{J^{\prime}}}c^{\prime}_{J,J^{\prime}}(\chi_{J},\chi_{J^{\prime}})[a^{\ast}_{J}(\chi_{J})a_{J^{\prime}}(\chi_{J^{\prime}}]_{N},$

where

$\displaystyle
c^{\prime}_{J,J^{\prime}}(\chi_{J},\chi_{J^{\prime}})=\sum_{K\in{\mathcal{K}}_{N}({\mathcal{L}})}\sum_{\chi_{K}}c_{J,K}(\chi_{J},\chi_{K})\tilde{c}_{K,J^{\prime}}(\chi_{K},\chi_{J^{\prime}}).$

For the second statement, we recall that $\pi_{\eta}^{N}$ sends the entire
${\rm GI}_{N+1}({\mathcal{L}})$ to zero. Hence, $\pi_{\eta}^{N}$ descends on
the quotient space and Proposition 4.24(ii) shows that this descent coincides
with the map (4.23).∎

##  5\. Interacting Fermions: Dynamics

This section formalizes the dynamics of the gauge invariant local physical
observables. Our main goal here is to identify a large enough set of
derivations that can serve as the core of an algebra associated with the
dynamics of the fermions. The starting point is the class of derivations with
finite interaction range, which are defined over and return values in a fixed
dense subalgebra ${\mathcal{D}}({\mathcal{L}})\subset{\rm
CAR}({\mathcal{L}})$. These derivations can be composed with each other, hence
they generate a subalgebra of ${\rm End}({\mathcal{D}}({\mathcal{L}}))$.
Subsequently, we explore the dependence of the derivations on the pattern
${\mathcal{L}}$ and the constraints imposed by the (assumed) Galilean
invariance. For guidance, we introduce a large class of Hamiltonians
satisfying these constraints, inspired from the physics literature and
generated from many-body potentials. In the process, we demonstrate that the
lattice deformations and the fermion permutations cannot be separated. This
leads us to the construction of the many-body covers of the space of Delone
sets, which supply the natural and rightful domain for the Hamiltonian
coefficients. It also enables us to give a precise formulation of a core
$\ast$-algebra of derivations that can be actually mapped from real
experiments. We show that this algebra admits representations by uniformly
bounded operators on the Fock sectors with finite number of particles.

###  5.1. Hamiltonians with finite interaction range

We consider a strongly continuous dynamics of the local physical observables

$\bm{\alpha}:{\mathbb{R}}\to{\rm Aut}\big{(}{\rm CAR}({\mathcal{L}})\big{)}$

and denote by $\delta_{\bm{\alpha}}$ the generator of this dynamics. In the
majority of the studied physical systems, the domain of $\delta_{\bm{\alpha}}$
is

${\rm
Dom}(\delta_{\bm{\alpha}})={\mathcal{D}}({\mathcal{L}}):=\mathop{\cup}_{k}{\rm
CAR}({\mathcal{L}}_{k}),$

where $\\{{\mathcal{L}}_{k}\\}$ is the net of finite lattices used in section
4.1 to define ${\rm CAR}({\mathcal{L}})$. In the above conditions,
$\delta_{\bm{\alpha}}$ is an inner-limit derivation [17, p. 26] and,
throughout, we will restrict to these cases. We recall that we are seeking a
core for the algebra of generators, which subsequently can be closed in many
different ways (see section 6.1). It is natural to start from the inner-limit
derivations because the laboratory reality is that a finite team of
experimenters can only map this type of generators in a finite amount of time.

######  Remark 5.1.

Note that ${\mathcal{D}}({\mathcal{L}})$ is not just a linear space but a
dense subalgebra of ${\rm CAR}({\mathcal{L}})$, closed under the
$\ast$-operation. Obviously, ${\mathcal{D}}({\mathcal{L}})$ is not the whole
${\rm CAR}({\mathcal{L}})$ and, in general, $\delta_{\bm{\alpha}}$’s are not
uniformly bounded over this domain. This is another fundamental qualitative
difference between interacting fermion systems and the systems studied in
section 3, where the generators were bounded. $\Diamond$

Derivations are linear maps over ${\mathcal{D}}({\mathcal{L}})$ and a
derivation that leaves ${\mathcal{D}}({\mathcal{L}})$ invariant is an element
of the algebra ${\rm End}\big{(}{\mathcal{D}}({\mathcal{L}})\big{)}$ of linear
maps over ${\mathcal{D}}({\mathcal{L}})$. A large class of such derivations is
supplied by finite interaction range Hamiltonians:

######  Definition 5.2.

A finite interaction range Hamiltonian is a formal sum

$H_{\mathcal{L}}=\sum_{J,J^{\prime}\in{\mathcal{K}}({\mathcal{L}})}\tfrac{1}{\sqrt{|J|!|J^{\prime}|!}}\sum_{\chi_{J},\chi_{J^{\prime}}}h_{J,J^{\prime}}^{\mathcal{L}}(\chi_{J},\chi_{J^{\prime\prime}})\,a^{\ast}_{J}(\chi_{J})a_{J^{\prime}}(\chi_{J^{\prime}}),$
(5.1)

where the coefficients $h^{\mathcal{L}}_{J,J^{\prime}}$ are bi-equivariant,
uniformly bounded, obey the constraints

$h^{\mathcal{L}}_{J,J^{\prime}}(\chi_{J},\chi_{J^{\prime}})=\overline{h^{\mathcal{L}}_{J^{\prime},J}(\chi_{J^{\prime}},\chi_{J})},$
(5.2)

and they vanish whenever the diameter of $J\cup J^{\prime}$ exceeds a fixed
value ${\rm R}_{\rm i}$, called here the interaction range. Also, the sum in
Eq. (5.5) runs only over pairs of sub-sets with $|J\cup J^{\prime}|$ even.

######  Remark 5.3.

The expression (5.1) is quite involved because we need to keep track of the
essential dependencies. Of course, if the lattice is fixed and a global
orientation is chosen, then the notation can be simplified, but this is not at
all the case here. Still, the notation will be simplified after we introduce
the many-body covers of the space of Delone sets (see Remark 5.27). $\Diamond$

######  Remark 5.4.

For reader’s convenience, we recall that the diameter of a subset $\Delta$ of
a metric space $(X,{\rm d})$ is the real number defined by

$d_{\Delta}:=\sup\big{\\{}{\rm d}(x,y),\,x,y\in\Delta\big{\\}}.$

In Definition 5.2, the metric space is ${\mathbb{R}}^{d}$ with its Euclidean
distance. $\Diamond$

######  Remark 5.5.

Note that in Definition 5.2, while the diameter of $J\cup J^{\prime}$ is
forced to be finite, there are no constraints on how far the set $J\cup
J^{\prime}$ can be from the origin. As such, the the formal series (5.1) does
not belong to ${\rm CAR}({\mathcal{L}})$, in general. $\Diamond$

######  Proposition 5.6.

Let $\\{{\mathcal{L}}_{k}\\}$ be the tower of subsets used to define ${\rm
CAR}({\mathcal{L}})$ in section 4.1, and let $H_{{\mathcal{L}}_{k}}$ be the
truncation of Eq. (5.1) to $J,J^{\prime}\in{\mathcal{K}}({\mathcal{L}}_{k})$.
Then

${\rm
ad}_{H_{\mathcal{L}}}(A):=\lim_{k\rightarrow\infty}\imath[A,H_{{\mathcal{L}}_{k}}],\quad
A\in{\mathcal{D}}({\mathcal{L}}),$ (5.3)

is a derivation that leaves ${\mathcal{D}}({\mathcal{L}})$ invariant.
Furthermore,

${\rm ad}_{H_{\mathcal{L}}}(A)^{\ast}={\rm
ad}_{H_{\mathcal{L}}}(A^{\ast}),\quad\forall\
A\in{\mathcal{D}}({\mathcal{L}}),$

hence ${\rm ad}_{H_{\mathcal{L}}}$ is a $\ast$-derivation.

###### Proof.

If $A\in{\mathcal{D}}({\mathcal{L}})$, then necessarily $A\in{\rm
CAR}({\mathcal{L}}_{p})$ for some $p\in{\mathbb{N}}$. Let ${\mathcal{L}}_{k}$
be the smallest subset from the tower which includes all the pairs
$(J,J^{\prime})$ with $d_{J\cup J^{\prime}}\leq{\rm R}_{\rm i}$ and $(J\cup
J^{\prime})\cap{\mathcal{L}}_{p}\neq\emptyset$. Since $|J\cup J^{\prime}|$ is
assumed to be even, we have
$[a^{\ast}_{K}a_{K^{\prime}},a^{\ast}_{J}a_{J^{\prime}}]=0$ for any pair
$(K,K^{\prime})$ with $(K\cup K^{\prime})\cap(J\cup J^{\prime})=\emptyset$. As
a result,

$[A,H_{{\mathcal{L}}_{k^{\prime}}}]=[A,H_{{\mathcal{L}}_{k}}]\in{\rm
CAR}({\mathcal{L}}),$

for all $k^{\prime}>k$, hence the limit in Eq. (5.3) exists and, in fact,

${\rm ad}_{H_{\mathcal{L}}}(A)=\imath[A,H_{{\mathcal{L}}_{k}}]\in{\rm
CAR}({\mathcal{L}}_{k}).$ (5.4)

The above shows that ${\rm ad}_{H_{\mathcal{L}}}$ is well defined over
${\mathcal{D}}({\mathcal{L}})$ and takes values in
${\mathcal{D}}({\mathcal{L}})$. Regarding the last statement, we have

${\rm
ad}_{H_{\mathcal{L}}}(A)^{\ast}=-\imath(H_{{\mathcal{L}}_{k}}^{\ast}A^{\ast}-A^{\ast}H_{{\mathcal{L}}_{k}}^{\ast}).$

With the stated assumptions on the Hamiltonian coefficients,
$H_{{\mathcal{L}}_{k}}^{\ast}=H_{{\mathcal{L}}_{k}}$, hence,

${\rm ad}_{H_{\mathcal{L}}}(A)^{\ast}={\rm
ad}_{H_{{\mathcal{L}}_{k}}}(A)^{\ast}={\rm
ad}_{H_{{\mathcal{L}}_{k}}}(A^{\ast})={\rm ad}_{H_{\mathcal{L}}}(A^{\ast})$

and the statement follows.∎

######  Definition 5.7.

A gauge invariant Hamiltonian, abbreviated as GI-Hamiltonian, is a finite-
range Hamiltonian as in Definition 5.2 with the first sum constrained on sub-
sets of equal cardinality. Hence, the formal expression of such a Hamiltonian
can be organized as

$H_{\mathcal{L}}=\sum_{n\in{\mathbb{N}}^{\times}}\tfrac{1}{n!}\sum_{J,J^{\prime}\in{\mathcal{K}}_{n}({\mathcal{L}})}\sum_{\chi_{J},\chi_{J^{\prime}}}h_{J,J^{\prime}}^{\mathcal{L}}(\chi_{J},\chi_{J^{\prime}})\,a^{\ast}_{J}(\chi_{J})a_{J^{\prime}}(\chi_{J^{\prime}}).$
(5.5)

######  Remark 5.8.

The derivations ${\rm ad}_{H_{\mathcal{L}}}$ corresponding to GI-Hamiltonians
commute with the gauge transformations on
${\mathcal{D}}({\mathcal{L}})\subset{\rm CAR}({\mathcal{L}})$. Our analysis
will be restricted from now on to GI-Hamiltonians. Note that, in this case,
$|J\cup J^{\prime}|$ is automatically an even number. $\Diamond$

We recall the unique faithful tracial state ${\mathcal{T}}$ of the ${\rm CAR}$
algebra. Since ${\rm ad}_{H_{\mathcal{L}}}$ are almost-inner, it follows
automatically that ${\mathcal{T}}\circ{\rm ad}_{H_{\mathcal{L}}}=0$ on
${\mathcal{D}}({\mathcal{L}})$. Then, according to Corollary 1.5.6 in [17],
the derivation ${\rm ad}_{H_{\mathcal{L}}}$ is closable. Furthemore,
${\mathcal{D}}({\mathcal{L}})$ is actually a set of analytic elements for
${\rm ad}_{H_{\mathcal{L}}}$, hence ${\rm ad}_{H_{\mathcal{L}}}$ is a pre-
generator of a 1-parameter group $\bm{\alpha}_{t}$ of $\ast$-automorphisms.
Clearly, ${\mathcal{T}}\circ\bm{\alpha}_{t}={\mathcal{T}}$. The GI-
Hamiltonians also have an intrinsic relation with the vacuum state:

######  Proposition 5.9.

We have $\eta\circ{\rm ad}_{H_{\mathcal{L}}}=0$ on
${\mathcal{D}}({\mathcal{L}})$, for any $H_{\mathcal{L}}$ as in Definition
5.7.

###### Proof.

We will use the notation from the proof of Proposition 5.6. If $A\in{\rm
CAR}({\mathcal{L}}_{p})$ for some $p\in{\mathbb{N}}$, then ${\rm
ad}_{H_{\mathcal{L}}}(A)=\imath[A,H_{{\mathcal{L}}_{k}}]$ for some $k\geq p$.
When evaluating $\eta$ on this commutator, the terms of the symmetric
presentation of $A$ that are not gauge invariant can be ignored. We can then
assume that $A$ is gauge invariant. Now, $H_{{\mathcal{L}}_{k}}$ belongs to
the ideal ${\rm GI_{1}}({\mathcal{L}})$, hence
$\imath[A,H_{{\mathcal{L}}_{k}}]$ also belongs to this ideal. As such, its
coefficient $c_{\emptyset,\emptyset}$ is null and the statement follows.∎

All the above are well established facts [18] and the inner-limit derivations
of the ${\rm CAR}$-algebra have been completely characterized in [29] using
its embedding into the Cuntz algebra. Our interest, however, goes well beyond
the individual derivations, as already stressed on several occasions.

###  5.2. Galilean invariant theories

As in section 3.2, let us imagine an experimenter sitting at the origin of the
physical space ${\mathbb{R}}^{d}$ and studying a system of fermions over the
lattice ${\mathcal{L}}$. From the dynamics of local observables, the
experimenter maps the Hamiltonian $H_{\mathcal{L}}$, which is equivalent to
the mapping of the equivariant coefficients $h_{J,J^{\prime}}^{\mathcal{L}}$.
The same experimenter then deforms the lattice, without changing the nature of
the fermions or resonators, and maps again the Hamiltonian. After repeating
this program for many Delone sets, the experimenter establishes a map

${\rm Del}_{(r,R)}({\mathbb{R}}^{d})\ni{\mathcal{L}}\mapsto H_{\mathcal{L}},$
(5.6)

which is entirely determined by the nature of the fermions. The notation
$H_{\mathcal{L}}$ can be interpreted as the evaluation of a global Hamiltonian
$H$ at ${\mathcal{L}}$ and this is the point of view we will adopt from now
on. Throughout, our working assumption is that $H_{\mathcal{L}}$’s in Eq. 5.6
have finite interaction ranges.

The formalism faces two immediate challenges. The first one is how to properly
define a continuity property for the maps in Eq. 5.6. The second one is
understanding the constraints imposed by the assumed Galilean invariance of
the theory. The latter is investigated below, while the first challenge is
addressed in the following subsections. For now, we will adopt the view that
the map (5.6) has been generated by the experiment. Now, during the lengthy
experimental process we just described, two lattices ${\mathcal{L}}$ and
${\mathcal{L}}^{\prime}$ may happen to enter the relation
${\mathcal{L}}^{\prime}={\mathcal{L}}-x$ for some $x\in{\mathbb{R}}^{d}$.
While the experimenter is pinned at the origin of the physical space at all
times, just for this situation, we can imagine the experimenter and the
lattice ${\mathcal{L}}^{\prime}$ being rigidly shifted until the experimenter
sits at position $x$. Then the experimenter is dealing with the same lattice
${\mathcal{L}}$ but from a difference location and, in the absence of
background fields, Galilean invariance of the physical processes involved in
the resonator couplings assures us that, up to a proper relabeling, the new
experiments return the same coefficients. Therefore, the Hamiltonian
coefficients must be subject to the following relations

$h_{J-x,J^{\prime}-x}^{{\mathcal{L}}-x}\big{(}\mathfrak{t}_{x}\circ\chi_{J},\mathfrak{t}_{x}\circ\chi_{J^{\prime}}\big{)}=h_{J,J^{\prime}}^{\mathcal{L}}\big{(}\chi_{J},\chi_{J^{\prime}}\big{)},$
(5.7)

for all ${\mathcal{L}}\in{\rm Del}_{(r,R)}({\mathbb{R}}^{d})$ and
$J,J^{\prime}\in{\mathcal{K}}({\mathcal{L}})$. These relations can be
expressed more concisely, as already explained in section 1 (see Eq. (1.2)).
Let us specify that Remark 3.1 applies here as well.

######  Remark 5.10.

Let us acknowledge that, among other things, Eq. (5.7) implies

$h_{J,J^{\prime}}^{\mathcal{L}}\big{(}\chi_{J},\chi_{J^{\prime}}\big{)}=h_{J-x_{1},J^{\prime}-x_{1}}^{{\mathcal{L}}-x_{1}}\big{(}\mathfrak{t}_{x_{1}}\circ\chi_{J},\mathfrak{t}_{x_{1}}\circ\chi_{J^{\prime}}\big{)},\quad
x_{1}=\chi_{J}(1).$ (5.8)

As one can see, the experimenter can archive the entire map
${\mathcal{L}}\mapsto H_{\mathcal{L}}$ by measuring just the coefficients