1. Introduction
It is a well-known fact that the spectrum of Laplacians or, more generally, Schrödinger operators with periodic potentials, on Abelian coverings, have band structure. These properties of the Laplacians are discussed, e.g., in [
1,
2] and the references therein. The spectrum consists of a continuous part (which is the union of intervals or spectral bands separated by gaps) and a set of eigenvalues with infinite multiplicity. The spectrum is described in terms of a so-called Floquet (or Bloch) parameter. This parameter is the dual of the Abelian group acting on the structure. If two consecutive spectral bands of a bounded self-adjoint operator
T do not overlap, then we say that the spectrum has a spectral gap, i.e., a maximal nonempty interval
that does not intersect the spectrum of the operator. The study of the spectral bands/gaps is a quite natural situation in several fields of mathematics and physics. In solid-state physics, where—for example in semiconductors or its optical counterparts, photonic crystals—the operators modeling the dynamics of particles have some forbidden energy regions (see, e.g., [
3,
4]). In band-gap engineering, a process to control de band/gap of some materials, for semiconductors is controlled for example with the composition of alloys [
5], and for the nanoribbons with temperature [
6], etc. Depending on the type of the periodic structure involved, spectral gaps may be produced by deformation of the geometry (cf., [
7,
8,
9]) or by a suitable periodic decoration of the metric or the discrete covering graph (see, e.g., [
10,
11,
12,
13,
14] and ([
15], Section 4)).
The study of energy-gaps has been widely studied. The gaps in nanoribbons as a function of the width can be found in [
16]. The gaps in the armchair structure can appear because quantum confinement and for the zigzag structure can appear because of an edge magnetization [
17].
In this article, we study the spectrum of discrete magnetic Laplacians (DMLs for short) on infinite discrete coverings graphs
where
is an (Abelian) lattice group acting freely and transitively on
(also the graph
is called as
-periodic graph with finite quotient
). We will present our analysis for graphs with arbitrary weights
m on vertices and arcs although the graphs presented in the examples of the last section will initially have standard weights which are more usual in the context of mathematical physics. Also, we consider a periodic magnetic potential
on the arcs of the covering graph
modeling a magnetic field acting on the graph.
We denote a weighted graph as , and a magnetic weighted graph (MW-graph for short) is a weighted graph together with a magnetic potential acting on its arcs. Any MW-graph with magnetic potential has canonically associated a DML denoted as . We say that with magnetic potential is a -periodic MW-graph if is a -covering and and are periodic with respect to the group action.
In this article, we generalize the geometric condition obtained in ([
18], Theorem 4.4) for
to non-trivial periodic magnetic potentials. In particular, if
is a
-periodic MW-graph with magnetic potential
, we will give in Theorem 3 a simple geometric condition on the quotient graph
that guarantees the existence of non-trivial spectral gaps on the spectrum of the discrete magnetic Laplacian
. To show the existence of spectral gaps, we develop a purely discrete spectral localization technique based on the virtualization of arcs and vertices on quotient
. These operations produce new graphs with, in general, different weights that allow localizing the eigenvalues of the original Laplacian inside certain intervals. We call this procedure discrete bracketing, and we refer to [
18] for additional motivation and proofs.
One of the new aspects of the present article is the generalization of results in [
18] to include a periodic magnetic field
on the covering graph
. In this sense,
may be used as a control parameter for the system that serves to modify the size and the regions where the spectral gaps are localized. We apply our techniques to the graphs modeling the polyacetylene polymer as well as to graphene nanoribbons. The nanoribbons are
-periodic strips of graphene either with an armchair or zig-zag boundaries (see, e.g., Figure 5). The graphic in
Figure 1 corresponds to an armchair nanoribbon with a width 3. It can be seen how a periodic magnetic potential with constant value
on each cycle (and plotted on the horizontal axis) affects the spectral bands (gray vertical intervals that appear as the intersection of the region with a line
) and the spectral gaps (white vertical intervals). We refer to
Section 5.2 for additional details of the construction.
In the case of the polyacetylene polymer, we find a spectral gap that is stable under perturbation of the (constant) magnetic field. Moreover, if the value of the magnetic field is , then the spectrum of the DML degenerates to four eigenvalues of infinite multiplicity. This discrete model suggests that a varying uniform magnetic field may drastically change the conductance of a material arranged as a periodic planar graph.
The article is structured in five sections as follows. In
Section 2, we collect the basic definitions and results on discrete weighted multigraphs (graphs which may have loops and multiple arcs). We consider discrete magnetic potentials on the arcs and define the discrete magnetic Laplacian on the graph, which will be the central operator in this work. In
Section 3, we present a spectral relation between finite MW-graphs based on an order relation between the eigenvalues of the corresponding DMLs. Moreover, we will present the basic arc and vertex virtualization procedure that will allow one to localize the spectrum of the DML on the infinite covering graph. In
Section 4, we extend the discrete Floquet theory considered in ([
18], Section 5) to the case of covering graphs with periodic magnetic potentials. In
Section 5, we apply the spectral localization results developed before in the example of
-periodic graphs modeling the polyacetylene polymer as well as graphene nanoribbons in the presence of a constant magnetic field.
2. Weighted Graphs and Discrete Magnetic Laplacians
In this section, we introduce the basic definitions and results concerning MW-graphs and also define discrete magnetic Laplacians. For further motivation and results, we refer to [
12,
18,
19] and references cited therein.
We denote by a (discrete) directed multigraph which in the following we call simply a graph; here is the set of vertices and the set of arcs. The orientation map is given by and is the pair of the initial and terminal vertices. Graphs are allowed to have multiple arcs, i.e., arcs with or as well as loops, i.e., arcs with . Moreover, we define
With this notation, the degree of a vertex is and a loop increases the degree by 2.
Given subsets , we define
Moreover, we put and .
To simplify the notation, we write instead of etc. Note that loops are not counted double in , in particular, is the set of loops based at the vertex . The Betti number of a finite graph is defined as
To study the virtualization processes of vertices, arcs and the structure of covering graphs, we will need to introduce the following substructures of a graph.
Definition 1. Let be a graph and denote by a triple such that , and .
- (a)
If , we say that is a partial subgraph in . We callthe set of connecting arcs of the partial subgraph in . - (b)
If , then is a subgraph of
Note that, in general, a partial subgraph
is not a graph as defined above, since we may have arcs
with
. We do exclude though the case that
and . The arcs not mapped into
under
are precisely the connecting arcs of
in
. Partial subgraphs appear naturally as fundamental domains of covering graphs (cf.,
Section 4) (Note that we use the name partial subgraph in a different sense as in usual combinatorics literature).
Let be a graph; a weight on is a pair of functions denoted by a unique symbol m on the vertices and arcs and such that is the weight at the vertex v and is the weight at . We call a weighted graph. It is natural to interpret m as a positive measure and consider for any . The relative weight is defined as
In order to work with bounded discrete magnetic Laplacians, we will assume that the relative weight is uniformly bounded, i.e.,
The most important and intrinsic examples of weights are
Standard weight:, , and , , so that .
Combinatorial weight:, , hence and .
Giving a weighted graph
, we associate the following two natural Hilbert spaces which we interpret as 0-forms and 1-forms, respectively.
with corresponding inner products
Let be a graph; a magnetic potential acting on is a -valued function on the arcs as follows, We denote the set of all vector potentials on just by . We say that two magnetic potentials and are cohomologous, and denote this as , if there is with
Given a , we say that a magnetic potential has support in if for all . We call the class of weighted graphs with magnetic potential MW-graphs for short.
It can be shown that any magnetic potential on a finite graph can be supported in many arcs. For example, if is a cycle, any magnetic potential is cohomologous to a magnetic potential supported in only one arc. Moreover, if is a tree, any magnetic potential acting on is cohomologous to 0.
The
twisted (discrete) derivative is the following linear operator mapping 0-forms into 1-forms:
Now, we present the following geometrical definition of Laplacian with magnetic field as a generalization of the discrete Laplace-Beltrami operator.
Definition 2. Let be a weighted graph with a vector potential. The discrete magnetic Laplacian (DML for short) is defined by , i.e., bywhere is the oriented evaluation and is the vertex opposite to v along the arc e, i.e., If we need to stress the dependence of the operator of the weighted graph , we will denote the DML as .
From this definition, it follows immediately that the DML is a bounded, positive and self-adjoint operator. Its spectrum satisfies and, in contrast to the usual Laplacian without magnetic potential, the DML depends on the orientation of the graph. If , then and are unitary equivalent; in particular, . Moreover, if then where denotes the usual discrete Laplacian (with vector potential 0). For example, if and is a tree, then for any magnetic potential .
3. Spectral Ordering on Finite Graphs and Magnetic Spectral Gaps
In this section, we will introduce a spectral ordering relation ≼, which is invariant under unitary equivalence of the corresponding operators. Moreover, we will introduce two operations on the graphs (virtualization of arcs and vertices) that will be used later to develop a spectral localization (bracketing) of DML on finite graphs. This technique will finally be applied to discuss the existence of spectral gaps for magnetic Laplacians on covering graphs. We refer to [
11,
12,
18] for additional motivation and examples. For proofs of the results stated in this section see ([
18], Sections 3 and 4).
Let be a weighted graph. Throughout this section, we will assume that . We denote the spectrum of the DML by , where we will write the eigenvalues in ascending order and repeated according to their multiplicities, i.e.,
Definition 3. Let and be two finite MW-graphs of order and , respectively, and magnetic potential . Consider the eigenvalues of the DMLs written in ascending order and repeated according to their multiplicities.
- (a)
We say that is spectrally smaller than (denoted by ), ifwhere we put for (the maximal possible eigenvalue). - (b)
Consider as above with . We define the associated k-th bracketing interval byfor .
Given an MW-graph, we introduce two elementary operations that consist of virtualizing arcs and vertices. The first one will lead to a spectrally smaller graph.
Definition 4 (virtualizing arcs). Let be a weighted graph with magnetic potential α and . We denote by the weighted subgraph with magnetic potential defined as follows:
- (a)
with for all ;
- (b)
with and for all ;
- (c)
, .
We call the weighted subgraph obtained from by virtualizing the arcs . We will sometimes denote the weighted graph simply by and we write the corresponding discrete magnetic Laplacian as .
The second elementary operation on the graph will lead now to a spectrally larger graph.
Definition 5 (virtualizing vertices). Let be a weighted graph with magnetic potential α and . We denote by the weighted partial subgraph with magnetic potential defined as follows:
- (a)
with for all ;
- (b)
with for all ;
- (c)
, .
We call the weighted partial subgraph obtained from by virtualizing the vertices
. We will denote it simply by . The corresponding discrete magnetic Laplacian is defined bywith It can be shown that the operator is the compression of onto a -subspace.
The previous operations of arc and vertex virtualization will be used to localize the spectrum of intermediate DMLs. Before summarizing the technique in the next theorem, we need to introduce the following notion of vertex neighborhood of a family of arcs.
Definition 6. Let be a graph and . We say that a vertex subset is in the neighborhood of if , i.e., if or for all .
Later on, will be the set of connecting arcs of a covering graph, and we will choose to be as small as possible to guarantee the existence of spectral gaps (this set is in general not unique).
Theorem 1. Let be a finite MW-graph with magnetic potential α and . Then, for any subset of vertices in a neighborhood of we havewhere with and with . In particular, we have the spectral localizing inclusion By construction, it is clear that the bracketing
depends on the magnetic potential
. In
Section 5, we show in some examples how the localization intervals
change under the variation of the magnetic potential (see, e.g., Figure 3). However, if the magnetic potential
has support on the virtualized arcs
, then
J will not depend on
because
.
Next, we make precise some notions concerning spectral gaps that will be needed when we study covering graphs. Recall that , where denotes the supremum of the relative weight, (cf., Equation (3)).
Definition 7. Let be a weighted graph.
- (a)
The spectral gaps set of is defined bywhere denotes the resolvent set of the operator . - (b)
The magnetic spectral gaps set of is defined by
where the union is taken over all the magnetic potential α acting on .
The following elementary properties follow directly from the definition: . In particular, if , then or, equivalently, if , then . Moreover, if is a tree, then , as all DMLs are unitary equivalent with the usual Laplacian .
Up to now, we have seen that arc/vertex virtualization will produce graphs that allows localizing the spectrum of the DML of any intermediate MW-graph satisfying
4. Periodic Graphs and Spectral Gaps
In this section, we will study the spectrum of the DML of an infinite covering graph with periodic magnetic potential in terms of its Floquet decomposition. In Proposition 2 the Floquet parameter of the covering graph is identified with a suitable set of magnetic potentials
on the quotient (cf., Definition 10). This approach generalizes results in ([
18], Section 5) to include Laplacians on the infinite covering graph with a periodic magnetic potential
. Finally, in Theorem 2, we state a bracketing technique to localize the spectrum.
4.1. Periodic Graphs and Fundamental Domains
Let be an (Abelian) lattice group and consider the -covering (or -periodic) graph
We assume that
acts freely and transitively on the connected graph
with finite quotient
(see also ([
20], Chapters 5 and 6) or [
18,
21]). This action (which we write multiplicatively) is orientation preserving, i.e.,
acts both on
and
such that
In particular, we have , , .
In addition, we will study weighted covering graphs with a periodic weight and periodic magnetic potential , i.e., we consider an MW-graph such that for any we have
Note that, by definition, the standard or combinatorial weights on a covering graph satisfy the invariance conditions on the weights. A -covering weighted graph naturally induces a weight m and a magnetic potential on the quotient graph , given by and .
We define next some useful notions in relation to covering graphs (see, e.g., [
18], Section 5 as well as ([
22], Sections 1.2 and 1.3) and [
8]).
Definition 8. Let be a Γ-covering graph.
- (a)
a vertex, respectively arc fundamental domain on a Γ
-covering graph is given by two subsets and satisfyingwith (i.e., an arc in has at least one endpoint in ). We often simply write D for a fundamental domain, where D stands either for or . - (b)
a (graph) fundamental domain of a covering graph is a partial subgraph (cf., Definition 1)where and are vertex and arc fundamental domains, respectively. We callthe set of connecting arcs of the fundamental domain in .
Remark 1. - (a)
Fixing a fundamental domain on the covering graph and the group Γ will be used to give coordinates (to the arcs and vertices) on the covering graph .
In fact, given a specific fundamental domain in a Γ
-covering graph , we can write any uniquely as for a unique pair . This observation follows from the fact that the action is free and transitive. We call the Γ
-coordinate of v (with respect to the fundamental domain ). Similarly, we can define the coordinates for the arcs: any can be written as for a unique pair . In particular, we have - (b)
Once we have chosen a fundamental domain , we can embed into the quotient of the covering bywhere and denote the Γ
-orbits of v and e, respectively. By definition of a fundamental domain, these maps are bijective. Moreover, if in , then also in , i.e., the embedding is a (partial) graph homomorphism.
Definition 9. Let be a Γ
-covering graph with fundamental graph . We define the index
of an arc as In particular, we have , and iff , i.e., the index is only non-trivial on the (translates of the) connecting arcs. Moreover, the set of indices and its inverses generate the group .
Since the index fulfils for all by (a) in Remark 1, we can extend the definition to the quotient by setting for all . We denote also .
4.2. Discrete Floquet Theory
Let be a weighted -covering graph and fundamental domain with corresponding weights inherited from . In this context one has the natural Hilbert space identifications
Floquet theory uses a partial Fourier transformation on the Abelian group that can be understood as putting coordinates on the periodic structure and allows to decompose the corresponding operators as direct integrals. Concretely, we consider
for
and where
denotes the character group of
. We adapt to the discrete context of graphs with periodic magnetic potential
the main results concerning Floquet theory needed later. See, e.g., ([
7], Section 3) or [
22] for details, additional motivation and references.
For any character consider the space of equivariant functions on vertices and arcs
These spaces have the natural inner product defined on the fundamental domains
and
:
The definition of the inner product is independent of the choice of the fundamental domain (due to the equivariance). We extend the standard decomposition to the case of the DML with periodic magnetic potential (see, for example, [
22,
23]).
Proposition 1. Let be a covering weighted graph where and is a periodic magnetic potential. Then there are unitary transformationssuch thatwhere equivariant Laplacian (fiber operators) are defined as . Proof. Consider the twisted derivative
specified in Equation (
4) and the equivariant twisted derivative on the fiber spaces defined by
It is straightforward to check that if
, then
and that
. Moreover, we will show that the unitary transformations
intertwine these two first order operators, i.e.,
In fact, this is a consequence of the following computation that uses the invariance of the magnetic potential. For any
and
This shows that
hence,
. □
4.3. Vector Potential as a Floquet Parameter
The following result shows that in the case of Abelian groups
, we can interpret the magnetic potential
on the quotient graph partially as a Floquet parameter for the covering graph
(see (b) in Remark 1). Moreover, recalling the definition of coordinate giving in (a) in Remark 1) we can define the following unitary maps (see also [
24] for a similar definition in the context of manifolds):
It is straightforward to check that and are well defined and unitary.
Definition 10. Let be a covering graph with periodic weights , periodic magnetic potential and fundamental domain . We denote by α a magnetic potential acting on the quotient . We say that α has the lifting property if there exists such that: We denote the set of all the magnetic potentials with the lifting property as .
Proposition 2. Consider a Γ
-covering graph with periodic magnetic potential , where , and is a fundamental domain. Then Proof. By Proposition 1, it is enough to show
To show the inclusion “⊂” consider a character
and define a magnetic potential on
as follows
On the other hand, we have
Therefore, the intertwining equation
holds if
or, equivalently, if
But this equation is true by definition of the magnetic potential on
given in Equation (
10). Finally, since
and
, then it is clear that these Laplacians are unitary equivalent.
To show the reverse inclusion “⊃” let
and
is such that
is a basis of the group
. Then define
and we can extend
to all
multiplicatively, so that
. As before, we can show then
and the proof is concluded. □
4.4. Spectral Localization for the DML on a Covering Graph
We apply now the technique stated in Theorem 1 to covering graphs.
Theorem 2. Let be a Γ
-covering graph and a periodic magnetic potential. Consider a fundamental domain and with magnetic potential β, where . The functions m and β are induced by and respectively. Letbe the image of the connectivity arcs on the quotient and in the neighborhood of . Define bythe corresponding arc and vertex virtualized graphs, respectively. Thenwhere the eigenvalues of and are written in ascending order and repeated according to their multiplicities. Proof. Now, by the bracketing technique of Theorem 1, we have for any potential with the lifting property
(cf., Definition 6):
Therefore, by Equation (
7)
since
has the lifting property, Equation (
8) implies that there exists
such that:
But for all
the index is trivial, i.e.,
(see Remark 1). Thus by
-periodicity we obtain that
for all arcs
. Since
and
are magnetic potentials acting on
, and
then then
. Similarly, for
with
in the neighborhood of
, we have that
. We obtain finally
Note that the last union does not depend anymore of and this fact concludes the proof. □
Note that the bracketing intervals
depends on the fundamental domain
. a right choice is one where the set of connecting arcs is as small as possible, providing high contrast between the interior of the fundamental domain and its boundary. In this case, we have a good chance that the localizing intervals
do not cover the full interval
. This choice is a discrete geometrical version of a “thin–thick” decomposition as described in [
12], where a fundamental domain of the metric and discrete graph has only a few connections to its complement.
The next theorem gives a simple geometric condition on an MW-graph
for the existence of gaps in the spectrum of the DML on the
-covering graph. We will specify which arcs and vertices should be virtualized in
to guarantee the existence of spectral gaps. This result generalizes the Theorem 4.4 in [
18].
Theorem 3. Let be a Γ-covering graph with a Γ-periodic magnetic potential . Denote by the quotient graph with induced magnetic potential β and induced weights m, respectively.
The spectrum of the DML has spectral gaps, i.e., , if the following condition holds: there exists a vertex and a fundamental domain such that the connecting arcs contain no loops, andwhere is the relative weight at and with . Proof. Consider the following arc and vertex virtualized weighted graphs:
Then by Theorem 2, we obtain
To prove that
, it is enough to show that the measure of
is positive and it can be estimated from below by:
Therefore it is enough to calculate
and
(see [
18], Proposition 3.3).
Step 1: Trace of . We define
where
. Recall that
,
; the weights on
and
coincide with the corresponding weights on
. The relative weights of
are
where
The trace of
is now
Step 2: Trace of . Let
, then the trace of
is given by
Combining Equations (
13)–(
15) we obtain
as defined in Equation (
12). This shows that if
, then the spectrum of the DML is not the full interval. □
Remark 2. - (a)
If the graph has the standard weights, the condition becomes:where denote the cardinality of the set . - (b)
If we have the combinatorial weights, the condition becomes simply: