Qualitative analysis of magnetic waveguides for two-dimensional Dirac fermions
Marie Fialová Vít Jakubský Matěj Tušek
cond-mat.mes-hall hep-th math-ph math.MP quant-ph





 Qualitative analysis of magnetic waveguides for two-dimensional Dirac fermions
    Marie Fialová^, Vít Jakubský^†, Matěj Tušek^

^†Department of Theoretical Physics,
Nuclear Physics Institute,
25068 Řež, Czech Republic

^Department of Mathematics, Czech Technical University in Prague, Trojanova 13, 120 00 Prague, Czech Republic 

E-mail: fialoma8@fjfi.cvut.cz, jakub@ujf.cas.cz,  tusekmat@fjfi.cvut.cz

 
    26.1. 2018
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We focus on the confinement of two-dimensional Dirac fermions within the waveguides created by realistic magnetic fields. Understanding of their band structure is of our main concern. We provide easily applicable criteria, mostly depending only on the asymptotic behavior of the magnetic field, that can guarantee existence or absence of the energy bands and provide valuable insight into the systems where analytical solution is impossible. The general results are employed in specific systems where the waveguide is created by the magnetic field of a set of electric wires or magnetized strips. 




§ INTRODUCTION

A great variety of condensed matter systems can host two-dimensional Dirac fermions. The list of the so called Dirac materials <cit.> contains not only graphene, but also other systems where relativistic quasi-particles were either predicted or even confirmed experimentally. Let us mention silicene, germanene, dichalcogenides <cit.>-<cit.>, or artificial crystal lattices synthesized in the lab with the use of optical traps  <cit.>-<cit.> or molecular manipulations <cit.>,  <cit.>. Due to their remarkable characteristics, Dirac materials (and graphene in particular) are hoped to be the building blocks of the post-silicon electronics. This sets a great motivation to understand and control their electronic properties.

Massless Dirac fermions can propagate through electrostatic barriers without being back-scattered. This phenomenon, Klein tunneling, challenges effective control of the quantum transport with the use of the electric field. Therefore, an alternative approach is needed. Cutting the sample into the required form or chemical adsorptions of other atoms (e.g. oxygen, fluoride) is demanding on precision and lacks flexibility. In this respect, control of the Dirac fermions by magnetic or magneto-electric barriers represents a more feasible option <cit.>.


Dynamics of two-dimensional (non-relativistic) electron gas (2DEG) in an inhomogeneous magnetic field was studied already in <cit.>, <cit.>, <cit.>, three-dimensional Dirac fermions in <cit.>. It was found that when otherwise homogeneous magnetic field changes its sign along a straight line, the electrons can propagate along the line with a non-vanishing group velocity. This is in coherence with the classical picture, where electrons propagate along the trajectories that are bending back and forth along the interface. These states are called snake states in the literature. 

Confinement of Dirac fermions in graphene by magnetic fields was proposed in <cit.>, <cit.>. Propagation both across and along the magnetic barriers was studied in great number of papers so that we have no hope to provide complete list of the related references. The wave vector filtering  known for 2DEG <cit.>, where the electrons bouncing on the magnetic barrier are totally reflected for a broad interval of angles, was analyzed for Dirac fermions e.g. in <cit.>, <cit.>. Existence of snake states in graphene was addressed in <cit.>, <cit.>. Effect of many-body interactions was discussed in <cit.>, spin snake states were discussed in <cit.>.
The existence of snake states in graphene was also studied both theoretically and experimentally in the systems  where both magnetic and electric fields  change abruptly along a straight line <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, see also the review article <cit.>.

The systems studied in the literature usually possess translational invariance in one direction. It implies conservation of the (longitudinal) momentum k along the barrier.  The energy spectrum of the two-dimensional system can be reconstructed  from the spectra of effectively one-dimensional subsystems with fixed k. 
In many of the mentioned works, the external fields are represented by rectangular barriers or step functions <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>. As the spectra of their one-dimensional subsystems are purely discrete, the spectrum of the full two-dimensional Hamiltonian consists of energy bands E_n(k) with integer n. The energy bands can coincide with Landau levels for large |k| (the case when the magnetic field is asymptotically constant and of the same sign at infinities) or there can also emerge dispersive states localized at the barrier and moving along it with a non-vanishing group velocity v_n=∂_kE_n(k), <cit.>, <cit.>. The solvable models with step-like vector potential were discussed e.g. in <cit.>, <cit.>. The effectively one-dimensional subsystem has non-vanishing essential spectrum and possibly also the discrete eigenvalues that correspond to the 
states confined at the barrier. The essential spectra build up the typical wedges of allowed energies in the spectrum  of the full Hamiltonian, while the discrete energy levels of the subsystem form the energy bands E_n(k) <cit.>. 



The energy bands E_n(k) can be associated with the existence of wave packets  that do not disperse in transverse direction <cit.>. 
When the group velocity v_n(k) acquires both positive and negative values, the barrier represents a waveguide for bidirectional quantum transport of the wave packets. When v_n(k) is strictly positive or negative, then a unidirectional quantum transport of the dispersionless wave packets along the barrier takes place <cit.>. 

The analysis of solvable models was not restricted to the potentials in the form of step-like functions. The techniques of supersymmetric quantum mechanics were used to study the systems with smooth vector potential <cit.>, <cit.>, electrostatic potential <cit.> or their combination <cit.>. (Quasi-) exactly solvable systems associated with the solutions of Heun equation were discussed in <cit.>.

In the current article, we focus on the qualitative analysis of the quantum transport in the magnetic waveguides. We aim at the systems where explicit solutions of the stationary equation do not exist analytically. Our goal is to provide as detailed as possible information on the spectrum of the considered systems. In particular, we will be interested in the existence and properties of the energy bands that are the hallmark of dispersionless wave packets in the systems with translational invariance. Our analysis is based on the variational principle and asymptotic behavior of the vector potential. We will analyze in detail realistic settings where magnetic field is generated either by a set of electric wires or magnetized strips with different direction of magnetization, posed in the proximity of the two-dimensional Dirac fermions. 

The work is organized as follows. In the next section, we rigorously define the considered model and explore its spectral properties in dependence on the asymptotic behavior of the vector potential. We review existing relevant results and introduce the new tools for spectral analysis in the form of statements together with their proofs. 
In the next section, we focus on the bundles of electric wires with zero net-current. Our motivation is two-fold there. First, the results should contribute to our understanding of the behavior of Dirac fermions in the proximity of electric circuits. Second, we show that the bundles of electric wires can be used as tunable magnetic waveguides, where the characteristics of the waveguide can be altered just by changing the electric current in the wires.
Section <ref> is devoted to the analysis of magnetized strips. We consider the cases of parallel and perpendicular magnetizations, and also discuss the situation where alignment of the vector of magnetization is not precise. The last section is left for discussion and outlook. 





§ THE MODEL

We consider a planar system of Dirac fermions in presence of magnetic field 𝐁 that is invariant with respect to translations along some axis. For definiteness, let it be the y-axis. Therefore, 𝐁 may be viewed as a function of x-variable only. Moreover, since the model is two-dimensional, 𝐁 has non-zero only its transverse z-component, which will be denoted by B. For our analysis, it is convenient to choose the following gauge

    𝐀_phys(x)=(0,A_phys,y(x),0),     A_phys,y(x):=∫_0^x B(s) s+const.

We fix units so that ħ=m_e=c=1. Then the elementary charge is just e=√(α), where α is the fine structure constant which measures the strength of the coupling. 

In the vicinity of the Dirac points, the system under consideration is described by the following Hamiltonian

    Ĥ_𝐀=σ_1(-i∂_x)+σ_2(-i∂_y+A_y(x))+σ_3 M,

where σ_i are the Pauli matrices, M is a non-negative constant and

    A_y(x):=√(α)∫_0^x B(s) s+const.

Remark that, for a concise presentation, we absorbed √(α) into the definition of A_y.  The domain of operator (<ref>) and its self-adjointness in particular will be discussed at the beginning of Section <ref>.

The translational symmetry of the system allows us to understand the system by analyzing one-dimensional subsystems with fixed longitudinal momentum k_y. Indeed, using the partial Fourier transform in y-variable,

    (ℱ_y→ k_yψ)(x,k_y)=1/√(2π)∫_e^-ik_y yψ(x,y)  y,

we infer that H_𝐀 is unitarily equivalent to

    H_𝐀:=σ_1(-i∂_x)+σ_2(k_y+A_y(x))+σ_3 M,

which decomposes into the direct integral,

    H_𝐀=∫^⊕_ H_𝐀[k_y],   H_𝐀[k_y]:=σ_1(-i∂_x)+σ_2(k_y+A_y(x))+σ_3 M,

where H_𝐀[k_y] acts in x-variable only (with k_y fixed).
More concretely, we have

    H_𝐀[k_y]=[                 M   -i(∂_x+k_y+A_y); -i(∂_x-(k_y+A_y))                -M ],

for the so-called fiber of the total operator H_𝐀. Once we find the spectrum of H_𝐀[k_y] for each k_y∈, we can reconstruct the spectrum of H_𝐀, cf. <cit.>. In particular, λ is an eigenvalue of H_𝐀 if and only  if the set of all k_y such that λ is an eigenvalue of H_𝐀[k_y] is not of zero measure.

To find the spectrum of H_𝐀[k_y], we will employ the fact that  the square of H_𝐀[k_y] reduces to  a direct sum of two Schrödinger operators,

    H_𝐀[k_y]^2=[ -∂_x^2+A_y'+(k_y+A_y)^2+M^2                           0;                           0 -∂_x^2-A_y'+(k_y+A_y)^2+M^2 ]=:
                [ h_+[k_y]        0;        0 h_-[k_y] ].

Then one can construct eigenstates of H_𝐀[k_y] in terms of those of h_+[k_y] and 
h_-[k_y], see e.g. <cit.>. For the reader's convenience, we will also briefly discuss the exact relation between the eigenvalues and associated eigenstates of h_±[k_y] and those of H_𝐀[k_y] in Section <ref>.

In this paper, we are primarily interested in the presence and location of discrete eigenvalues, i.e., the isolated eigenvalues of finite multiplicity, in the spectrum of H_𝐀[k_y] in the case when the magnetic field vanishes asymptotically, i.e., 

    √(α)lim_x→±∞B(x)=lim_x→±∞A_y'(x)=0.

It is worth noting that both  h_+[k_y] and h_-[k_y] act like

    h_Iw:= -∂_x^2+(k_y+A_y(x))^2 + V(x),

i.e., like the fiber of a two-dimensional spin-less Schrödinger operator with magnetic and electric fields that are invariant with respect to translations in y-direction. The case with V≡ const. has been studied intensively but only when the limits in (<ref>) are non-zero.  In that case, h_Iw has pure point spectrum, i.e., the spectrum consists of isolated eigenvalues of finite multiplicity (in fact, all of them are simple). See <cit.> for the original study of A. Iwatsuka and <cit.> for an extension to non-constant V's and for an overview of the known results. However, in the present case (<ref>), the spectrum possesses also a continuous part, cf. Theorem <ref>.



§ GENERAL OBSERVATIONS
 
Spectral properties of any operator are to a large extend determined by its domain of definition. Hence, its specification should precede any rigorous spectral analysis. 
Since A_y is continuous and vanishes at ±∞, (H_𝐀[k_y]-H_0[0]) is a bounded perturbation of H_0[0]. Therefore, by the Kato-Rellich theorem <cit.>, H_𝐀[k_y] is selfadjoint on any domain of selfadjointness of H_0[0], which we choose naturally as the first Sobolev space W^1,2(;^2)–roughly speaking the  space of ^2-valued square integrable functions with square integrable derivatives. See <cit.> for a proof of selfadjointness of the latter operator.  Consequently, H_𝐀 defined as a direct integral of H_𝐀[k_y] is also selfadjoint <cit.> and, due to unitarity, the same holds true for Ĥ_𝐀 defined on the partial Fourier preimage of the domain of H_𝐀. Remark that this preimage is just W^1,2(^2;^2).


Our starting point for the spectral analysis of H_𝐀[k_y] will be the main result of <cit.> which we reproduce here not in its full generality but under some special assumptions on the vector potential.


 Let A_y∈ C^1() satisfy  
 
    lim_x→±∞A_y(x)=0

 and (<ref>).  Moreover, let A_y(A_y+2k_y) be either integrable or there exists x_0 such that A_y(A_y+2k_y)≤ 0 for all x<x_0. Then
 
  
  * E∈σ(H_𝐀[k_y])⇔ -E∈σ(H_𝐀[k_y]) 
  and σ_ess=(-∞,-√(k_y^2+M^2)]∪[√(k_y^2+M^2),+∞),
  
  * if either 
  
    ∫_ A_y(x)(A_y(x)+2k_y) x<0

  or there exists a∈ and δ∈{±1} such that 
  
    ∫_-∞^a A_y(x)(A_y(x)+2k_y) x<-2/√(3)√(sup_x∈(a,+∞)(A_y(x)(A_y(x)+2k_y)+δ A'_y(x)))-δ A_y(a)

  then H_𝐀[k_y] has at least one discrete eigenvalue in (-√(k_y^2+M^2),√(k_y^2+M^2)), 
  
  * ± M is never an eigenvalue of H_𝐀[k_y],
  
  * when k_y=0 or if k_y≠ 0 and one of A_y(A_y+2k_y)± A_y' is non-negative then there are no discrete eigenvalues in the spectrum of H_𝐀[k_y].
 



Note that (<ref>) amounts to an appropriate choice of gauge that may or may not exist.
For example, if B is integrable, one can fix the constant in (<ref>) so that lim_x→+∞A_y(x)=0. Then lim_x→-∞A_y(x)=0 if and only if

    ∫_ B(x)  x=0,

which holds true, e.g., for all integrable odd magnetic fields B. 


 
 The condition (<ref>) can be modified for a neighbourhood of +∞ as follows
 
    ∫_a^+∞ A_y(x)(A_y(x)+2k_y) x<-2/√(3)√(sup_x∈(-∞,a)(A_y(x)(A_y(x)+2k_y)+δ A'_y(x)))+δ A_y(a),

 whenever A_y(A_y+2k_y) is either integrable or there exists x_0 such that A_y(A_y+2k_y)≤ 0 for all x>x_0.




Let us stress that from now on we will only deal with magnetic fields that satisfy the general assumptions of Theorem <ref>, i.e.,  (<ref>) and (<ref>). 





 §.§ Basic properties of energy bands
 

Clearly, the square of any eigenvalue of H_𝐀[k_y] is an eigenvalue of H_𝐀[k_y]^2. Conversely, the eigenvalues of H_𝐀[k_y]^2 are given exactly as the union of the eigenvalues of h_±[k_y]. Let us investigate their point spectra in more detail.
It is convenient to rewrite h_±[k_y] in factorized form, h_+[k_y]=pp^†+M^2 and h_-[k_y]=p^† p+M^2 with p:=-i(∂_x+k_y+A_y). As h_±[k_y]≥ M^2, their eigenvalues cannot be smaller than M^2.  Moreover, by direct inspection, one can verify that assuming (<ref>) no non-zero solution f of the eigenvalue problem h_±[k_y]f=M^2f is square integrable. Therefore, M^2  may not be an eigenvalue of h_±[k_y]. The point spectra of the two operators are identical. Indeed, if h_+[k_y]ψ=λψ (with λ>M^2) then h_-[k_y](p^†ψ)=λ p^†ψ, and if h_-[k_y]φ=λφ then h_+[k_y](pφ)=λ pφ.  As h_±[k_y] is a Sturm-Liouville operator, the 
wronskian of two solutions to the eigenvalue problem is constant.  Since these solutions have to belong to the domain of h_±[k_y], which, in particular, contains only functions vanishing at x=±∞ together with their first derivatives, this constant must be zero.  Consequently, one solution is a constant multiple of the other. Hence, all eigenvalues of h_±[k_y] are simple. 

The spinors Ψ_±:=(ψ,(M±√(λ))^-1p^†ψ)^T and Ψ̃_±:=((-M±√(λ))^-1pφ,φ)^T are eigenstates of H_𝐀[k_y] with eigenvalue ±√(λ). These eigenvalues are simple, i.e.,  Ψ_± and Ψ̃_±  may differ only by a multiplication constant. Indeed, as λ is a simple eigenvalue of both  h_±[k_y], we can choose  φ=p^†ψ. Consequently, ψ=(λ-M^2)^-1pφ and Ψ̃_±=(M±√(λ))Ψ_±. 
We conclude that having the point spectrum of h_+[k_y] (or h_-[k_y]) we can completely reconstruct the point spectrum of H_𝐀[k_y]. Moreover, the latter  is simple and localized symmetrically in (-√(k_y^2+M^2),-M)∪(M,√(k_y^2+M^2)).

Due to the minimax principle, the eigenvalues of h_+[k_y] may only accumulate at the bottom of its essential spectrum, which was derived in <cit.> to be σ_ess(h_+[k_y])=[k_y^2+M^2,+∞). Therefore, we can enumerate them in increasing order as E_n[k_y],  n=1,2,… N, where N may be possibly infinity and in general it varies with k_y. Remark that there are no embedded eigenvalues in the essential spectrum because we already know that every eigenvalue is simple. Therefore, every eigenvalue is necessarily discrete. The energy bands of H_𝐀 are then given as the functions k_y↦±√(E_n[k_y]) on some open intervals. Above, we deduced that they may not intersect each other (it would contradict non-degeneracy of the eigenvalues) and cannot pass via the gap ×[-M,M] or cross the curves k_y↦±√(k_y^2+M^2), which enclose the essential spectra of H_𝐀[k_y]. Moreover, since H_𝐀[k_y] depends analytically on k_y (in fact, it 
forms the so-called analytic family of type (A), see <cit.>), the energy bands of H_𝐀 are 
analytic on their domains of definition.

The  Hellmann-Feynman formula says that

    / k_yE_n[k_y]=2∫_(k_y+A_y(x))|ψ_n[k_y](x)|^2 x,

where ψ_n[k_y] stands for the normalized eigenfunction of h_+[k_y] associated with E_n[k_y]. Therefore, for all k_y sufficiently positive,  E^±_n[k_y] is strictly increasing, and for all k_y sufficiently negative, E^±_n[k_y] is strictly decreasing. This implies that the energy bands of H_𝐀 tend either to +∞ or -∞ as |k_y|→∞. In particular, no energy band  may be constant; not even locally, due to analyticity. Consequently, there are never eigenvalues in the spectrum of the total, two-dimensional, operator H_𝐀.

Finally, if, for any k_y>0, there are infinitely many eigenvalues of H_𝐀[k_y] then all respective energy bands are defined on the whole positive half-axis, i.e. they emerge only from k_y=0 and may not disappear in the essential spectrum ( which is (-∞,-√(k_y^2+M^2)]∪[√(k_y^2+M^2),+∞), in view of Theorem <ref>) for some positive value of k_y. Indeed, assume, e.g., that at some definite k_y>0 a new pair of  energy bands emerges from the endpoints of the essential spectrum, i.e., ±√(k_y^2+M^2). If we increase k_y slightly then the energy bands that existed also for smaller values of k_y are sandwiched between the new pair. Since there is infinitely many of them in a bounded interval, they have to accumulate somewhere within its closure. But the only possible accumulation points given by ±√(k_y^2+M^2) are separated from the interval, which is a contradiction. The same holds true on the negative semi-axis.





 §.§ On existence and number of energy bands


In this section, we present several conditions for existence or non-existence of eigenvalues of H_𝐀[k_y], which are chiefly derived from Theorem <ref> but are much easier to apply then those of Theorem <ref> directly. Furthermore, in some cases we will be able to decide whether there is either finite or infinite number of them. Let us start with a criterion for slowly decaying vector potentials.

 
 Let A_y fulfill the general assumptions of Theorem <ref>. 


  * If there exists x_0∈ such that for all x<x_0, A_y(x)≥ 0 or A_y(x)≤ 0, respectively, and A_y is not integrable on (-∞,x_0) then for any k_y<0 or any k_y>0, respectively, H_𝐀[k_y] has infinite number of discrete eigenvalues. 

  * If there exists x_0∈ such that for all x>x_0, A_y(x)≥ 0 or A_y(x)≤ 0, respectively, and A_y is not integrable on (x_0,+∞)
then for any k_y<0 or any k_y>0, respectively, H_𝐀[k_y] has infinite number of discrete eigenvalues. 



For definiteness, we consider the case A_y(x)≥ 0 for all x<x_0, and we choose k_y<0. Firstly, take ε>0 such that (ε+2k_y)<0. Due to (<ref>), one can find x_1∈ such that x_1<x_0 and, for all x<x_1,
0≤ A_y(x)<ε.  We have

    ∫_-∞^x_1A_y(A_y+2k_y)≤(ε+2 k_y)∫_-∞^x_1A_y=-∞.

Therefore, (<ref>) with a=x_1 is clearly fulfilled, and so there is at least one eigenvalue of H_𝐀[k_y] in (M,√(k_y^2+M^2)) (and, due to the spectral symmetry, at least one eigenvalue in (-√(k_y^2+M^2),-M)). Inspecting the proof of Theorem <ref>, which is based on an explicit construction of appropriate test function for h_±, one can see that it is even possible to construct infinite number of  test functions with   pair-wise disjoint supports. Consequently, there are infinitely many discrete eigenvalues. The other cases are treated similarly.


If the vector potential is integrable, i.e. ∫_ |A_y|<+∞, the same holds true for its square, because A_y is bounded under our assumptions. Then (<ref>) may be 
reformulated as follows.

 
Suppose A_y fulfills the general assumptions of Theorem <ref> and is integrable. 

 
  * If ∫_ A_y>0 then H_𝐀[k_y] has some discrete eigenvalues for all k_y<-∫_ A_y^2/2∫_ A_y.

  * If ∫_ A_y<0 then H_𝐀[k_y] has some discrete eigenvalues for all k_y>∫_ A_y^2/2|∫_ A_y|.



If A_y does not change its sign then we have a more explicit, though generally rougher, estimate for the threshold value of k_y. 

 
Let us suppose that  A_y fulfills the general assumptions of Theorem <ref>.

 
  * If A_y≥ 0 and A_y≢0 then H_𝐀[k_y] has some discrete eigenvalues for all k_y<-1/2max A_y.

  * If A_y≤ 0 and A_y≢0 then H_𝐀[k_y] has some discrete eigenvalues for all k_y>1/2|min A_y|.



 The proof follows directly from (<ref>), where we require either A_y+2k_y<0 or A_y+2k_y>0, respectively.



If one does not care about an explicit range of the values of k_y for which  discrete eigenvalues of  H_𝐀[k_y] exist then one arrives to the following conclusion.

 
 Let A_y fulfill the general assumptions of Theorem <ref>. 


  * If, for some a∈, ∫_-∞^a A_y<0 or ∫_-∞^a A_y>0, respectively, then there exists K>0 such that for all k_y>K or k_y<-K, respectively, H_𝐀[k_y] has some discrete eigenvalues.

  * If, for some a∈, ∫_a^∞ A_y<0 or ∫_a^∞ A_y>0, respectively, then there exists K>0 such that for all k_y>K or k_y<-K, respectively, H_𝐀[k_y] has some discrete eigenvalues.


 Let us prove the first statement as the proof of the second one is almost identical.
 One can observe that the left-hand-side and the right-hand-side of (<ref>) behave differently as |k_y|→+∞. Indeed,
 we have
 
    ∫_-∞^a A_y(A_y+2k_y)=2(∫_-∞^a A_y)k_y+𝒪(1)

 as k_y→±∞. On the other hand, 
 
    0≤sup_x∈(a,+∞)(A_y(x)(A_y(x)+2k_y)+δ A'_y(x))≤sup_x∈(a,+∞)A_y(x)^2+2sup_x∈(a,+∞)|A_y(x)||k_y|+sup_x∈(a,+∞)|A'_y(x)|.

 Hence, there is a lower bound for the right-hand-side of (<ref>) with the asymptotic expansion
 
    -2√(2/3) √(sup_x∈(a,+∞)|A_y(x)|)√(|k_y|)+𝒪(1)

 as k_y→±∞. If k_y is of the opposite sign than ∫_-∞^a A_y then, for all |k_y| sufficiently large, (<ref>) is surely below (<ref>).   


Remark  that whenever B≢0, there exists a∈ such that ∫_-∞^a A_y≠ 0.
If we had ∫_-∞^a A_y= 0 for all a∈ then by taking the derivative with respect to a of the  integral we see that A_y≡ 0, and so B=A_y'≡ 0. Therefore, there are  always some energy bands in the spectrum of H_𝐀.

Beside our criteria let us reproduce here a known result on the Schrödinger operators that can be found in <cit.> and that we applied on h_±[k_y]. Recall that k_y^2+M^2 is the minimum of the essential spectrum of h_±[k_y].

  Suppose A_y obeys (<ref>) and (<ref>). If

    lim sup_|x|→+∞(x^2 (A_y(A_y+2k_y)± A'_y(x)))<-1/4

then h_±[k_y] has infinitely many discrete eigenvalues below k_y^2+M^2, and
if 

    lim inf_|x|→+∞(x^2 (A_y(A_y+2k_y)± A'_y(x)))>-1/4

then h_±[k_y] has finitely many (and possibly none) discrete eigenvalues below k_y^2+M^2. 


Recall that if λ is an eigenvalue of h_±[k_y] then ±√(λ) are eigenvalues of H_𝐀[k_y]. Thus, we can use  (<ref>) and (<ref>) directly as sufficient conditions for the existence of infinite or finite (possibly zero) number of eigenvalues of H_𝐀[k_y]. To evaluate these conditions it is sufficient to know the asymptotic expansions of A_y and A_y' at ±∞. In fact, the same holds true  when using Proposition <ref> or <ref>. Although in some situations our criteria may overlap with Proposition <ref>, they are not a special case of the latter (and vice versa). For instance, notice that in Proposition <ref> we require some type of asymptotic behavior at both endpoints ±∞, whereas in Proposition <ref> or <ref> we are interested only in either of the endpoints. Also beware that employing  (<ref>) we cannot distinguish between none and finitely 
many eigenvalues, whereas Proposition <ref> can guarantee their existence. 

In addition to the criteria on existence of discrete eigenvalues, it is useful to present a sufficient condition on emptiness of the point spectrum of H_𝐀[k_y].


 
Suppose A_y obeys (<ref>) and (<ref>).


  * If A_y≥ 0 then, for all k_y≥ 0, there are no eigenvalues in the spectrum of H_𝐀[k_y].

  * If A_y≤ 0 then, for all k_y≤ 0, there are no eigenvalues in the spectrum of H_𝐀[k_y].




We will prove only the first statement, as the proof of the second one is almost identical.
From Section <ref>, we already know that H_𝐀[k_y] has eigenvalues if and only if  h_+[k_y] has some. The eigenvalues E_n[k_y] of the latter operator may only be found in the interval (M^2,M^2+k_y^2) and the corresponding energy bands E_n:k_y↦ E_n[k_y] can only emerge from the bottom of essential spectrum. Suppose there is such energy band emerging at k̃_y≥ 0 from the bottom of the essential spectrum M^2+k̃_y^2. By (<ref>), we have

    E_n/ k_y[k̃_y]≥ 2k̃_y,

because A_y≥ 0 and ψ_n[k_y] is normalized. But in the same moment, we get

    / k_ymin(σ_ess(h_+[k_y]))=2k_y.

This means that the energy band E_n  grows at least as fast as the threshold of the essential spectrum and, hence, it is out of the allowed region, which is a contradiction.



At the end of the section,  several remarks are in order. 
As we mentioned briefly in the introduction, the spectral bands can be associated with dispersionless wave packets. They are constructed as follows. 
For definiteness, let us consider a positive energy band √(E_n) that exists for any fixed k_y from an open interval J⊂. There is an associated normalized bound state Ψ_n[k_y] that satisfies 

    (H_A[k_y]-√(E_n[k_y]))Ψ_n[k_y]=0,  ∀ k_y∈ J.
 
Now, we can construct a wave function Ψ_n=Ψ_n(x,y) using the bound states Ψ_n[k_y]=Ψ_n[k_y](x), 

    Ψ_n(x,y)=1/√(2π)∫_I_ne^ik_y yβ_n(k_y)Ψ_n[k_y](x) k_y,   I_n⊂ J.

The coefficient function β_n(k) determines the actual form of the wave packet.  We suppose ∫_I_n|β_n(k_y)|^2dk_y=1 which guarantees normalization of Ψ_n. The wave function Ψ_n(x,y) is dispersionless in transverse direction, i.e., its density of probability along x-axis remains intact during the evolution, 

    ∫_|^-itĤ_𝐀Ψ_n(x,y)|^2 y=∫_I_n|^-itH_𝐀Ψ_n(x,k_y)|^2 k_y=∫_I_n|β_n(k_y)^-it√(E_n[k_y])Ψ_n[k_y](x)|^2 k_y
    
     =∫_|Ψ_n(x,y)|^2 y.
 
The speed of the wave packet can be then approximated by the averaged group velocity 

    v=∫_I_n(√(E_n[k_y]))' k_y/|I_n|,

cf. <cit.>.

Now, let us suppose that A_y does not change sign. When it is everywhere positive, then Proposition <ref> or  Proposition <ref>  guarantee existence of discrete energies for any k_y<k_-<0, where the explicit value of k_- is specified in different manner by each of the criteria. Simultaneously, Proposition <ref> tells that there are no discrete energies for all k_y≥0.  Hence, the band structure of the system with such A_y is substantially asymmetric with respect to k_y. As it is suggested by the formula (<ref>),  the energy bands are decreasing for sufficiently large negative values of k_y. The speed of the associated dispersionless wave packets is then negative. Hence, there are wave packets moving in one direction but one cannot construct similar wave packets moving in the other direction. (Notice that the wave packets assembled from the bound states corresponding to the negative energies are composed from holes, and, therefore, contribute to the electric current in the same direction as the wave packets assembled from the bound states with positive energies.) A similar spectral
asymmetry can be found in the systems with negative A_y. We conclude that when A_y does not change sign, the system can host unidirectional dispersionless wave packets moving along y-axis. 

On the other hand, if the vector potential A_y changes sign asymptotically, the Proposition <ref> tells us that there are energy bands for all sufficiently large |k_y|. In  that case, we can construct dispersionless wave packets that will propagate in both directions along the wave guide. The system will host bidirectional dispersionless wave packets. 

When the system possesses additional symmetries, they can be reflected in its spectrum. Let us take the system with even magnetic field, and, hence, with odd A_y. Then h_±[k_y] is unitarily equivalent to h_±[-k_y] via the parity transform P:ψ(x)↦ψ(-x). Therefore, their spectra coincide. Consequently, the eigenvalues of H_𝐀[k_y] coincide with those of H_𝐀[-k_y], too. Putting this together with the first point of Theorem <ref>, we conclude that the family of energy bands of H_𝐀 is symmetric with respect to reflections over horizontal and vertical axes.





§ WIRES

Let us have an infinite wire of negligible radius that runs parallelly with the y-axis. It can be parametrized as y↦(x_0,y,z_0) where x_0∈ and z_0∈∖{0}. The magnetic vector potential along the sheet of the Dirac material (placed in (x,y)-plane) induced by the wire carrying the current I in the negative y-direction can be computed as 𝐀=(0,A_y,0) with

    A_y(x)=β Iln((x-x_0)^2+z_0^2),

where

    β:=μ_0√(α)/4π.

Thus we have

    B(x)=1/√(α)A_y'(x)=μ_0 I/2π(x-x_0)/(x-x_0)^2+z_0^2.

With this choice of A_y, the potential term in h_±[k_y] tends to infinity as x→±∞. Hence, the spectrum of h_±[k_y] and, consequently, that of  H_𝐀[k_y] is purely discrete <cit.>. So there are always infinitely many eigenvalues for any value of k_y∈.

We will now consider a configuration of  parallel wires with tunable asymptotics of A_y. Each of the wires will be parametrized as y↦(x_j,y,z_j) and will carry the current I_j. We will always require that the total electric current is zero, i.e., 

    ∑_j=1^NI_j=0.

Under this condition, the total vector potential diminishes at x=±∞. Violating (<ref>) leads to the purely discrete spectrum as in the case of the single wire.


If (<ref>) holds then the vector potential associated with the system of the wires obeys

    A_y(x)=β∑_j=1^N  I_jln((x-x_j)^2+z_j^2)=α_1/x+α_2/x^2+α_3/x^3+…

as x→±∞. Here the the coefficients α_k are functions of the coordinates (x_j,z_j) and currents I_j, j=1,… N. For instance, we have 

    α_1=-2β∑_j=1^N I_j x_j,  α_2=β∑_j=1^N I_j(z_j^2-x_j^2).

Since (<ref>) is analytic at ±∞, we can differentiate it term by term. In this way, we obtain

    B(x)=1/√(α)A_y'(x)=-1/√(α)(α_1/x^2+2α_2/x^3+3α_3/x^4+…)

as x→±∞.

Firstly, let α_1≠ 0. Then A_y is not integrable near infinities and it is positive near +∞ and negative near -∞ (or vice versa, depending on the sign of α_1). Using Proposition <ref> we immediately infer that, for any k_y≠ 0, H_𝐀[k_y] has infinite number of discrete eigenvalues.


Secondly, let α_1=0 but α_2≠0. For definiteness, take α_2<0. By Proposition <ref>, H_𝐀[k_y] has discrete eigenvalues for all sufficiently large k_y>0. If α_2>0 then the discrete eigenvalues are guaranteed for all large enough negative k_y's. In both cases, the number of eigenvalues may be finite or infinite, depending on the particular values of α_2 and k_y, see Proposition <ref>. As the vector potential does not change sign asymptotically, the system is a good candidate for hosting the unidirectional dispersionless wave packets. However, we are able to prove their existence only when A_y is of constant sign everywhere, cf. Proposition <ref>. In Section <ref>, we will discuss a specific example with A_y of such type.


Finally, for α_1=0, α_2=0, and α_3≠ 0, we have

    A_y(x)=α_3/x^3+𝒪(1/x^4)

as x→±∞. Therefore, due to Proposition <ref>, there may be, in principle, at most finitely many eigenvalues in the spectrum of H_𝐀[k_y]. Employing Proposition <ref> we see that there are definitely some eigenvalues of H_𝐀[k_y] for all k_y above some positive threshold and all k_y below some negative threshold.

In Section <ref>, we will propose a simple configuration of wires which will make it possible to switch between different types of asymptotic behavior of A_y just by setting the currents appropriately.
Notice that in real experiments, it is rather impossible to place the wires at exact positions and tune the currents so that α_k=0 for all required k's. Imprecision in positioning of the wires prevents the coefficients α_k from vanishing, the vector potential behaves asymptotically as α_1/x, and H_𝐀[k_y] has infinite number of discrete eigenvalues for any k_y≠ 0. Despite these facts, the situation is actually not that bad. In the next section we will deal with a particular configuration of wires that may host unidirectional  dispersionless wave packets. Although this will change even under a small perturbation of the configuration, we will demonstrate that the eigenstates that emerge due to the perturbation are well localized very far from the wires.




 §.§ Pair of wires 


Let us now focus on situation when N=2, i.e., we have pair of wires. The condition (<ref>) then reads I_1=-I_2. Hence, equivalently, one can think about an infinite single loop. 
The vector potential generated by the loop is given by 𝐀=(0,A_y,0) with

    A_y(x,z)=β I_1ln(x-x_1)^2+(z-z_1)^2/(x-x_2)^2+(z-z_2)^2.

We fix the distance between the wires to be 2d for some d>0. It is convenient to choose the coordinate frame so that

    x_1=dcosγ,   x_2=-dcosγ,   z_1=-dsinγ,   z_2=dsinγ,

where γ∈[0,2π). If we identify the sheet of the Dirac material with the plane z=z_0, then γ measures the angle between the sheet and the plane defined by the loop, see Figure <ref>. Substituting z=z_0 and (<ref>) into (<ref>) we obtain the vector potential along the sheet,

    A_y(x)=β I_1ln(x-dcosγ)^2+(z_0+dsinγ)^2/(x+dcosγ)^2+(z_0-dsinγ)^2.







One can calculate that

    A_y(x)=β I_1(-4dcosγ/x+4dz_0sinγ/x^2+𝒪(1/x^3))

as x→±∞. If γ∉{π/2,3π/2}, i.e., 
the plane of the loop is not perpendicular to the sheet, the vector potential 
is not integrable. Using Proposition <ref> we infer that there are infinitely many discrete eigenvalues of H_𝐀[k_y] for any k_y≠ 0.

This changes dramatically if γ∈{π/2,3π/2}. In that case, A_y is of constant sign and the leading term in (<ref>) is proportional to x^-2. We can prove that there are discrete eigenvalues for all k_y from some half-line and there are no discrete eigenvalues for all k_y from another half-line. To be more specific, let us start by putting 
γ=π/2, i.e., with

    A_y(x)=β I_1lnx^2+(z_0+d)^2/x^2+(z_0-d)^2.

Suppose I_1>0 and z_0>d. The first assumption amounts only to choosing a definite direction of the current, the latter means that the whole loop lies below the sheet.

Since A_y is everywhere positive, H_𝐀[k_y] has no eigenvalues for all k_y≥ 0, due to Proposition <ref>. As A_y is also integrable, we can use the first criterion of Proposition <ref> to prove the existence of discrete eigenvalues.
After a rather tedious calculation we arrive at

    ∫_ A_y(x) x=4πβ I_1 d,
       ∫_ A^2_y(x) x=8πβ^2 I_1^2 (z_0lnz_0^2-d^2/z_0^2+dlnz_0+d/z_0-d).

Therefore, there are discrete eigenvalues in the spectrum of H_𝐀[k_y], whenever

    k_y<-β I_1(lnz_0+d/z_0-d+z_0/dlnz_0^2-d^2/z_0^2).

Remark that the first term is the same as that we would obtain if we used Proposition <ref>, the second term further enlarges the range of admissible k_y's.

 


The other case, γ=3π/2, is equivalent to the previous one, γ=π/2, after changing the orientation of the currents.  Therefore, for all 
 
    k_y>β I_1(lnz_0+d/z_0-d+z_0/dlnz_0^2-d^2/z_0^2),

there are some and, for all
 
    k_y≤0,

there are none eigenvalues in the spectrum of H_𝐀[k_y], respectively.


We found that when the plane of the loop and the sheet of the Dirac material are perfectly perpendicular, then  the waveguide might host unidirectional dispersionless wave packets. However, when the matching of the two planes is not perfect, the system may host bidirectional dispersionless wave packets. We will analyze in more detail  a special case of the loop that is slightly rotated away from its perpendicular position, i.e., we take γ=π/2+δ, where δ is very small. Let us denote the potential term of h_±[k_y] with constants subtracted by V_±. We have

    V_±(x)=A_y(x)(A_y(x)+2k_y)± A_y'(x),

and so

    V_±(x)=8dβ I_1  k_y(sinδ/x+z_0cosδ/x^2)+(4dβ I_1 sinδ)^2∓ 4dβ I_1 sinδ/x^2+𝒪(1/x^3)

as |x|→∞.

Now, for δ=0, if k_y  is sufficiently large then V_± is everywhere positive. In fact, V_± remains positive on a chosen interval (-K,K) for all sufficiently small δ's. Let us fix K so that (<ref>) is a good approximation outside (-K,K). Then up to the second order in x and the first order in δ, the inequality V_-<0 is satisfied for δ>0, provided that

    δ/x+z_0/x^2<0,

i.e. x<-z_0/δ. For δ<0, V_+>0 is satisfied, provided that x>z_0/|δ|. Therefore, for small perturbations of the perfectly perpendicular configuration, the potential well emerges from infinity. The associated bound states are presumably localized near the potential well, i.e., far from the position of the wires. 




 §.§ Tunable waveguide
 

In this section, we propose a configuration of five wires such that their positions are fixed whereas by tuning the currents that they carry we can achieve different types of asymptotic behavior for the associated vector potential

    A_y(x)=β∑_j=1^5I_jln((x-x_j)^2+z_j^2).

Some of the currents may be zero. In that case, we effectively deal with a lesser number of wires. Let us denote the jth wire by W_j and fix their positions as follows,

    x_1=x_4=-x_2=-x_5,   x_3=0,   z_1=z_2,   z_4=z_5,

where 0>z_1>z_4. The wires W_1, W_2, W_4, and W_5 form an infinite cuboid below the plane z=0 of the sheet of the Dirac material, and W_3 is placed in the plane of vertical symmetry of this cuboid. See Table <ref> for figures of some typical cross-sections (y=const.).

Firstly, we choose  the currents in the following manner,

    I_1=I_2=-I_4=-I_5,   I_3=0.

This corresponds to a pair of loops with common direction of currents that are both perpendicular to the sheet. We have

    A_y(x)=β I_1 ln(1+(z_1^2-z_4^2)(2x^2+2x_1^2+z_1^2+z_4^2)/((x-x_1)^2+z_4^2)((x+x_1)^2+z_4^2))=α_2/x^2+O(1/x^4)

as x→±∞.
Due to the additivity of vector potential, one can expect similar result as in the case of a single loop that was treated within Section <ref>. In particular, the system might host unidirectional  dispersionless wave packets. Their direction may be switched  by changing the orientation of the currents. 


If we switch the current orientation in one of the loops, i.e., we set 

    I_1=I_5=-I_2=-I_4,   I_3=0,

then the situation changes dramatically, because the leading terms in the asymptotic expansions of vector potentials generated by single loops cancel out. Indeed, after some manipulations we obtain

    A_y(x)=β I_1ln(1-4xx_1(z_4^2-z_1^2)/((x-x_1)^2+z_1^2)((x+x_1)^2+z_4^2))=α_3/x^3+O(1/x^5)

as x→±∞.
Therefore, by Proposition <ref>, the system possesses   bidirectional dispersionless wave packets, as there are discrete eigenvalues in the spectrum of H_𝐀[k_y] for any k_y with sufficiently large |k_y|. In view of Proposition <ref>, there is finitely many of them.

Finally, we will present another simple configuration that makes the vector potential to decay even faster. We switch the current off in W_1 and W_2, and switch it on in W_3,  i.e., we set

    I_1=I_2=0,   I_4=I_5=-1/2I_3.

Additionally, we require that

    z_4=-√(x_4^2+z_3^2).

As we suppose that the relative positions of the wires are fixed, the condition (<ref>) restricts the distance of the system of wires from the sheet. The vector potential then acquires the following form,

    A_y(x)=-β I_4ln(1-4x_4^2(x_4^2+z_3^2)/((x-x_4)^2+x_4^2+z_3^2)((x+x_4)^2+x_4^2+z_3^2))=α_4/x^4+O(1/x^6)

as x→±∞.
Clearly, A_y is of constant sign, which depends on the current orientation. In view of Propositions <ref> and <ref>, the system may host unidirectional dispersionless   wave packets. Taking Proposition <ref> into the account, we infer that if  H_𝐀[k_y] has some eigenvalues then there is only finitely many of them. 

We put the derived properties of the discussed system together with the illustration of the wire configuration into Table <ref>.





 

plain




§ MAGNETIZED STRIPS
 
We will study here spectral properties of H_𝐀[k_y] with the magnetic field generated by infinitesimally thin magnetized strips of constant width 2d with the vector of magnetization directed either parallelly or perpendicularly to the strip.
Since the chosen gauge (<ref>) is always transverse, i.e., div𝐀=0, the vector potential associated with a generic magnetized material with the magnetization 𝐌(𝐫) can be written as 

    𝐀(𝐫)=μ_0√(α)/4π∫_ℝ^3𝐌(𝐫_0)× (𝐫-𝐫_0)/|𝐫-𝐫_0|^3𝐫_0.





 §.§ Perpendicularly magnetized strip


Firstly, consider the situation when the perpendicularly magnetized strip is parallel with the sheet of the Dirac material. For definiteness, we take [-d,d]××{ 0} for the strip and ^2×{z_0} with z_0>0 for the sheet, see Figure <ref>. 



The magnetization vector 𝐌 is then given by

    𝐌(𝐫)=(0,0,m χ_[-d,d](x)δ(z)),

where m>0.
Here, the indicator function χ_[-d,d] is defined as χ_[-d,d](x)=1 for all x∈[-d,d] and it is zero elsewhere. 
Substituting (<ref>) into (<ref>), we get A_x=A_z=0, and, for all z≠ 0,

    A_y(x,z)=β̃/2∫_-d^d x_0∫_ y_0x-x_0/((x-x_0)^2+y_0^2+z^2)^3/2=β̃/2ln(z^2+(x+d)^2/z^2+(x-d)^2),

where we defined

    β̃:=μ_0 m√(α)/2π.


We studied the same vector potential (with the current -I_1  substituted for the magnetization m) in the case of the current loop that consists of two infinite wires {± d}××{0}, cf. (<ref>) with γ=0 and z_0=z. Rotating the strip along the y-axis by angel γ is equivalent to rotating the wires to the position given by (<ref>). Hence, we can use the results of Section <ref> literally. In particular, one can expect the unidirectional dispersionless wave packets only when the strip is exactly perpendicular to the sheet.






 §.§ Parallelly magnetized strip

Firstly, let us assume again that the strip is parallel with the sheet and choose the same coordinates as in the previous section. If we take

    𝐌(𝐫)=(m χ_[-d,d](x)δ(z),0,0)

with m>0 for the magnetization then we get A_x=A_z=0 and, for all z≠ 0,

    A_y(x,z)=-β̃/2∫_-d^d x_0∫_ y_0 z/((x-x_0)^2+y_0^2+z^2)^3/2=β̃(arctanx-d/z-arctanx+d/z).


Next, let us rotate the strip around the y-axis by angle γ∈[0,2π) measured from the positive x-axis towards the positive z-axis. This amounts to a natural change of coordinates,

    𝐫'=R𝐫,   with R=[  cosγ     0 -sinγ;     0     1     0;  sinγ     0  cosγ ].

The vector potential 𝐀' associated with the rotated strip is given by (<ref>), where we substitute (x',z') for (x,z). Therefore, going back to the original variables, we obtain

    𝐀(𝐫)=R^-1𝐀'(R𝐫)=β̃(0,arctanxcosγ-zsinγ-d/xsinγ+zcosγ-arctanxcosγ-zsinγ+d/xsinγ+zcosγ,0).

In particular, along the sheet placed at z=z_0 we get A_x=A_z=0 and

    A_y(x)=β̃(arctanxcosγ-z_0sinγ-d/xsinγ+z_0cosγ-arctanxcosγ-z_0sinγ+d/xsinγ+z_0cosγ).

The magnetic field along the sheet is then given by

    B(x)=1/√(α)A_y'(x)=μ_0 m/2π 2 d sinγ(x^2+d^2-z_0^2)+4 d x z_0 cosγ/(x^2+z_0^2)^2+2 d^2 ((z_0^2-x^2)cos(2 γ)+2 x z_0 sin(2γ))+d^4.


Looking at asymptotic expansions at x=±∞,

    A_y(x)=β̃(-2dsinγ/x-2dz_0cosγ/x^2)+𝒪(1/x^3)

and

    B(x)=μ_0 m/2π 2dsinγ/x^2+𝒪(1/x^3),

we infer that the general assumptions of Theorem <ref> are fulfilled.

Formula (<ref>) suggests that A_y is not integrable except for the angles γ=0 or γ=π, which correspond to the case when the strip is parallel with the sheet. Except this specific case, H_A[k_y] has infinite number of discrete eigenvalues for any k_y≠ 0, which follows from Proposition <ref>.


Let us now focus on the case when the strip is parallel with the sheet. For definiteness, put γ=0.  Using (<ref>) and (<ref>) with γ=0, we obtain

    V_±(x)=A_y(x)(A_y(x)+2k_y)± A_y'(x)=-4β̃d z_0 k_y/x^2+𝒪(1/x^3)

as x→±∞. Therefore, due to Proposition <ref>, 
H_𝐀[k_y] has infinitely many discrete eigenvalues, whenever

    k_y>1/16β̃d z_0,

and at most finite number of them (possibly none), provided that 

    k_y<1/16β̃d z_0.



In the same moment, we can apply Proposition <ref>, because, for γ=0, A_y is everywhere negative.  It says that,
for all k_y satisfying

    k_y>1/2|min A_y|=-1/2A_y(0)=β̃arctand/z_0,

there are discrete eigenvalues in the spectrum of H_𝐀[k_y].
In fact, we can also use Proposition <ref>, to provide a sharper, though more difficult to evaluate, bound on k_y. After some tedious calculation, one arrives at

    ∫_ A_y(x) x=-2πβ̃d,
       ∫_ A_y^2(x) x=2β̃^2 z_0∫_arctanx(arctanx-arctan(x+2d/z_0)) x.

Hence, another sufficient condition for the existence of discrete eigenvalues reads

    k_y>β̃z_0/2π d∫_arctanx(arctanx-arctan(x+2d/z_0)) x.


We conclude that, for any k_y that obeys (<ref>) or (<ref>) (or its more explicit, though weaker, version (<ref>)), there are discrete eigenvalues in the spectrum of H_𝐀[k_y]. There are infinitely many of them if (<ref>) is fulfilled, and finitely many if (<ref>) holds together with (<ref>). On the other hand, for k_y≤ 0, H_𝐀[k_y] has no eigenvalues, because A_y is everywhere negative and, consequently, one can apply Proposition <ref>.

The other case γ=π amounts to changing the orientation of the magnetization, i.e. changing β̃ for -β̃<0. Consequently, there will always be discrete eigenvalues for all k_y sufficiently negative, and there will not be any eigenvalues for all k_y≥ 0 . One can find  a threshold value of k_y for the presence of eigenvalues in exactly the same manner as in the case γ=0.


Finally, let us remark that when the strip is tilted very slightly from its parallel position, i.e., γ≠0 but |γ|<<1, one can perform an analysis analogous to that from the end of Section <ref> to conclude that tilting the strip produces a potential well that can accommodate  bound states but is localized far from the strip.




§ SUMMARY AND OUTLOOK

Magnetic field can be used to create waveguides for two-dimensional Dirac fermions. 
The aim of the current paper was to get insight into spectral properties of both massive and massless Dirac fermions in presence of realistic magnetic fields. As the explicit solution of the associated stationary equation is not possible in these cases, we resolved to the qualitative spectral analysis. We arrived at general results applicable for a broad class of magnetic fields and employed them in the settings where the waveguides are created by placing the Dirac material into the proximity of a set of parallel wires with zero net current or of a magnetized strip.  

The analysis was facilitated by the translational symmetry of the system. The spectral properties of the two-dimensional Dirac Hamiltonian (<ref>) were fully deduced from those of the class of one-dimensional Hamiltonians H_𝐀[k_y], see (<ref>), which is obtained by direct integral decomposition of the total Hamiltonian (<ref>) and describes dynamics of the system with the fixed longitudinal momentum k_y. We focused on a wide class of magnetic fields where the vector potential can be fixed so that it vanishes asymptotically together with its first derivative, see  (<ref>) and (<ref>). Discrete energies  of H_𝐀[k_y] were of our main interest;  they give rise to energy bands of the total Hamiltonian. These energy bands can be associated with dispersionless wave packets in the systems with translational symmetry  <cit.>. 

In Section <ref>, we discussed general spectral properties of H_𝐀[k_y]. In particular,  we presented a set of practical criteria that make it possible to guarantee existence or absence of discrete energies. They can also tell whether there is finite or infinite number of discrete energies in the spectrum of H_𝐀[k_y] for a given k_y. We employed these qualitative tools extensively in the analysis of the magnetic waveguides generated by a set of wires with zero net electric current or magnetized strips.

In Section <ref>, we showed that magnetic field generated by a set of parallel wires with zero net current, placed in the proximity of  the Dirac material (e.g. graphene), can create magnetic  waveguide with tunable transport properties. Depending upon the currents in the wires and their positions, the waveguide can host either unidirectional or bidirectional dispersionless wave packets. We proposed a simple system with five wires whose transport properties are easily controllable by switching the currents in particular wires, cf. Table <ref>. 
In Section <ref>, we analyzed spectral properties of Dirac fermions in the magnetic waveguide produced by the magnetized strip. We considered both  the perpendicular (<ref>) and the parallel (<ref>) magnetization of the strip. In both cases, the waveguide can again host either unidirectional or bidirectional dispersionless wave packets, depending upon the tilt of the strip.

Note that in our current model, no energy can enter the interval (-M,M). It is worth mentioning in this context that there exist models of Dirac fermions with constant mass M on the  half-plane where a specific energy band can cross the interval (-M,M), see <cit.>. Existence of these bands is related to the boundary condition along the half-plane.


Our qualitative approach is complementary to the existing literature where the models offering explicit analytical  solutions are mostly discussed. 
It does not rely on the concrete form of the potential and it can provide valuable information on the spectrum in the cases where the explicit solutions are unknown. 
The presented results are applicable to two-dimensional massive or massless Dirac fermions in  presence of a magnetic field with translational symmetry. Their extension  to systems with other symmetries or with electric fields  would be desirable and should be addressed in the future.



§ ACKNOWLEDGMENTS

MF and MT were supported by GAČR grant No. 18-08835S and No. 17-01706S, respectively. VJ was supported by No. 15-07674Y.

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