ABSTRACT (original article): Chiral anomalies resulting from the breaking of classical symmetries at the quantum level are fundamental to quantum field theory and gaining ever-growing importance in the description of topological materials in condensed matter physics. Here we present analytical solutions of the Dirac equation for massless 3+1 fermions confined to an infinite stripe and placed into a background gauge field forming a triangular potential well across the width of the stripe. Such an effective 1+1 system hosts zero-energy modes resulting in the gauge field-dependent chiral anomaly structure. This problem has a direct relation to a half-bearded graphene nanoribbon placed into an in-plane external electric field and offers it an exact solution in terms of new special functions that are similar but not reducible to Airy functions. CITATION (original article): Renebeth B. Payod, Vasil A. Saroka; On a solution to the Dirac equation with a triangular potential well. J. Math. Phys. 2024; 65 (11): 113505. https://doi.org/10.1063/5.0214567
Preprint arXiv: https://arxiv.org/abs/2409.04595
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PacletInstall["https://github.com/vasilsaroka/TBpack/releases/download/v0.5.0/TBpack-0.5.0.paclet"]
In case you want to uninstall TBpack use
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Needs["TBpack`"]

Section II

Basic equations

We start from the Dirac equation featuring linear scalar potential (considering only one valley K):
∂
x
ϕA+kyϕA-(ϵ+Fx)ϕB==0​​
∂
x
ϕB-kyϕB-(ϵ+Fx)ϕA==0
with a Dirichlet boundary condition:
ϕB[-L/2]==ϕB[L/2]==0
which, after a series of transforms and substitutions, leads us to the final equation:
∂
ξξ
g+νξ+2ky-
1
ξ
∂
ξ
g+λξg==0
(
1
)
where
ν=2F
,
μ=2ky
,
λ=2kyF
and
ϕB(ξ)=g(ξ)exp-
2
Fξ
2exp(kyξ)
with
ξ=
ϵ
F
+x
.
This final Eq. (1) has an analytic solution in terms of power series,
g(ξ)=
∞
∑
n=0
a
n
n
ξ
, defining two special functions ℛ and  via the following recursive relation:
a
n
==-μ
n-1
n(n-2)
a
n-1
-
ν
n
a
n-2
-
λ
n(n-2)
a
n-3
(
2
)
with
a
0
=1;
a
2
=0
for ℛ and
a
0
=0;
a
2
=1
for , while
a
1
=0
for both cases.
Thus defined ℛ and  are somewhat similar to Airy functions, see for example https://mathworld.wolfram.com/AiryDifferentialEquation.html.

Functions

ℛ and  special functions


ℛ and  special functions compiled for performance


MyPlotGrid

05/05/2019: this function is based on plotGrid function above. It uses explicitly provided values for the imagepadding parameters. In contrast to the plotGrid, MyPlotGrid does not distort subplots for non - equal left and right or top and bottom image padding parameters.
Clear[MyPlotGrid];​​​​MyPlotGrid[l_List,w_,h_,leftPadding_,rightPadding_,bottomPadding_,topPadding_]:=Module[​​{​​nx,ny,​​widths,heights,​​dimensions,positions,​​frameOptions=FilterRules[Options[Graphics],Except[{ImagePadding,Frame,FrameTicks}]]​​},​​{ny,nx}=Dimensions[l];​​widths=(w-leftPadding-rightPadding)/nxTable[1,{nx}];​​widths[[1]]=widths[[1]]+leftPadding;​​widths[[-1]]=widths[[-1]]+rightPadding;​​heights=(h-topPadding-bottomPadding)/nyTable[1,{ny}];​​heights[[1]]=heights[[1]]+bottomPadding;​​heights[[-1]]=heights[[-1]]+topPadding;​​positions=Transpose@Partition[Tuples[Prepend[Accumulate[Most[#]],0]&/@{widths,heights}],ny];​​Graphics[Table[Inset[Show[l[[ny-j+1,i]],ImagePadding{{If[i1,leftPadding,0],If[inx,rightPadding,0]},{If[j1,bottomPadding,0],If[jny,topPadding,0]}},AspectRatioFull],positions[[j,i]],{Left,Bottom},{widths[[i]],heights[[j]]}],{i,1,nx},{j,1,ny}],PlotRange{{0,w},{0,h}},ImageSize{w,h},Evaluate@Apply[Sequence,frameOptions]]]

Section III. D

Airy differential equation and Airy functions

The Airy equation is
∂
zz
y-zy==0.
Its power series solutions are defined via a recursive relation
a
n+2
==
a
n-1
(n+2)(n+1)
(
3
)
with
a
0
=1;
a
1
=0
for Ai(z) and
a
0
=0;
a
1
=1
for Bi(z), while
a
2
=0
for both cases.

ℛ - function and  - function vs Airy Functions [series representation]

Note that our definition of the Airy functions, given above in Eq. (3), is different from that built into Wolfram Mathematica and used in mathematical
literature; see, for instance, Sec. 10.4 in Ref.
1
. In notations of Ref.
1
our Airy functions Ai(z) and Bi(z) should be denoted as
f (z) and g(z), respectively. Therefore, we redefine these functions below:
In[]:=
sol=SolveA==c1f-c2g,B==
3
(c1f+c2g),{f,g}1
Out[]=
f-
-3A-
3
B
6c1
,g-
3A-
3
B
6c2

In[]:=
c1sub=c1->AiryAi[0]​​c2sub=c2->(-D[AiryAi[z],z]/.z->0)
Out[]=
c1
1
2/3
3
Gamma
2
3

Out[]=
c2
1
1/3
3
Gamma
1
3

In[]:=
fgsub=sol/.{c1sub,c2sub,A->AiryAi[z],B->AiryBi[z]}
Out[]=
f-
-3AiryAi[z]-
3
AiryBi[z]Gamma
2
3

2
1/3
3
,g-
3AiryAi[z]-
3
AiryBi[z]Gamma
1
3

2
2/3
3

In[]:=
myAi[z_]=f/.fgsub​​myBi[z_]=g/.fgsub
Out[]=
-
-3AiryAi[z]-
3
AiryBi[z]Gamma
2
3

2
1/3
3
Out[]=
-
3AiryAi[z]-
3
AiryBi[z]Gamma
1
3

2
2/3
3
Now we can have a close up look at both types of functions by expanding them into series:
In[]:=
Series[myAi[z],{z,0,8}]​​Series[myBi[z],{z,0,8}]​​Series[ℛ[z,μ,ν,λ,MaxIterations->15],{z,0,5}]​​Series[[z,μ,ν,λ,MaxIterations->15],{z,0,5}]
Out[]=
1+
3
z
6
+
6
z
180
+
9
O[z]
Out[]=
z+
4
z
12
+
7
z
504
+
9
O[z]
Out[]=
1-
λ
3
z
3
+
1
8
λμ
4
z
+-
λ
2
μ
30
+
λν
15
5
z
+
6
O[z]
Out[]=
2
z
-
2μ
3
z
3
+
1
4
(
2
μ
-ν)
4
z
+
1
15
(-λ-
3
μ
+3μν)
5
z
+
6
O[z]
We can now see some similarities and differences between our ℛ and  functions and the Airy functions. The power series of ℛ
function has a missing
2
z
-term, while  function is missing a constant term equal to 1. These are linearly independent monomials. The power series
of Ai(z) is missing z-term, while Bi(z) is missing constant term equal to 1. Globally, Airy functions are missing
2
z
- terms and all dependent ones as
follows from Eq. (3):
0=
a
2
=
a
5
=
a
8
=···=
a
3n+2
. In contrast, ℛ and -functions are generally missing z-terms in their power
series representation. When
µ=ν=0
, it follows from Eq. (2) that the series of missing terms is
0=
a
1
=
a
4
=
a
7
=...=
a
3n+1
​
for both ℛ and .

Figure 2 (a): ℛ - and  - functions vs Airy Functions

mfun=MaTeX[#,FontSize12]&;​​tfun=mfun[Round[#]]&;​​​​{μ,ν,λ}={0,0,-1};​​funR[x_?NumericQ]:=ℛc[(1.0+0.0)x,μ,ν,λ];​​funV[x_?NumericQ]:=c[(1.0+0.0)x,μ,ν,λ];​​​​tlen={0.02,0};​​ticks=LinTicks[-6,6,2,5,MajorTickLength->tlen,MinorTickLength->tlen/2,TickLabelFunction->tfun];​​tticks={#〚1〛,"",#〚3〛,#〚4〛}&/@ticks;​​fticks={{tticks,tticks},{ticks,tticks}};​​​​p1=ComplexPlot[funR[z],{z,-5.0-5.0,5.0+5.0},​​FrameStyle->BlackFrame,​​FrameTicks->{{ticks,tticks},{ticks,tticks}},​​PlotRange->{{-5,5},{-5,5}},​​ColorFunction"CyclicLogAbsArg",​​FrameLabel->mfun/@{"\\mathrm{Re}(z)","\\mathrm{Im}(z)"},Epilog->Text[Framed[mfun@"R(z,0,0,-1)",Background->White,RoundingRadius->7],Scaled[{0.5,0.8}]],​​PlotPoints30];​​p2=ComplexPlot[myAi[z],{z,-5.0-5.0,5.0+5.0},​​FrameStyle->BlackFrame,​​FrameTicks->fticks,​​PlotRange->{{-5,5},{-5,5}},​​ColorFunction"CyclicLogAbsArg",​​FrameLabel->{mfun@"\\mathrm{Re}(z)",None},​​Epilog->Text[Framed[mfun@"Ai(z)",Background->White,RoundingRadius->7],Scaled[{0.5,0.8}]],​​PlotPoints30];​​​​​​p3=ComplexPlot[funV[z],{z,-5.0-5.0,5.0+5.0},​​FrameStyle->BlackFrame,​​FrameTicks->fticks,​​PlotRange->{{-5,5},{-5,5}},​​ColorFunction"CyclicLogAbsArg",​​FrameLabel->{mfun@"\\mathrm{Re}(z)",None},Epilog->Text[Framed[mfun@"V(z,0,0,-1)",Background->White,RoundingRadius->7],Scaled[{0.5,0.8}]],​​PlotPoints30];​​p4=ComplexPlot[myBi[z],{z,-5.0-5.0,5.0+5.0},​​FrameStyle->BlackFrame,​​FrameTicks->fticks,​​PlotRange->{{-5,5},{-5,5}},​​ColorFunction"CyclicLogAbsArg",​​FrameLabel->{mfun@"\\mathrm{Re}(z)",None},​​Epilog->Text[Framed[mfun@"Bi(z)",Background->White,RoundingRadius->7],Scaled[{0.5,0.8}]],​​PlotPoints30];​​​​leftPadding=50;​​rightPadding=20;​​bottomPadding=50;​​topPadding=20;​​width=700;​​height=220;​​fig=MyPlotGrid[{{p1,p2,p3,p4}},width,height,leftPadding,rightPadding,bottomPadding,topPadding]

Figure 2 (b): ℛ-function panel

Figure 2 (c): -function panel

 - function [JMP Highlight Image]

References

[1] Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables. (United States. Government Printing Office, Washington, D.C., 1972). doi:10.1115/1.3625776.

CITE THIS NOTEBOOK

On a solution to the Dirac equation with a triangular potential well​
by Renebeth B. Payod, Vasil A. Saroka​
Wolfram Community, STAFF PICKS, October 29, 2024
​https://community.wolfram.com/groups/-/m/t/3311017