Universality and stability of the edge states of chiral nodal topological semimetals; Luttinger model for $j=\frac32$ electrons as a 3D topological semimetal

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by  Maxim Kharitonov

Universality and stability of the edge states of chiral nodal topological semimetals;
Luttinger model for j = 3 electrons as a 3D topological semimetal
2
Maxim Kharitonov, Julian-Benedikt Mayer, and Ewelina M. Hankiewicz
Institute for Theoretical Physics and Astrophysics,
University of Wu¨rzburg, 97074 Wu¨rzburg, Germany
We theoretically demonstrate that the chiral structure of the nodes of nodal semimetals is re-
sponsible for the existence and universal local properties of the edge states in the vicinity of the
nodes. We perform a general analysis of the edge states for an isolated node of a 2D semimetal,
protected by chiral symmetry and characterized by the topological winding number N. We derive
theasymptoticchiral-symmetricboundaryconditionsandfindthatthereareN+1universalclasses
7
ofthem. Theclassdeterminesthenumbersofflat-bandedgestatesoneithersideoffthenodeinthe
1
1D spectrum and the winding number N gives the total number of edge states. We then show that
0
2 the edge states of chiral nodal semimetals are robust: they persist in a finite-size stability region of
parametersofchiral-asymmetricterms. Thissignificantlyextendsthenotionof2Dand3Dtopolog-
n ical nodal semimetals. We demonstrate that the Luttinger model with a quadratic node for j = 3
a 2
electrons is a 3D topological semimetal in this new sense and predict that α-Sn, HgTe, possibly
J
Pr2Ir2O7, and many other semimetals described by it are topological and exhibit surface states.
6

] Introduction. Edge states in 2D nodal semimetals extension of the notion of a topological nodal semimetal
l
l have been demonstrated in numerous theoretical calcu- inboth2Dand3D.Asanimportantexample,wedemon-
a
h lations1–16, mainly for models describing monolayer and strate that the Luttinger model17 with a quadratic node
- bilayergraphene. Theirexistenceisattributed15,16tothe for j = 3 electrons, describing materials like α-Sn19–21,
s 2
e topologicalinvariantscharacterizingthenodes,thewind- HgTe19–21, and Pr2Ir2O722, exhibits surface states and
m ingnumbersN, whicharewell-definedinthepresenceof is a 3D topological semimetal in this new more general
. chiral symmetry. Still, up to now, the general structure sense.
at oftheedgestatesofchiral-symmetric2Dnodalsemimet- 2D chiral-symmetric nodal semimetal. First, we con-
m als and its relation to the winding numbers have not yet sider an isolated chiral node of a 2D semimetal arising
been explicitly established for arbitrary N. at some point in the BZ from a degeneracy of two elec-
-
d InthisLetter,wecarryoutthistasklocallyintheBril- tron levels, to be denoted a and b. We assume that
n louinzone(BZ),byperformingageneralanalyticalanal- the Hamiltonian for the two-component wave function
o ysis of the edge states in the vicinity of an isolated node. ψˆ=(ψ ,ψ )T has the form
c a b
As the main advancement, we derive the most general
[
form of the asymptotic boundary conditions (BCs) that (cid:18) 0 pN (cid:19)
1 respect chiral symmetry. We find that there are N +1 HˆN(px,py)= pN 0− , p± =px±ipy, (1)
v +
discrete universal classes of them. These classes describe
3
allpossibleuniversalstructuresoftheedgestates(Fig.1) totheleadingorderinmomentum(p ,p )inthevicinity
5 x y
andestablishtheirconnectiontothewindingnumberN. of the node; N is a positive integer.
5
1 We then address the stability properties of the edge The Hamiltonian has chiral symmetry16: under the
0 states and show that they are robust under the effects of transformation
. chiral symmetry breaking. This allows for a significant
1
ψˆ→τ ψˆ, τ =diag(1,−1), (2)
0 z z
7
1 ϵ ϵ ϵ ϵ ϵ it changes its sign, τzHˆN(px,py)τz† =−HˆN(px,py). Due
: tochiralsymmetry,Hˆ ischaracterizedbyawell-defined
v N
i topologicalinvariant,thewindingnumberN,16relatedto
X
the Berry phase πN.
r px px px px px WestressthatthewindingnumberN isalocaltopolog-
a
ical characteristic of the node in the BZ. The properties
(1;0) (0;1) (2;0) (1;1) (0;2)
ψa=0 ψb=0 ψa,∂yψa=0 ψa,ψb=0 ψb,∂yψb=0 oftheedgestatesthatwestudyarealsolocal,andwedo
not address the reasons (symmetry, topology, or other)
(a)N=1 (b)N=2
for the existence of the node. We assume that the node
FIG. 1: Universal local structure of the edge states of 2D is isolated from possible other nodes at different points
chiral-symmetricnodalsemimetals,withtheHamiltonianHˆ in the BZ, also in the presence of the edge; the latter re-
N
[Eq. (1)] and boundary conditions (9) of classes (Na,Nb), il- quires preserved translational symmetry along the edge.
lustrated for (a) N = 1 and (b) N = 2. There are Nb,a flat Universal asymptotic chiral-symmetric boundary con-
edge-state bands (red) at px ≷ 0, respectively; for (2,0) and ditions. WefirstderivethemostgeneralformoftheBCs
(0,2), the degenerate bands are split for visibility. for the Hamiltonian Hˆ that satisfy chiral symmetry.
N
2
This derivation is free of any microscopic assumptions The expressions for the probability current for Hˆ
N
and,besidechiralsymmetry,invokesonlytwonaturalre- [Eq. (1)] read [see Supplementary Material (SM)]
quirements: long-wavelength limit and vanishing of the
probability current perpendicular to the boundary. N(cid:88)−1
j =−i(j −j∗), j = (pˆN−1−nψ )∗pˆnψ . (8)
We assume the sample occupies the half-plane y > y + + + + a − b
0. Since the Schr¨odinger equation Hˆ (pˆ ,pˆ )ψˆ = (cid:15)ψˆ n=0
N x y
(pˆx,y = −i∂x,y) is a differential equation of order N in The bilinear form (7) must vanish identically for any ψˆ.
each component ψ , BCs at the boundary y = 0 are
a,b Inspecting Eq. (8) (see SM), we find that, for a given
a set of N linear homogeneous (meaning that the lin-
group (N ,N ), only one BC among (5) is allowed, the
a b
ear combinations are equated to zero) relations for the
one with the lowest-order derivatives nullified:
derivatives ∂nψ ≡ ∂nψ (x,y = 0), n = 0,...,N −1
y a,b y a,b
(∂y0ψa,b = ψa,b being the components themselves); we ψa,...,∂yNa−1ψa,ψb,...,∂yNb−1ψb =0. (9)
drop the arguments of the functions in the BCs formulas
for brevity. These are all possible asymptotic chiral-symmetric
The long-wavelength limit means the following. Any current-conserving BCs for the chiral-symmetric Hamil-
linearrelationinvolvingderivativesofdifferentordernec- tonian HˆN [Eq. (1)]. There are N +1 classes (Na,Nb)
essarily contains spatial scales. Consider, for example, a of them. This is the first key result of this work. For
relation∂yψa+l∂y2ψa =0,characterizedbyaspatialscale N =1,2, all BC classes (Na,Nb) are shown in Fig. 1.
l. In the long-wavelength limit, at spatial scales larger Edge states and the winding number N. Further, the
thanl,thesecondterml∂y2ψa becomesnegligibleandthe edge states for the Hamiltonian HˆN [Eq. (1)] and BCs
relation reduces to ∂ ψ = 0. Thus, in the BCs satisfy- (9) can be found explicitly (see SM). Taking the plane-
y a
ing the requirement of the long-wavelength limit, to be wave form ψˆ(x,y) = ψˆ(px,y)eipxx with momentum px
referredtoasasymptoticBCs,onlythederivativesofthe along the edge, we look for solutions to the Schr¨odinger
sameordercanbepresentin one relation. Therefore,for equation Hˆ (p ,pˆ )ψˆ(p ,y) = (cid:15)ψˆ(p ,y) that decay into
N x y x x
a given order n, there is either one BC the bulk as y →+∞. We find N edge-state solutions
b
can∂ynψa+cbn∂ynψb =0 (3) ψˆn(px >0,y)=(1,0)Tyne−pxy, n=Na,...,N −1,
with dimensionless coefficients can,bn, or two BCs at px >0 and Na edge-state solutions
∂ynψa =0 and ∂ynψb =0 (4) ψˆn(px <0,y)=(0,1)Tyne+pxy, n=Nb,...,N −1,
with both derivatives vanishing individually.
at p <0. All solutions have zero energy (cid:15)=0 and thus
x
Demanding the chiral symmetry, we find that BC (3)
represent flat bands.
remains invariant under the transformation (2) only if
Thus, we have shown that for a 2D chiral-symmetric
one of the coefficients c is zero, so that BC (3) re-
an,bn nodal semimetal with both the bulk Hamiltonian
duces to either
[Eq. (1)] and BCs [Eq. (9)] obeying chiral symmetry, a
set of flat-band edge states always exists asymptotically
∂nψ =0 or ∂nψ =0.
y a y b in the vicinity of an isolated node. The sum of the num-
Combinedwiththepossibility(4),wefindthatunderthe bersNa,b oftheedge-statebandsonbothsidespx ≷0off
chiralsymmetrythemostgeneralformoftheasymptotic thenodeinthe1Dedgespectrumisequaltothewinding
BCs is when some N out of 2N derivatives are individu- number N, Eq. (6) and Fig. 1. This is the second key
ally nullified: result of this work.
And so, the total number of the edge-state bands is
∂nψ =0, (λ,n)∈Λ. (5) determined solely by the local in the BZ bulk character-
y λ
istic of the node, the winding number N, irrespective of
Here λ=a,b and n=0,...,N −1, and Λ is a subset of
the chiral BC class (N ,N ), which determines the num-
a b
size N of 2N indices (λ,n) labelling the said derivatives.
bers of the edge-state bands on either side off the node.
These CN types of BCs can be sorted into N + 1
2N The specific class (Na,Nb) is in general determined by
groups (N ,N ) with N =0,...,N, such that
a b a,b the bulk Hamiltonian also away from the node, as well
as by the orientation and microscopic structure of the
N +N =N, (6)
a b edge. Still, (N ,N ) are also topological numbers, since
a b
accordingtothenumbersN ofconstraintsimposedon they cannot be changed by continuously changing the
a,b
the derivatives ∂nψ of a given component. systemparametersorsurfaceorientationwhilepreserving
y a,b
Finally, the hermiticity of the full Hamiltonian de- the chiral symmetry. Changes in the numbers (Na,Nb)
mands that the probability current perpendicular to the of the edge-state bands can occur by changing the sur-
boundary must vanish at the boundary face orientation only when projections of different nodes
onto the surface collapse, which necessarily requires the
j (x,y =0)=0. (7) presence of more than one node in the BZ. The BC class
y
3
(a) ββ⟂0 O(3)LM (b) 2 εϵ+(px) (c) 1 ϵ (d) ϵ0e.2V Freogrimβ02e<we|βw⊥i|l2l+foβcuz2,stohne,swysitthempaisrtiincltehaenndodhaollesebmanimdse.tal
1 Hα-gTSen(c) ε+(px) 0.1 We calculate the edge states for chiral-symmetric BCs
1 ℰ+(px)ℰ-(px) -0.8-0.4 0.4 px
ℰ-(px) px px nm-1 ψa,ψb =0 (11)
2 (b) 0 ε-(px) Hy=gT0e -0.1 of class (1,1), Fig. 1(b). We obtain (see SM) the edge-
0 1 1 βz ε-(px) pz=0 -0.2 state dispersion relations
3 β⟂
(cid:112)
β |β |±β |β |2+β2−β2
FIG. 2: (a),(b),(c) Edge states of the chiral-asymmetric E (p )=2|β | 0 ⊥ z ⊥ z 0p2, (12)
quadratic Hamiltonian Hˆβ [Eq. (10)] with chiral-symmetric ± x ⊥ |β⊥|2+βz2 x
2
BCs(11)ofclass(1,1): (a)Edge-statestabilityphasediagram
√ at p ≷ 0, respectively, Fig. 2(a),(b),(c). In the plane
intheparameterplane(β ,β );dashedline|β |/|β |=1/ 3 x
z 0 z ⊥
(β ,β ) of the chiral-asymmetry parameters, Fig. 2(a),
corresponds to the O(3)-symmetric Luttinger model (LM); z 0
(cid:112)
(b),(c)Edgestates(12)(red)for(βz,β0)/|β⊥|= √13(1,0)and the semimetal region |β0| < |β⊥|2+βz2 consists of
√13(1,1.7) in the stability regions 2 and 1 with two [E±(px)] tbherreseofsutbhreegeidognes-s2ta,1t,e0,balanbdesl:lesdtaabcicliotrydrineggiotno 2th(egrneuenm)-,
andone[E (p )]bands,respectively. (d)Surfacestates(red)
− x
of the LM HˆL(p) [Eq. (13)] with BCs (14) for y >0 sample which contains the point of chiral symmetry β0,z = 0,
atpz =0fortheparametersofHgTewithneglectedinversion and in which both bands E±(px) at px ≷ 0, originating
asymmetry. from the chiral-symmetric edge states of the BC class
(1,1)[Fig.1(b)], persist, Fig.2(b); stabilityregion1(or-
ange),whereonlyoneofthebandsE (p )ononesideoff
± x
(N/2,N/2), however, possible for even N, could be real- px = 0 exists, Fig. 2(c); and region 0 (magenta), where
ized for just a single node in the whole BZ; the BC class the edge states are absent. Regions 2 and 1 and regions
(1,1) will be relevant below. 1 and 0 are separated by the curves |β0| = ±√|β⊥|2−βz2 ,
Stability of chiral-symmetric edge states under break- |β⊥|2+βz2
respectively; these boundaries are where the edge states
ing of chiral symmetry. The found edge states of a 2D
merge with the bulk states, E (p )=ε (p ).
chiral-symmetric nodal semimetal [Eqs. (1) and (9)] are ± x ± x
Extended notion of 2D and 3D topological nodal
stable under the effects of chiral-symmetry breaking. In
semimetals. The above findings offer a significant exten-
general,boththebulkHamiltonianandBCsmaycontain
sion of the notion of a 2D topological nodal semimetal:
terms that break chiral symmetry, making them deviate
onemayregarda 2Dnodalsemimetal astopologicalif it
fromtheirchiral-symmetricforms(1)and(9). Aschiral-
belongs to the stability region of some chiral-symmetric
asymmetrictermsareintroduced,theedgestatescandis-
2Dnodalsemimetal. Theedgestatesofthelatterareen-
appear only by merging with particle or hole continua of
suredbyawell-definedtopologicalinvariant,thewinding
the bulk states. Since for preserved chiral symmetry the
number N. Yet exact chiral symmetry is not required,
flat edge-state bands are positioned at (cid:15) = 0, it takes fi-
and the edge states will persist in the former as long as
nitestrengthofchiral-asymmetrictermstoforcetheedge
chiral-symmetric terms are dominant. The above exam-
statesmergewiththebulkstates. Wethusintroducethe
ple, Eqs. (10) and (11), Fig. 2, is a 2D topological nodal
notionofthestabilityregionofthechiral-symmetricedge
semimetal in this sense. This definition is then readily
states: it is a finite-size region in the parameter space of
extendedto3D:onemayregarda3Dnodalsemimetalas
chiral-asymmetric terms around the point of chiral sym-
topological, if its 2D reductions to the planes in momen-
metry, within which the edge states persist. This is the
tumspacepassingthroughthenode(s)are2Dtopological
third key result of this work.
nodalsemimetalsintheabovesense. Inthiscase,the3D
We illustrate the effect of the chiral asymmetry of the
nodalsemimetalwillexhibitsurfacestatesoftopological
bulk Hamiltonian for N = 2. We assume that the N- origin18.
node of Hˆ is “preserved” in the sense that terms of
N This viewpoint has wide-reaching implications, since
lower order than N in momentum are not allowed (they
chiral terms are ubiquitous in the Hamiltonians of 2D
would lead to a gap opening or splitting of the N-node
and 3D nodal semimetals, though exact chiral symme-
into lower-order nodes). So, as the simplest example, we
try is not necessarily present. It allows one to prove
consider the Hamiltonian
the existence and topological origin of the edge or sur-
(cid:18)(β +β )p p β p2 (cid:19) face states in semimetal systems by relating them to 2D
Hˆβ(p ,p )= 0 z + − ⊥ − (10) chiral-symmetric models, even in cases when a precise
2 x y β p2 (β −β )p p
⊥ + 0 z + − topological invariant may be hard or impossible to de-
fine.
with complex β and real β . The terms due to β
⊥ 0,z 0,z Luttinger model for j = 3 electrons as a 3D topological
breakthechiralsymmetry;atβ =0,Hˆβ reducestothe 2
0,z 2 semimetal. One important example of a 3D semimetal
chiral-symmetric form Hˆ2 [Eq. (1)]. The bulk spectrum with a quadratic node that may be regarded as topolog-
of Hˆβ is ε (p )=(β ±(cid:112)|β |2+β2)p2, p2 =p2+p2. ical in this new sense is the 4-band Luttinger model17
2 ± ⊥ 0 ⊥ z ⊥ ⊥ x y
4
(LM) for electrons with j = 3 angular momentum: Since for spherical symmetry the same holds for any
2
other orientation of the momentum plane, we conclude
HˆL(p)=(α0+52αz)p2ˆ14−2αz(Jˆ·p)2+α(cid:3)Mˆ(cid:3)(p). (13) that the O(3)-symmetric 3D LM exhibits 2D surface
states in the whole nodal semimetal regime |α |<2|α |,
0 z
Here,ˆ14istheunitmatrixoforder4,andJˆ =(Jˆx,Jˆy,Jˆz) two bands for |α0|<|αz| and one band for |αz|<|α0|<
are the angular-momentum matrices. 2|αz|, for any orientation of the surface and any direc-
It describes the local electron band structure around tion of 2D momentum along the surface. According to
the Γ point of a material with full cubic point group theabovestabilityarguments,thisresultalsoholdsupon
O with inversion and time-reversal symmetry. It is the includingthecubic-anisotropytermα(cid:3)Mˆ(cid:3)(p),aslongas
h
most general form up to quadratic order in momentum α(cid:3) is small enough. We have thus proven that the Oh-
p = (p ,p ,p ) allowed by these symmetries. All four symmetric LM HˆL(p) [Eq. (13)] with the BCs (14) is a
x y z
j = 3 states are degenerate at p = 0 due to O sym- 3Dtopologicalnodalsemimetalinthesenseofthiswork.
2 h
metry. Odd-p terms are prohibited by inversion. The This is the fourth key result of this work.
termsp2ˆ1 and(Jˆ·p)2 areinvariantsofthefullspherical Our results on the LM are relevant to a multitude of
4
symmetry group O(3) with inversion. Their linear com- realmaterialseitherwithexactcubicsymmetryO orin
h
bination HˆL(p)| , characterized by two parameters whichdeviationsfromitaresmall. Amongmaterialswith
α0,z, is the O(3)α-(cid:3)sy=m0metric LM; its bulk spectrum has Ohthatexhibitaquadraticnodeisα-Sn19–21;aprimeex-
twodouble-degeneratebandsεL,O(3)(p)=(α ±2|α |)p2, ample of a material with weakly broken Oh is HgTe19–21
± 0 z with a tetrahedral point group T . The LM parameters
p=|p|; |α |<2|α | is the nodal semimetal regime. The d
0 z
for α-Sn and HgTe with neglected inversion asymmetry,
additionaltermMˆ(cid:3)(p)=Jˆx2p2x+Jˆy2p2y+Jˆz2p2z−25(Jˆ·p)2− extracted from Ref. 21, are (α ,α ) = (9.31,5.94)/m
15Jˆ2p2 with parameter α(cid:3) is a cubic anisotropy term, and (α0,αz,α(cid:3)) = (7.28,4.29,−00.4z4)/me, respectivelye,
which arises from lowering the symmetry from spherical
where m is the electron mass. For O(3) symmetry, they
to cubic, O(3)→O . e
h both belong to region 1, as indicated in Fig. 2(a), and
The LM Hamiltonian HˆL(p) must be supplied thus exhibit one band of 2D surface states. Fig. 2(d)
by proper physical BCs. We find (see SM) that shows the surface states for y > 0 sample at p = 0 for
z
the asymptotic BCs for the wave function ψˆL = the parameters of HgTe including the cubic anisotropy
(ψ+L3,ψ+L1,ψ−L1,ψ−L3)T (subscripts indicate jz) of the α(cid:3). Recently, a quadratic node was predicted and likely
LM2follow2ing f2rom t2he 6-band Kane model with hard- observedinPr2Ir2O722; accordingtoourfindings,oneor
wall BCs, describing an interface with vacuum, have the two bands of surface states can be anticipated for this
form material, although a separate analysis would be desir-
able. We thus predict that α-Sn, HgTe, and many other
ψˆL =ˆ0. (14) semimetalmaterialsdescribedbytheLMaretopological
in the sense of this work.
The Kane model describes materials like α-Sn and HgTe Deviations from cubic symmetry O due to breaking
h
(see below). of inversion, rotational (strain, confinement), or time-
At p = 0, HˆL(p ,p ,0) is block-diagonal: the pairs reversal (magnetism) symmetries modify the low-energy
z x y
(ψL ,ψL ) and (ψL ,ψL ) of states decouple; the BCs band structure, causing the quadratic node of the LM to
+3 −1 +1 −3
(14)2lead2 to the ch2iral-s2ymmetric BCs (11) of class gap out or split into linear nodes. A variety of resulting
(1,1) for each pair. For O(3) symmetry, the respec- topological phases, such as a topological insulator24–26,
tive 2×2 blocks of HˆL(p ,p ,0)| are of the form Weyl semimetal27, and quantum anomalous Hall insula-
x y α(cid:3)=0 tor28, has been predicted or observed. According to our
Hˆβ(p ,p ) [Eq. (10)] of opposite chiralities, with pa-
2 x y √ findings,theO -symmetricquadraticnodalsemimetalof
h
rameters β = α and β = − 3α . The LM is
0,z 0,z ⊥ √ z the LM [Eqs. (13) and (14)] can be regarded as the par-
thus on the line |βz|/|β⊥| = 1/ 3 in the parameter ent, highest-symmetry topological phase for these phases,
plane (β ,β ) of Hˆβ, Fig. 2(a), and always belongs to with its own surface states of topological origin.
0 z 2
the stability regions 2 or 1, as determined by the ra- Future directions. Relations between the topological
tio |α0|/|αz|; |α0| = |αz| is the transition point between properties of these phases is an interesting future direc-
regions 2 and 1. For y > 0 sample, the surface-state tion. Among other possible applications and extensions
dispersion relations for (ψ+L3,ψ−L1) are E±L,O(3)(px,pz = ofthisworkare: relationofthelocalintheBZproperties
0)= √3(√3α ±sgnα (cid:112)4α22−α22)p2 at p ≷0, respec- of the edge and surface states established here to their
2 0 z z 0 x x globalproperties,suchasthoseof3DWeylsemimetals16;
tively [Eq. (12)], shown in Fig. 2(b) and Fig. 2(c) for
addressing the edge states in graphene and similar 2D
α = 0 and the parameters α /α = 1.7 of HgTe (see
0 0 z systems1–16 within this framework; the role of electron
below) belonging to regions 2 and 1, respectively. When
interactions for the edge and surface states in quadratic
combined with the dispersion relations for (ψL ,ψL ),
+1 −3 nodalsemimetalsin2D(bilayergraphene)and3D(LM),
2 2
EL,O(3)(p ,p = 0) at p ≷ 0, respectively, full surface- whereinteractionsarepredicted29–35toresultininterest-
∓ x z x
state bands of the LM are obtained. ing physics in the bulk.
5
Acknowledgements. We appreciate discussions with cialsupportfromtheDFGviaSFB1170“ToCoTronics”
B. Trauzettel and S. Juergens. We acknowledge finan- andtheENBGraduateSchoolonTopologicalInsulators.
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6
Supplementary Material
I. PROBABILITY CURRENT
Here, we derive the probability current
j=(j ,j )
x y
for the chiral-symmetric Hamiltonian Hˆ [Eq. (1)]. The procedure is standard and analogous to the one for the
N
conventional quadratic Hamiltonian1. The probability density
ρ=ψˆ†ψˆ=ψ∗ψ +ψ∗ψ (S1)
a a b b
of a wavefunction ψˆ=(ψ ,ψ )T satisfying the time-dependent Schr¨odinger equation
a b
(cid:26)i∂ ψ =pˆNψ ,
i∂ ψˆ=Hˆ ψˆ⇔ t a − b (S2)
t N i∂ ψ =pˆNψ
t b + a
must satisfy the continuity equation
∂ ρ+∇·j=0, ∇=(∂ ,∂ ).
t x y
In the integral form
(cid:90) (cid:90)
dr∂ ρ+ dsj=0, (S3)
t
V ∂V
the rate of change of the probability in a 2D space region V must be compensated by the current flow through the
(cid:82)
boundary ∂V of the region. Substituting the expressions (S2) for ∂ ψ into ∂ ρ and integrating dr∂ ρ in parts
t a,b t V t
sufficient number of times, we arrive at
j =j +j , j =−i(j −j ), j =j∗, (S4)
x + − y + − − +
with j given in Eq. (8).
+
In particular,
N =1: j =ψ∗ψ ,
+ a b
N =2: j =ψ∗pˆ ψ +(pˆ ψ )∗ψ .
+ a − b + a b
The formulas (8) and (S4) can naturally be understood as follows. For a plane-wave function ψˆ ∼ ei(pxx+pyy) the
current is given by the derivatives of the Hamiltonian over momentum,
∂Hˆ ∂Hˆ
j = N,ab =NpN−1, j = N,ba =NpN−1,
+ ∂p − − ∂p +
− +
and Eqs. (8) and (S4) represent the properly symmetrized operator version of this.
II. BOUNDARY CONDITIONS, CURRENT CONSERVATION CONSTRAINTS
Here, we prove in more detail how the current conservation constraint (7) restricts the allowed asymptotic chiral-
symmetric BCs from the most general form (5) to the final form (9).
OnlysuchformofBCsisallowedthat,foranywavefunctionψˆ=(ψ ,ψ )T satisfyingthem,thecurrentcomponent
a b
j (x,0)=0[Eq.(7)]perpendiculartotheedgevanishesidenticallyattheedgey =0. Thecurrentcomponentj (x,0)
y +
in j [Eq. (8)] is a sum of the terms
y
∂xnax∂ynayψa∗(x,0)∂xnbx∂ynbyψb(x,0) (S5)
7
with nonnegative integers n =0,...,N −1 such that
ax,ay,bx,by
n +n +n +n =N −1
ax ay bx by
and j (x,0)=j∗(x,0) is the sum of the corresponding conjugate terms. Since for chiral symmetry the only allowed
− +
forms(5)ofBCsarewhensomeindividualderivatives∂nψ (x,0)vanish(whilelinearrelations(3)betweendifferent
y a,b
components are prohibited), the current j (x,0) can be nullified only if all the terms (S5) vanish individually.
y
The terms (S5) involve derivatives both perpendicular to (∂ ) and along (∂ ) the edge. Since these terms must
y x
vanish at any point (x,0) along the edge, this is equivalent to vanishing individually of the terms
∂ynayψa∗(x,0)∂ynbyψb(x,0) (S6)
identically for all x for all n =0,...,N −1 such that
ay,by
n +n ≤N −1.
ay by
This is possible for chiral-symmetric BCs (5) only if, for given (N ,N ), the lowest-order derivatives are nullified at
a b
the edge y =0, as expressed in Eq. (9).
III. EDGE STATES FOR CHIRAL SYMMETRY, DETAILS
Here, we provide details of the derivation of the edge states for chiral-symmetric Hamiltonian (1) and BCs (9).
Taking the plane-wave form
ψˆ(x,y)=ψˆ(p ,y)eipxx
x
with momentum p along the edge, we first look for the general solution to the Sch¨odinger equation
x
Hˆ (p ,pˆ )ψˆ(p ,y)=(cid:15)ψˆ(p ,y).
N x y x x
We find the edge states at energy (cid:15)=0 and have checked that there are no other edge states at (cid:15)(cid:54)=0. At (cid:15)=0, the
components are decoupled and we get the equations
(p +∂ )Nψ (p ,y)=0, (p −∂ )Nψ (p ,y)=0.
x y a x x y b x
There are N independent solutions for each component:
ψ (p ,y)=yne−pxy, ψ (p ,y)=yne+pxy, n=0,...,N −1.
an x bn x
At p > 0, ψ (p ,y) decay and grow into the bulk, y → +∞, respectively, and so, only the solutions with finite
x an,bn x
ψ (p ,y)andvanishingψ (p ,y)≡0componentsareallowed. ApplyingthechiralBCs(9),wegetthatN boundary
a x b x a
conditions for ψ (p ,y) yield N −N =N independent edge-state solutions ψˆ (p >0,y), n=N ,...,N −1, with
a x a b n x a
(cid:15)=0. Similarly, at p <0, we find N −N =N edge-state solutions ψˆ (p <0,y), n=N ,...,N −1, with (cid:15)=0,
x b a n x b
provided in the Main Text.
IV. EDGE STATES FOR CHIRAL-ASYMMETRIC HAMILTONIAN Hˆβ(p ,p ), DETAILS
2 x y
Here,weprovidedetailsofthederivationoftheedgestatesforchiral-asymmetricquadraticHamiltonianHˆβ(p ,p )
2 x y
[Eq. (10)] and chiral-symmetric BCs (11). Taking the plane-wave form
ψˆ(x,y)=ψˆ(p ,y)eipxx
x
with momentum p along the edge, we first look for the general solution to the Sch¨odinger equation
x
Hˆβ(p ,pˆ )ψˆ(p ,y)=(cid:15)ψˆ(p ,y). (S7)
2 x y x x
Its characteristic equation
det[Hˆβ(p ,p )−(cid:15)ˆ1 ]=0
2 x y 2
8
has four momentum solutions
(cid:114) (cid:15)
p =±i p2 − ,
y x β
±
(cid:112)
where β = β ± |β |2+β2 are the curvatures of the electron and hole bulk bands ε (p ) = β p2; we consider
± 0 ⊥ z ± ⊥ ± ⊥
the nodal semimetal regime, where β >0 and β <0, and (cid:15) such that p2 > (cid:15) .
+ − (cid:113) x β±
The partial solutions to Eq. (S7) corresponding to the pair p =i p2 − (cid:15) of momentum solutions are
y x β±
(cid:32) (cid:113) (cid:33)
ψˆ±(px,(cid:15))e−(cid:113)p2x−β(cid:15)±y, ψˆ±(px,(cid:15))= 2p2x− β(cid:15)(cid:15)±(1+−2pβx0+βpz2x)− β(cid:15)± .
β⊥ β±
For the sample at y > 0, these solutions decay into the bulk and are admitted, while the partial solutions with
(cid:113)
p =−i p2 − (cid:15) grow into the bulk and are prohibited.
y x β±
Applying the BCs (11) to the linear combination
ψˆ(px,y)=C+ψˆ+(px,(cid:15))e−(cid:113)p2x−β(cid:15)+y+C−ψˆ−(px,(cid:15))e−(cid:113)p2x−β(cid:15)−y
of the decaying solutions, we find that a nontrivial solution with nonzero C exists when
±
ψ (p ,(cid:15))ψ (p ,(cid:15))−ψ (p ,(cid:15))ψ (p ,(cid:15))=0.
+a x −b x +b x −a x
Solving this equation with respect to (cid:15), we obtain the edge-state dispersion relations E (p ) [Eq. (12)] at p ≷ 0,
± x x
respectively, and find the phase diagram in the plane (β ,β ) of chiral-asymmetry parameters, presented in Fig. 2(a).
0 z
V. LUTTINGER MODEL FROM KANE MODEL
A. 6-band Kane model
For studying the surface states of the 4-band LM for j = 3 states, its bulk Hamiltonian HˆL(p) [Eq. (13)] must be
2
supplemented by proper physical BCs. In this paper, we derive the asymptotic BCs for the LM that follow from a
more general Kane model2,3 (KM) with “hard-wall” BCs. The 6-band KM includes, in addition to j = 3 quartet, a
2
j = 1 doublet of opposite inversion parity and describes a large family of semiconductor materials2–4, in which j = 3
2 2
statesoriginatefromaporbitalinthepresenceofspin-orbitinteractionsandj = 1 statesoriginatefromansorbital.
2
Considering the KM is instructive from a more general standpoint, for the purpose of demonstrating a systematic
“folding”procedure,wherethehigh-energyj = 1 statesofalargerHilbertspaceoftheKMareconsistentlyeliminated
2
to generate the effective bulk Hamiltonian and BCs of the LM with the smaller Hilbert space that contains only the
low-energy j = 3 states.
2
So, the Hamiltonian and the wave function of the KM have the general block structure
(cid:32) (cid:33)
Hˆ (p) Hˆ (p)
HˆK(p)= Hˆ1212(p) Hˆ1232(p) , p=(px,py,pz), (S8)
31 33
22 22
 Ψ 
Ψˆ =(cid:32)ΨΨˆˆ231 (cid:33), Ψˆ12 =(cid:18)ΨΨ121,,−+121 (cid:19), Ψˆ32 =ΨΨ32233,,,−++31221 , (S9)
2 2 2 Ψ2 2
3,−3
2 2
in the space of j = 1 and j = 3 states; here, j =±1,±3 denote the angular momentum projections on the z axis.
2 2 z 2 2
LiketheLM,theKMdescribesthelocalelectronbandstructurearoundtheΓpointp=0. Forfullcubicsymmetry
O with inversion and time reversal symmetry, the most general form up to quadratic order in p reads
h
Hˆ (p)=(∆+γ p2)ˆ1 ,
11 1 2
22 2
9
 (cid:113) 
Hˆ2132(p)=v−√102p+ −√1236ppz+ (cid:113)√1623pp−z √120p− , Hˆ3212(p)=Hˆ†2132(p),
Hˆ33(p)=γ0p2ˆ14+γzMˆ(p)+γ(cid:3)Mˆ(cid:3)(p),
22
√ √
p p −2p2 − 32p p − 3p2 0 
+√− z − z − √
Mˆ(p)= 25p2ˆ14−2(Jˆ·p)2 = −−√323pp+2+pz −p+p√−0+2p2z −p√+p−0+2p2z √−32p3−p2−pz ,
0 − 3p2 32p p p p −2p2
+ + z + − z
2 1
Mˆ(cid:3)(p)=Jˆx2p2x+Jˆy2p2y+Jˆz2p2z− 5(Jˆ·p)2− 5Jˆ2p2,
√
0 3 0 0  +3 0 0 0 
2
Jˆ =(Jˆx,Jˆy,Jˆz), Jˆ± =Jˆx±iJˆy, Jˆ+ =00 00 20 √03, Jˆ− =Jˆ+†, Jˆz = 00 +012 −01 00 , Jˆ2 = 145ˆ14.
2
0 0 0 0 0 0 0 −3
2
Here and below, ˆ1 denotes the unit matrix of order n.
n
This form follows from the method of invariants2,3 (k·p method). The j = 3 and j = 1 states form a four- and
2 2
a two-dimensional (projective) irreducible representation of O , respectively. They correspond to four- and two-fold-
h
degenerate levels at p=0, which we take to be at energies (cid:15)=0 and (cid:15)=∆, respectively. Due to opposite inversion
parities of the j = 3 and j = 1 states, the cross-product block Hˆ (p) contains only odd powers of p, while the
2 2 13
22
self-productblocksHˆ (p)andHˆ (p)containonlyevenpowersofp. Thecross-productblockHˆ (p)containsone
11 33 13
22 22 22
linear-in-p invariant, with the velocity coefficient v, which is real due to time-reversal symmetry. Within the j = 1
2
states, the block Hˆ (p) contains one invariant ˆ1 p2 quadratic in p. Within the j = 3 states, the block Hˆ (p)
11 2 2 33
22 22
contains three invariants p2ˆ14, (Jˆ·p)2, Mˆ(cid:3)(p) of Oh and time-reversal symmetry quadratic in p. Understandably,
the block Hˆ (p)=HˆL(p)| has the structure of the LM (13), since this is the most general form allowed
2332 α0,z,(cid:3)=γ0,z,(cid:3)
by symmetry.
In fact, all the above invariants of Oh, except for Mˆ(cid:3)(p), are also invariants of the full spherical symmetry group
O(3) with inversion. Therefore, the KM HˆK(p)|γ(cid:3)=0 without the Mˆ(cid:3)(p) term is the most general form allowed by
O(3) and time-reversal symmetries. The term Mˆ(cid:3)(p) thus represents cubic anisotropy, which arises from lowering
the symmetry O(3)→O ; it transforms as a linear combination of the states of angular momentum 4.
h
B. Effect of hybridization between j = 3 and j = 1 states
2 2
Our main interest is the behavior of j = 3 states at energies |(cid:15)| (cid:28) |∆| close to the j = 3 level (cid:15) = 0. Exactly at
2 2
p = 0, the j = 3 and j = 1 states are decoupled. However, even at small momenta hybridization to j = 1 states
2 2 2
affects the properties of j = 3 states. Therefore, simply neglecting the hybridization Hˆ (p) to j = 1 states in the
2 13 2
22
KM (S8) and considering the block Hˆ3232(p)=HˆL(p)|α0,z,(cid:3)=γ0,z,(cid:3) with “bare” parameters γ0,z,(cid:3) as the Hamiltonian
for j = 3 states would be incorrect.
2
The effect of hybridization is best illustrated by considering momenta p = (0,0,p ). For clarity, we also consider
z
the case of full spherical symmetry O(3) with inversion, putting γ(cid:3) =0; the corresponding quantities will be labeled
withO(3)superscript. TheO(3)-symmetricKaneHamiltonianHˆK,O(3)(0,0,p )≡HˆK(0,0,p )| atagivenp (cid:54)=0
z z γ(cid:3)=0 z
possesses axial symmetry with respect to rotations about the z axis and the states with different j are decoupled.
z
The j =±1 states are present for both j = 3 and j = 1 and there is hybridization between them. For both pairs
z 2 2 2
(Ψ ,Ψ ), the 2×2 Hamiltonian has the form
1,±1 3,±1
2 2 2 2
 (cid:113) 
∆+γ p2 v 2p
Hˆ|Ojz(|3=)21(0,0,pz)= v(cid:113)2p21 z (γ +23γz)p2 . (S10)
3 z 0 z z
10
DuetoO(3)symmetry,thespectrumisisotropic;so,diagonalizingEq.(S10)andreplacingp2 →p2 ≡p2 =p2+p2+p2,
z x y z
we get two double-degenerate bands
(cid:40) (cid:114) (cid:41)
1 8
εK,O(3)(p)= ∆+(γ +γ +2γ )p2± [−∆+(−γ +γ +2γ )p2]2+ v2p2 . (S11)
a,b 2 21 0 z 12 0 z 3
At small momenta, the band originating from the j = 3 level (cid:15)=0 at p=0 has the form
2
2v2
εK,O(3)(p)=(γ +2γ − )p2+O(p4) (S12)
a 0 z 3 ∆
We see that, indeed, due to hybridization, the spectrum is modified compared to the respective band (γ +2γ )p2 of
0 z
the Hˆ (p)| block with bare parameters γ .
2323 γ(cid:3)=0 0,z
On the other hand, the j =±3 states are present only for j = 3 and thus they do not hybridize to j = 1 states.
z 2 2 2
For both Ψ , the scalar Hamiltonian reads
3,±3
2 2
HˆO(3) (0,0,p )=(γ −2γ )p2. (S13)
|jz|=32 z 0 z z
It gives one double-degenerate band
εK,O(3)(p)=(γ −2γ )p2, (S14)
c 0 z
exactly equal to that of the Hˆ (p)| block.
3232 γ(cid:3)=0
C. Folding procedure, effective Hamiltonian for the Luttinger model for j = 3 states
2
To account for the effect of hybridization, a systematic “folding” procedure3 must be performed for both the bulk
Hamiltonian and BCs, where the high-energy j = 1 states are consistently eliminated from the Hilbert space, while
2
the effect of virtual transitions to them is taken into account.
For the bulk Hamiltonian, the procedure is as follows. Excluding Ψˆ [Eq. (S9)] from the Schr¨odinger equation
1
2
HˆK(p)Ψˆ =(cid:15)Ψˆ, we obtain the equation
(cid:32) (cid:33)
1
Hˆ (p)+Hˆ (p) Hˆ (p) Ψˆ =(cid:15)Ψˆ (S15)
3232 3212 (cid:15)ˆ1 −Hˆ (p) 2132 23 32
2 11
22
for Ψˆ . At |(cid:15)|(cid:28)|∆| and γ p2 (cid:28)|∆|, the energy (cid:15) and momentum p should be set to zero in the denominator in the
3 1
2 2
left-hand side. After that, Eq. (S15) becomes an effective Schr¨odinger equation
HˆL(p)ψˆL =(cid:15)ψˆL
for j = 3 states only with the 4-component wave function
2
 ψL 
+3
2
 ψL 
ψˆL = +12 , (S16)
ψL 
 −1 
2
ψL
−3
2
for which
Ψˆ →ψˆL (S17)
3
2
needs to be substituted.
Expectedly, the effective Hamiltonian
1
HˆL(p)=Hˆ (p)+Hˆ (p) Hˆ (p) (S18)
3223 2312 0ˆ1 −Hˆ (0) 1232
2 11
22