The Nielsen-Ninomiya theorem, PT-invariant non-Hermiticity and single 8-shaped Dirac cone

2.6 MB·

by  M. N. Chernodub

The Nielsen-Ninomiya theorem, PT -invariant non-Hermiticity
and single 8-shaped Dirac cone
M. N. Chernodub
CNRS, Laboratoire de Math´ematiques et Physique Th´eorique UMR 7350, Universit´e de Tours, 37200 France and
Laboratory of Physics of Living Matter, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia
The Nielsen-Ninomiya theorem implies that any local, Hermitian and translationally invariant
latticeactionineven-dimensionalspacetimepossessanequalnumberofleft-andright-handedchiral
fermions. We argue that if one sacrifices the property of Hermiticity while keeping the locality and
translation invariance, while imposing invariance of the action under the space-time (PT) reversal
7 symmetry,thentheexcitationspectrumofthetheorymaycontainanon-equalnumberofleft-and
1 right-handedmasslessfermionswithreal-valueddispersion. Weillustrateourstatementinasimple
0 1+1 dimensional lattice model which exhibits a skewed 8-figure patterns in its energy spectrum.
2 A drawback of the model is that the PT symmetry of the Hamiltonian is spontaneously broken
n implyingthattheenergyspectrumcontainscomplexbranches. WealsodemonstratethattheDirac
a coneisrobustagainstlocaldisordersothatthemasslessexcitationsinthisPT invariantmodelare
J not gapped by random space-dependent perturbations in the couplings.

5
2
I. INTRODUCTION However, it was later found that a large special class
] of non-Hermitian Hamiltonians, H (cid:54)= H†, unexpectedly
l
al Chiral fermions play un increasingly important role possess entirely real energy spectrum [11]. These local
and translationally invariant Hamiltonians are required
h in modern physics. One can mention (nearly) massless
to be invariant under a combined action of the space
- neutrinos in (astro)particle physics and cosmology [1],
s (P) and time (T) reversals, [PT,H] = 0. Their energy
e light quarks in quark-gluon plasma which emerges in
spectrumstaysrealapartfromcertaincasesinwhichthe
m early Universe and heavy ion collisions, condensed mat-
PT symmetry is broken spontaneously [12].
ter physics of graphene [2], Weyl/Dirac semimetals [3]
.
t A rapid development of experimental technology led
a andliquidhelium[4]. Themasslessfermionicexcitations
toasurgeofinterestinPT-invariantnon-Hermitiansys-
m may appear in discretized spaces such as naturally or-
tems, bothintheoreticalandexperimentalcommunities,
dered crystal structures of real materials in solid state
- covering broad areas of photonic crystals [13–16], non-
d physicsorinthelatticemodelsthatareusedtosimulate
n nonperturbative theories of particle physics. topological superconductors [17], ultracold atoms [18] to
o mention a few.
One of the most fundamental statements in physics
c In our paper we attempt to circumvent the Nielsen-
[ of lattice chiral fermions is the Nielsen-Ninomiya theo-
Ninomiya theorem by constructing a PT-invariant non-

1 remwhichstatesthatineveryphysicallymeaningfuldis- Hermitian Hamiltonian that possesses fermionic excita-
cretized theory in even space-time dimensions the chiral
v tionsofthesamehandedness. Ourchoiceissupportedby
fermions should always come in pairs of left- and right-
6 the following three arguments. First, the non-Hermitian
2 handed chiralities thus keeping the net chirality of the nature of the Hamiltonian makes it impossible to apply
4 lattice fermions equal to zero [5]. In many cases the
theNielsen-Ninomiyatheoremsothatthenetnumberof
7 fermion doubling is an undesirable property that blocks
right-andleft-handedfermionsmaybenonzero. Second,
0 investigation of certain interesting systems that possess
the PT-invariance may guarantee that (at least, a part
.
1 excitations (particles) of only one chirality (for example,
of) the energy spectrum of these fermions is real. And,
0 neutrinos, which are always left-handed).
finally, third, we notice that many PT systems natu-
7
Since the Nielsen-Ninomiya no-go theorem is applied rally possess unidirectional transport [14, 15, 19] what
1
: to a broad class of chiral lattice Hamiltonians that are matches perfectly with expected properties of the sys-
v (i)translationallyinvariant, (ii)localand(iii)Hermitian tems with the fermions of the same handedness.1 The
Xi operators,therewerevariousattemptstocircumventthe non-Hermiticity is often associated with unidirectional-
theorem by abandoning the Hermiticity property of the ity [20].
r
a model while keeping the other requirements. The structure of our paper is as follows. In Sect. II
The early ideas to avoid the fermion doubling [6, 7] we briefly review the PT symmetry in a continuum the-
with the help of various non-Hermitian Hamiltonians ory following Ref. [12] and then we discuss its realiza-
weredismissedeitherbecauseatasubsequentcloserlook
the models turn out to exhibit doubling [8, 9] or due
to emerging inconsistencies at the level of perturbation
1 For example, in (1+1) dimensions right- and left-handed fermi-
theory [10]. Moreover, the non-Hermitian Hamiltonians
ons are associated with, respectively, right- and left-movers. If
usually have only complex energy spectra thus making thespectrumcontainsfermionsofthesame(say,right)handed-
their treatment and interpretation of the results rather ness then the fermion-mediated transport in the system would
difficult. beunidirectional(inourexample,intherightdirectiononly).
2
tion in two simplest tight-binding chain Hamiltonians in with one type of lattice sites A labelled by the index l,
l
one spatial dimension. In Sect. III we calculate the en- Fig. 1. Here a† and a are, respectively, creation and an-
l l
ergyspectrumofasuitableinfinite-chainmodelandshow nihilation operators satisfying the fermionic anticommu-
that in certain parameter range the spectrum contains a tation relation {a ,a†} = δ . The hopping parameters
l l(cid:48) ll(cid:48)
tiltedDiracconewithlinesclosedinafigure-8curve. We t+ and t− correspond to the hops of electrons in positive
show that the fermion solutions corresponding to linear and negative directions along the chain lattice, Fig. 1.
dispersions have the same handedness. In addition to
the8-typeDiraccurvetheenergyspectrumcontainstwo t+
complex arcs which connect upper and lowers parts of A A A A A
-1 0 1 2 3
the “8”. Our conclusions are given in the last section.
t–
II. HERMITICITY VS PT-INVARIANCE FIG.1. Uniformone-dimensionalchainwithonetypeofsites
described by the Hamiltonian (7).
A. PT symmetry in continuum theory
Applying a Hermitian conjugation to the Hamilto-
nian (7) one gets:
TheparityoperatorP flipsallspatialcoordinatesand,
consequently,invertsthesignsofcoordinateandmomen- (cid:88)(cid:104) (cid:105)
H† =− (t+)∗a† a +(t−)∗a† a , (8)
tum operators, 1 l−1 l l+1 l
l
PxˆP =−xˆ, PpˆP =−pˆ, (1)
wherewerearrangedthevariablel→l±1appropriately.
According to Eqs. (7) and (8) the Hermiticity of the
thus leaving intact the commutation relation:
Hamiltonian, H† = H , implies the following simple re-
[xˆ,pˆ]=i. (2) 1 1
lation between the hopping parameters:
Thetime-reversaloperatorP flipsthetemporalcoordi- Hermiticity: t− =(t+)∗. (9)
nate, t→−t. It leaves the spatial coordinate unchanged
while reversing the sign of the momentum: Theparitytransformation(1)andthetimereversal(3)
and (4) act on creation/annihilation operators and the
TxˆT =xˆ, TpˆT =−pˆ. (3)
hopping parameter as follows:
InordertokeepcompatibilityoftheT operation(3)with Pa P =a , Pa†P =a† , Pt±P =t±, (10)
thecommutationrelation(2),thetime-reversaloperation l −l l −l
must also change the sign of the complex number i: TalT =al, Ta†lT =a†l , Tt±T =(t±)∗. (11)
TiT =−i. (4) The PT transformed Hamiltonian (7) then reads,
(cid:88)(cid:104) (cid:105)
In particular, for a complex number c one gets: HPT =− (t+)∗a† a +(t−)∗a† a , (12)
1 l−1 l l+1 l
TcT =c∗. (5) l
where we have inverted the summation variable l→−l.
Relations (4) and (5) demonstrate that the time-reversal
ComparingtheHamiltonians(12)and(8)weconclude
T is not, actually, a linear operator (it is often called as
that for the simplest chain model (8) the PT invariance
“anti-linear” operator).
imposes the following condition on the hopping parame-
The operators P and T commute with each other,
ters:
[P,T] = 0, and, obviously, P2 = T 2 = 1. For brevity, a
PT transformationofanoperatorOisdefinedasfollows: PT invariance: t− =(t+)∗. (13)
OPT =(PT)O(PT). (6) which coincides with the condition for Hermiticity (9).
Therefore, in the simplest uniform model (7) the PT in-
A PT-invariant Hamiltonian obeys H =HPT. variance implies Hermiticity and vice versa, HPT ≡H†.
Below we will examine a difference between the prop- 1 1
In other words, it is impossible to construct simultane-
erty of Hermiticity and the PT-invariance in tight-
ously non-Hermitian and PT invariant model using only
binding lattice systems. For simplicity, we work with
one (sub)lattice of sites.
one-dimensional lattice Hamiltonians.
C. Chain model with two sublattices
B. Chain model with one type of sites
1. The model
Consider first a simplest chain Hamiltonian,
H =−(cid:88)(cid:16)t+a† a +t−a† a (cid:17) , (7) Now let us consider a model with sublattices of alter-
1 l+1 l l−1 l
nating lattice sites, A and B, as shown in Fig. 2.
l
3
t+ t+ eters:
BB AA
B-1 tB–BA0 B0 ttAB+–BA A1 tA–AB1 A2 ttBA–+AB B2 Hermiticity: (cid:0)(cid:0)(cid:0)(cid:0)tttt++B+A+ABAA(cid:1)(cid:1)(cid:1)(cid:1)∗∗∗∗ ==== tttt−−A−A−BABA . (16)
BB BB
FIG. 2. Uniform one-dimensional chain with two types of
The hopping parameters corresponding to the hops for-
sites described by the Hamiltonian (14).
ward and backwards are related by the complex conju-
gation similarly to the simplest uniform model (13).
We allow for the electron to hop between the nearest-
neighbor sites, both in positive and negative directions, 3. PT invariance
from an A site to a nearest B site, and vise versa. The
corresponding hoping parameters are t± and t± , re-
AB BA The P parity inversion (1) works slightly differently
spectively. Here the superscript (±) marks the direction
for the A and B sublattices. From Fig. 3 we deduce that
of the hop and the superscript (AB or BA) corresponds
the parity P flip (1) acts on the creation/annihilation
to the starting and ending sites of the hop. We also
operators as follows:
allow to an electron to make hops between sites of the
same type, by adding into consideration the next-to-the- PalP =a−l, Pa†lP =a†−l, (17)
nearesthoppingparameterst±AA andt±BB,whichdescribe Pb P =b , Pb†P =b† , (18)
nearest-neighbor motion within the same sublattice. l −l−1 l −l−1
whilethetime-reversaloperationT,Eq.(3),leavesthem
The corresponding Hamiltonian is:
intact:
H2 =−(cid:88)(cid:104)t+ABb†lal+t−ABb†l−1al TalT =al, Ta†lT =a†l , (19)
l Tb T =b , Tb†T =b†. (20)
+t+ a† b +t− a†b l l l l
BA l+1 l BA l l
+t+AAa†l+1al+t−AAa†l−1al B-2 A-1 B-1 A0 B0 A1 B1
(cid:105)
+t+ b† b +t− b† b , (14) P
BB l+1 l BB l−1 l
B A B A B A B
1 1 0 0 -1 -1 -2
wherea (a†)andb (b†)aretheannihilationandcreation
l l l l
operators at the A and B sublattices, respectively. FIG. 3. The parity transformation P, Eq. (1), applied to the
alternating lattice of Fig. 2.
The four lines in the Hamiltonian (15) describe the
nearest-neighbor hops, respectively, (i) from the sublat-
Similarly to the case of the uniform Hamiltonian H ,
tice A to B, (ii) from the sublattice B to A; (iii) within 1
the complex hopping parameters are not affected by the
the A sublattice and (iv) within the B sublattice. First
parity transformation while they are changed to their
(second) terms in each line corresponds to hops in posi-
complexconjugatedafterthetime-reversaloperation(5)
tive (negative) direction.
is applied:
Pt± P =t± , Tt± T =(cid:0)t± (cid:1)∗, (cid:96)(cid:96)(cid:48) =A,B. (21)
(cid:96)(cid:96)(cid:48) (cid:96)(cid:96)(cid:48) (cid:96)(cid:96)(cid:48) (cid:96)(cid:96)(cid:48)
ThePT transformedHamiltonian(14)takestheform:
2. Hermiticity
(cid:88)(cid:104)
HPT =− (t+ )∗b† a +(t− )∗b† a (22)
2 AB −l−1 −l AB −l −l
Applying the Hermitian conjugation to the Hamilto-
l
nian (14) one gets: +(t+ )∗a† b +(t− )∗a† b
BA −l−1 −l−1 BA −l −l−1
H2† =−(cid:88)(cid:104)(t+AB)∗a†lbl+(t−AB)∗a†lbl−1 ++((tt++AA))∗∗ab††−l−1ba−l++(t−A(tA−)∗)a∗†−bl†+1ba−l (cid:105).
l BB −l−2 −l−1 BB −l −l−1
+(t+ )∗b†a +(t− )∗b†a
BA l l+1 BA l l Changing the summation variable l → −l or l → −l+1
+(t+AA)∗a†lal+1+(t−AA)∗a†lal−1 for certain lines in Eq. (22), we get:
(cid:105)
+(t+BB)∗b†lbl+1+(t−BB)∗b†lbl−1 , (15) H2PT =−(cid:88)(cid:104)(t+AB)∗b†l−1al+(t−AB)∗b†lal
l
wherewerearrangedthevariablel→l±1appropriately. +(t+BA)∗a†lbl+(t−BA)∗a†l+1bl
According to Eqs. (14) and (15) the requirement of +(t+ )∗a† a +(t− )∗a† a
HtheermfoiltloicwitiyngosfetthoefrHealamtiioltnosnbiaentw(e1e4n),thHe2†ho=ppHin2g, pimarpalmies- +(t+BAAB)∗b†ll−−11bll+(t−BABA)∗b†ll++11bll(cid:105). (23)
4
Next,wecomparetheHamiltonians(14)and(23),and B. Energy spectrum
concludethatthePT invarianceHPT =H imposesthe
2 2
following condition on the hopping parameters: We look for eigenstates of the Hamiltonian (27) in the
 (cid:0)t+ (cid:1)∗ = t− standard form:
 (cid:0)tA+B(cid:1)∗ = tA−B |ψ(cid:105)=(cid:88)(cid:16)α a†+β b†(cid:17)|0(cid:105), (29)
PT invariance: (cid:0)tB+A(cid:1)∗ = tB−A . (24) l l l l
(cid:0)t+AA(cid:1)∗ = tA−A l
BB BB where |0(cid:105) is the vacuum state with a |0(cid:105)=b |0(cid:105)=0.
l l
The constrains that are imposed on the hopping pa- ApplyingEq.(27)to(29)wesolvetheeigenstateequa-
rameters by the Hermiticity (16) and by the PT invari- tion H|ψ(cid:105)=(cid:15)|ψ(cid:105) using the ansatz
ance (24) are not equivalent. In more details, the intra-
α =αeipl, β =βeipl, (30)
latticehoppingparametersbetweenthesitesofthesame l l
sublattices (t± and t± ) are, unsurprisingly, obeying where p is a quasimomentum and α and β are complex
AA BB
the same conditions both for Hermitian and PT sym- numbers. Theeigensystemisdeterminedbythefollowing
metricHamiltonians: thehopsforwardandbackwardare matrix equation:
related to each other by the complex conjugation. How-
H Ψ =(cid:15)(p)Ψ , (31)
ever, the parameters that control inter-lattice hopping p p p
(t± and t± ) satisfy different relations for Hermitian
AB BA where
and PT invariant Hamiltonians. Therefore the model
(cid:32) (cid:33)
with two sublattices (14) allows us to construct a chain H =−2 tAsinp g2e−i2p sinp2 , (32)
Hvaarmiainlttoantiatnhewshaimcheitsimnoet. Hermitian while being PT in- p g1ei2p sinp2 tBsinp
and
(cid:18) (cid:19)
α
III. THE MODEL Ψ = . (33)
p β
A. PT invariant non-Hermitian Hamiltonian In accordance with Eq. (28) the Hamiltonian H in
p
Eq. (32) is indeed a Hermitian operator, H† = H , pro-
p p
We are interested in a PT symmetric non-Hermitian vided g1 =g2.
Hamiltonian. We have eight complex parameters t±(cid:96)(cid:96)(cid:48) The eigenenergies (cid:15)±(p) can be readily determined
with (cid:96),(cid:96)(cid:48) =A,B, which are relatedto each other by four from Eq. (31):
equations (24). Thus we have four independent hopping
(cid:15) (p)=−(t +t )sinp
parameters, or eight real parameters that describe our ± A B
(cid:114)
model. Forthesakeofconveniencewereducetheparam- ±sign(p) (t −t )2sin2p+4hsin2 p, (34)
eter space and choose the following set of parameters: A B 2
t+ =−t− =it , t+ =−t− =it , (25) where
AA AA A BB BB B
t+AB =−t−AB =ig1, t+BA =−t−BA =ig2, (26) h=g1g2. (35)
where tA, tB, g1 and g2 are real-valued numbers. For Hermitian Hamiltonians one always has h(cid:62)0 while
The Hamiltonian (14) with the hopping parame- the region h<0 corresponds to a non-Hermitian case.
ters (25) and (26) reads as follows:
H =−i(cid:88)(cid:104)g (cid:0)b†a −b† a (cid:1)+g (cid:0)a† b −a†b (cid:1)
1 l l l−1 l 2 l+1 l l l C. Weyl modes in a single closed Dirac cone
l
(cid:105)
+t (a† a −a† a )+t (b† b −b† b ) . (27) For a generic set of parameters (t ,t ,h) our model
A l+1 l l−1 l B l+1 l l−1 l A B
(27) possesses gapless solutions (cid:15) (p) = (cid:15)(0)(p)+O(p3)
By construction, the Hamiltonian (27) is a PT in- ± ±
with a linear dispersion relation at the origin p=0:
variant but not Hermitian operator. The special choice
of parameters (25) and (26) allows us to determine the (cid:15)(0)(p)=v p. (36)
± ±
Hermiticity criteria in the simple way: the intra-lattice
coupling parameters (25) of the PT-invariant Hamilto- According to Eq. (34) these modes are propagating with
niansatisfytheHermiticityconditions(16)automatically the following velocities:
while the inter-lattice hopping parameters (26) satisfy (cid:112)
v =−(t +t )± (t −t )2+h. (37)
the Hermiticity conditions (16) if and only if g =g : ± A B A B
1 2
(cid:26)g = g , Hermitian, However, what is most interesting is the global behav-
Hamiltonian: 1 2 (28) iorofthedispersions(34)intheBrillouinzone. Foracer-
g (cid:54)= g , non-Hermitian.
1 2 tain region of the parameters the energy spectrum (34)
5
� =�|� |� �=-���
� � �
�
�
/�� �
-�
-� -� -� � � � �
�
FIG. 4. The energy spectrum (34) with t =1, t =1/4 and h=−1/4. The real-valued branches of energy form the 8-type
A B
closedcurve(shownbytheblueline)whilethecomplexvaluedenergyhasatopologyofthecircle(shownbytheorangeline).
has a real-valued dispersion relation which has a shape tensor. The analogue of the γ5 matrix in (1+1) dimen-
of a skewed 8-figure as it is shown by the blue solid line sions is the following [22]:
in an example of Fig. 4. The upper and lower parts of
(cid:18) (cid:19)
1 0
the 8-shaped energy curve are attached to each other by γ¯ =γ0γ1 =σ ≡ . (41)
z 0 −1
two a complex-valued energy arks (shown by the dashed
orange lines in Fig. 4).
Theprojectorsonpositive(P )andnegative(P )chi-
Intriguingly,the8-shapedspectrumdescribesmassless + +
ralities are:
fermions, and there is precisely one Dirac cone in the
whole Brillouin zone with the real-valued energy disper- 1±γ¯
P = , (42)
sion. This fact means that if massless excitations at the ± 2
conehavethesamehandednessthentheycannotbecom-
or, explicitly,
pensated by excitations from another cone with a differ-
(cid:18) (cid:19) (cid:18) (cid:19)
ent handedness. Below we explicitly demonstrate that 1 0 0 0
P = , P = . (43)
the gapless modes (36) are indeed fermions which can + 0 0 − 0 1
be described by a 1+1 dimensional Dirac equation, and
The projectors satisfy the relations P P = P P = 0
that these modes have indeed the same handedness in a + − − +
and P2 =P .
certain range of the model parameters. ± ±
We diagonalize the Hamiltonian (45),
We expand the Hamiltonian around the origin, H =
p
Hp(0)+O(p3), and get the linearized Hamiltonian H(0) =U†HdiagU, (44)
p p
(cid:18) (cid:19)
2t g
H(0) =− A 2 p. (38) where U is an SU(2) rotation matrix and
p g 2t
1 B
(cid:18) (cid:19)
v 0
that acts on two-component wavefunction (33): Hdiag = + p, (45)
p 0 v
−
H(0)Ψ(0) =(cid:15) Ψ(0). (39)
p p p p is the diagonalized Hamiltonian with the velocities (37).
The corresponding linear dispersions (cid:15) ≡ (cid:15)(0) given in Then we substitute Eq. (44) into (39), and multiply the
p ± resultbythematrixγ0U/(v v )wherev v =4t t −
Eqs. (36) and (37). The superscript “(0)” indicates that + − + − A B
h. We get for the eigenvalue equation (39) the following
Eq. (39) is valid in the vicinity of the origin p=0.
expression:
Equation (39) can be interpreted in terms of a Dirac
equationinwhichthefermionspin,similarlytographene, (cid:2)γµ∂+P −γµ∂−P (cid:3)ψ (t,x)=0, (46)
µ + µ − p
corresponds to an internal degree of freedom associ-
ated with the presence of two sublattices A and B. where the wave function
We choose the following representation for the two-
ψ (t,x)=χ e−i(cid:15)t+ipx, (47)
dimensional Dirac matrices: p p
γ0 =(cid:18)0 1(cid:19) , γ1 =(cid:18) 0 1(cid:19) . (40) is expressed via the spinor χp = UΨ(p0), and the deriva-
1 0 −1 0 tives are defined as follows:
(cid:18) (cid:19)
The matrices (40) satisfy the commutation relations 1 ∂ ∂
∂± = , . (48)
{γµ,γν} = 2gµν, where gµν = diag(1,−1) is the metric µ v ∂t ∂x
±
6
The solutions with positive and negative chiralities, E. The parameter space of the model
ψ =P ψ, (49)
± ±
The parameter space of the model contains five differ-
satisfy the following two-component Weyl equations, entregionsclassifiedbyqualitativefeaturesoftheenergy
γµ∂±ψ (t,x)=0, (50) spectrum. In Fig. 5 we show the energy spectra in dif-
µ ± ferent regions of the parameter space of the model (27)
or, in the explicit form: labelled by the couplings t and h ≡ g g (in units of
B 1 2
(cid:18) (cid:19) coupling t ). The essential features of the energy spec-
1 ∂ ∂ A
γ0v ∂t +γ1∂x ψ±(t,x)=0. (51) tra are as follows:
±
Thus, in each Dirac cone we have two Weyl fermions (cid:73) The model is Hermitian at h (cid:62) 0 and non-
withthedispersions(36),(cid:15) =v pcorrespondingtothe Hermitian at h<0.
± ±
propagation with velocities (37).
(cid:73) In the upper, Hermitian semiplane (the unshaded
region I in Fig. 5) the energy dispersions are given
D. Handedness and chirality of the (1+1) modes by purely real-valued functions. The typical ex-
amples of the energy spectra across this region are
shown in the insets (b), (i), (j) and (h).
Before proceeding further it is important to clarify
the issue of handedness of fermions in (1+1) dimensions.
(cid:73) Thelower(non-Hermitian)semiplanecontainsfour
The fermions have right- ofleft-handed helicity provided
distinct regions:
the projection of the fermion’s spin onto direction of
fermion’s momentum is a positive or negative quantity, (cid:46) In region II (shaded by the blue color) the Dirac
respectively. Strictly speaking, in (1+1) dimensions the cone at p = 0 hosts both left-handed and right-
fermionsdonotcarryadynamicalspindegreeoffreedom. handed fermions. The spectrum possesses two
For example, a fermionic field theory can be bosonized complex-valued arcs that connect the parts of the
and, consequently, represented by a field theory of spin- Diracconeovertheboundaryatp=±π. Anexam-
less bosons. pleisshowninplot(c). RegionIIisdeterminedby
In the absence of spin we cannot, in a strict sense, the couplings satisfying 0>h>4t t and h<0.
A B
define the helicity in (1+1) dimensions. However, it
is customary to associate right-handed and left-handed (cid:46) Inregion IV(shadedbytheredcolor)thespectrum
particles with particles moving in positive (right-movers, does not possess purely real-valued eigenvalues at
v > 0) and negative (left-movers, v < 0) directions. To all. The Dirac cones are located in the imaginary
justifythischoice,onecanimagine,forexample,atheory space as seen from plots (e) and (f). Region IV is
of chiral fermions in 3+1 dimensions subjected to suf- determined by the couplings h<−(t −t )2.
A B
ficiently strong magnetic field directed along the x axis.
In these conditions (cid:46) Inregion III(shadedbytheyellow-greencolor)the
clockwiseskewed8-typeclosedDiracconepossesses
(i) the spin of the fermions is pointed along the posi- twofermionswiththesameright-handedhelicityas
tive direction of the z axis; seen in plot (d). The upper and lower parts of the
figure-8spectrumareconnectedwitheachotherby
(ii) the fermions occupy the lowest Landau level;
thecomplexarksextendingviatheboundaryofthe
(iii) the motion of the fermions is restricted along the Brillouinzoneatp=±π. RegionIIIissandwiched
same axis. in between regions II and IV with the couplings
satisfying 4t t >h>−(t −t )2 and t <−t .
As a consequence, right- and left-handed chiral fermions A B A B B A
in(3+1)dimensionsbecome,respectively,right-andleft- (cid:46) Region V (shaded by the green color) possesses
movers in the reduced (1+1) dimensions in the (t,z) the counterclockwise skewed 8-type spectrum that
planewiththe(1+1)dimensionalenergydispersionrela- hosts two fermions with the same left-handed he-
tion ε=|pz|. licity, an example is shown in plot (g). The region
The Dirac equation (51) implies that the states with isdeterminedbytherelations0>h>−(t −t )2
A B
positiveandnegativechiralitiespropagatewiththeveloc- and t >−t .
B A
ities v and v , respectively, x = v t. Their dispersion
+ − ±
relations are given by Eq. (36). If the velocities v and (cid:73) The energy level E = 0 at p = 0 is always dou-
+
v havethesamesign,thenthehandednessofthecorre- ble degenerate and the spectrum of the model al-
−
sponding branches of solutions is the same because they ways possesses a Dirac cone at the origin p = 0.
propagate in the same direction. Therefore, the positive The Dirac cone corresponds to the real-valued dis-
and negative solutions of the chiral γ operator may be persions in all regions except for region IV where
5
both right-handed or left-handed from the point of view the Dirac cone is entirely located in the imaginary
of their helicity. space.
7
(a) (j)
(b) (i)
(c) (h)
(g)
(e) (f)
(d)
FIG. 5. Energy spectrum in the space of couplings t and h (in units of t > 0). The model is Hermitian at h (cid:62) 0 and
A A
non-Hermitianath<0,theboundaryh=0betweenthesesemiplanesismarkedbyhorizontaldashedgreenline. Theskewed
8-type spectrum (with fermions of the same helicity) appears only in the regions III and V shaded by the greenish colors.
The Dirac cone hosts either two left-handed fermions [examples (a) and (g) in the green area V] or two right-handed fermions
[example (d) in the yellow-green area III]. In the red-shaded region IV the spectrum has no real-valued eigenenergies [(e) and
(f)], while the blue shaded region II possesses excitations with the 8-type spectrum where a Dirac cone hosts fermions of both
helicities (c). In the Hermitian part I the energy spectrum has only real eigenvalues [(b), (h), (i) and (j)]. The regions II
and III are separated by the dashed blue line h = 4t t , while the boundary of the region IV is determined by the relation
A B
h=−(t −t )2 (the red solid curve).
A B
The skewed-8 spectrum with two Weyl excitations of appear and the net helicity of the lattice fermion states
the same helicity are realized in the non-Hermitian re- is zero.
gions III and V.
Ontheotherhand,themodelbecomesnon-unitarybe-
An unwanted property of the model is the presence
cause ofthe appearance ofthe complex energy branches.
of the complex branches in the energy spectrum which
Alternatively, one also can remark that two fermionic
implies that the system is not unitary and the total par-
excitations of the same handedness with real-valued dis-
ticlenumberisnotaconservedquantity. Thissituationis
persions at p = 0 are “compensated” by two fermionic
quitetypicalinopensystemsdescribedbynon-Hermitian
modes of the other handedness with complex energy dis-
Hamiltonians [12].
persionsatp=±π [cf. Figs.5(d)and(g)]. Thecompen-
If the whole energy spectrum were real then, topolog- sation modes are an unstable mode with Im(ω)>0 and
ically, the two states of the same helicity at the central a decaying dissipative mode with Im(ω)<0 which come
Dirac cone would be complemented with other states of always in pairs.
oppositehelicityatanotherconezonesothatthenethe-
licityof thelatticestatesin thecompactBrillouinwould Concluding our survey of the parameter space we no-
be zero. However, in our model example the energy tice that that the most interesting physics is realized in
curves go into the imaginary dimension that allows then the non-Hermitian regions III and V in which the en-
to avoid crossing and form another Dirac cone. Conse- ergyspectrumcontainstheclosedreal-valuedDiraccones
quently,nocompensatinghelicitystateswithrealenergy which are connected by the complex energy arcs.
8
F. Robustness of the single Dirac cone against
local disorder
We are going to demonstrate that the closed 8-shaped
Dirac cone is essentially robust feature of our model.
Firstofall,wenoticethattheveryexistenceoftheDirac
pointinthePT–invariantHamiltonian(27)doesnotde-
pend on global variations of any of the parameters of
the model (27). Indeed, the changes of parameters do
not shift the position of the Dirac point in the Brillouin
zone which always appears at the point (p,E) = (0,0).
The above statement can be supplemented by analytical
calculations that demonstrate the double-degeneracy of
eigenvalues at p=0. This fact can also be seen from the
examplesoftheenergyspectrainFig.5acrossthewhole
FIG. 6. The eigenvalues of the locally disordered Hamilto-
parameter space: the Weyl branches of the Dirac cones
nian (52) with the degrees of disorder d = 0.1,0.2,0.5 and 1
come always in pairs and they are never gapped. The
for the chain of the length 2N with N = 16 for the non-
cones are real-valued everywhere except for region IV in Hermitian central parameters t =4t and g =−g =2t
B A 1 2 A
whichtheDiracconeisentirelylocatedintheimaginary correspondingtothecounterclockwise“figure-8”coneinplot
space. (g)ofFig.5(regionVoftheparameterspace). Thecomplex-
WehavealsocheckedtherobustnessoftheDiracspec- and real-valued energies are shown in the orange and blue
trum against the local disorder. To this end we have colors,respectively. Thegreenopencirclescorrespondtothe
disordered the Hamiltonian (27) at each site, undisordered Hamiltonian with d=0.
N
Hdisord =−i(cid:88)(cid:104)gl,1(cid:0)b†lal−b†l−1al(cid:1)+gl,2(cid:0)a†l+1bl−a†lbl(cid:1) complex-valued arcs (the orange dots) as compared to
l=1 the corresponding undisordered values (the green open
(cid:105)
+t (a† a −a† a )+t (b† b −b† b ) , (52) circles). In other words, the excitations with the same
l,A l+1 l l−1 l l,B l+1 l l−1 l helicity are not affect by the moderate disorder at all.
where the disorder in the site-dependent couplings Asexpected,atlargerdisorderwithd=0.5andd=1,
(cid:112) thecentralDiracconeturnsinthecomplexplanesothat
t = t (1+δ ), g =+ |h|(1+δ ),
l,A A l,A l,1 l,1 thespectrumshiftsfromregionVtothetotallyunstable
(53)
t = t (1+δ ), g =−(cid:112)|h|(1+δ ) region IV.
l,B B l,B l,2 l,2
issimulatedby4N independentfactorsδ whichfluctu-
l,i
aterandomlyandindependentlyateachsitel=1,...,N
IV. CONCLUSIONS
at each type of the coupling i = A,B,1,2. The fluctua-
tions are constrained to the range
We demonstrated that one-dimensional PT-invariant
−d(cid:54)δ (cid:54)d, i=A,B,1,2, l=1,...,N, (54)
l,i non-Hermitian tight-binding systems may contain
where the nonnegative parameter d (cid:62) 0 determines the fermionic excitations with a nonvanishing net handed-
degree of the local disorder. ness and with the real-valued energy spectrum. We pro-
In Fig. 6 we show the distribution of the energy eigen- vided an explicit example (27) of a local PT-invariant
values of the disordered Hamiltonian (52) for various de- non-HermitianHamiltonianwiththedesiredenergyspec-
grees of disorder d at the periodic chain lattice of the trum. In particular, we demonstrated that the Bril-
length 32. The parameters fluctuate randomly around louin zone may contain only one Dirac cone with two
the central values t = 4t and g = −g = 2t which Weyl fermions of the same handedness. The real-valued
B A 1 2 A
correspondtothenon-HermitianregionVoftheparame- branchesofthespectrumformaclosedDiracconewhich
terspace,Fig.5whichexhibitstheskewedfigure-8Dirac resembles visually a skewed figure-8 curve.
cone [a relevant example is shown in plot (g) of Fig. 5]. An example of the closed Dirac cone with two chiral
The zero-energy eigenvalue E = 0 at p = 0 is always modes of the same handedness is visualized in Fig. 4.
present at all disordered configurations of the couplings The upper and lower parts of the real spectrum are con-
so that the presence of the Dirac cone is not affected nected by two complex energy arks that automatically
by the disorder. While the disordered couplings do not exclude the appearance of a compensating Dirac cone
open the gap they may turn the Dirac cone from the with a real-valued energy dispersion. The complex en-
real-valued plane to the complex values plane. ergy arks appear in a region of the Brillouin zone where
One can see that the moderate disorder with d = 0.1 the PT symmetry of the Hamiltonian is broken sponta-
and d=0.2 perturbs only slightly both the central real- neously.
valued Dirac cone (the blue dots) and the connecting ThemodelhastherichphasediagramshowninFig.5.
9
The single closed Dirac cones are realized in two pockets atedisorderdoesnotopenthegapinthe8-shapedDirac
of the parameters space (regions III and V in Fig. 5) at cone which stays therefore protected.
the non-Hermitian part of the phase diagram. In other
words, the Hamiltonian must be non-Hermitian in order
for the figure-8 dispersion to appear. ACKNOWLEDGMENTS
We have also shown that the presence of the two chi- The author is very grateful to Mar´ıa Vozmediano for
ralmodesofthesamehandednessisrobustagainsalocal communications, valuable comments and suggestions,
randomdisorderoftheHamiltoniancouplings. Amoder- and encouragement.
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