A physical interpretation for the non-Hermitian Hamiltonian

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by  L. Jin

A physical interpretation for the non-Hermitian Hamiltonian
L. Jin, and Z. Song
School of Physics, Nankai University, Tianjin 300071, China
Weexploreawayoffindingthelinkbetweenanon-HermitianHamiltonianandaHermitianone.
Based on theanalysis of BetheAnsatzsolutions fora class of non-HermitianHamiltonians and the
scattering problems for the corresponding Hermitian Hamiltonians. It is shown that a scattering
stateofanarbitraryHermitianlatticeembeddedinachainasthescatteringcentersharesthesame
wave function with the corresponding non-Hermitian tight binding lattice, which consists of the
Hermitian lattice with two additional on-site complex potentials, no matter the non-Hermitian is
broken PT symmetry or even non-PT. An exactly solvable model is presented to demonstrate the
main pointsof this article.
1
PACSnumbers: 03.65.Ge, 05.30.Jp,03.65.Nk,03.67.Bg
1
0
2
I. INTRODUCTION certain tight-binding lattice shares the same wave
n functionPwTith a resonanttransmission state of the corre-
a
spondingHermitianlattice. Insuchaframework,further
J In general, a non-Hermitian Hamiltonian is said to be
questions to ask are whether the requirements of the en-
1 physical when it can have an entirely real energy spec-
tireness of the real eigenvalunes and the symmetry
trum. Muchefforthasbeendevotedtoestablishaparity- PT
of the non-Hermitian system are really necessary.
] time ( ) symmetric quantum theory as a complex ex-
h PT In this paper, we propose a physicalinterpretation for
tension of the conventional quantum mechanics [1–8]
p a general non-Hermitian Hamiltonian based on the con-
- since the seminal discovery by Bender [1]. It is found
t figurations involving an arbitrary network coupled with
n that non-Hermitian Hamiltonian with simultaneous
PT the input and output waveguides. Relevant to our pre-
a symmetry has an entirely real quantum mechanical en-
vious discussionis the interpretationof the imagiarypo-
u ergyspectrumandhasprofoundtheoreticalandmethod-
q tentials. Based on this, we make a tentative connection
ological implications. Reseaches and findings relevent to
[ between a non-Hermitian system and the corresponding
the spectra of the symmetric systems are presented,
PT largeHermitiansystem. Itis shownthatforanyscatter-
1 such as exceptional points [9], spectral singularities for
ing state of such a Hermitian system, the wavefunction
v complex scattering potentials [10], complex crystal and
within the center lattice alwayscorrespondsto the equal
1 other specific models [11]have been investigated. At the
5 energy eigenfunction of the non-Hermitian Hamiltonian,
sametime the symmetryis alsoofgreatrelevanceto
3 PT no matter it is symmetric or not. Our formalism is
the technologicalapplications based on the fact that the PT
0 generic and is not limited to the pseudo-Hermitian sys-
imaginary potential could be realized by complex index
1. tem.
in optics [12–16]. In fact, such opticalpotentials can
0 PT This paper is organized as follows. Section II is the
berealizedthroughajudiciousinclusionofindexguiding
1 heartofthis paper which presents a formulismto reduce
andgain/lossregionsandthemostinterestingaspectsas-
1 ascatteringprocessofaHermitioniansystemtotheeigen
: sociatedwith symmetricsystemareobservedduring
v PT problem of the non-Hermitian system. Section III con-
dynamic evolution process [17–20].
i sists of two exactly solvable examples to illustrate our
X Thus one of the waysof extracting the physicalmean- main idea. Section IV is the summary and discussion.
r ing of a pseudo-HermitianHamiltonian with a real spec-
a
trum is to seek for its Hermitian counterparts [21–23].
The metric-operator theory outlined in Ref. [6] provides II. NON-HERMITIAN REDUCTION OF A
a mapping of such a pseudo-Hermitian Hamiltonian to HERMITIAN SYSTEM
anequivalentHermitianHamiltonian. Thus,mostofthe
studies focused on the quasi-Hermitian system, or un-
A typical scattering tight-binding network is con-
broken symmetric region. However, the obtained
structed by a scattering-center network and two semi-
PT
equivalent Hermitian Hamiltonian is usually quite com-
infinite chains as the input and output leads. The well-
plicated [6, 24], involving long-rangeor nonlocal interac-
established Green function technique [26–28] can be em-
tions, which is hardly realized in practice.
ployed to obtain the reflection and transmission coeffi-
Toanticipatetheseproblems,alternativeproposalsfor cients for a givenincoming plane wave. The correspond-
theconnectionbetweenapseudo-HermitianHamiltonian ing wave function within the scattering center should
andarealphysicssystemhavebeensuggestedinthecon- be obtained via Bethe ansatz method. In the following
text of scattering problems [25]. Central to that analy- we will show that this can be done by solving a finite
sis was the recognition that the Hamiltonian may non-Hermitian Hamiltonian. In our previous work [25],
PT
be used to depict the resonant scattering for an infinite a symmetric non-Hermitian Hamiltionian has been
PT
system. It is shown that any real-energy eigenstate of connected to a physical system in the manner that any
2
A B
real-energyeigenstate of a tight-binding lattice with
PT
on-site imaginary potentials shares the same wave func-
tionwitharesonanttransmissionstateofthecorrespond-
ingHermitianlatticeembeddedinachain. Themainaim
of this article is to answer the question of whether such
a statement still holds for broken non-Hermitian or (a)
PT
non- lattice. In the following, we will show that a
PT
scatteringstate of the Hermitian systemalwayshas con-
nection to the eigenstate of its non-Hermitian reduction. U U
Foracertainincidentplanewave,thescatteringproblem (cid:1) (cid:0)
ofthe wholeinfinite Hermitiansystemcanbe reducedto
the eigen problem of a finite non-Hermitian system. (b)
The Hamiltonian of a typical scattering tight-binding
network has the form FIG. 1. Schematic illustration of the configuration of the
concerned network. It consists of an arbitrary graph of a
Hermitian tight-bindingnetwork(shadow) connectingtotwo
H =H +H +H (1) semi-infinite chains L and R as the waveguides. The wave
A B c
function within thescattering centerfor a scattering state of
where thewholesystemisidenticaltoanequal-energyeigenfunction
ofthenon-HermitianHamiltonianwhichisconstructedbythe
centerHermitiannetworkwithimaginarypotentialsaddedat
−∞
HA =−JiX= 1b†i−1bi−gAb†−1aA+H.c. (2) thejoint sites A and B.
−
+
∞
HB =−J b†ibi+1−gBb†1aB +H.c. (3)
Xi=1 Jf Jf =Ef ,
j 1 j+1 j
− − −
represent the left and right waveguides and (j ( , 2] [2,+ )) (8)
∈ −∞ − ∪ ∞
Jf g h =Ef
2 A A 1
N − − − −
Jf g h =Ef .
Hc =− κija†iaj +H.c. (4) − 2− B B 1
iX,j=1 From Eq. (8), we obtain E = 2Jcos(k) and
−
describesanarbitraryN-sitenetworkasascatteringcen-
J
ter. SitesAandBarearbitrarywithinthenetwork. Here h = eik+re ik , (9)
A −
bi, ai, are boson(or fermion) operators, κii (we denote gA (cid:0) (cid:1)
−
V = κ V = κ only for the sake of simplicity) J
reApres−entAsAtheBpote−ntiBalBat site i. Fig. 1(a) represents a hB = g te−ik. (10)
B
schematic scattering configuration for an arbitrary net-
Vanishing h (h ) is beyond of our interest. From Eqs.
work. A B
(9) and (10), one can express the wavefunctions of two
ForanincidentplanewaveincomingfromwaveguideA
joints (A, B) as,
with energy E = 2Jcos(k), the scattering wave func-
−
tion can be obtained by the Bethe ansatz method. The
g 1+r
wave function has the form f = A h , (11)
−1 J eik+re−ik A
|ψki=Xl flb†l |vaci+Xl hla†l |vaci (5) f1 = gBeikhB. (12)
J
where TheexplicitformoftheSchr¨odingerequationsforH can
c
be written as,
eik(l+1)+re ik(l+1), l ( , 1]
fl =(cid:26) teik(l−1−), ∈l −[1∞, −) (6)
∈ ∞ N
h =h , l [1,N].
l l ∈ κijhi =Ehj, (j =A,B)
− 6
Here r, t are the reflection and transmission coefficients. Xi
The explicit form of the Schr¨odinger equations for the N
κ h g f =(E V )h , (13)
waveguides H and H are iA i A 1 A A
A B −iX=A − − −
6
H ψk =E ψk (7) N
| i | i
κ h g f =(E V )h .
iB i B 1 B B
− − −
admits iX=B
6
3
Substitutingtheexpressionforf andf fromEqs. (11)
1 1
and (12), to the above Eqs. (1−3), we get the following v 3 -J v
A 2 N B
Schr¨odinger equations for the center network, A B
-g -g
2N N+2
(a)
N
κ h =Eh , (j =A,B)
ij i j
− 6
Xi
3 -J
N 2 N
κ h =(E U )h , (14) UA A B UB
iA i A A
− −
iX=A
6 2N N+2
N
(b)
κ h =(E U )h .
iB i B B
− −
iX=B
6 FIG. 2. (Color online) Schematic illustration of the concrete
configurationforascatteringsystem. Aringasthescattering
with
center,connectstotwosemi-infinitechainsLandRaswaveg-
uideswithcoupling−g. Theon-sitepotentialsattheconnec-
g2 1+r tionsareVAandVB. Thewavefunctionwithinthescattering
UA =VA− JAeik+re ik, (15) centerforascatteringstateofthewholesystemisidenticalto
− an equal-energy eigen function of the non-Hermitian Hamil-
U =V gB2 eik. tonianwhichisconstructedbythecenterHermitianringwith
B B − J imaginary potentials UA and UB added at the joint sites A
and B.
This is equivalent to the effective non-Hermitian Hamil-
tonian
mentioned scattering process. The corresponding Bethe
=H +(U V )n +(U V )n . (16) ansatz wave function has the form
c A A A B B B
H − −
Withoutlosinggenerality,wetaker = r eiδ with r <1. e−ik(l+1)+r∗eik(l+1), l ∈(−∞,−1]
| | | |  t e ik(l 1), l [1, ) (18)
This leads to ∗ − −
 h , l∈ [1,∞N]
∗l ∈

(gAgBsink)2(1 r 2) with energy E = −2Jcosk. The above conclusion still
Im(U )Im(U )= −| | <0, holds. Straightforward algebra shows that the corre-
A B
−J2 1+2 r cos(δ 2k)+ r 2 sponding non-Hermitian reduction is . In the frame-
h | | − | | i(17) workof non-HermitianquantummechHan†ics, † takesan
H
which means that the imaginary part of the additional importantroletoconstructacompletebiorthogonalbasis
potentialshaveoppositesigns,oneprovidinggainandthe set, which has no physical correspondence. In the con-
other loss. This is in accordance to the conservationlaw text of our approach, † has the same physics as , in
H H
of the current. It is important to stress that magnitude describing the scattering problemof the same Hermitian
of the two imaginary potentials may not equal, which system.
deviatesfromthe generalunderstandingofanimaginary
potential.
III. ILLUSTRATIVE EXAMPLES
The existence of the scattering solution of the Hermi-
tian system H ensures that there must exist at least one
real solution of with eigenvalue equals to the incident In this section, we investigate simple exactly solvable
H
energyE. It possessesthe identicalwavefunctionas that systems to illustrate the main idea of this article. We
ofthe scatteringstate withinthe regionofthe scattering will discuss two examples which correspond to a
PT
center. Thenascatteringproblemisreducedtotheeigen and a non- non-HermitianHamiltonian, respectively.
PT
problem of a non-Hermitian Hamiltonian. This conclu- The advangtage of these examples are that the non-
sion is an extension of our previous result [25]. In this Hermiltian Hamiltonians are exactly solvable.
work, our formalism is generic: The central network is
not limited to the linear geometry and the scattering is
notrestrictedtoberesonanttransmission. Thusthescat- A. Exactly solvable PT Hamiltonian
tering interpretation for the non-Hermitian Hamiltonian
isnotlimitedtopseudo-Hermitiansystem. Thisrigorous To exemplify the previously mentioned analysis of re-
conclusionhasimportantimplicationsinboththeoretical lating the stationary states of a non-Hermitian -
PT
and methodological aspects. symmetric Hamiltonian to a scattering problem for a
Likewise, if we consider the inverse scattering pro- Hermitian one, we take the center network to be a sim-
cess,i.e.,takingthetime-reversaloperationontheabove ple network: a uniform ring system. We start with the
4
scattering problem for a class of symmetric systems, the exists a solution in ε to match the energy ε of the
j n
{ }
Hamiltonian can be written as incident wave.
From Eqs. (22, 23), on can see that a pair of imagi-
naryeigenvaluesappear,i.e.,the symmetryisbroken
2N ε g2 when g > √2J. In general, a nPonT-Hermitian Hamilto-
Hss =−J a†iai+1+H.c.− 2J2 (n1+nN+1) nian with a broken symmetry is unacceptable be-
Xi=1 PT
cause its complex energy eigenvalues make a hash of the
+
−∞ ∞ physicalinterpretation. Ontheotherhand,the sym-
−JiX= 1b†i−1bi−JXi=1b†ibi+1+H.c. (19) metry breakingwas observedin optics realm exPpeTrimen-
− tally [29]. In theoretical apects, symmetry in non-
−gb†−1a1−gb†1aN+1+H.c., Htheermpoitiinatnosfpviniewchoafinthsiyssatermticlwe,aswPedTinscoutesstehdat[3e0v].enFro[mn]
where we denote the connection sites as a = a and H
A 1 possessesabroken symmetry,thespectrum ε still
j
a =a . PT { }
B N+1 contains the state with the energy ε = ε . It is worth
j n
The corresponding non-Hermitian Hamiltonian de-
mentioning that the broken symmetry does not contra-
pends on the energy E of the incident plane wave as
dicttheinterpretationofthenon-HermitianHamiltonian
well as the parameters ε and g. To be concise, as an
(20). This idencates that even the symmetry is bro-
illustrative example, we would like to present the ex- PT
kenthenon-HermitianHamiltonianstillhasphysicalsig-
actly solvable model, which are helpful to demonstrate
nificance.
our main idea. Therefore, we will focus on the following ii) g = √2J, E = ε [ 2J,2J]. Here we do not
configurations: ∈ −
restrict the energy of the incident plane wave but the
i) g = √2J, E = ε = ε = 2Jcos(nπ/N) where
6 n − magnitue of g. Straight forward algebra shows that the
n [1,N 1]. Here we restrict the energy of the inci-
∈ − problemofsolvingtheSchrodingerequationisreducedto
dent plane wavesince it will leads to the pure imaginary
theeigenproblemofthefollowingnon-HermitianHamil-
potential, thus ensures the existence of the exact solu-
tonian
tion. Straightforwardalgebrashowsthattheproblemof
solving the Schrodinger equation is reduced to the eigen
problem of the following non-Hermitian Hamiltonian 2N
H[ε] =−J a†iai+1+H.c.+iγεa†1a1−iγεa†N+1aN+1
Xi=1
2N (24)
H[n] =−J a†iai+1+H.c.+iγna†1a1−iγna†N+1aN+1 with the imaginary potential
Xi=1
(20)
γ = 4J2 ε2. (25)
with the imaginary potential ε −
p
g2 nπ From Appendix A, the solution of the Hamiltonian [ε]
H
γn = J sin(N ). (21) has the same form of Eqs. (22, 23) with γn replaced by
γ . Here we would like to see the relation between the
ε
Obviously, this Hamiltonian depicts a 2N-site ring with Hamiltonians [n] and [ε]: Both of them come from
H H
twoimaginarypotentialsattwosymmetricalsites,which the same model with different coupling constants (with
isa -invariantHamiltonian. Notethatthemagnitude g = √2J and g = √2J) and different incident plane
PT 6
oftheimaginarypotentialisdiscreteinordertoobtained waves (with discrete and continuous spectra). However
the exact solutions. In Appendix A, it is shown that they have the same structure but different values of the
such lattices can be synthesized from the potential-free imaginary potentials. In Appendix A, we provide the
lattice by the intertwining operator technique generally universal solution contains that of [n] and [ε].
H H
employed in supersymmetric quantum mechanics. The Obviously, Hamiltonian [ε] is always exact sym-
eigen spectrum of [n] consists of metric. All the eigenvalueHs are real. AmongPthTem we
H
can find that ε = 4J2 γ2 = ε, one of ε equals
εj =−2Jcos(jπ/N), (22) to the energy o±f inc±idpent pl−aneεwav±e ε and thus±verifies
( j [1,N 1],2-fold degeneracy) the above mentioned conclusion. Furthemore, the solu-
∈ −
tion of it has the following peculiar feature: in the case
and two additional levels
of ε = ε , i.e., the incident wave has wave vector nπ/N
n
ε = 4J2 γ2. (23) (n [1,N 1]), the exceptional points appear in [ε].
± ± − n It i∈s shown−inAppendix A that the correspondingeiHgen-
p
The eigenstates with eigenvalue ε can be decomposed functions of ε (ε ) and ε (ε ) coalesce.
j + n N n
− −
into two sets: bonding and antibonding, with respect According to non-Hermitian quantum mechanics, in
to the spatial reflection symmetry about the axis along general, [ε] has the Hermitian counterpart H[ε] which
H
the waveguides. For the scattering problem, only the possessesthe samespectrum. When the potential γ ap-
ε
bonding states are involved. It shows that there always proches γ , the similarity transform that connects [ε]
εn H
5
and H[ε] becomes singular. The Hamiltonian [ε] be- Hamiltoian. In this sense, one can conclude that a non-
H
comes a Jordan-block operator, which is nondiagonaliz- non-Hermitian Hamiltonian still has physical signif-
PT
ableandhasfewerenergyeigenstates(N 1)thaneigen- icance.
−
values(N+1),(i.e.,thelackofcompletenessoftheenergy
eigenstates.) SuchaHamiltonianhasnoHermitiancoun-
terpart [31]. According to our analysis, one can see that IV. CONCLUSION
even at the exceptional points [9] the coalescing eigen-
states still has physical significance.
Insummary,wehavestudiedtheconnectionbetweena
non-Hermitian system and the corresponding large Her-
mitian system. We propose a physical interpretation
B. Exactly solvable non-PT Hamiltonian
for a general non-Hermitian Hamiltonian based on the
configurations involving an arbitrary network coupled
Now we turn to exemplify the previously mentioned with the input and output waveguides. We employed
analysis of relating the stationary states of a non- the Bethe ansatz approach to the scattering problem to
Hermitian non- -symmetric Hamiltonian to a scatter- showthatforanyscatteringstateofaHermitiansystem,
PT
ingproblemforaHermitianone. Westilltakethecenter the wavefunction within the scattering center lattice al-
networkasasimplenetwork: auniformringsystemwith ways corresponds to the equal energy eigenfunction of
uniform coupling but none on-site real potentials. The thenon-HermitianHamiltonian. Itisimportanttostress
corresponding Hamiltonian can be written as thatsuchaphysicalinterpretationforthenon-Hermitian
Hamiltonian is not limited to the pseudo-Hermitian sys-
tem. As an application, we examine concrete networks
2N
Has =−JXi=1a†iai+1+H.c. (26) csoolnustiisotninsgfoorfasurcihngtylaptetsicoefacsotnhfieguscraatttioernisngarceenotbetra.inEexdatcot
demonstrate the results. Such results are expected to be
+
−∞ ∞
−JiX= 1b†i−1bi−JXi=1b†ibi+1 ntheecensosanr-yHearnmditiinasnigHhtafmulilftoornitahne. physical significance of
−
−Jb†−1a1−Jb†1aN+1+H.c.
We consider the incident plane wave with energy E = ACKNOWLEDGMENTS
2Jcosϑ, where ϑ (π, π) without any restriction.
− ∈ −
Straightforwardalgebrashowsthatthe problemofsolv- WeacknowledgethesupportoftheCNSF(GrantNos.
ingtheSchrodingerequationisreducedtotheeigenprob- 10874091and 2006CB921205).
lem of the following non-Hermitian Hamiltonian
Appendix A: Construction of PT-Hamiltonian by
2N
H[ϑ] =−J a†iai+1+H.c.+UAa†1a1+UBa†N+1aN+1, Interwining operator technique
Xi=1
(27) In this Appendix, we will derive the central formula
where the complex potentials are for studying the eigen problem of the ring system.
PT
eiϑNcos[(N 1)ϑ]+isinϑ
U = J − 2sinϑ, (28) 1. Linear Transformation
A − eiϑNsin[(N 1)ϑ] sinϑ
− −
U = Jeiϑ.
B First of all, the Hamiltonian can be decomposed into
−
We cansee that, in general, Hamiltonian [ϑ] is not two independent sub-Hamiltonians
H PT
symmetric, except in some special cases. It is hardly
= + (A1)
to get the analytical solution of such a Hamiltonian in H Hα Hβ
general cases. Fortunately, what we need to do is to
prove that the incident energy E = 2Jcosϑ is always
N 1
− −
osinnegloe-fptahreticeliegebnavsailsuetsheofmtahterixHarmepirlteosneniatna.tioInnMfac[ϑt,] ionf Hα =−J Xi=2 α†iαi+1−√2J(cid:16)α†1α2+α†NαN+1(cid:17)+H.c.
the Hamiltonian (27) satisfies
+UAα†1α1+UBα†N+1αN+1, (A2)
det [ϑ]+2Jcosϑ =0, (29)
(cid:12)M (cid:12)
(cid:12) (cid:12) N 1
arecscuolrtdidnogntoottdheepdeen(cid:12)rdivoantiotnhegipvseenudi(cid:12)no-AHperpmenitdicixityB.ofTthhies Hβ =−J Xi=−2 βi†βi+1+H.c. (A3)
6
UA UB this construction explicitly. We start with the following
(N 1) (N 1) Hamiltonian
A B − × −
1 2 3 N N+1 N 2
−
H = (n n+1 +H.c.) (A5)
1
− | ih |
nX=1
which depicts an (N 1)-site uniform chain. The spec-
2 3 N −
trum of H can be expressed as
1
FIG. 3. (Color online) Schematic illustration of the reduc- nπ
ε = 2cos( ),n [1,N 1]. (A6)
tion for a ring system by linear transformation. The top n − N ∈ −
panel (black) represents the Hamiltonian Hα, while the bot-
tom panel (red) shows the Hamiltonian Hβ On the other hand, H1 can be written in the form
.
H = µ (A7)
1
QR−
with [ α, β]=0, by using the following linear tranfor- where
H H
mation:
N 1
−
α1 =a1,αN+1 =aN+1, Q= (qn|nihn|+q¯n|nihn+1|) (A8)
1 nX=1
αj = (aj +a2N+2 j), j [2,N], (A4) N 1
√2 − ∈ −
= (r n n +r¯ n+1 n)
n n
1 R | ih | | ih |
β = (a a ), j [2,N]. nX=1
j j 2N+2 j
√2 − − ∈
and
We will focus on the solution of the Hamiltonian .
α
H µ=2cosκ,(κ>0)
Typically, the solutioncan be obtained via Bethe ansatz
method as shown in Ref. [28]. In this Appendix, we rn =qn = e−iκ/2 . (A9)
−
will use the intertwining operator technique to get the r¯n =q¯n =eiκ/2
solutions in order to reveal their characteristic features.
ThentheHamiltonianH canbeconstructedintheform
2
H = µ (A10)
2
2. Interwining operator technique RQ−
N 1
−
= (n n+1 +H.c.) eiκ 1 1 e−iκ N N ,
The intertwining operator technique is generally em- −nX=1 | ih | − | ih |− | ih |
ployed in supersymmetric quantum mechanics, which
provides the universal approach to creating new exactly whichpossessesanextraeigenvalue 2cosκbasedonthe
−
solvable models. Recently, it is applied to discrete sys- spectrum εn.
tems inorderto constructthe model whichsupports the Next step, we repeat the above procedure based on a
desirable spectrum [32, 33]. new Hamiltonian H2′, which is obtained from H2 under
Thecriticalideaoftheintertwiningoperatortechnique parity operation P, i.e.,
is as the following: Consider an N N Hamiltonian H
which has the form H = +×µ , where and1 H2′ =P−1H2P (A11)
1 1 1 1 1
R1 represent N ×(N +1) anQdR(N +1)×N matQices, re- = N−1(n n+1 +H.c.) e iκ 1 1 eiκ N N
spectively. One can construct an (N +1) (N +1) new −
− | ih | − | ih |− | ih |
HamiltonianH (H = +µ )byint×erchangingthe nX=1
2 2 1 1 1
R Q
operators and . The spectrum of H is the same
1 1 1 where
R Q
as that of H except for the energy level µ . Iterating
2 1
this method results in a series of Hamiltonians H3, H4, Pij =δi,N+1 j (A12)
H , whose energy spectra differ from that of H ow- −
5 1
···
ing to the additionof the discrete energy levels µ1,µ2 , isthematixrepresentationofmirrorreflection. Notethat
{ }
{µ1,µ2,µ3}, {µ1,µ2,µ3,µ4}, ···. H2′ and H2 have identical spectra. Accordingly, H2′ can
Our aim is to construct a -invariant Hamiltonian be written as the form
PT
H by adding two energy levels E = µ and E = µ
3
−
(0 6 µ 6 2, the obtained conclusion will be extended H2′ =Q′R′+µ (A13)
beyond this region later) into the energy spectrum of a
uniform chain system. We will show the processes of where
7
where µ = 2coshω > 2. On the other hand, if κ is
replaced by a complex number κ = π/2 iω, one can
N −
obtain a non-Hermitian symmetric Hamiltonian in
Q′ = (qn′ |nihn|+q¯n′ |nihn+1|) (A14) the brokenphase. In thisPcaTse the two added eigenstates
nX=1
have pure imaginary eigenvalues µ= 2isinhω.
N ±
R′ = (rn′ |nihn|+r¯n′ |n+1ihn|)
nX=1
4. Coalescence of eigenstates
and
Now we investigate the eigenfunctions in the case of
r =q =ie iκ/2, κ = k (k = nπ/N,n [1,N 1]). In this situation, all
n′ n′ − ∈ −
r¯ =q¯ =ieiκ/2 the eigenfunction can be written explicitly as
n′ n′ . (A15)
r =q =i√2e iκ/2
1′ 1′ − ψ (1)= i√2( 1)nsink
r¯N′ =q¯N′ =i√2eiκ/2 ψ (nj)= 2−ie (N+−1 j)iksink , (A20)
n − −
Finally,thetargetHamiltonianH3 canbe constructed  ψn(N−+1)=−i√2sink
in the form 
H3 =R′Q′+µ (A16) ψN n(1)=i√2( 1)N( 1)nsink
=−NX2−1|nihn+1|−√2|1ih2|−√2|NihN +1|+H.c.  ψN−n(−j)ψN=−−n((−N1+)(N1−)−j=)2ii√e−i2(Nsi+n1k−j)ksink , (A21)

+2isinκ(1 1 N +1 N +1),
| ih |−| ih |
and the energy spectrum (22) and (23) [n] and [ε]can ψN(1)=i√2( 1)ne−ik/2
be obtained by adding the unit J. H H  ψN(j)=2ie−(N+−1−j)ike−ik/2 , (A22)
 ψ (N +1)=i√2e ik/2
N −

3. Eigenfunctions of H3
ψ (1)=( 1)je ikj/√2
N+1 −
−
Now we turn to derive the eigen functions of H3. The  ψN+1(j)=( 1)je−ikj . (A23)
eigenfunctions of the Hamiltonians H1, H2 and H3 are ψ (N +1)=(−1)je ikj/√2
N+1 −
denoted by φ , ϕ and ψ , respectively. The eigenfunc- −
n n n 
tions of a uniform chain can be readily written as For odd N, we have ψ ψ and ψ ψ , which
n N N n N+1
means the coalescence o∝f eigenstates. −Als∝o the norms of
the above four eigenstates vanish. For even N, we have
2 nπj
φ (j)= sin ,(n [1,N 1]), (A17) the same conclusion except when n=N/2. In this case,
n rN (cid:18) N (cid:19) ∈ −
we have ψ ψ = ψ ψ , which means the
n N N n N+1
coalescence o∝f the three e−ige∝nstates. Also the norms of
According to the interwining operator technique of su-
the above three eigenstates vanish.
persymmetry theory, we have
ϕ (j)= φ ,n [1,N 1] (A18)
n n
R ∈ −
ϕ (j)=e iκj,j [1,N], Appendix B: Zero determinant
N −
∈
and
In this appendix we will prove the Eq. (29). Applying
ψn(j)= ′Pϕn,n [1,N] (A19) the lineartransformationintroducedin Appendix A, the
R ∈
1 ,j =1,N +1 2N-dimensionalmatrix [ϑ]canbewritteninadiagonal
ψN+1(j)=( 1)je−iκj √2 . block form, i.e., M
− (cid:18) 1,j [2,N] (cid:19)
∈
Note that the eigenfunctions are not normalized.
0
In the above, we restricted µ in the region [0, 2] for [ϑ]+2Jcosϑ= D (B1)
the purpose of obtaining H as a non-Hermitian unbro- M (cid:20) 0 (cid:21)
3 A
ken symmetric Hamiltonian with imaginary poten-
tialsPaTt the edges in the form of Eq. (A16). However, where is (N +1)-dimensional, while is N-
D A
dimensional. Then we have
the obtained result can be extended beyond the region.
Actually, one can simply replace κ by iω in all the ex-
−
pressions. Then one can obtain a Hermitian Hamil-
tonian with two added bound staPtTes with energy µ, det [ϑ]+2Jcosϑ =det det . (B2)
± (cid:12)M (cid:12) |D| |A|
(cid:12) (cid:12)
(cid:12) (cid:12)
8
Consider the the (N +1)-dimensional matrix , which where D is the j j determinant
j
D ×
determinant D =det has the form
|D|
E 1
− −
(cid:12) 1 E 1 (cid:12)
(cid:12) − − − (cid:12)
UA−E −√2 (cid:12)(cid:12) 1 ... ... (cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) −√2 −E1 −E1 1 (cid:12)(cid:12)(cid:12) Dj =(cid:12)(cid:12)(cid:12) − ... ... 1 (cid:12)(cid:12)(cid:12). (B5)
(cid:12) − − − (cid:12) (cid:12) − (cid:12)
(cid:12)(cid:12) 1 ... ... (cid:12)(cid:12) (cid:12)(cid:12) −1 −E −1 (cid:12)(cid:12)
D =(cid:12) − (cid:12). (cid:12) 1 E (cid:12)
(cid:12)(cid:12) ... ... 1 (cid:12)(cid:12) (cid:12)(cid:12) − − (cid:12)(cid:12)
(cid:12)(cid:12) 1 −E 1 (cid:12)(cid:12) Such kinds of de(cid:12)terminants follow the recursi(cid:12)on formula
(cid:12) − − − (cid:12)
(cid:12) 1 E √2 (cid:12)
(cid:12) − − − (cid:12)
(cid:12) √2 U E (cid:12)
(cid:12)(cid:12) − B −(B3(cid:12)(cid:12)) Dj =−EDj−1−Dj−2, (B6)
(cid:12) (cid:12)
Using cofactor expansion along the first and last rows,
which leads to
we obtain
1 e2(j+1)ik
Dj = −1 e2ik e−jik, (j <N). (B7)
−
D = E2+UAUB−E(UA+UB)−4 DN−1 (B4) Substituting the expressions for UA, UB, from Eq. (28),
(cid:2) 2(U +U )D , (cid:3) we get that D =0. Thus, Eq. (29) is proved.
A B N 2
− −
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