Entanglement and Majorana edge states in the Kitaev model

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by  Saptarshi Mandal

Entanglement and Majorana edge states in the Kitaev model
Saptarshi Mandal,1, Moitri Maiti,2, and Vipin Kerala Varma3,
∗ † ‡
1Institute of Physics, Bhubaneswar-751005, Orissa, India
2BLTP, JINR, Dubna, Moscow region, 141980, Russia
3The Abdus Salam ICTP, Strada Costiera 11, 34151, Trieste, Italy
(Dated: July 26, 2016)
Weinvestigatethevon-NeumannentanglemententropyandSchmidtgapinthevortex-freeground
state of the Kitaev model on the honeycomb lattice for square/rectangular and cylindrical subsys-
tems. We find that, for both the subsystems, the free-fermionic contribution to the entanglement
entropy SE exhibits signatures of the phase transitions between the gapless and gapped phases.
Howeverwithinthegaplessphase,wefindthatSE doesnotshowanexpectedmonotonicbehaviour
asafunctionofthecouplingJz betweenthesuitablydefinedone-dimensionalchainsforeithergeom-
etry;moreover thesystem generically reaches a point of minimum entanglement within thegapless
phasebefore theentanglement saturates orincreases again untilthegapped phaseis reached. This
6
maybeattributedtotheonsetofgaplessmodesinthebulkspectrumandthecompetition between
1
0 the correlation functions along various bonds. In the gapped phase, on the other hand, SE always
2 monotonicallyvarieswithJz independentofthesub-regionsizeorshape. Finally,furtherconfirming
theLi-Haldaneconjecture,wefindthattheSchmidtgap∆definedfromtheentanglementspectrum

l alsosignalsthetopologicaltransitionsbutonlyiftherearecorrespondingzeroenergyMajoranaedge
u
states that simultaneously appear or disappear across the transitions. We analytically corroborate
J
someofourresultsonentanglemententropy,Schmidtgap,andthebulk-edgecorrespondenceusing
3
perturbation theory.
2

PACSnumbers: 75.10.Jm
]
l
e
- I. INTRODUCTION hasalsobeen foundto characterizequantumphasetran-
r
t sitions 18,19. Furthermore it has been found that for two
s
. dimensions and above entanglement entropy can char-
t
a Entanglement of quantum states is a non-local phe- acterize the topological properties of the system as well
m nomenonandisamanifestationofthesuperpositionprin- 20,21. For non interacting systems, there are well defined
- ciple of quantum states. In a quantum many-body sys- prescriptions to compute the various entanglement mea-
d tem entanglement of the constituent particles or states sures discussed above; however, investigation of entan-
n
essentially dictates its macroscopic properties, viz. su- glement properties for interacting systems are severely
o
c perconductivity,superfluidity,quantumphasetransitions challenged by the complexity and the size of the Hilbert
[ etc. Recentlytherehasbeenanactiveinteresttocharac- space as the system size is increased. Evaluating the
terizeandquantifythe notionofentanglementdue toits entanglement entropy is not straightforward in general
2
fundamental applicability in quantum information the- and powerful numerical algorithms exist mostly for one-
v
5 ory, topological quantum computations, and also to nu- dimensional systems 22–26.
9 merous physical systems such as black hole and generic Thismotivatesustoinvestigateentanglemententropyin
6 quantum many body systems 1–3. The usual method to an exactly solvable 2D quantum spin model, namely the
1 measure the entanglement of a part (called subsystem) Kitaev model27,28 which has attracted a lot of attention
0 of a composite system (called full system) is to examine fromresearchersincondensedmatter,quantuminforma-
1. thereduceddensitymatrixofthesubsystem. Tothisend tion, and specific high energy theorists alike. The model
0 manydefinitionshavebeenintroducedforcharacterizing realizes an unusual quantum spin-liquid ground state
6 the entanglement viz. von-Neumann entropy, Rényi en- with short-range and bond-dependent correlation func-
1 tropy4,5,entanglementspectrum,Schmidtgap6–10, and tions 29, topological degeneracies for all eigenstates30,
:
v suchlike. and displays a tunable phase transition from a topologi-
i These different measures of entanglement are seen to cally trivial phase to a topologically nontrivial phase as
X
characterize physical systems according to their univer- the parameters of the model Hamiltonian are varied. It
ar sality classes, follow certainscaling laws, and also detect is known that this model reduces to a problem of nonin-
topologicalorders. Forexample,entanglemententropyof teracting Majorana fermions hopping in the presence of
one-dimensional critical systems with size L is found to a background Z2 gauge field; this reduces an otherwise
vary logL 11,12. For gapped systems in one dimension quartic fermionic interaction to an effective quadratic
scalin∼g of the predictions of the massive field theory 12 fermionic interactions exactly28. All of these intrigu-
hasalsobeenrecentlyverifiedindimerizedfreefermionic ing facts motivated a series of important studies tak-
modelshostingtopologicalphases13 andinspinsystems ing the Kitaev model as a test-bed for understanding
14. In two and higher dimensions bosons and fermions many fundamental theoretical concepts such as quench-
follows different area law. 15–17. Entanglement entropy ing and defect production31, phase transitions32, braid-
2
ing statistics33,34, dynamics of hole vacancies35, to name Nx=Ny Ny Nx
a select few. Ny Ny
Studies ofthe entanglemententropyina particularlimit A B
of this model were undertaken early on36,37. Zanardi
and coauthors verified the area law for the model and Ny 2
by taking different partitions of spin configurations had
argued that the entanglement entropy can be used as a Nx
probe to detect topological order. However this work
lacksadetailedanalysisofpossiblesubsystemconfigura- D Jz = 1 Nx
tionsquitepossiblyduetothenumericalcomplexitiesas-
a
sociatedwithaquantumspinmodel. Thisissuehasbeen Jx = 0 b Jy = 0
recently circumventedinaninteresting workby Yao and a a y−link x−link
Qi38 where they have shown that for the Kitaev model Jy = 1 Jz = 0 Jx = 1 C z−link
the entanglement entropy of a given subsystem can be
separated into two parts: one part due to the Z gauge
2
fields and the other due to the free Majorana fermions. FIG. 1. (Color online) Top panel: geometry of subsys-
tems chosen with square/rectangular block (plot A) and
Following Yao and Qi’s work,in this paper, we calculate
cylindrical/half-region(plotB);periodicboundaryconditions
theeigenvaluesofthereduceddensitymatrixforasystem
are imposed on the full system so that a torus is formed.
offree Majoranafermionsinthe twodimensionalKitaev
While considering the entanglement entropy and gap for the
model,andinvestigatetheentanglementpropertiesinde- square block, we have chosen the torus such that Nx = Ny;
tailfor this systemwhichhavenot beenreportedbefore. on the other hand for the half-region Ny > Nx. Bottom
Moreover for the extended Kitaev model, a recent den- panel: therealspacelinksofthehoneycomblattice thatcor-
sity matrix renormalization group study concluded that respond totheJx,Jy,Jz couplings (plot C); schematic phase
the entanglement entropy and Schmidt gap may only be diagramoftheKitaevmodelshowingthethreegappedphases
occasionally employed as a good indicator of the phase surroundingthe central gapless phase, with the vertical lines
transitions between the various phases harboured in the illustrating the contours which we study in this paper (plot
D).
extended system, such as the Kitaev spin-liquid phase
and certain magnetic phases39.
In this study, we present results of the entanglement en-
tropy and Schmidt gap for subsystems with square or
J ,J ,andJ . Forsimplicityinourstudywehavetaken
rectangular block geometry, and one half of the torus x y z
all the coupling parameters to be positive, although the
(herein called the half-region). For a square/rectangular
results presented here do not depend on the sign of the
blocksubsystem,thefullsystemisatorusofsizeN N
x× y couplingparameters. Wenotethatallthegappedphases
(with N =N ); for the half-regionthe dimension ofthe
x y aretopologicallyequivalent: theirgroundstatedegenera-
torus is taken such that N < N and the half-region is
x y cies and excitations are of the same topological nature.
definedbydividingthe torusinthey-directioni.echoos-
With this background let us summarize our results.
ing a length Ny in y direction. The subsystems under
2 Half-region geometry: Forthe Kitaevmodelinthehalf-
consideration are sketched in the upper panels of Fig.1.
regiongeometrytheentanglemententropyS grosslyin-
We first consider the entanglement properties for a cou- E
creases as J is increased,until in the Toric code limit it
pledchainsysteminthe Kitaevmodelfollowingthe geo- z
decreases and saturates to a finite value in the large J
metrical setup adopted in the study of a transverse field z
limit. However within the gapless phase there is a non-
Ising model 9. We view our 2D system as N coupled
y monotonicdependenceofS onthestrengthofJ ,man-
one-dimensional periodic chain and consider the subsys- E z
ifesting itself as oscillations; the height and number of
tem as the first N /2 chains. The interchain coupling
y theseoscillationswithinthe gaplessphasearedependent
between two neighbouring chain is given by J . Also
z onthesystemsizeschosen. Atthegapless gappedtran-
each one-dimensional chain is characterized by alternat- −
sitions a cusp in S is generically visible. The Schmidt
ing bond interaction parameter J and J . E
x y gap ∆ in this subregion shows a first order jump from
Let us briefly summarize the phases and physics of this
a finite value in the gapped weakly-coupled chain limit
model27,28, as pertinent for understanding our results.
to zero in the gapless phase; finite size effects for ∆ are
ForsmallvaluesofJ ,i.e forweaklycoupledchainlimit,
z almost absent for this topologicalphase transition. This
the system is gapless only at the point J = J and
x y is in contrast to the case of the 2D transverse field Ising
gapped for J = J . For J = J , as J is increased
x 6 y x 6 y z model on a similar geometry where ∆ displays logarith-
the system enters into a gapless phase for some critical
micscalingwithsystemsize const./ln(N /π)9. More-
value of Jz. However when Jz exceeds another critical over as we further increase t∼he inter-coupliyng chain and
value,thesystemagainentersintoanothergappedphase
reachthelargeJ gappedregimei.etheToriccodelimit,
characterized by large values of J . This limit is usually z
z the entanglement gap still remains zero. Thus we find
known as Toric code limit. The condition for gapless
that although the two gapped phases are topologically
phase is J J + J and cyclic combinations of
| x| ≤ | y| | z| identical, the Schmidt gaps of a givensubsystem are dif-
3
ferent in the two gapped phases indicating that other where, J ’s are dimensionless coupling constants, σα
α k
properties of the subsystem compared with the full sys- is the α-component of the Pauli matrices. Following
templayanimportantrole. Inparticularweascribethis Ref. 28, we introduce a set of four Majorana fermions
tothepresenceorabsenceofMajoranaedgestatesinthe bx,by,bz,c at a given site ‘k’ to represent the Pauli
{ k k k k}
twogappedphasesofthesystemandinferthatavanish- operators. Notice that this definition implies that the
ing Schmidt gap does not necessarily imply topological spin operators live in an enlarged Hilbert space. We de-
order. finethePaulioperatorsinthisenlargedHilbertspaceas,
Square/Rectangularblockgeometry: Inthiscasethequal- σ˜α = ibαc (α = x,y,z). We have used σ˜ instead of σ
k k k
itative behaviour of the entanglement entropy depends to denote the fact that these operators are defined in an
crucially on the details of the connectivity of the sys- enlarged Hilbert space. To get the physical spin opera-
tem; nevertheless a nonmonotonic behaviour of SE per- tors σ one has to enforce the projection in the physical
sists within the gapless phase here as well. The cusps in Hilbert space using the operator D = bxbybzc = 1, at
k k k k k
SE are conspicuous, as in the half-region, at the phase a given site ‘k’. Substituting the above definition, we
transitions. TheSchmidtgapis vanishinglysmallwithin can rewrite the Hamiltonian in Eq.(1) (in the enlarged
the gapless phase and shows minute nonmonotonic vari- Hilbert space) as
ations, especially close to the conformal critical point;
i
in the gapped phases, where the entanglement entropy H˜ = J uˆ c c , (2)
satisfies the area law i.e (S ∼ α˜Ld−1), these variations 2<jX,k>α αjk jk j k
areproportionallywashedawayandthe coefficientα˜ de-
whereuˆ =ibαjkbαjk arethelinkoperatorsdefinedon
pends nontriviallyonthe underlying systemparameters. jk j k
a given link < jk >. The remarkable fact which makes
The organization of the rest of the paper is as follows.
the Kitaev model an integrable system is that these uˆ
In Sec.II, we further explicate on the physics and phase jk
operators defined on each link mutually commute with
diagram of the Kitaev model. We also outline the for-
each other and also commute with the Hamiltonian in
malism(followingRef. 40)thatweemploytoinvestigate
Eq.(2). They play the role of static Z gauge field op-
the entanglement properties of this model. In Sec.III, 2
erators as one may readily check that uˆ2 = 1. Here we
we present the results of the entanglement entropy and jk
have α = x,y,z depending on whether the ‘j’ and ‘k’
Schmidt gap for the half-region which has a cylindrical jk
indices form an x,y or z link.
geometry; in Sec.IV we discuss the case of square and
The original Hamiltonian as given in Eq.(1) has been
rectangular blocks. In Sec.V we present some analyti-
transformed into an equivalent problem as described by
cal perturbative calculations to corroborate some of our
Eq.(2); now onehas to solvefora free Majoranafermion
numericalfindingspresentedinSecs.IIIandIV.Wesum-
hoppingprobleminthepresenceofstaticZ gaugefields.
marize our primary results in Sec.VI. 2
Forthe detailedsolutionandthe phase diagramwe refer
the reader to the original work by A. Kitaev 28; a pic-
torial summary is presented in the lower panel of Fig.1.
II. KITAEV MODEL AND ENTANGLEMENT
Working in the extended Hilbert space of the spin oper-
PROPERTIES
ators, a general eigenstate of the Hamiltonian in Eq.(2)
can be written as Ψ˜ = φ(u) u 28. φ(u) is obtained
Due to the recent interest in the Kitaev model, a vast | i | i| i | i
by considering the Hamiltonian identical to Eq.(2) with
literature is already available on various aspects of this
uˆ replaced by its eigenvalues u = 1. u describes
system. However,inthissection,weintroducethemodel jk ij ± | i
the gaugefield configurations. The actualeigenstate Ψ
brieflyforcompletenessandself-sufficiencyofthearticle. | i
belonging to the physical Hilbert space is then obtained
Theoriginalmodelisdefinedonahexagonallatticewith
by projecting Ψ˜ onto the physical Hilbert space and is
eachsiteassociatedwithaspin 21 object. Eachspininter- given by38 | i
actswithitsnearest-neighbourandthecouplingstrength
depends on the directionality which is in contrast to the 1
Ψ = φ(u) u , (3)
Heisenberg interaction ~si.~sj. In the hexagonal lattice, | i √2N+1 Xg Dg| i| i
therearethreedifferentorientationsofthebondsandwe
labelthemas"x-bonds","y-bonds"and"z-bonds". Two whereN isthetotalnumberofsitespresentinthesys-
nearest-neighbourspinsjoinedbyanα(α=x,y,z)bond tem. Dg = kDk where g denotes a set of sites and the
interactwiththeαcomponentoftheirspinscontributing product runQs over all the sites ‘k’ within a given set ‘g’.
atermσασα totheHamiltonian. Herej andk represent The sumover‘g’isover2N possiblecombinationofsets.
the site ijndikces for two nearest-neighbour spins situated Thesum √2N1+1 gDg representsthegaugeaverageover
at the two ends of the bond. The model Hamiltonian 28 equivalent copiePs in the extended Hilbert space. In Yao
is given by andQi’swork38ithasbeenshownthatduetothespecial
structureoftheHamiltonianinEq.(2),theentanglement
entropy(S )ofasubsystem‘A’foragiveneigenstate Ψ
A
| i
H = J σασα, (α = x, y, z), (1) consists of two contributions and may be written as
− αjk j k
<jX,k>α SA =SA,F +SA,G log2. (4)
−
4
In the above expression, the contribution S comes whereQisthematrixwhosecolumnsarecomposedal-
A,G
exclusively from the Z gauge fields and equals L log2 ternatively of the real and imaginary parts of the eigen-
2
when the subsystem shares L number of bonds with the vectors of the Majorana Hamiltonian. The eigenvalues
rest of the system. The quantity log2 is a topologi- λ ,...,λ ,...λ of P˜ P lie between [ 1,1], where P˜
− 1 i NA ⊂ −
cal quantity known as the topological entanglement en- denotes the correlation matrix restricted to the subsys-
tropy which is same for both the gapless and gapped tem and N denotes the total number of sites inside the
A
phases of the Kitaev model. S is the contribution subsystem; then the eigenvalues of the reduced density
A,F
from the free Majorana fermions and is obtained from matrix obtained from φ(u) can be written as41,42,
| i
the reduced density matrix ρ = Tr φ(u) φ(u) as
A,F B
| ih |
S = Tr[ρ logρ ]. In this article we calculate
A,F A,F A,F
− 1+s λ
SovAe,rFafoprltahqeugetrtoeunisd1stai.tee. stehcteovrowrhteexr-efrteheespercotdour;ctinodfeueidj Γ(s1,...si,...sNA)= 2i i. (8)
i=Y1,NA
S is a gauge independent quantity. Thus the Hamil-
A,F
tonian which is central to our investigation is obtained si can take values 1 and thus we obtain in total 2NA
from Eq.(2) by replacing u =1 and is given by, ±
ij eigenvaluesofthedensitymatrix. Thevon-Neumannen-
i tanglement entropy is thus given by 38,
H = J c c . (5)
′ 2 αjk j k
Xj,k NA 1+λ 1+λ 1 λ 1 λ
h i i i i i
S = log + − log − . (9)
For a subsystem with boundaryof linear size L, it can A,F 2 2 2 2
been shown36,38 that in the limit L , an area law1 Xi=1
→ ∞
SA,F α˜L holds, where α˜ is a non-universal constant Entanglement gap or Schmidt gap is another useful
∼
that contains necessary information of the entanglement quantity to characterize the entanglement properties; in
between the subsystem and the rest. particularthisgaphasbeenconjecturedtobe capableof
Wenotehere,thattheentanglemententropycanbewrit- detecting topologicalorder6 and phase transitions9. The
tenasEq.(4)foranyshapeandsizeofthesubsystemand entanglement gap may be defined as 6,9
thus the entanglement entropy of a given subsystem can
in principle be calculated by applying the formalism of ∆A =−(logΓM−logΓ′M), (10)
Ref. 40. As mentioned earlier, one of the main objec-
whereΓ andΓ arethe largestandthe secondlargest
tive of this article is to investigate the behaviour of α˜ in M ′M
eigenvalues of the reduced density matrix.
the parameter space of the Kitaev model. Specifically,
we wouldlike to know,apartfromanarealaw,whatad- From Eq.(8), it is clear that ΓM is obtained when the
ditional information might be extracted from the entan- contribution 1+s2iλi is maximized i.e one has to consider
glement entropy of the system. For this we examine the the contribution Max 1+2λi,1−2λi for a given λi. The
entanglement entropy and entanglement gap as a func- (cid:16) (cid:17)
second largest eigenvalue is obtained by changing (1+
tion of the model parameters J ,J ,J . For simplicity
x y z λ)/2to(1 λ)/2 forthesmallestvalue of λ. Withthis,
we choose certain particular contours in the phase space − | |
entanglement gap can be simplified to
ofKitaevmodeltoascertainhowtheentanglementprop-
ertyvariesalongthesepaths. ThefreeMajoranafermion (1+ λ )
min
∆ =log | | . (11)
hopping Hamiltonian given by Eq.(5) has two different A
(1 λ )
min
topological phases, gapless and gapped phases as shown −| |
in lower panel of Fig.1. This Hamiltonian can easily be We evaluate the quantities given by Eq.(9) and Eq.(11)
diagonalizedbyaFouriertransformationandtheground numerically and discuss the results in Sec.III.
state correlation functions can be subsequently calcu- Before we end this section we show how the spectrum,
lated. Following Ref. 40 this now reduces the calcula- eigenvalues and eigenvectors of the Hamiltonian given
tionofthevon-Neumannentanglemententropyofagiven in Eq.(5) may be constructed, required for our analyti-
subsystem ‘A’ to simply calculating the two-pointcorre- cal computations in Sec. VI. This Hamiltonian can be
lation functions in order to obtain the eigenvalues of the easily diagonalised by using a Fourier transformation,
Trehdeucceodrrdeelantsiiotny mmaattrriixx;othfitshies dfuelslcrsiybsetdemashfaoslloewlesm28e,4n0t.s ccj’s=a√re1NMPa~kjoeria~kn·~rajcf~ke,rmwiiothnsc.~kT=hec†−H~kawmhilitcohneiannsutrheesnthraet-
j
given by
duces to
Pij =hΨ|cjci|Ψi−δij. (6) H′ = if(~k)c† c +h.c, (12)
This can be rewritten as kxX,ky≥0 ~k,a ~k,b
P =iQ01 −01 ... ... QT, (7) H′ =kxX,ky≥0ψ~k†h˜kψ~k,ψ~k† =(c~†k,a,c~k,b), (13)
 ... ... 01 −01 ˜hk =(cid:18) if0∗(~k) if0(~k) (cid:19). (14)
  −
5
a and b correspond to the two sublattice points. The for each k -mode in y-direction. We consider the case
x
dispersion f(~k) is given by Jy/Jx = 0.3, such that Jx = 1.0 and Jy = 0.3. Thus,
in the plots of entanglement properties as shown in Fig.
f(~k) =Jxeikx +Jyeiky +Jz, (15) 2 as a function of Jz, Jz is also scaled in units of Jx.
| | In the upper left panel of Fig.2, we have plotted the en-
with k and k being the components of ~k along the x tanglement entropy for the half-region as a function of
x y
and y bonds respectively. kx and ky can be rewritten Jz. It may be worthwhile to note that a similar region
as the Cartesian components of~k along x and y axes as on a torus had previously been considered for studying
the entanglement properties 38, where the entanglement
k = qx + √3qy and k = qx + √3qy. With the unitary
x 2 2 y − 2 2 entropy was studied for each kx-mode for a given set of
transformations
parametervaluesinthegaplessandgappedregime;how-
iexpiθ everthatworklackedacomprehensiveanalysisoftheen-
c = ~k(α +β ),
~k,a √2 ~k ~k tanglement entropy, in particular the physics contained
in the non-universal parameter α˜ appearing in the area
1
c = (α β ), (16) law alongwith its variationwith the system parameters,
~k,b √2 ~k− ~k
which we provide in this work.
the Hamiltonian in Eq.(14) may be diagonalised as As may be seen from the plot, for each of these systems
a cusp is prominent at J = J +J , which is the tran-
z x y
H′ =kxX,ky≥0|f(~k)|(cid:16)α~†kα~k−β~k†β~k(cid:17). (17) soviavtleiiordnawtpeiotdhi.nintHftrohowmeegvtahepre,pgeiandpslipedshesasttheosethtgheaepgalaeprspesaepdlhapawhseai,sset;chmleeaoerrnley--
c ’s are defined in the first half of the Brillouin zone tanglement entropy has intermittent peaks whose num-
~k
(HBZ)andeiθ~k = ff((~~kk)) . Thegroundstate|Giisobtained bpeerctinracrteioa.seTshweiethntianncgrleeamsienngtseynsttreompysfizoer tfohreahaglifv-reengiaosn-
| |
by filling all the β~k modes, i.e, |Gi = ~k HBZβ~k†|0i with Jx =Jy (not shown) is similar to that obtained for
where 0 denotes the fermionic vacuum. Q ∈ Jx = Jy with the initial monotonic increase for Jx = Jy
| i 6 6
The discretization of k and k employed here for a fi- being absent. We note that the intermittent peaks we
x y
nite size lattice is as follows. For the hexagonal lat- find in the gapless phase for the entanglement entropy
tice unit vectors are chosen as ~a = ~e , ~a = 1~e + is very similar to results for the fidelity-susceptibility43.
1 x 2 2 x
√3~e . These yield the reciprocal vectors as G~ = The number of peaks is found to increase linearly with
2 y 1 the number of chains in the total subsystem.
2π(cid:16)~ex− √13~ey(cid:17), G~2 = √4π3~ey. Any ~k vector is dis- Wenowprovideapossibleexplanationfortheoscillations
cretized as ~k = nxG~ + nyG~ . k = ~k ~a with oftheentanglemententropyinthishalf-region. Weknow
Nx 1 Ny 2 x/y · x/y that the fundamental quantity which determines the en-
~a = 1~e + √3~e . Here ~a are the unit vectors
x/y ±2 x 2 y x/y tanglement entropy in a free-fermion system is the two
joining two z-bonds by x and y interactions respectively, point correlationfunction40. To elucidate the relation of
and ~e are the unit vectors along x and y-directions
x/y theoscillationspresentintheentanglemententropywith
kreysp=ec−tiv2Neπlnxyx. +T2hNπneyrye;foNrex aonnde Nobytadiennsotkexth=e n2uNπmnyyberanodf tTnheheaertewesxot-p-npeeocitignahttbicooonurrrveaclaloutreiroenilsaftutiaonkncetinfounninsc,ttwiohenechoCnazzlsfi-dfi≡elrlehtdchzieMczzi+a−j1oiz-.
unitcellstakeninthexandy-directionrespectively,and
rana system for many different system sizes keeping the
n varies from 1 to N .
x/y x/y aspect ratio identical in accordance with the upper left
panelofFig.2. This maybe seento be expressedanalyt-
ically as
III. ENTANGLEMENT IN CYLINDRICAL
SUBREGIONS
Re(f(~k))
= cosθ = , (18)
In this section we discuss the entanglement entropy Czz Xk k |f(~k)|
andgapforthehalf-regionwhichhasacylindricalgeom-
etry. Here we have taken a torus of N N unit cell where f(~k) is the dispersion of the Kitaev model
x y
× | |
which can be thought of as N coupled one-dimensional given in Eq.(15). In the lower left panel of Fig.2 we
y
chainswhereeachchainhasN sitessuchthatN =N 9. have plotted the derivative of C with respect to J ,
x x y zz z
6
We have kept the aspect ratio N /N = 10; our quali- where we observe the appearance of similar oscillations.
y x
tative results are independent of this ratio. The subsys- The oscillations in the correlation function produces
tem is defined such that it contains N /2 coupled one- the oscillations in the derivative of it as well, but of
y
dimensional chains. This is because the system is peri- higher magnitude, and explain the oscillations in the
odic in the x-direction, hence one can employ a Fourier entanglement entropy (the oscillations in the derivative
transform in the x-direction which reduces the entangle- of correlation function are more pronounced than the
ment study to that of decoupled one-dimensional chains oscillations in the correlation function itself due to the
6
1.6
SE/Nx 0.7
6 ∆
1.2 Ny=60
Ny=80 0.6
0.8 Ny=100
0.4 ny = 30 0.5 E
0 Gapless 4 nnyy == 2100 0.4
_∂(<cziczi+1>) 0.3
0.8 ∂Jz 2
0.2
0.4
0.1
Gapless Gapless
0 0 0 Jz
0 0.4 0.8 1.2 0 0.4 0.8 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
FIG. 2. (Colour online) Entanglement properties of cylindrical subregions for Jy/Jx = 0.3 as a function of Jz in the Kitaev
system; gapless phase is shaded blue. Left panel: top plot shows the entanglement entropy for various sizes of the full system
with the aspect ratio fixed at Ny/Nx = 10, with the arrow denoting the saturation value in the large-Jz limit; bottom plot
showsthederivativeofcorrelation functionsobtainedanalytically, withtheoscillations presentat thesamevaluesofJz asthe
entanglement entropy, the former resulting in the latter. Middle panel: Schmidt gap for the half-region is plotted against Jz.
Initially whenthesystem isinthegappedphase,theentanglement gap gradually decreasesasJz increases, butitgoes tozero
in a first order jump when it enters the gapless regime. However the gap remains zero when Jz is increased further and the
system transits to the large Jz gapped phase, consistent with zero energy edge-modes in the latter two phases (right panel);
inset shows the variation of the gap for different number of chains in the subregion ny as indicated. We have taken a system
withNx =6,Ny =60. Thegapisagenericfeatureofasubsystemcomprisinganynumberofparallelchains. Howeverthefirst
orderjumpbecomessharperasny isincreased. Rightpanel: positivespectrumoftheMajoranaHamiltonianforthehalf-region
withJy/Jx =0.3asafunctionofJz fora10×100system. Energymodesbecomezeroenergymodesatthetransitionfromthe
gappedtogaplessphase;wehavecheckedthattheseareindeededgestateslocalized ontheboundary. HoweverinthelargeJz
limit where the system again becomes gapped, theedge modes persist and remain gapless.
presence of an additional factor f(~k) in the denomina- enters the other gapped regime (when J J + J ).
z x y
| | ≥
tor which becomes vanishingly small for some values of This is a new result and we conclude that the zero
k ,k .). Schimdt gap is not a sufficient condition to signal the
x y
The oscillations in C may also be understood alter- presence of a gapless phase as was done in an earlier
zz
natively from the form of Eq.(18). In the gapless phase study38. To ascertainthe originofthe zerogap,we have
the dispersion f(~k) vanishes at certain ~k points in the investigated the edge mode spectrum of the half-region
Brillouin zone; however due to the discrete numerical becausethepresenceoftopologicalorderandedgestates
evaluation of ~k-points, the dispersion may take values are intimately linked to the entanglement spectrum6,44.
that are arbitrarily close to zero depending on the We have found that for the half-region geometry, which
systemsize and the value of the model parameters. This haszig-zagx y chainsattheends,thesystemharbours
−
can occur only in the absence of a bulk gap, eventually gaplessedgemodesinthelargeJz phaseaswellasinthe
resulting in the oscillations of the entanglement entropy, gaplessphase. This has been shownin the rightpanelof
the magnitude of which depends on the full system size; Fig.2. However the system has a gapped edge spectrum
we point out that this is not a violation of the area law, in the weakly coupled gapped phase. Our findings of
the fulfilment of which is more conspicuously visible in the Majorana edge mode spectrum agree with earlier
the gapped phase for the same reason. analyticalresultsonthesame45 which,togetherwithour
findings on the Schmidt gap, corroborates the previous
observation44,46 that a gapless edge state is associated
Let us turn now to the entanglement gap. In the with a gapless entanglement spectrum.
middle panel of Fig.2, we have plotted the Schmidt gap Our results for entanglement entropy and Schmidt
for the half-region for J = J . We have found that as gap may be compared with that of a previous study9,
x y
long as we are in the weak6 ly coupled chain limit and in where the authors studied the entanglement entropy
the gapped phase (J J +J ) the Schmidt gap and Schmidt gap of the two-dimensional transverse field
x(y) y(x) z
is finite, but it abruptly g≥oes to zero when it enters the Ising model. Both the entanglement entropy and the
gaplessregion(when J J +J where i, j, k couldbe gap have different behaviour than what we have found
i j k
x ,y ,z and its cyclic co≤mbinations). The gap continues here. Thecharacteristicscalingofentanglementgapand
to remain zero even in the large J limit as the system the crossing of entanglement gap at the transition point
z
7
when the system goes from gapped to gapless regime mains almost constant before entering the gapped phase
is absent in the Kitaev model. This may be attributed where the entanglement entropy decreases uniformly ir-
to the fact that the transition in the transverse model respective of subsystem sizes and shapes. Thus we find
is continuous and for the Kitaev model the transition that the entanglement entropy may be nonmonotonic or
from gapless to gapped one is a topological one. Thus monotonicdepending onthe systemparameters. We be-
we can see that the entanglement gap does not depend lieve that the nonmonotonic variation of the entangle-
on the gapless or gapped nature of the bulk spectrum mententropyacrossphasetransitionsandalsowithinthe
but rather it is a property of the gapless edge mode: givengaplessphase (implying that there is point of min-
when such gapless edge modes exists, the entanglement imal entanglement in the gapless phase which depends
gap is zero. However the entanglement gap can can also on the subregion geometry) is a fact worth highlighting.
be zero without the presence of a gapless edge mode, Indeedanoscillatorynonmonotonicitywasalsoobserved
for instance by breaking time reversal symmetry but fortheentanglemententropyinthehalf-regiongeometry
preserving the inversion symmetry46, the aspect which withinthe gaplessphase;thatwashoweverattributedto
we do not pursue here. the onset of gapless~k modes in the bulk.
We note that the above behaviour of entanglement We explain the above variation of entanglement entropy
entropyandgapaspresentedinFig.2isalsotrueforany qualitatively again with the correlation function. We
number of chains taken periodic in the x-direction. We note that an exact quantitative determination of entan-
demonstratethisintheinsetofthemiddlepanelofFig.2 glement entropy requires an analytical diagonalization
wherewehaveplottedtheentanglementgapfordifferent of correlation matrix which is beyond the scope of any
number of chains ny; it is evident that the entanglement known technique to our knowledge. However we can ex-
gap remains qualitatively unchanged except that the pect that the boundary bonds shared between the sub-
change from a finite gap to a zero gap becomes sharper systemandthesystemwoulddeterminequalitativelythe
as the number of chains in the subsystem increases. entanglement entropy. In the middle panel of Fig.3, we
have plotted the two point Majorana fermionic correla-
tion function for x-bonds and z-bonds. Any rectangular
or square block shares x-bonds with the system at left
andrightboundary. Ontheotherhanditsharesz-bonds
IV. ENTANGLEMENT IN
SQUARE/RECTANGULAR SUBREGIONS at the upper and lower boundary bonds. As we see from
themiddle panelofFig.3,the twobondcorrelationfunc-
tions behave differently as we increase the value of J .
In this sectionwe presentnumericalresults for the en- z
As the ratio of number of z-bonds and x-bonds are dif-
tanglement entropy and gap of a finite block of square
ferentfordifferentsubsystems,theentanglemententropy
or rectangular geometry as shown in Fig.1. Firstly, we
in turn shows a nonmonotonic behaviour. Thus we see
analyze the variation of the entanglement entropy and
that by manipulating the ratio of length to breadth of
gap as we vary J for fixed values of J ,J (i) along the
z x y a rectangular subsystem, one can go from monotonic to
contour J =J when the system changes from the gap-
x y nonmonotonic dependence of the entanglement entropy
less to the gapped phase and (ii) along the J = J line
x 6 y within the gapless phase.
whenthesystemchangesfromthegappedtogaplessand
For J /J = 0.3, entanglement entropy is plotted in the
againtogappedphase. Andsecondly,westudy the vari- y x
lowerleftpanelofFig.3,wherewealsoobservesimilarbe-
ation of the entanglement entropy and gap as a function
haviour;irrespective ofthe subsystem size,the entangle-
ofsystemsizeandshapes,mainly fromsquaretorectan-
ment entropy shows monotonic behaviour in the gapped
gular geometry. Our primary motivation in undertaking
phases. For this case, in the gapless phase, as explained
thesetwoapproachesistodiscernsignaturesofthetran-
in the previous paragraph and for the half-region of the
sitions,individualphases,andMajoranaedgestatesfrom
previous section, the entanglement entropy or may not
the non-universalconstant α˜.
show monotonic behaviour depending on the values of
In the left panels of Fig.3 we plot the variation of the
J /J andJ /J . Indeedforagivenb /b n /n ratio
entanglemententropyasafunctionofJz forvarioussub- (wyhexreb anzdbxarethenumberofx-xbonzd∝sanydz-xbonds
systemsizesforJ =J (upperplot)andforJ /J =0.3 x z
x y y x sharedbetweenasquare/rectangularsubsystemwiththe
(lower plot). For J /J = 0.3, similar to the cylindrical
y x system, and n n is the subsystem size) the scaled
subregions, Jx = 1.0 and Jy = 0.3. Thus, in the suc- entanglement exnt×ropyy curves seem to collapse on top of
cessive plots of entanglement properties for J /J = 0.3
y x each other both for J = J and J = J . This may be
as shown in Fig. 3 as a function of Jz, Jz is also scaled explained qualitativelyy as foxllows: tyhe6 prximary contribu-
in units of J . The results are scaled by the length of
x tiontotheentanglemententropycomesfromthenearest-
the boundary in each case. In the first case we observe
neighbour bond-correlation between the system and the
thatinitiallyasweincreaseJ ,theentanglemententropy
z subsystem. Then let S(n ,n ) denote the entanglement
decreases for all subsystem sizes. However depending on x y
entropyofacertainrectangularsubsystemgeometryhav-
the subsystem shapes, the entanglement entropy either
ing a total of b ( n ) and b ( n ) boundary x-bonds
starts to increase after some value of Jz (the particular and z-bonds resxp∝ectivyely. Forzs∝impxlicity we may assume
valueofJ dependsontheshapeofthesubsystem)orre-
z
8
1 .12 SE/(Nx + Ny) 1.0 <cαicαi+1> ∆
0.8
0.6 12x12 6e-06
12x18 0.8 0.4
0.4 18x12 3e-02
18x18
18x30
0.2 30x18
0 0.6 Squares: α = x
Circles: α = z 4e-06 2e-02
1.4
0.2
1.2 0.4
1 1e-02
0.8 2e-06
0.6 0.2
0.0
0.4
0.2
0 0.0 0 Jz
0 0.4 0.8 1.2 1.6 2 2.4 0 1 2 3 4 0 0.5 1 1.5 2 2.5 3
FIG.3. (Colouronline)EntanglementpropertiesofgeneralsquareandrectangularsubregionsasafunctionofcouplingJz ina
36×36Kitaevsystem;gaplessphaseisshadedblueasinFig.2. Leftpanel: Entanglemententropyofvarioussubregionsacross
thephasetransitionsforJy/Jx =1(top)andJy/Jx =0.3(bottom);atthetransitionto/fromthegaplessphasethereisacusp
in the free fermion entanglement entropy. However within the gapless phase the entropy does not monotonically change with
varyingJz,anddependsonthetransversecouplingandgeometryofthesubregion. Middlepanel: nearest-neighbourcorrelations
for x−x and z−z bonds in the ground state for Jy/Jx = 0.3 (closed symbols),1.0 (open symbols), the nonmonotonicity of
which in turn suggests the nonmonotonicity of the entanglement entropy in the left panel. Right panel: Schmidt gap ∆ for
varioussubsystemgeometrieswith Jy/Jx =0.3. Withinthegaplessphasetherearestrongervariationsin∆thanwhenwithin
or closer to the transition to the gapped phases. Inset displays similar results for Jy/Jx = 1 (left y-axis corresponds to the
largest subsystem30×18). TheMajorana edgestates(notshown) arenotstrongly localized along theboundariesin thiscase
having finite extension in the bulk as well. Thus the putative correspondence between the gapless edge mode and the gapless
entanglement spectrum might not be straightforward here.
S(n ,n ) = κ n +κ n where κ and κ are the con- thesystemisgapless;∆graduallybecomessmallerasJ
x y x y z x x z z
tributions due to a boundary x-bond and z-bond; then approaches deep within the gapless phase, and remains
S(n ,n )/(n +n ) takes on a given value for a given zero thereafter. We have checked the lowest eigenfunc-
x y x y
n /n . However deviations from this qualitative argu- tions andfoundthatalthoughitis primarilylocalizedat
x y
ment are expected as contributions come from all pos- theedgeithasconsiderableextensioninthebulkaswell;
sible correlation functions beyond the nearest-neighbour in such a situation the precise relation between the edge
correlations. mode and the entanglementgap ∆ becomes complicated
Now we turn to the right panel of Fig.3, where we have and is unclear to us.
plotted the entanglement gap or Schmidt gap ∆ for
J /J = 0.3. We find in the gapped region that the en-
y x
tanglemententropyisvanishinglysmallbutinthegapless V. ANALYTICAL INSIGHTS
regimetheentanglemententropyfluctuatesasafunction
of Jz. However given the small value of ∆, we can as- In this section we provide an analyticalinsight to cor-
sume that it is essentially zero. We have also looked at roborate our numerical findings in Secs. III and IV. It
the zero energy eigenmode and found that they are es- maybenotedherethattheresultspresentedintheprevi-
sentiallyconfinedattheedgeforsmallandlargeJz,with ous sections are exact since in the derivation of Eqs. (9)
smallextensionstothebulk;thisisverydistinctfromthe and(11),noapproximationshavebeenused40. However,
case of the half-region where no gapless edge states were because of the complexity of obtaining an analytic ex-
found in the small Jz gapped phase. ∆=0isaconfirma- pressionfor the correlationmatrix andits spectralprop-
tionofRefs. 6and44whichpositthatforasystemwith erties, one has to resort to numerical diagonalization of
a gappedbulk, the Schmidt gapdepends onthe zero en- the same.
ergyedgemodes. Moreover,analogoustothe half-region Using perturbative approximations in certain regimes
of the previous section, although the system is gapless of J one may obtain some analytical insights into the
z
in the bulk for the shaded region, we can attribute the nature of the entanglement entropy and gap, and may
gapless edge modes as the cause of vanishingly small ∆. thus better understand the numerical results.
Now let us consider the Schmidt gap for Jx =Jy, which For completeness we first enlist our key results so far.
isshownintheinsetoftherightpanelofFig.3. Wefinda Firstly,inthe caseofthe half-region,wehavefoundthat
strongdependenceof∆onthesubsystemgeometry. But theentanglemententropyincreasesparabolicallywithJ
z
they are more or less finite in the small Jz limit where in the gapped phase (where Jz is small i.e in the weakly
9
coupled chain limit) and becomes oscillatory in the gap- where θ = tan 1 Jysink . Using (24), the inter-
lessregime. Theentanglementgapisfinite intheweakly k − (cid:16)Jx+Jycosk(cid:17)
chain perturbation governedby J can be written as,
z
coupled chain limit where the system is gapped. It goes
traoetmtrzaiebirnuosteogdnacpteolettshhseeinpsyrteshsteeenmlacreegoneftJezrzesrgotahepenpegerdagpyplehMsasasejro,ergwaimnhaiecheadwngdee Hpm,m+1 =kX≥0Jze4iθk(cid:16)αmk†αmk−1−αmk†βkm−1
astactreitsi.caSlecvoanludely,Jwe=foJundfotrhtahtefosrquJaxr=e bJlyo,ctkheernetaenxgisltes- +βkm†αmk−1−βkm†βkm−1(cid:17)+h.c, (25)
z c
ment entropy, upto which it decreases and then either where the summation is over all the m chains.
increases or remains constant. Thirdly, for the square Groundstateofthesystemcanbewrittenastheprod-
block we found that the entanglement gap shows small uct of the individual ground states of each chain. Let
fluctuations for small J . We begin our discussion in g,m denote the ground state of mth chain and de-
z | i |Gi
the small J limit i.e weakly coupled chain limit in Sec. note the ground state for the complete system, then we
z
VA where we qualitatively explain the behaviour of the can write,
entanglement entropy and gap for the half-region. The
Ny
blaerhgaevJioulrimoifttwhiellebnetaenxgplleamineendtiennStreocp.yVaBn.d gap in the |Gi= |g,mi, |g,mi= βkm†|0i. (26)
z mY=1 Yk
Our next task is to find the perturbed ground state
A. Weakly coupled chain limit: small Jz when we take into account Eq.(25) as the perturbation.
For simplicity, we limit ourselves up to to second order
inJ forthecorrectionsintheentanglemententropyand
The weakly coupled chain limit can be addressed by z
gapandneglectsubsequenthigherordercorrections. For
perturbationtheorywheretheunperturbedHamiltonian
this purpose it is sufficient to limit up to the first order
consists of one-dimensional chains. The complete two-
corrections of the ground state which is given as,
dimensional system may be considered as a composition
of many one-dimensional chains coupled by a small J .
z
Hrthioeerdeificrwsbetoccuohnnadsianidr.yerTsNhoeythcnoaumtmptbhleeetreNoHfythcaomcuhiplatleoindniiacshnacfioonnrsntwehciettehtdwptoeo-- |G1i=|Gi(cid:16)1−NyXk J64z2ǫ12k(cid:17)
dimensional lattice is written as (−1)Ny−mJz e−iθk 0,0;m 1 1,1;m :m,m 1
= 0+ ′, (19) −Xm 8ǫk (cid:16) | − i| i|G − i
H H H
Ny eiθk 0,0;m+1 1,1;m :m+1,m . (27)
H0 = H0m, H′ = Hm′ ,m+1, (20) − | i| i|G i(cid:17)
Xm Xm Where, : a,b,c,... = g,m and
|G i m=a,b,c,..| i
m = iJ cm cm +iJ cm cm , (21) p,q;m = (α )p(β )q 0 , where pQ, q6can only take val-
H0 Xn (cid:16) x n,a n,b y n,b n+1,a(cid:17) u|es0ori1. Th†meredum†ced| diensitymatrixofthe half-region
Hm′ ,m+1 = iJzcmn,acmn,−b1. (22) ccaonrrebcetioenastiolythcaelcgurloautnedd sbtyatceownsaivdeefruinncgtitohneitfisresltf.oTrdheisr
Xn
contains two diagonal terms: the first comes from the
Here H0m is the unperturbed Hamiltonian for the mth first term of Eq.(27) and is proportional to unity. The
one-dimensionalchainand ′denotestheinterchaincou- secondtermofEq.(27)yieldsanotherdiagonaltermpro-
H
pling and forms the perturbation to the unperturbed portionalto J2. The off-diagonaltermsareoftwotypes,
z
Hamiltonian H0. H can be diagonalised using a Fourier terms which are proportionalto Jz and is obtained from
transform,cmn,γ = ~kei~k.~ri,γcmn,γ forthemth chainwhere the product of first and third terms of Eq.(27). These
γ = a,b. Notice tPhat for cmn to be Majorana fermions, off-diagonal terms can be treated as perturbation to the
wemusthavecmk,γ =cmk,γ† whichimplies thatinmomen- first diagonal terms of the reduced density matrix which
tum space only−the fermion operators belonging to first is proportional to unity. While the other off-diagonal
half of Brillouin zone are independent i.e~k summation terms coming from the second term yields a correction
is over (0,π). Using this definition and afte−r subsequent ∼ Jz4. Thus considering only the terms upto Jz2, we
diagonalizationwe can write, get the following expressions for the largest and second
largest eigenvalues of the reduced density matrix,
H0m =X|ǫk|(cid:16)αmk†αmk −βkm†βkm(cid:17), (23)
wdeifithneǫdka=s, |Jx +Jyeik|. The new fermionic modes are λ1 =λ0+Xk (N′3−2ǫ12k)Jz2(cid:16)λ0− 6J4zǫ22k(cid:17)−1, (28)
cmk,a = 1 ieiθk ieiθk αmk , (24) λ = Jz2 = Jz2 , (29)
(cid:18) cmk,b (cid:19) √2(cid:18) 1 −1 (cid:19)(cid:18) βkm (cid:19) 2 (cid:16)64ǫ2k(cid:17)min 64(Jx−Jy)2
10
x,y, is then given by
χ
ψ1 χ1 ψ2 |G1i=(cid:18)1−N˜8JJ2z2(cid:19)|Oi+<i,Xi+δα>2JJαz|1n,1n+δαi,
2 (31)
where 1 ,1 = 1 1 anddenotesthefilledstates
n m n m
| i | i ×| i
at the dimer ‘n’ and ‘m’ and N˜ is the total number of
z-bonds in the system. We retain the expansion of the
FIG. 4. (Colour online) In the large Jz limit we associate groundstate upto first order as we intend to find the re-
each z-bond with a ψ fermion in order to compute the re- duceddensitymatrixuptosecondorderinJ. Weseethat
duced density matrix. The boundary is composed of the up- the two Majorana fermions may be grouped together to
per, lower, and the two side boundaries. For the upper and defineasinglecomplexfermionatagivenz-bond. While
lower boundaries a ψ fermion is shared between the system calculatingthereduceddensitymatrixoneneedstointe-
andthesubsystem. Forthisreasonweregroupedthetwoad-
grate out the Majorana fermion outside the subsystem.
jacent ψ fermions to define two χ fermions (as shown at the
Tothisendweintroduceapairofχfermions(χ ,χ )out
right side of the figure) such that χ1 belongs to the system 1 2
ofthetwoadjacentψ fermionsasshownintheFig.4and
and χ2 belongs to thesubsystem.
thencalculatethereduceddensitymatrixofanextended
subsystemwhichincludes boththe sites ofthe boundary
z-bonds. The reduced density matrix for the half-region
whereλ0 =1−Ny k 3J2zǫ22k. Inthe aboveN′ isthe num- can then be obtained easily by taking a trace of the χ1
ber of chains in thePsubsystem. From the expression of fermions. The detailed mapping between the Fock space
Eq.(28), we find that for small values of J , the entan- of ψ and χ fermions is as follows:
z 1,2 1,2
glement entropy has a parabolic dependence on J as is
z 1
found in the upper left panel of Fig.2. 0,0 = (1,0 +i0,1 ),
| iψj √2 | iχj | iχj
1
1,0 = (0,0 i1,1 ),
| iψj √2 | iχj − | iχj
B. Dimer limit: strong Jz 1
0,1 = (i0,0 i1,1 ),
| iψj −√2 | iχj − | iχj
In this section we present the perturbative results for
1
entanglement entropy and gap in the large J limit for 1,1 = (0,1 +i1,0 ), (32)
the half-region. In the limit J the Hazmiltonian | iψj √2 | iχj | iχj
z
→ ∞
consistsofisolatedz-bondsonlyandtheHamiltonianfor where j = 1,2. We can now write down the reduced
eachz-bondisJzick,1ck,2 wherek denotesaparticularz- density matrix of the extended subsystem in the ψ basis
bond and ‘1’ and ‘2’ refer to the sites inside and outside as follows
of the subsystem. It is straightforward to see that inte-
J2 J2
grating out one Majorana fermion yields a contribution ρ = 1 N + ˜,1 ˜,1 +
log2totheentropy. ThusifthereareintotalNz number E (cid:18) − 4Jz2(cid:19)|OihO| Xi (cid:16)4Jz2|O iihO i|
of Jz bonds shared between the subsystem and the sys- J2 J
teenmtr,opwyeaosbtNaiznlotgh2e. lTimhietinpgervtualrubeastioofnthheereenatmanogulenmtsentot Xi,j (cid:20)4Jz2|O˜,1i,1jihO˜,1i,1j|+ 2Jz|OihO˜,1i,1j|(cid:21)(cid:17).
switching on the hopping of Majorana fermions between (33)
nearest-neighbourdimers. The Hamiltonian can then be
written as In the above expression denotes the vacuum of the
|Oi
extendedsubsystemonlyand ˜,1 ,1 ... denotesastate
i j
|O i
withfilleddimeronsitesi, j,etc. Inthesecondterm,the
H = J ic c + J ic c , α=x,y. (30)
z n,1 n,2 α n,1 n+δα,2 index ‘i’ refers only to the boundary z-bonds. In third
Xn Xn
and fourth terms ‘i’ and ‘j’ denotes nearest-neighbours.
Nowemployingthe transformationinEq.(32)betweenψ
In the above expression n+δ refers to the nearest-
α
neighbour dimer connected by an α-bond with the nth andχbasis,weimmediatelyseethatfirsttermofEq.(33)
dimer. We may now define a ψ fermionic basis using the yieldsNz numberofdegenerateeigenvalues 21(1−N˜4JJ2z2)
substitution c =ψ +ψ , c = i(ψ ψ ), such where N is the number of boundary z-bonds. This is a
n,1 n n† n,2 − n− n† z
thatψ 0 =0,with 0 being the vacuumstate ofthe vital point and corroborates the vanishing Schmidt gap
n n n
| i | i
dimer at site ‘n’; the ground state of the full system is observed in the dimer limit. This result, taken together
then obtained as = where = 0 . To with the analytical computations of Ref. 45 where zig-
|G0i |Oi |Oi i| ii
calculate the reduced density matrix, as in tQhe weak Jz zag Majorana edge states are unveiled in the large-Jz
limit, we begin with the perturbed ground state. The limit, provides an analytical confirmation of the bulk-
perturbed ground state to second order in J =J , α= edge correspondence6,44.
α