Perturbative approach to an exactly solved problem: the Kitaev honeycomb model

0.64 MB·English

by  J. Vidal

Perturbative approach to an exactly solved problem : the Kitaev honeycomb model
Julien Vidal,1,∗ Kai Phillip Schmidt,2,† and S´ebastien Dusuel3,‡
1Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, CNRS UMR 7600,
Universit´e Pierre et Marie Curie Paris 06, 4 Place Jussieu, 75252 Paris Cedex 05, France
2Lehrstuhl fu¨r theoretische Physik, Otto-Hahn-Straße 4, D-44221 Dortmund, Germany
3Lyc´ee Saint-Louis, 44 Boulevard Saint-Michel, 75006 Paris, France
WeanalyzethegappedphaseoftheKitaevhoneycombmodelperturbativelyintheisolated-dimer
limit. Ouranalysis isbased onthecontinuousunitarytransformations methodwhich allows oneto
compute the spectrum as well as matrix elements of operators between eigenstates, at high order.
9
Thestartingpointofourstudyconsistsinanexactmappingoftheoriginalhoneycombspinsystem
0
onto a square-lattice model involving an effective spin and a hardcore boson. We then derive the
0
low-energy effective Hamiltonian up to order 10 which is found to describe an interacting-anyon
2
system, contrary to the order 4 result which predicts a free theory. These results give the ground-

n state energy in any vortex sector and thus also the vortex gap, which is relevant for experiments.
a Furthermore, we show that the elementary excitations are emerging free fermions composed of a
J
hardcore boson with an attached spin- and phase- operator string. We also focus on observables

2 and compute, in particular, the spin-spin correlation functions. We show that they admit a multi-
1 plaquetteexpansion that we deriveup to order6. Finally, we study thecreation and manipulation
ofanyonswithlocaloperators,showthattheyalsocreatefermions,anddiscusstherelevanceofour
] findingsfor experiments in optical lattices.
r
e
h PACSnumbers: 75.10.-b,75.10.Jm,03.65.Vf,05.30.Pr
t
o
.
t I. INTRODUCTION tracoldatoms28,29,30,31 orJosephsonjunctions32. It thus
a
m constitutes an appealing candidate for the observation
of anyons. Nevertheless, the presence of fermions in the
- Elementaryparticlescanbeclassifiedintwocategories
d spectrum may spoil the detection process ; a point com-
accordingtothevalueoftheirspin. Half-integerspinpar-
n pletelymissedinarecentproposal(seeRef.33forexpla-
ticles obeyFermi-Diracstatisticsandarecalledfermions
o nation and Ref. 16 for details).
whereas integer-spin particles obey Bose-Einstein statis-
c
[ tics and are known as bosons. However, some quantum The goal of the present paper is to investigate the
objects may obey other (fractional) statistics describing gapped phase of the Kitaev honeycomb model7. Indeed,
2
nontrivial braiding as initially suggested by Leinaas and in his remarkable seminal paper, Kitaev mainly focuses
v
Myrheim more than thirty years ago1, and by Wilczek on the special subspace of the Hilbert space to which
3
5 in the eighties2,3. Despite numerous theoretical works, the ground state belongs to and the low-energy spec-
5 these so-calledanyonsarestillwaitingfor adirectobser- trumofothersubspaceshasonlybeendiscussedlately13.
1 vation although recent experimental proposals are very Our aim is to bridge this gap by providing a high-order
9. promising (see Ref. 4). perturbative analysis, in the isolated-dimer limit, of the
0 In the last years, anyons have drawn much atten- spectrum as well as some interesting results about the
8 tion because of their interest for topological quantum creation and the manipulation of anyons which is of rel-
0 computation5. In this perspective, several models have evance for experiments30,31. Part of our results have al-
v: been proposed among which the celebrated toric code6 readybeengivenintwoshortpapers13,16 andthepresent
i which is a spin-1/2system whose elementary excitations papermaybeconsideredasanextendedanddetailedver-
X
behave as semions. However, the experimental realiza- sionoftheseworks. Howevermanyotherresultsarepre-
ar tion of this system is rather tricky since it involves four- sentedhereamongwhichthe interplaybetweenfermions
spininteractions. Here,weshallfocusonanothersystem and anyonsunder string operationsdiscussed in Sec. IX.
originally proposed by Kitaev7 which only involves two- Thispaperisorganizedasfollows. Inthenextsection,
spininteractions. Thismodelisveryrichsinceitcontains we introduce the model as well as its main properties.
Abelianandnon-Abeliananyonicaswellasfermionicex- In particular, we discuss the importance of the bound-
citations. Thus, it has been the subject of many recent aryconditions andinsistonthe roleplayedbyconserved
studies concerning the spectrum,8,9,10,11,12,13,14,15,16 the quantities15 and the constraints resulting from them. In
correlation functions and the entanglement16,17,18,19,20, Sec. III, we show how to map the Kitaev model involv-
or the quench dynamics21,22. Let us also mention sev- ing spins onthe honeycomblattice onto aneffective spin
eral extensions23,24,25 among which the analysis of time- andhardcorebosonona squarelattice. This mapping is
reversal symmetry breaking terms26,27 which may give the starting point of the perturbation theory presented
rise to a chiral spin liquid. in this work. In Sec. IV, we explain how to diagonal-
Furthermore,thismodelissusceptibletoberealizedin ize the Hamiltonian order by order using the perturba-
variousexperimentalsystemssuchaspolarmolecules,ul- tive continuous unitary transformation (PCUT) method
2
. The study of the low-energy(zero-quasiparticle)sector x (a) (b)
is the subject of Sec. V, a large part of which is devoted z
to a pictural (and hopefully pedagogical) analysis and y
construction of the eigenstates of the toric code model n2 n1 U n2 n1
which naturally emerges from this problem. There, we
givethe perturbativeexpansionformofthegroundstate 2 3 4
L p R
energy for any vortex configuration. The effective low- p
energy theory is found to be described by interacting
1 6 5 D
anyonscontrarytothelowest-orderresultwhichpredicts
free anyons7. Sec. VI focuses on the study of the one-
FIG. 1: (color online). Mapping of the honeycomb (brick-
quasiparticle subspace, where the physics is shown to be
wall) lattice onto a square lattice with unit basis vectors n
1
that of a particle hopping in a magnetic field with zero
and n . Each z−dimer with four spin configurations is re-
2
or half a flux quantum per elementary plaquette. The placed by a single site with four degrees of freedom : the
demonstration of the fermionic nature (known from ex- occupation numberof hardcoreboson (0or1),and theeffec-
act solutions) of the quasiparticles is briefly sketched in tivespin(⇑,or⇓)whichischosenasthespinoftheblacksite
Sec. VII. In Sec. VIII, we provide some checks of our of the considered z− dimer. The numbering of the sites of a
results by analyzing simple vortex configurations which plaquettep is shown in both cases.
allow for an exact solution. The spin-spin correlation
functions and the manipulation of anyons are tackled
(a) (b)
in Sec. IX, which is devoted to the renormalization of
observables. Finally, we discuss several issues and give
some perspectives. Technical details as well as all rele-
vant coefficients involved in the perturbative expansions n n
2 1
are gathered in appendices.
In what follows, we tried to be as pedagogical as pos-
sible and always favored simple demonstrations on con-
2pn 2pn
creteexamplesratherthanlengthyproofsforgeneralsit- 2 1
uations. We hope that it will help the reader to under-
stand the richness of this model.
FIG.2: (coloronline). Theperiodicboundaryconditionsused
in this work, (a) on the original brick-wall lattice and (b) on
the effective square lattice of z-dimers (see Fig. 1). In the
II. THE MODEL
figure p=1. In both cases, the finite-size system is put on a
torusobtainedbyidentifyingtheoppositesidesofthedashed
A. Hamiltonian and boundary conditions (magenta) square. Forclarity, thesite at thepoint chosen as
the origin has been depicted bigger (and in magenta). Half
thesquareplaquettesin(b)havebeencolored incyan(gray)
Themodelconsideredinthisworkisaspin-1/2system
to show that the periodic boundary conditions allow us to
proposed by Kitaev7 in which spins are located at the
bi-color the lattice.
vertices of a honeycomb lattice. Since the honeycomb
latticeistopologicallyequivalenttothebrick-walllattice,
we shall always represent it as shown in Fig. 1a. In this anticipatewhatfollowsandmentionthattheseboundary
lattice, one distinguishes three types of links (x, y, and conditions are such that the lattice of z-dimers (Fig. 1b)
z) to which one associates three different couplings and can be bi-colored as shown in Fig. 2b).
interactions. The Hamiltonian of the system is
H = J σασα, (1) B. Conserved quantities
− α i j
α=x,y,zα−links
X X
A remarkable property of Hamiltonian (1), is that its
wfohlleorweinσgiα wareeatshseumuseu,awl iPthauoulitmloastsricoefsgaetnesirtaeliti.y7I,nththaet elementaryoperatorsKij =σiασjα commutewithplaque-
tte operators W so that [H,W ]=0. For the plaquette
p p
J 0 for all α and J J ,J .
α z x y p shown in Fig. 1a, such an operator is defined as
≥ ≥
We will either work with an infinite system and open
boundary conditions (a plane), or with a finite (or infi- W =K K K K K K =σxσyσzσxσyσz. (2)
p 12 23 34 45 56 61 1 2 3 4 5 6
nite)systemandperiodicboundaryconditions(atorus).
Inthelattercaseandforreasonsthatwillbecomeclearer Let us mention that in the expression of W in terms
p
in the following (in particular, see Sec. VB), we shall of the K’s, one could have started at any site instead
restrict ourselves to the periodic boundary conditions of site 1 and/or one could have taken the product of
(PBC) depicted in Fig. 2. The number of sites N is K′s anti-clockwise instead of clockwise. Furthermore,
s
N =2(2p)2 =8p2,with p N (p=1 in Fig.2a). Letus theexpressionintermsofσ’scouldalsobewrittenW =
s p
∈
3
(a) (b)
p p ′
p C σ C
x
σ
x
σ
y
σ
y
σ
z
σ
z
FIG. 4: (color online). Illustration of therelation L′=WpL,
FIG.3: (coloronline). Illustrationoftheconservedplaquette with L = Qi∈Cσiout(i), Wp already shown in Fig. 3 and
quantityWp. Thethickyellow(lightestgray)linedelimitates L′ = Q σout(i). The thick yellow (lightest gray) line in
i∈C′ i
the plaquette p. The thick red (gray), green (light gray) or (a)representsthecontourC andtheonein(b)representsC′.
blue(darkgray)segmentsrepresentthePaulimatricesσout(i). As in Fig. 3, the thick red (gray), green (light gray) or blue
i
(dark gray) segments represent the Pauli matrices σout(i).
i
σout(i) where i runs over the set of six spins around
i i
the plaquette p, and where the notation out(i) means ′
Qthe “outgoing” direction at site i, with respect to the (a) (b) Cb
b
plaquette’s contour. An illustration of the Wp operator 8
is given in Fig. 3. a4 a5 a6 b6 b7 Cb
Since W2 = I, the eigenvalues of the plaquette oper- a a b b
p 2 3 4 5
ators are wp = 1. Note that [Wp,Wp′] = 0, as can a b b
± 1 a 2 3
be shown from the usual Pauli matrices algebra. As a C
b
consequence, H andthe W ’s canbe diagonalizedsimul- 1
p σ
taneously. Following Kitaev, we will call a vortex sector x
a subspace of the Hilbert space with a given map of the σ
y
w ’s. By definition a vortex is a plaquette for which
p
wp = 1, so that for example, the vortex-free sector is σ
− z
defined by w =+1 for all p’s.
p (c) (d)
In fact, all loop operators made of “outgoing spins”
d
(see Figs. 4 and 5) are conserved and all commute with C
each other, which can be verified in the same way as for
the W ’s. However, not all of them can be set indepen-
p
dently to 1. Some relations among them arise from
±
the following fact (which can be checked by studying all
possible cases) : the product of Wp and a nearby loop Cd
operator givesanewloopoperator ′ =W ,asillus- c
p C
L L L
trated on a particular example in Fig. 4.
As an illustration of other relations involving loop op-
erators around the torus, with the loops of Fig. 5 one FIG.5: (color online). Examplesofconservedloop operators
has L=Q σout(i) in a finite-size system with PBC.
i∈C i
6
= W , (3)
La an taken overthe complementary set of plaquettes. Indeed,
nY=1 onthetorustherelationsamongloopoperatorsyieldthe
8
following constraint
′ = W , (4)
Lb Lb bn
nY=1 Wp =I, (6)
= , (5)
Ld −LaLbLc alYlp′s
out(i)
where we havedenoted, for example, = σ . showing in particular that the number of vortices has to
La i∈Ca i
Theminussigninthelastequationabovecomesfromthe be even in a system with PBC.
Q
crossing of and . In the three expressions above, From examples shown in Fig. 5, one can deduce that
b c
L L
the product of plaquette operators could also have been all W ’s [except one, because of Eq. (6)], and can
p b c
L L
4
besetindependentlyto 1,whichthenimposesallother can be described by a spin 1/2, indicating which of the
±
conserved quantities. two configurations is realized. There are many possible
parametrizations but here we choose the following
= 0 , = 0 , = 1 , = 1 . (7)
C. Some results from the exact solution |↑↑i |⇑ i |↓↓i |⇓ i |↑↓i |⇑ i |↓↑i |⇓ i
Theleft(right)spinistheoneoftheblack(white)siteof
The above discussed local conserved quantities are the dimer ( = , etc). Double arrows represent
• ◦
|↑↓i |↑ ↓ i
not sufficient to fully diagonalize the Hamiltonian. In- the state of the effective spin which is here the same as
deed, if N is the number of sites, then there is a total the state of the left (black) spin.
s
of N = N /2 plaquettes, but only N 1 independent Within such a mapping, effective spins and hardcore
s
−
ones. With the two cycles around the torus, this gives bosonsliveontheeffectivesquarelatticeofz-dimers(see
N +1 independent conserved quantities, which is obvi- Fig. 1). This lattice is shown again in Fig. 2b, together
ously smaller than N . with the PBC, which are such that it can be bi-colored.
s
However, H has a crucial property : it can be trans- In what follows, the sites of the effective lattice will be
formed into a free Majorana fermions Hamiltonian and denoted with bold letters, such as i.
is thus exactly solvable. Let us also mention that an- LetusnowwritetheHamiltonian(1)inthislanguage.
other solution based on the Jordan-Wigner transfor- Therefore, we first translate the action of the spin oper-
mation maps the spin Hamiltonian H onto a spinless ators in the effective-spin boson (ESB) formalism. It is
fermions system with p-wave pairing9,11,12. easy to check that one has
AsshownbyKitaev7,thegroundstateofH liesinthe
σx =τx(b†+b ) , σx =b†+b ,
vortex-freesectorandthephasediagramcontains,inthis i,• i i i i,◦ i i
sector,twophases: agappedphaseforJ >J +J and σy =τy(b†+b ) , σy =iτz(b† b ), (8)
z x y i,• i i i i,◦ i i − i
a gapless phase for Jz ≤ Jx +Jy. In the gapped phase σiz,• =τiz , σiz,◦ =τiz(1−2b†ibi).
the low-energy excitations are Abelian anyons (semions)
whereas in the gapless phase, the low-energy excitations The operators τiα (α = x,y,z) are the Pauli matrices
are fermionic. The gapless phase acquires a gap in the acting on the effective spin at site i, while b and b†
i i
presence of a magnetic field and then contains gapped are hardcore bosonic annihilation and creation opera-
non-Abelian anyon excitations. The phase diagram has tors, satisfying the usual on-site anticommutation rela-
alsobeeninvestigatedinothervortexconfigurationssuch tion b ,b† = I (and operators on different sites com-
{ i i}
as the vortex-full sector and similar phases have been mute). Setting once for all J = 1/2 so that creating a
z
obtained. More precisely, one has a gapped phase for boson costs an energy 1 in the isolated-dimer limit, the
J2 >J2+J2 and a gaplessphase inthe opposite case14. Hamiltonian (1) reads
z x y
Ourgoalhereistodeterminethelow-energyspectrum
N
for anyvortexconfiguration. Ofcourse,one may use the H = +Q+T +T +T , (9)
0 +2 −2
fermionic Hamiltonian mentioned above but, it can only − 2
be exactly diagonalizedfortranslation-invariantconfigu- where N is the number of z-dimers (or, equivalently, of
ration. Here, we follow an alternative route by focusing square plaquettes), and
on the isolated-dimer limit J J ,J .
z x y
≫ Q = b†b , (10)
i i
i
X
III. MAPPING ONTO AN EFFECTIVE SPIN T = J ti+n1 +J ti+n2 +h.c. , (11)
BOSON PROBLEM 0 − x i y i
Xi (cid:16) (cid:17)
A. Mapping of the Hamiltonian T+2 = − Jxvii+n1 +Jyvii+n2 =(T−2)†.(12)
Xi (cid:16) (cid:17)
The very first step of our analysis consists in mapping These operators are built from local hopping and pair
thefourpossiblestatesofthetwospinsofaz-dimeronto creation operators
those of an effective spin and a hardcore boson. More
ti+n1 =b† b τx , ti+n2 = ib† b τy τz,
precisely, denoting ( ) the eigenstate of σz with i i+n1 i i+n1 i − i+n2 i i+n2 i
eigenvalue +1 ( 1),|a↑niis|ol↓aitedz-dimer canbe in one of vi+n1 =b† b†τx , vi+n2 =ib† b†τy τz.
− i i+n1 i i+n1 i i+n2 i i+n2 i
the two low-energy states , with energy J , (13)
z
{|↑↑i |↓↓i} −
or in one of the two high-energy states , with Weemphasizethatthemapping(8)explicitlybreaksthe
{|↑↓i |↓↑i}
energy+J . Keepinginmindthatouraimistoperforma symmetry between white and black sites of the original
z
perturbationtheoryinthelimitJ J ,J ,itisnatural brick-wall lattice. This is responsible for the apparent
z x y
≫
to interpret the change from a ferromagnetic to an anti- breakingofsymmetrybetweenthex/n andy/n direc-
1 2
ferromagnetic configuration as the creation of a particle tions in Eq. (13). However,for all the physically observ-
withanenergycost2J . Byconstruction,suchaparticle ableresults,thissymmetryremainsintact(seeforexam-
z
is a hardcore boson. The remaining degree of freedom pletheseriesexpansionofeigenenergiesinAppendix C).
5
Note however that the n +n and n n directions
1 2 1 2
− (a) (b)
are not equivalent, as can be seen from the underlying
z z z
brick-wall lattice.
y y
x
z z −
x x x
B. Conserved quantities 1 −
x x x
−
a x
Letusnowrephrasetheconservedoperatorsdiscussed C z −
in Sec. IIB in the effective language. Using the nota- b
C
tions depictedin Fig.1b, as wellas the mapping (8), the
plaquette operators transform into
Wp =(−1)b†LbL+b†DbDτLyτUzτRyτDz. (14) (c) (d)z z z
x Cd
Note that (−1)b†ibi =1−2b†ibi. In the same vein,for the x z z z z
cycles around the torus shown in Figs. 5b-5c, which are
x
reproducedfortheeffectivelatticeinFigs.6b-6c,onehas
Lb = i∈Cb −(−1)b†ibiτix = i∈Cb (−1)b†ibiτix (since x z
there is an ehven numberiof sites onh the contouir with
Q Q
the PBC chosen here), as well as = τx. The Cc Cd
Lc i∈Cc i
expression for (see Fig. 5d), namely = ω ,
Ld LQd i∈Cd i
is a bit more complicated, but it should be clear from FIG. 6: (color online). On the four figures, the thick yellow
Fig.6dwhatthe ω ’s are. Finally,forthe contoQurshown (lightest gray) line represents the contours C. The operators
i
in Fig. 6a (which is in correspondencewith Fig. 5a), one ωi which are such that the loop operators read L=Qi∈Cωi
has W = ω = with the ω ’s indicated areindicatedinthefigures. Inthepresentcase,theycantake
in thepfi⊂gCuare,pand wi∈itCha pi Lameaning thie plaquettes thefollowing valuesx=τx,x=(−1)b†bτx (andthesamefor
a
p enQclosed in theQcontour ⊂.CWith these notations, one y and z), and −x=−(−1)b†bτx. These figures are the same
a
C
can easily check that Eq. (5) still holds. as theones of Fig. 5, but on the effectivelattice.
The elementary hopping and pair creation operators,
namely tj and vj with i and j nearest neighbors, have
i i limit,thespectrumismadeofequidistantanddegenerate
a very remarkable property : they all commute with the
levelsseparatedbyanenergygap∆=2J =1. Tocom-
W ’s, as well as with any other loop operator z
p pute the perturbative spectrum, there are of course sev-
eral methods among which the Green’s function formal-
[tj,W ]=[vj,W ]=[tj, ]=[vj, ]=0. (15)
i p i p i L i L isminitiallyusedbyKitaev7. However,ifthisapproachis
efficient to obtain the first nontrivial (nonconstant) cor-
The originalspin problem on the honeycomblattice is
rection,itbecomes tricky to implementathigher orders.
thus mapped onto a quadratic hardcore boson problem
Here, following Ref. 13, we use an alternative ap-
on an effective square lattice, with conserved plaquette
proach based on continuous unitary transformations
and loop operators. Let us underline that this mapping
(CUTs)conjointlyproposedbyWegner34andG lazekand
is exact and just provides an alternative description of
Wilson35,36. We refer the interested reader to Ref. 37
thespinproblem. TheresultingHamiltonian(9)remains
for a recent pedagogical introduction. Its perturbative
difficult to diagonalize (except, of course, if one remem-
version denoted PCUTs is especially well-suited to the
bersthatthemodelcanbefermionized),since(i)bosons
problem at hand. This technique is detailed in sev-
are hard core which prevents the use of a Bogoliubov
eral works38,39. Let us simply mention that the CUTs
transformation, (ii) bosonic and spin degrees of freedom
methodrequiresthe choiceofageneratorthatdrivesthe
arecorrelated. Theconservedplaquetteoperatorswillof
flow of the operators. All the results given here have
course be useful in simplifying and solving the problem
been obtained with the so-called quasiparticle number-
as recently underlined in Ref. 15.
conservinggeneratorfirstproposedbyMielke40 forfinite
matricesandgeneralizedtomany-bodysystemsbyKnet-
ter and Uhrig39.
IV. PERTURBATION THEORY IN THE
The latter have computed the perturbative expansion
GAPPED PHASE
for any Hamiltonian of the form
A. Effective Hamiltonian from PCUTs H =Q+T +T +T +T +T , (16)
−2 −1 0 +1 +2
provided two hypothesis are satisfied:
The starting point of the present perturbation theory
is the isolated-dimer limit, namely J = J = 0. In this theunperturbedHamiltonianQhasanequidistant
x y
•
6
spectrum bounded from below ; the linear size 2p of the lattice. Such loop operators are
associated to contours as the ones shown in Figs. 5 and
• t[Qhe,Tpne]rt=urnbiTnng.Hamiltonian +n=2−2Tn issuchthat 6,′.nTamheelyprLesbe,ncLecoafnsducLh′blofoopr otpheeracotonrtsouinrsthCeb,effCecctainvde
P Cb
Hamiltonian shows that the eigenstates of the Hamilto-
Clearly,theHamiltonian(9)meetsthesetwocriteria(up
nian are also eigenstates of these loop operators. Their
to a constant term) noting that in the present case, one
effect is to lift the degeneracies between states (which
has T = 0. Here, we have included the ”small” pa-
±1
for each energy is at least four in the thermodynamical
rameters, namely J and J , in the definition of the T
x y n
limit, since some of the excitations are Abelian semions
operators, which is not the convention usually adopted
and the genus of a torus is 1, see Ref. 5). We shall not
in the CUTs community.
dive into the details of such finite-size corrections, since
The CUTs method together with the quasiparticle
our approach allows us to directly tackle with the most
number-conserving generator unitarily transforms the
interestingthermodynamicallimit. However,letusmake
Hamiltonian (16) into an effective Hamiltonian H =
eff
U†HU commuting with Q, U being a unitary operator. a remark about a numerical check of this statement for
smallsystemsizes. Foratoruswhoselinearsizeisstrictly
We give the first terms of the expansion up to order 4
smallerthan4,the loopoperatortermsaroundthetorus
in Appendix A. As can be seen in Table I, the number
dominatetheexpansionovertheW ’s,andforasizeof4
oftermsappearingintheperturbativeexpansionquickly p
bothtypesoftermsstartcontributingatthesameorder.
increases with the order. For instance, at order 2, the
One should thus not be surprised to find a ground-state
effective Hamiltonian reads in our case
for p=1 which is not in the vortex-free sector15.
N 1 1
H = +Q+T T T + T T , (17)
eff 0 −2 +2 +2 −2
− 2 − 2 2
whereas at order 10, there are more than 104 operators B. Counting of states
to consider.
Writing Heff this way is only the very first part of the Before we turn to a detailed analysis of each QP sub-
job since one next has to (i) determine its action in each space, let us show that we do not miss any state us-
subspace of a given quasiparticle (QP) number q ; (ii) ing simple counting arguments. We have already seen
diagonalize Heff in each of these subspaces. This is the in Sec. II, that one has N +2 conserved Z2 quantities
object of the next sections : we first study the lowest- (two loop operators, and N plaquette operators), with
energy states (q =0 QP) which is the main contribution the constraint W = I. There is in fact one more re-
p p
of our work ; then we turn to the q = 1 QP states and lationbetweentheW ’s,involvingthenumberofbosons,
p
recoverthe high-energy gapfromthe QP dispersion; we which reads Q
end by q >2 QP states, whose properties determine the
statisticsoftheQPs,andwewillseethattheQPsbehave Wp =( 1)Pib†ibi =( 1)Q = Wp,
asfermionswhichare,furthermore,noninteracting. This − −
p∈wYhite p∈cyaYn (gray)
fact is at the originofa tremendous simplificationof the
(19)
effectiveHamiltonian. Indeed,wefoundthatH canbe
eff showing that the parity of the number of vortices living
written, at all orders and in the thermodynamical limit,
onwhiteplaquettes(seeFig.2)hastobethesameasthe
as
parityofthe number ofbosons. The lastequalitysimply
comesfromthepreviouslymentionedconstraint(6). The
H = E +µQ C W ...W (18)
eff 0 − p1,...,pn p1 pn firstequality can be checkedusing the expression(14) of
{p1X,...,pn} theW ’s. Indeed,forasiteihavingawhiteplaquetteon
p
−{j1X,...,jn}Dj1,...,jnSj1,...,jnb†jnbj1. ittwsoleafsts,oacniadteadnoWthpe’srwoinllegoinveitτsiyr×ig(h−t,1)tbh†iebipτiryo=du(c−t1o)fb†itbhie.
In the same way, for a site i having a white plaquette
We shall discuss each term in detail in the following
above it, and another one under it, the product of the
sections, but let us mention that E , µ, the C’s and
the D’s are coefficients whose series e0xpansion are com- twoassociatedWp’swillgiveτiz×(−1)b†ibiτiz =(−1)b†ibi.
puted. The S operators are strings of spin operators τα Let us note that (19) has a meaning in the two bases we
j are working in, the initial one and the unitarily trans-
andofphasefactors(−1)b†jbj onthecluster{j1,...,jn}. formedone. Indeed, in the initial basis,the Hamiltonian
Thisveryspecialformofmulti-particleterms[remember H commutes with the parity operator ( 1)Q ; in the
(−1)b†jbj =1−2b†jbj], leading to phase factors and spin- rotated basis, Heff commutes with Q. −
strings only, is responsible for the emergence of fermions Wethus seethatinasubspacewithagivennumberof
in the model. QP’s, there are N independent conserved Z quantities.
2
For a finite-size system with PBC, new terms appear Thus, N being the number of effective spins τ , there is
i
intheeffectiveHamiltonian. Theyinvolveloopoperators no remaining effective spin degree of freedom once the
around the torus, and appear at a minimal order being Z quantitiesarechosen. Asaconclusion,theq-QPsub-
2
7
N
space has dimension d = 2N , with the usual no-
q q
(cid:18) (cid:19)
tation for binomialcoefficients. This shows that we miss
no state, since
N N
N
d =2N =22N =2Ns, (20)
q q
q=0 q=0(cid:18) (cid:19)
X X
where N is the total number of spins in the brick-wall
s
lattice.
Thisdiscussionfurthermoreshedslightonthefactthat
inordertocompute eigenenergies,aperturbativeexpan-
sion of the Kitaev model (as opposed to exact numerics)
is really of interest only in the 0-QP subspace. Indeed,
we have just seen that there are N independent Z con-
2
served quantities. It is thus clear that as soon as we
will have written down the effective Hamiltonian in the
0-QP subspace, the Hamiltonian will already be diago-
nal, whatever the vortex configuration, although writing
down the eigenstates of the Z quantities in the basis of
2
effective spin operators still has to be done. However,in
the1-QPsubspace,onewillhavetodiagonalizeanN N
×
matrix (numerically in the case of a nonperiodic vortex
configuration), which is identical to what one has to do
when solving the problem exactly as Kitaev did. For
q >2, the perturbative expansion looks even more com-
plicated than the exact solution, but this is an artifact,
since we recover free fermions.
FIG. 7: (color online). Two-anyon configurations (gray cen-
tral plaquette and one of the numbered plaquettes) on a
vortex-free background. ∆E (∆E ) is the energy cost (at
V. EFFECTIVE HAMILTONIAN IN THE 0-QP 1v 2v
order 10) for adding one vortex (two vortices) to the vortex-
SUBSPACE
free state. ∆E reads ∆E = J4+8J6 +75J8 +784J10.
1v 1v
For simplicity, we haveset here Jx =Jy =J but the results,
A. Effective Hamiltonian and eigenenergies in the general case, are easily obtained from the coefficients
given in AppendixC.
In the 0-QP sector, and in the thermodynamical limit
the effective Hamiltonian (18) simplifies and reads
The lowest nontrivial order involving the W ’s (order
p
H =E C W ...W , (21) 4) has been derived by Kitaev7 (C = J2J2/2) and led
eff|q=0 0− p1,...,pn p1 pn p x y
{p1X,...,pn} him to identify the effective low-energy theory with the
toric code6. One of the main results of our work is to
where p1,p2,...,pn denotes a set of n plaquettes and show that, at order 6 and beyond, one obtains a multi-
{ }
the Wp’s are the conserved plaquette operators intro- plaquette expansion in the effective low-energy Hamilto-
duced in Sec. II. Note than when restricted to the 0- nian. In other words vortices interact, though they re-
QP sector they simplify to Wp|q=0 = τLyτUzτRyτDz [see mainstaticastheyhavetosincethe Wp’s areconserved.
Eq. (14)]. Theinteractionenergiesbetweenvorticesarenotdirectly
As mentioned at the end of the previous section, ob- the C coefficients. One shouldwrite w =1 2n where
p p
−
taining eigenenergies only requires a minimal amount of n is the number ofvorticesatplaquettep (0 or1), then
p
work, namely replacing each W by numbers w = 1, look at coefficients in the expansionin terms of the n ’s.
p p p
±
and doing the same with loop operators, without for- The results of such an analysis for two-vortex interac-
getting about the constraints among these quantities. tion energies in the case J = J = J are illustrated in
x y
The perturbative expansion of the coefficients E and Fig.7whichshowsthattheinteraction(i)lowerstheen-
0
C are given in Appendix C. Let us note that ergy and is therefore attractive, (ii) is anisotropic even
p1,...,pn
p ,p ,...,p does not need to be a linked cluster of for J = J = J which is clear from the structure of
1 2 n x y
{ }
plaquettes (as seen for C that is nonvanishing at the underlying brick-walllattice, (iii) decreases with the
p,p+2n1
order10),andthattranslationalinvarianceoftheHamil- distance d between vorticesas expected in a gappedsys-
tonianimplies that theC coefficients only depend tem. Note that for a finite-size system with PBC, the
p1,...,pn
on n 1 relative positions of the plaquettes. two-vortex configurations with a central vortex and an-
−
8
(a) (b)
τ s
e
e →
m m
FIG.8: (color online). Illustration ofthetwodifferentpoints
of view one can have of a bi-colored lattice : sites are at FIG. 9: (color online). Two of the states entering the equal-
vertices in (a) and on the bonds in (b). The plaquette m in weight superposition in (24). Crosses on the sites indicate
(a) remains a plaquette in (b), while the plaquette operator a spin flip, with respect to the reference state |⇑i where all
e transforms into a star operator. spins point upwards (the Ae operators are still pictured by
thick crosses at the vertices). The loops, in dotted lines, are
obtained byjoining theflipped spins.
other vortex at one of 1, 5, 6 and 7 sites (see Fig. 7) are
forbidden since they violate the constraint (19). A one-
vortexconfigurationisalsoforbiddensinceitviolatesthe of sites
constraint(6). These configurationswould be allowedin τx =sy , τy =sz , τz =sx,
aninfinitesystem,orinafinitesystemwithopenbound- τ•x = sy , τ•y =sx , τ•z =sz. (22)
ary conditions. ◦ − ◦ ◦
Amostremarkablepointwhichemergesfromtheanal- This way, a cyan (gray) (resp. white) plaquette such
ysisofH isthatits eigenstatesarethoseofthe W ’s. as m (resp. e) in Fig. 8a transforms into a plaque-
eff 0 p
|
wTofihtethhyteahreteoprteihcrutscuortbdhaee6tsi,oanamlotehradotueragnh(wyteohreedmirepreh(ia≥gsein4z)ee,nwaeenrgdayraerctehaatlhnkoignseges tAtee (=sta(r−)1t)ebr†LmbL+Bb†DmbD=sxLs(xU−s1xR)bs†LxDb)L,+ba†DsbDshszLowszUnszRwsitzDh(trhesicpk.
(red) lines in Fig. 8b. We have kept track of the phases
about eigenstates of H , not of the original Hamilto-
eff|0 involving boson numbers because our construction will
nianH). Wegraphicallysketchtheconstructionofthese
be neededfor (q >1)-subspaces,but it is clearthat they
eigenstates in the next subsection, which will also prove
can be dropped in the 0-QP subspace. Let us mention
to be useful for the (q > 1) QP sectors, and show ex-
that the distinction between plaquette and star terms is
plicitly that they obey anyonic,more precisely semionic,
purely conventional. The letters m and e refer to the
statistics. Our discussion of the toric code focuses on
magnetic and electric vocabulary also used by Kitaev7,
peculiarities relatedto our wayof studying the problem,
although we emphasize there is absolutely no difference
thatisnotrestrictedtothe0-QPsubspace. Formorede-
betweenanAandaBoperator,whicharebothdisguised
tails about the toric code model, we refer the interested
W operators. Up to an additive constant term, the ef-
reader to Refs. 5,6,7.
fective Hamiltonian in the 0-QP subspace, and at order
4 finally reads (with J =J2J2/2)
eff x y
B. The Toric Code in a nutshell
H = J A + B . (23)
eff q=0 eff e m
| − !
1. Mapping to the toric code Xe Xm
We work with this lowest (nontrivial) order Hamilto-
As we have seen in the previous sections, the eigen- nian,becausetheeigenstatesofHeff q=0 remainthesame
|
states of the effective Hamiltonian in the 0-QP subspace whatever the order in perturbation. One should simply
are the eigenstates of the W ’s and of the ’s. We re- rememberthattheeigenstatesofH alsohavetobeeigen-
p
L
call that in this subspace, the plaquette operators read states of operators. If the PBC are not of the type we
L
W =τyτzτyτz (see Eq. (14) and Fig. 1b). A sim- use (see Fig. 2), the sites can usually not be bi-colored
p|q=0 L U R D
ilar simplification occurs for the ’s. As mentioned by andtherotations(22)cannotbeperformed,whichmakes
L
Kitaev7 (for the Hamiltonian at order 4), the effective the construction much more complicated, and we shall
Hamiltoniancouldbestudieddirectly,butitismucheas- refer the interested readers to Ref. 41.
ier to visualize the eigenstates by performing some spin
rotations, and bring the Hamiltonian to the one of the
toric code (generalized by multi-vortex terms). Thanks 2. Construction of the ground state(s)
to the special PBC we have chosen, the lattice sites can
be bi-colored in black and white as illustrated in Fig. 8. As a warming up, let us construct a ground-state of
Then, one performs a differentrotationonthe twokinds H , i. e., an eigenstate of all B and A operators
eff 0 m e
|
9
x y
L L
y y y y
− −
y y
y − y
−
FIG. 10: (color online). The two loop operators Lx and Ly
and theircontoursin yellow (lightest gray) thick lines, corre-
FIG.11: (coloronline). Twoofthestatesenteringtheequal-
sponding to C and C in Fig. 6. As in the latter figure, but
b c weight(inabsolutevalue)superpositionin(25),involvingthe
for s spins instead of τ spins, y means sy, etc. In the 0-QP Lx loop operator. Graphical conventions are the same as in
sector, y=(−1)b†bsy is thesame as y. Figs. 9 and 10.
with eigenvalue 1 (there are actually four of these). An
Let us note that the preceding construction relies on
eigenstateofallB ’sisforexamplethe”reference”state
m the state and the fact that it is an eigenstate of the
whereallspinss pointinthe +z-direction,suchthat |⇑i
|fo⇑riall i one has sz = . This state is not an eigen- Bm’s, etc. However, one could also have started with a
i|⇑i |⇑i state where all spins point in the x-direction, which
state of the Ae operators yet, but a simple projection is an|e⇒igienstate of the A ’s and then follow a similar
yields the desired state e
route.
I+A
2N/4−1/2 e 0 , (24)
b
2 |⇑i⊗| i
e (cid:18) (cid:19)
Y 3. Construction of excited states
whose normalizationfollows from the number N/2 of e’s
andtheproperty A =I[seeEq.(19)]. Thestate 0
e e | ib We now have to see how to construct excited states,
indicatesthatthereisnoquasiparticle,i.e. nohardccore
Q i.e. statescontainingvortices(eorm)butstillnoquasi-
boson. A graphical interpretation can be given of the
particle.
state (24) : it is an equal-weight superposition of multi-
Constructing a state with some B ’s being minus one
loopconfigurationsproducedbytheA operators,asthe m
e
(”magnetic vortices”) is easy, once one has noticed that
ones shown in Fig. 9.
sx anticommutes with two B ’s, and thus changes their
One next has to get an eigenstate of two independent i m
valuestotheiropposite. SincesxcommuteswithallA ’s,
loop operators, which we choose to be and (see i e
Lb Lc as wellas with x and y wheni does notbelong to the
Figs. 5 and 6) and which, from now on, will be denoted L L
corresponding contours, sx w =1 ,lx,ly is an eigen-
x and y, with eigenvalues lx and ly. The expressions i|{ p } i0
L L stateoftheeffectiveHamiltonian,withtwovorticesliving
of these operators in the s-spin language are given in
on the plaquettes touching the bond to which i belongs.
Fig. 10.
(Note that since sx also anticommutes with x and y
Note that in the 0-QP subspace, one could also have i L L
whenibelongstothecorrespondingcontours,oneshould
used other conserved loop operators which are products
use a string of sx going around the torus without cross-
of sx or of sz on the contours defining x and y. Such j
operators resemble more the ones usedLby KitaLev6, but ingLx andLy insteadofsxi). The correspondingstateis
again an equal-weight (in absolute value) superposition
they are conserved only in the 0-QP subspace (in con-
trast to x and y), and so will not prove to be very of states, but now with all possible open strings joining
L L the created vortices, as well as all possible closed loops.
useful in the following. As can be seen in Fig. 10,
x
L This is illustrated in Fig. 12.
and perform spin flips, with respect to on their
y
assocLiated contours. The four ground-states|⇑oif (23) are Creating ”electric” vortices is as easy, since one can
then obtained with another projection and proper nor- replace e I+2Ae inEq.(25)by e I+a2eAe ,withae =
malization S±u1c,hreaspcQhecatn(cid:0)ignegctah(cid:1)neaclosnostbreaionbttaineQeade(cid:0)v=ia1th[ese(cid:1)aecEtiqo.n(o1f9)s]z.
wp =1 ,lx,ly 0 =2N/4+1/2 (25) operators. Indeed, each sz opeQrator anticommutes withi
|{ } i i
I+lx x I+ly y I+A twoA ’s,andthuschangestheirvaluestotheiropposite.
e e
× 2 L 2 L 2 |⇑i⊗|0ib. The fluctuation of the strings induced by szi operators
(cid:18) (cid:19)(cid:18) (cid:19)Ye (cid:18) (cid:19) is however hard to see with the construction we have
Thesefourstatesareequal-weight(inabsolutevalue)su- given, which relies on the reference state . To see
| ⇑i
perpositionofallpossiblemulti-loopconfigurations,pro- this,one shouldconstructstatesfromthe referencestate
duced by the A operatorsas in Fig. 9, as wellas the x where all spins point in the x direction, and then
e
and y operators, as illustrated in Fig. 11 for x. L |u⇒seiprojectors involving B ’s instead of A ’s. This is
m e
L L
10
m m m m
X X′
Z
e e e e
m m m m
FIG.12: (coloronline). Twoofthestatesenteringtheequal- FIG. 13: (color online). Illustration of operators involved in
weight (in absolute value) superposition for the state having thebraidingofa”magnetic”vortexaroundanelectricvortex.
two”magnetic”vortices. Thelatterarerepresentedwithlittle X istheproductQ sxforibelongingtotheblue(darkgray)
i i
gray squares marked with the letter m, and are linked with thick path linking the two m vortices (orthogonal to bonds).
a string of spin-flips [blue (dark gray) thick line]. The other ZistheproductQ szforibelongingtothegreen(lightgray)
i i
graphical conventionsare the same as in Figs. 9. thick path linking the two e vortices (drawn on the bonds).
X′ is the product Q sx for i belonging to the dashed loop
i i
around the leftmost e vortex, and is also equal to a product
of the Ae’s operators encircled by the loop (denoted as thick
notuseful forour purposesowe let the interestedreader crosses).
doing it on his own.
VI. EFFECTIVE HAMILTONIAN IN THE 1-QP
SUBSPACE
4. The statistics of vortices
A. Form of the Hamiltonian
Forcompleteness,letusnowshowthat”magnetic”and
The spectrum we obtainedinthe 0-QPsubspacegives
”electric”vorticesbehaveassemionswithrespecttoeach
the lowest eigenenergies for each configuration of the
other. This isdoneby firstcreatinga pairof”magnetic”
W ’s. In this section, we explain how to compute the
vortices,then a pair of ”electric” vortices, and finally by p
high-energy spectrum for states with one quasiparticle,
movingoneofthe ”magnetic”vorticesaroundoneofthe
foreachW ’s configuration,andhowto buildthe associ-
”electric”vorticesasshownin Fig.13(one couldalso do p
ated eigenstates. This is achieved by diagonalizing H
the contrary, but then one should work with the refer- eff
in the 1-QP subspace (whose dimension is d = N2N,
ence state to see things more easily). With the no- 1
|⇒i see the end of Sec. IV). In this subspace, the effective
tationsofthisfigure(seealsoitscaption),letusconsider
Hamiltonian (18) reads
the state ψ = ZX w = 1 ,lx,ly with two e and
p 0
| i |{ } i
m vortices. Then the repeated application of spin-flips
H = E +µ C W ...W
along the loop X′ (in any direction) moves the down- eff|q=1 0 − p1,...,pn p1 pn
most m vortexaroundthe leftmost e vortex. The result- {p1X,...,pn}
ing state is X′ ψ . But as Z and X′ have one (and only D S b† b , (26)
one)commons|itei,theyanticommute,whereasX andX′ − j1,...,jn j1,...,jn jn j1
commute, so that X′ ψ = ZXX′ w = 1 ,lx,ly . {j1X,...,jn}
p 0
Now, X′ which is a p|roiduct−of sxi o|p{erators f}ormingia where the second sum is performed over all non self-
closed loop is nothing but a product of Ae’s operators retracingpathsoflengthnstartingatsitej1 andending
(the ones enclosed in the loop). As |{wp = 1},lx,lyi0 is at site jn, with possibly jn =j1 when working at order
an eigenstate of the Ae’s with eigenvalue one, we finally 4 or higher. This is the reason why we give the expan-
obtain that X′ ψ = ψ : braiding a magnetic vortex sion up to this order but we would like to emphasize
| i −| i
around an electric vortex yields a nontrivial phase of π thatobtainingordersupto10forH is ofthe same
eff q=1
( 1=eiπ), which proves the semionic statistics. complexityasforH . Self-retracin|gpathsarerenor-
− eff|q=0
Let us mention that the magnetic vortices behave as malizing the chemical potential µ. Note that a hopping
bosonsamongthemselves,andsodotheelectricvortices. processofonequasiparticlearoundaloopisnothing but
This is easily seen by noticing that creating and moving the product of the Wp’s enclosed in the loop, as can be
mvorticesforexample,onlyrequiressx operators,which easilychecked. Thisexplainswhyatorder4,oneobtains
all commute with one another. To end this discussion some terms proportionalto b†b W , where the plaquette
i i p
about the statistics of vortices, let us also remark that p shares site i (see Appendix D).
a compound object made of an electric and a magnetic From now on (q > 1) the phase factors appearing in
vortex is a fermion (see Ref. 5). the W ’s [see Eq. (14)] must be taken into account. The
p