Table Of ContentExact results on the Kitaev model on a hexagonal lattice: spin states, string and
brane correlators, and anyonic excitations
Han-Dong Chen1 and Zohar Nussinov2
1Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801
8 2Department of Physics, Washington University, St. Louis, MO 63160, USA
0 (Dated: February 3, 2008)
0
2 In this work, we illustrate how a Jordan-Wigner transformation combined with symmetry con-
siderations enables a direct solution of Kitaev’s model on the honeycomb lattice. We (i) express
n
the p-wave type fermionic ground states of this system in terms of the original spins, (ii) adduce
a
that symmetry alone dictates the existence of string and planar brane type correlators and their
J
composites, (iii) compute the value of such non-local correlators by employing the Jordan- Wigner
7
transformation,(iv)affirmthatthespectrumisinconsequentialtotheexistenceoftopologicalquan-
1
tum order and that such information is encoded in the states themselves, and (v) express the the
localsymmetriesofKitaev’smodelandtheanyoniccharacteroftheexcitationsintermsoffermions.
]
h
c PACSnumbers:
e
m
I. INTRODUCTION tion. The spin basis form of the ground state is also
-
t studied. It is shown that the ground state can be writ-
a
ten as a projection of a given reference state over a sec-
t Topological quantum order (TQO) [1] [2] is a new
s tor of Hilbert space that has a chosen set of topological
. paradigm that lies beyond the realm of Landau’s theory
t numbers which are similar to, yet slightly more compli-
a [3][4]oflocalorderparameters. TQOisintuitivelyasso-
m ciatedwithinsensitivitytolocalperturbations: theorder catedthan,othersystemswhichexhibitTQOsuchasthe
Rokhsar-Kivelson dimer model[11] and the Kitaev toric
- istopological. Assuch,TQOcannotbedescribedbylocal
d order parameters. A quintessential example of a system code model[2]. The phase diagram and Bogliubov exci-
n tations are also obtained. In Section(VI), we show how
with TQO is Kitaev’s model on the honeycomb lattice
o the known results about the anyonic vortex excitation
[5] [6] [7]. In this article, we will study this model. Our
c
in gapped state can be easily studied in the fermionic
[ newcentralresultsare(a)anexplicitformfortheground
representation. In section VII, a symmetry argument is
states in real space which extends and complements the
5 put forward concerning the vanishing correlation func-
worksof[5]and[6]aswellas(b)ourfindingofnon-local
v tions, which is also recently derived anew by a Majo-
correlations[twodimensionalstringorbranetypecorrela-
3
rana fermionization construction[12]. We also show that
3 tors]in this system. We further showhow the (c) known
string correlators naturally appear in Kitaev’s model.A
6 anyonic excitations of Kitaev’s model can be examined
briefsummaryinsectionVIIIconcludesthiswork. Inthe
3 anew by using a direct Jordan-Wigner transformation.
0 Beyond providing a direct solution which highlights cer- appendices, we provide technical details concerning the
7 determination of the ground state and review Elitzur’s
tain previously overlooked aspects, our results will flesh
0 theorem.
out some of the more general ideas [8], [9] regarding the
/
t general character of TQO.
a
m
III. FERMIONIZATION
-
d
II. OUTLINE We begin with a Fermionization of the Kitaev model
n
o on the hexagonal lattice [5] [6] which is defined by the
c This article is organized as follows: In section III, following S =1/2 Hamiltonian
:
v we review the fermionization of the Kitaev model on a
i hexagonal lattice[9, 10]. The original model of spins on H = −Jx σRxσRx′ −Jy σRyσRy′
X
a hexagonallattice is mapped to a model of p-waveBCS x−bonds y−bonds
X X
ar mferomdeiolnwsitohnsaitseq-dueapreenladtetnictec.hIenmsieccatliopnotIeVn,tiwaelfdoirscsupsinsltehses −Jz σRzσRz′, (1)
z−bonds
fermionic representation of symmetries embodied in the X
model. It is shown that the local conserved quantities with R and R′ lattice sites. This systemcan be fermion-
on the plaquettes studied by Kitaev[5] are equivalent to izedbyaJordan-Wignertransformation[9,10]. Thisone-
the conservedbond quantities in the fermionic represen- dimensional fermionization is made vivid by deforming
tation, up to a gauge fixing. In section V, the ground the hexagonal lattice into a “brick-wall lattice” which is
stateconfiguration,whichisvortexfree inspinrepresen- topologically equivalent to it on which we may perform
tationandwithauniformchemicalpotentialinfermionic a one dimensional Jordan-Wigner transformation. The
representation, is exactly solved in fermionic representa- schematicsareshowninFig.1. Inthe upandcoming,we
2
will mark all sites by “white” or “black” (w/b)in order
to denote to which sublattice which they belong to. The
distance between two nearest-neighboring sites on this
lattice will be set to unity.
x y x y x
z z z
y x y x y
z z z
x y x y x
z z z
y x y x y
ey ex
z z z
x y x y x
1 2 3 FIG. 2: Schematics of the contour for the Jordan-Wigner
z z z transformation that we employ in the deformed hexagonal
6 5 4 lattice- see text and Eq.(2) in particular.
y x y x y
for the black sites. With all of these transformations in
FIG.1: Deformedhexagonallatticeandthreetypesofbonds.
tow, the Hamiltonian now reads
Throughout, we will consider the system with open
boundary conditions unless stated otherwise. The di-
H = −i J A A − J A A
agonal directions eˆx and eˆy shown in Fig.(1) will be of x w b y b w
paramount importance in our final solution. Let us de- x−Xbonds y−Xbonds
note by (i,j)the Cartesiancoordinatesofeachsite R on −J J (BA) (BA) . (6)
z z b w
the lattice of Fig. 1. Let us next consider the Jordan-
z−bonds
X
Wigner transformation defined by a simple one dimen-
sionalcontourwhichthreadstheentirelattice[SeeFig.2]: Itis easytoseethatBB alongthez-bondisaconserved
quantity [10]. Thus, the Z operator
2
σi+j = 2j′<j i′ σiz′j′"i′<iσiz′j#c†ij αr =iBbBw, (7)
Y Y Y
σz = 2c† c −1. (2) with r the coordinate of the midpoint of the bond con-
ij ij ij necting the black and red sites, is fixed for each vertical
This path goes through each lattice site exactly once as bond. The Hamiltonian of Eq.(1) now reads
showninFig.(2). InEq.(2), σ+ =(σ +iσy)is twice the
x
spin raising operator at a given site- hence the factor of
H({α}) = −i J A A − J A A
two. x w b y b w
The Kitaev model of Eq.(1) now becomes x−Xbonds y−Xbonds
−iJ α A A . (8)
H = J c†−c c†+c z r b w
x w b z−bonds
x−bonds X
X (cid:0) (cid:1) (cid:0) (cid:1)
−J c†+c c†−c Here, r denote the centers of the vertical bonds. In
y b w Section(IV), We will show that {α } are intimately re-
y−bonds r
X (cid:0) (cid:1) (cid:0) (cid:1) lated to the local symmetries present in Kitaev’s model
−J 2c†c−1 2c†c−1 . (3)
z b w of Eq.(1). This identification, combined with Reflection
z−Xbonds(cid:0) (cid:1) (cid:0) (cid:1) Positivityarguments,[5]willallowustoinferthat,upto
Henceforth, the subscripts b andw willdenote the white (d = 1 [13]) symmetry operations, αr = 1 for all r. The
and black sites of a bond as illustrated in Fig.1. ground state does not contain any “vortices” which are
Let us next introduce the Majorana fermions marked by one dimensional in the Ising variables {αr}
along a row. Similar Reflection Positivity arguments re-
Aw =(c−c†)w/i Bw =(c+c†)w (4) garding the absence of vortices in other systems and a
bound on the energy penalties that they entail are. e.g.,
for the white sites and
given in [14]. This, in turn, will allow us to explicitly
B =(c−c†) /i A =(c+c†) (5) diagonalize the Hamiltonian.
b b b b
3
IV. FERMIONIC REPRESENTATION OF
+ I=1 + I=1 + I=1 + I=1 + I=1
LOCAL SYMMETRIES
+ I=1 + I=1 + I=1 + I=1 + I=1 + I
As shown by Kitaev[5], the Hamiltonian of Eq.(1) has
+ I=1 + I=1 + I=1 + I=1 + I=1
one conserved quantity for each plaquette (or hexagon)
h, + I=1 + I=1 + I=1 + I=1 + I=1 + I
I =σy σz σx σy σz σx . (9)
h 1w 2b 3w 4b 5w 6b (a)
Here,1−6denote the sites ofa givenplaquette, as illus-
trated in the plaquette of the lower corner in Fig.1. In + I=+1 + I=+1 + I=+1 + I=+1 + I=+1
the subscripts, we also label the (w/b) sublattices of the
sites. The conserved quantity I on plaquette is equiva- + I=+1 + I=+1 + I=+1 + I=+1 + I=+1 + I=
h
lenttothebondconservedquantityαdefinedinprevious
+ I=-1 - I=+1 - I=+1 - I=+1 - I=+1 -
section,uptoagaugefixingasweshallshowbelow. This
canbeshownbyfermionizingI usingthetransformation + I=+1 + I=+1 + I=+1 + I=+1 + I=+1 + I=
h
weintroducedinprevioussection. Letusfirstfermionize
the product of first three spins (b)
1
σy σz σx = c†−c σz σz σz c†+c FIG. 3: (a) Vortex free configuration. (b)Reversing I in a
1w 2b 3w i 1w 2b 1w 2b 3w plaquette corresponds to flip the chemical potential along a
= i (cid:0)c†+c (cid:1) c†+c (cid:0) (cid:1) string.
1w 3w
= iB B . (10)
(cid:0)1w 3w(cid:1) (cid:0) (cid:1)
Similarly, reverses the sign of [α]i while leaving others untouched
σ6xbσ5zwσ4yb =iB4bB6b. (11) (Uiw)†H(...,[α]i−1,[α]i,[α]i+1,...)(Uiw)
=H(...,[α] ,−[α] ,[α] ,...). (14)
Therefore, i−1 i i+1
I =α α . (12) Therefore,wehaveextrafreedomtofixoneαineachsub-
h 34 16
set [α] . This degree of freedom is closely related to the
i
This model has an extra degeneracy that links differ-
string-likeconservedquantity inthe originalspin model,
ence sectors parameterized by different sets of α’s. The
namely,
vortex[oranyon]variablesaretheproductoftwoconsec-
utive Ising bond variables. In other words, anyons (I )
h P = σz, (15)
are none other than domain walls in the Ising variables j ij
({αr})thatoursystemcontains. Invertingallofthe val- Yi
ues of α for all sites r which lie along a horizontal row
r whichrotates allspins onj-th line by 180degreearound
leaves the system unchanged. Physically, effecting the
z. Under the gauge fix in which one α is fixed to +1 in
transformation α → −α for all bonds r along a row
r r each [α] , the set of Kitaev conserved quantities {I } is
i h
does not change the vorticity content of the system: all
equivalent to {α}.
domainwallsalongthechainremaininvariant(andaswe
show so does the spectrum). In the notation of [8] and
[13], this corresponds to a d = 1 dimensional operator
[as it involves spin operators on a (d = 1 dimensional) V. GROUND STATE
line]. Forsimplicity,letusdivideαintodifferentsubsets:
{α} = ∪i[α]i. Here, [α]i denotes the set of bonds that A. Diagonalization
are connected to white sites of the i-the horizontal line
of the brick-wall lattice. We next explicitly write down
Armed with a physical meaning of the bond variables
thesed=1symmetryoperators(see[8],[13]foradefini-
αfromSection(IV),wenowproceedtosolvetheproblem
tion of d dimensional symmetry operators)in both their
posed by Eq.(6) and diagonalize the problem.
fermionic and original spin language form. Towards this
It is interesting to introduce a fermion on the z-bond
end, we construct the unitary operator Uw
i
Uw = A . (13) d=(Aw+iAb)/2, d† =(Aw−iAb)/2, (16)
i n
j≥in∈j
YY whereA andA aretheMajoranafermionsonthewhite
w b
In Eq.(13), n is a site index. The product is taken over and black site of a given z-bond. We thus have a model
all sites in the rows j ≥i. We notice that Uw effectively for fermions on a square lattice with a site dependent
i
4
chemical potential
H = J d† +d d† −d
x r r r+eˆx r+eˆx
Xr (cid:0) (cid:1)(cid:16) (cid:17)
+J d†+d d† −d
y r r r+eˆy r+eˆy
+JzXr α(cid:0)r(2d†rdr(cid:1)−(cid:16)1). (cid:17) (17) Jx Jy
r
X
Theunitvectoreˆ connectstwoz-bondsandcrossesay-
y
bond,see Fig.(1). A similar definition holds for eˆ . Note -q q
x x y
that
[αr,dr]=[αr,d†r]=0. (18) Jz
For sufficiently large systems, the ground state config-
FIG. 4: A geometric interpretation of Eqs.(20) for the exis-
urations are bulk vortex-free configurations [5] in which
tenceof gapless excitations. The two conditions ǫq =∆q =0
Ih =1forallplaquettes(hexagons)h. Thegroundstates imply that qx,y can be regarded as angles in the triangle
ofthefermionicproblemofEq.(17)hasαr =1everwhere formed by {Jx,Jy,Jz} as shown. The law of cosines gives
[and all sectors of {α } related to it by the d = 1 oper- Eqs.(22). Thisplotimmediatelyleadstothetriangleinequal-
r
ation of Eqs.(13,15)] which corresponds to I = 1 for ity condition of Eq.(27). When the triangle inequality is vi-
h
all plaquettes h. As shown in Fig.(3), reversing Ih to olated (the gapped phase), qx,y are imaginary and lead to a
I = −1 leads to an inversion of the chemical potentials finitecorrelation length as we will discuss later [Eq.(67)].
h
{α } along a horizontal string.
r
The exact solution for groundstate is now easy to ob-
energies which determine whether or not TQO exists.
tainforthebulksystembyaFouriertransformation. Up
[8], [9] Inother words,the same Hamiltonianin different
to innocuous additive constants, the Hamiltonian of the
representations which are related to one another by uni-
vortex-free configuration now reads,
tary transformations and thus preserve the same set of
∆ eigenvalues (the energy spectrum) [8] can describe both
Hg = ǫqd†qdq+i 2q d†qd†−q+H.c. , (19) topologicallyquantumorderedsystems(suchasKitaev’s
Xq (cid:20) (cid:16) (cid:17)(cid:21) model)orsystemswithnotopologicalorder.[8][9]Inthe
current context, these mappings are the Jordan-Wigner
where
transformations that we apply. [9]
Let us now proceed to study when the spectrum of
ǫ = 2J −2J cosq −2J cosq ,
q z x x y y
Eqs.(20, 21) describes a system with a spectral gap be-
∆ = 2J sinq +2J sinq . (20)
q x x y y tween the ground and the next excited states and de-
lineatetheseboundariesbetweenthegappedandgapless
The fermionized Hamiltonian (17) describes a p-wave
phases. Towardsthisend-seeFig.(4)-westudythenom-
type BCS pairing model with site-dependent chemical
inal solution to E =0 which is
potential. AfteraBogliubovtransformation,this Hamil- q
tonian can be diagonalized and the quasiparticle excita- J2+J2−J2
tion is q = ±arccos x z y ,
x
2J J
" x z #
Eq = ǫ2q +∆2q. (21) J2+J2−J2
q = ±arccos y z x . (22)
q y 2J J
" y z #
B. Spectrum
This solution makes sense only when the conditions
−2|J ||J |≤J2+J2−J2 ≤2|J ||J |, (23)
Theenergyspectrum,inthelowenergyvortex-lesssec- y z y z x y z
tor αr = 1 [and all sectors of {αr} related to it by the −2|Jx||Jz|≤Jx2+Jz2−Jy2 ≤2|Jx||Jz| (24)
d = 1 operation of Eqs.(13,15)], which we found to be
given by Eq.(20, 21) obviously does not encode infor- are satisfied simultaneously. These inequalities can be
mation about the topological nature of Kitaev’s model. easily solved to obtain the condition for gapless excita-
The equivalence of the lowest eigenvalues of the Hamil- tions
tonians in both (i) a simple BCS type problem given by
|J |≤|J |+|J |, (25)
Eqs.(20,21) and (ii) the Kitaev model of Eq.(1) vividly x y z
illustratesthemaximthatthestatesthemselvesinapar- |Jy|≤|Jx|+|Jz|, (26)
ticular (operator language) representation and not their |J |≤|J |+|J |. (27)
z x z
5
This result is in agreement with Kitaev’s original solu- In the spin basis, a fermionic vacuum corresponds to
tion and Pachos’ analysis. [5] [6] We can now view q
x,y
as (up to an inversion q → −q for one of the compo- 1 1
a a |0i=N (1+B ) (1+I )
ncoeunptslinogfs~q)Ja,sJthaenadnJgle-ssiene tFhieg.(t4ri)a.nEglqe.(f2o2rm) iesdthbye ltahwe Yrw (cid:20)2 rw (cid:21)! Yh (cid:20)2 h (cid:21)!
x y z 1
of cosines in this triangle which automatically satisfies × (1+αr∗ ) |φi. (31)
∆q =ǫq =0 of Eq.(20) and consequently has Eq =0. It (cid:16)Yj (cid:20)2 jw (cid:21)(cid:17)
is also interesting to notice that the cyclic symmetry be-
InEq.(31),theproductoverhisthatoverallelementary
tween J ,J ,J is explicitly restoredin the groundstate
x y z
hexagons, |φi is an arbitrary reference state: e.g. in the
solution although its explicitness is lost in the fermion-
σz basis,we maychooseit to be the fully polarizedstate
ized model.
|φi=|↑↑...↑i, the operator
Next, we briefly remark on not only the low energy
spectrumofpertinencetothezerotemperatureproblem, B ≡−σx σz ...σz σx (32)
rw rw rw+1 rb−1 rb
butratherexaminetheentirespectrumofthetheory. Af-
extends over all sites lying between (and including) r
ter tracingoverthe fermionic degreesof freedompresent w
andr as labeled by the one dimensional Jordan-Wigner
in Eq.(17), we obtain an effective two dimensional Ising b
contour that connects r with r , N is a normalization
typeHamiltonian(in{α })withlongrangeinteractions. w b
r factor,andthehexagonaloperatorI asgivenbyEq.(9).
The full spectrum of this long range Ising type Hamilto- h
Similarly,invertingthe fermionizationcarriedearlier,we
nian is identical to that of Kitaev’s model.
find that
αrw = σlz −σrxw + σlz′ (−iσryb) (33)
(cid:16)l<Yrw (cid:17)h (cid:16)rw≤Yl′<rb (cid:17) i
C. Real space form of the ground states
is the spin representation of the operator α of Eq.(7).
r
Eq.(33)definesanoperatorα foranywhite latticesite
The BCS type ground state corresponding to Eq.(21) r . In the last product in Eqr.(w31), we have a product of
w
is 1(1+α ) over one lattice site r in every row j. The
2 rw w
symbol r∗ denotes the first leftmost white lattice site
jw
|gi= uk+vkd†kd†−k |0i, (28) rw in the jth row. In combination with Eq.(12) [valid
Yk (cid:16) (cid:17) for any plaquette], the conditions αrj∗w = 1 and Ih = 1
ensure thatα =1 for allr. AderivationofEqs.(31, 32)
r
where is given in the appendix. It is noteworthy that
|vk|2 = 12(cid:20)1− Eǫkk(cid:21), |uk|2 = 21(cid:20)1+ Eǫkk(cid:21). (29) [Ih,Brw]=[=Ih[,IIhh,′α]=rj∗w[]B=rw[,BBrrww′,]α=rj∗w[I]h=,d†k0]. (34)
{I }and{B }leadtodisjointZ algebras. Thus,when
h rw 2
Let us now invert the transformations that we have per- combined, Eqs.(20, 21, 28, 29, 30, 31, 32, 33) give us the
formeduntilnow(Jordan-Wignerandothers)inorderto explicit form of the ground state wavefunctions for the
express the Fermionic operators and the Fermionic vac- S =1/2 system defined by Eq.(1).
uum state in terms of the original spin degrees of free-
dom. Inwhatfollows,wewill,whenneeded,keepexplicit
track of the (w/b) sublattices of eachof the sites of each D. Comparison to ground states of other
vertical bond whose center is at ~r. Undoing all of the topological ordered systems
transformations that we employed thus far, we have
Thespinbasisformofthegroundstateofthistopolog-
ically ordered system can be compared to that of other
1
d†k =−2Xrw "σryw r′Y<rwσrz′!−iσrxb rY′<rbσrz′!# sthysetRemokshesxahri-bKitiivneglstoonp[oRloKg]ic[1a1l]qaunadnttuhmeKoritdaeerv. tEo.rgi.cbcoodthe
×e−i~k·(~rw−12eˆz).(30) models[2](aswellasWen’splaquettemodel[21]whichis
equivalent to Kitaev’s toric code model) [8] have ground
states of the form
[For a definition of eˆ see Fig.(1).] The product
corresponds to the pzroduct of all lattice sites ~r′ wr′h<ircwh |ψi=N |χii (35)
appearbeforerw ontheJordan-WignercontourofQEq.(2) Xi
which traverses all sites of the two dimensional lattice.A withN anormalizationconstantandthesumoveriper-
similardefinitionappliesto : itistheproductover formed over all states |χ i which belong to a given topo-
r′<rb i
all lattice sites which appear before r in the Jordan- logicalsectorwhichisevidentbytherealspacerepresen-
b
Q
Wigner product. tation (e.g. an even or odd number of dimers/positive
6
bondsintheRokhsar-Kivelson/Kitaevtoriccodemodels
respectively). InboththeRKandKitaevtoriccodemod-
els, these ground states may be expressed as a product
of projection operators (related to a Z algebra) acting
2
on a given reference state. For instance, in the Kitaev’s
toric code model on the square lattice [2]
H =− A − B (36)
s p
s p
X X
withA = σx andB = σz,asstatedearlier,
s i is p hiji∈p ij
all ground states may be expressed as
Q Q
1
|gtoric codei = N (1+As) FIG. 5: The geometry of three hexagons that span a total of
2
s ! thirteenvertices. ByKramers’theorem,groundstatesofthis
Y
system that contains an odd number of S = 1/2 spins must
1
× (1+B ) |φi, (37) break time reversal invariance.
p
2
!
p
Y
with |φi a reference state. Similarly, the general Eq.(35) thenEq.(38)isnottimereversalinvariant. Similarly,the
can be written as a projection of a given reference state operator B of Eq.(32) contains (L+1) spin operators
rw
over a sector of Hilbert space that has a chosen set andisnottimereversalinvariantforevenL. Areference
of topological numbers (those corresponding to a given state |φi of Eq.(31) on anevensize lattice canbe chosen
topologicalsector). AsseenfromoursolutionofEq.(28), to be time reversal invariant. For the odd size lattice,
the groundstate ofKitaev’smodelonthe hexagonallat- the considerations are more immediate.
tice (Eq.1) [5] is more complicated. This is so as it in- As a concrete example, let us consider three hexagons
volves the fermionic operators of Eq.(30). The Fermi that span a total of thirteen vertices as shown in Fig.5.
vacuum of Eq.(31) indeed already has a form similar to This example serves to illustrate the presence of time
Eq.(37)but,asseenfromEq.(28),itneedstobeactedon reversalsymmetry breaking even without doing any cal-
by Fermionic string operators. This leads to a far more culations.Itisimpossibletowritedownagroundstateof
nontrivial state. Written longhand in the original spin Kitaev’s model of thirteen S=1/2 spins of this thirteen
basis, we have that the fermion pair creation operator site “disk like geometry” that is non-degenerate. The
same considerations apply also to instances in which we
1
d†d† = eik(rw2−rw1) havevarioustopologies. Forinstance,wecanputthefig-
k −k 4
Xrw1Xrw2 ure aboveof three hexagonswith a totalofthirteensites
on a three-fold torus– that of three handles (the genus
× σryw1(r′Y<rw1σrz′)−iσrzb1(r′Y<rb1σrz′)! nofumeabcehrogf=the3)h.eHxaegreo,ntsh.eOrenewiollf btheeasihtoesleisatcothmemcoenntteor
three ordinary tori and three other sites are common to
× σy ( σz )−iσz ( σz ) . (38) two ordinary tori. There is periodicity around each of
rw2 r” rb2 r” ! the hexagons (hence the designation of g = 3) in this
r”Y<rw2 r”Y<rb2
case. The thirteen site system of Fig.(5) on the plane
needs to be augmented. Here, in the product signs we corresponds to a system with open boundary conditions
madeexplicitthatalongtheJordan-Wignercontourthat while if it is a g = 3 torus is corresponds to a system
we employ here rb =rw+L. When Eq.(38) is combined thathasnon-trivialperiodicityaround3hexagonalrings
with Eqs.(28, 31, 32), we have a form for the real space (along with three other trivial periodicity directions of
spin states of the Kitaev model. periodone). Inbothcases,astheHamiltonianH istime
reversalinvariant(becauseT−1S T =−S withT the
a;i a;i
time reversal operator and S the a = x,y,z compo-
a;i
E. Time reversal symmetry breaking nent of the spin operator at spin i and the Hamiltonian
is quadratic in the spins) and there is an odd number of
Our expressions show that the state |gi of the open spins (n =13), by Kramers’ theorem, the ground state–
boundaryconditionsystemisgenerallynottime reversal andin factany energyeigenstate–must be, atleast, two
invariantforanysizesystem-whetheritcontainsanodd fold degenerate. General considerations for constructing
or even number of spins. In the expressions above for a torus of genus g from individual ordinary (g = 1) tori
the ground state this is seen by taking σa →−σa for all are outlined in [16]. We emphasize that this degeneracy
components a=x,y,z under time reversal. If any ofthe ismandatedforanoddsizesystembyKramers’theorem
indices in Eq.(38) aresuchthat(r −r )orL areodd foranytime reversalinvariantHamiltonian. Time rever-
w1 w2
7
sal symmetry breaking similarly occurs in extensions of read
Kitaev’s model that allow for odd cycles [5] [15] (e.g. a
triangular decoration of the lattice). Hboundary = Jxσ1x,2jσLxh,2j +Jyσ1y,2j+1σLyh,2j+1 (.39)
In any half integer spin system in which the number Xj h i
of particles is odd, the ground states must, by Kramers’
Here, L is the size of the sytem along the horizontal
h
theorem, exhibit time reversalsymmetry breaking. This
direction. After the Jordan-Wigner transformation (2),
conclusion may be fortified for general gapped systems
the coupling strength of the boundary terms aquires a
withshortrangeHamiltonians. Insuchsystems,there is
phase
an exponential clustering of correlations. [17] This clus-
tering, in turn, implies that the ground state can, up to H = [J φ(2j)A A
boundary x 1,2j Lx,2j
exponentially small corrections, [18] be written as a ma-
j
X
trixproductstateconstructedoflowlyingstatesonfinite
+J φ(2j+1)A A ] (40)
sizeblocks. Theblocksizemay,inprinciple,betakenar- y 1,2j+1 Lx,2j+1
bitrarily large so long as it is smaller than the system. with the phase term given by
Consider a general Hamiltonan H = h with R la-
R R
bellingdifferentblocksandh aHamiltonianthathasits φ(j) = σz . (41)
R i,j
P
supportonR. SpinscanbesharedbydifferentblocksR. 1≤Yi≤Lh
By Hastings’ theorem [18], the ground state of H is up
ThisphasedoesnotcommutewiththeZ bondoperator
to exponential corrections a matrix product state. Let 2
α on the z-bonds attached to j-th horizontal line. The
us now examine what this implies when combined with
phase factor φ(j) reverses all α of the z-bonds attached
time reversal. If all blocks R are chosen to have an odd
to the sites on the j-the horizontal line. The boundary
number of sites in the decomposition of any short range
terms thus lift the degeneracy characterized by differ-
S = 1/2 Hamiltonian H, then all states of the different
ent choices of α bonds. The gauge freedom discussed in
blocks R willbe, atleast, doubly degenerate. This is be-
section IV is now fixed by the boundary terms and the
cause of the (at least) two fold degeneracy of each level
ground state degeneracy acquires a topological depen-
in the odd size blocks R implied by Kramers’ theorem.
dence. Ontheotherhand,theboundarytermcommutes
We see here that the matrix product construction then
with the plaquette quantity defined by Kitaev. There-
implies that the ground state of large systems (also if
fore, the discussion about the correlator in later section
they are of even size) must also be, at least, two-fold de-
is still valid when boundary term is included. The mo-
generate up to correctionswhich are exponentially small
mentum components (k ,k ) employed in the solution
in the size of the system. This degeneracy is dictated by x y
of the previous subsections aboveare relatedby a 45 de-
thetimereversalnoninvarianceofthelocalenergyeigen-
greerotationtothediscretevalues(k ,k )forthesystem
states in each of the odd sized blocks R. The considera- h z
withperiodicboundaryconditionsalongtheverticaland
tions above are general and apply to both systems with
horizontal directions discussed here.
or without topological order. Topological order pertains
We conclude with brief remarksabout the symmetries
to systems such as Kitaev’s which donot display global
of Eq.(15) in the case of periodic boundary conditions
symmetry breaking associated with local order parame-
andonacandidatetrialstate. ThespinsystemofEq.(1)
ters. [1] The existence of degeneracy in Kitaev’s model
on the torus has the φ(j) operators of Eq.(15, 41) as
can also be seen by the non-commutativity of existant
exactsymmetries. Onthetorus,thesymmetryofEq.(15)
symmetries: this non-commutativity lies at the heart at
corresponds to the product of spin operators along an
Kitaev’s inception of this model. We will detail symme-
entire toric cycle. These operators satisfy the following
try considerations for this model in a later section.
identity
φ(j)φ(j+1)= I . (42)
h
F. Boundary terms
h∈S
Y
InEq.(42),S isaribbonofwidthoneonthetoruswhich
For infinite open systems, the previous discussion will
containsallhexagons(plaquettes)lying betweenthe two
be sufficient for the analysis of bulk properties. How-
consecutive rows j and j +1. As the ground states of
ever, in a closed system, boundary terms will lead to a
the Kitaev model are vortex free (I = 1 for all h), the
topologicaldependencyofthegroundstates. Inthissub- h
righthandside of Eq.(42) is one. Thus, for allrowsj, we
section,weshallconsiderthetoridalgeometry,whichcan
have that φ(j) = φ(j +1). A variational state for the
beviewedasperiodicboundaryconditionsalongbothdi-
periodic system is given by
rections. FortheJordan-Wigertransformationdefinedin
Eq.(2), the periodic boundary condition along the verti-
|ψvari=N |gα′i=N [1+φ(j)] |gi (43)
cal direction will has no effect since the phase terms of
two nearestneighboring terms alongx or y bonds cancel Xα′ (cid:16)Yj (cid:17)
out and there is no phase term for the z bond coupling. withN anormalizationconstant,and{α′}allconfigura-
Fortheperiodicboundarycondition,theboundaryterms tions, which up to the symmetry operations of Eqs.(13,
8
15),correspondtotheuniformsectorα =1forallbonds Here, we have droppedthe sublattice subscripts b andw
r
r. Here, |gα′i is the ground state in a given α′ sector of for the A fields. If we introduce a dual spin on vertical
theopenboundarysystem. Suchastateisrelatedtothe bonds,
general uniform α = 1 ground state which we derived
r
earlier (the state |gi of Eqs.(28, 29, 30, 31, 32)) by the µzr =2d†rdr −1, (46)
application of the symmetry operators φ(j) of Eqs.(15,
then the effective Hamiltonian H will be nothing but
41). The variational state of Eq.(43) is an eigenstate of eff
an Ising coupling between two neighboring spins along
all of the symmetries {φ(j)} of Eq.(15, 41) [with a uni-
horizontal direction H(4) ∝ µz µz . The to-
form eigenvalue which is equal to one]. eff i,j+1/2 i+2,j+1/2
tal perturbative Hamiltonian thus reads
VI. ANYONS IN THE GAPPED PHASE H = 2J α µz − Jx2Jy2µz µz +const. (47)
z i i 16J3 ij i+2,j
ij " z #
We will now derive and study anyons in the gapped X
phase of Kitaev’s model by relying directly on the on the lattice of vertical bonds. The coefficient −1/16is
Jordan-Wigner transformation. The existence of anyons worked out by Kitaev[5]. As one would expect, the J
x
in this phase has long been recognized [5] and, due to and J terms lift the degeneracy and the ground state is
y
theprospectoffaulttolerantquantumcomputing,isone the one with α = 1 in our aforementioned gauge fixing
i
of the main motivations for studying this system. The of the extra freedom in section IV. The interesting ex-
current appendix illustrates how anyons these may be citation state that corresponds to the anyon studied by
directly studied and derived within our framework. Kitaev[5] is thus the ground state of the Hamiltonian of
Besides the Bogliubov quasiparticles, other the configuration sketched in Fig.3(b). This is exactly
excitations- the vortex excitation illustrated in Fig.3(b)- thesamevortexstateinplaquettepdefinedbyKitaev[5]
are also manifest. It is more interesting to study these whereoneconservedquantityI isreversedintheplaque-
excitations within the gapped state. In this section, tte p. In this state, to save (positive) J contributions
z
we shall follow Kitaev’s original argument[5] and now to the energy, the effective spins {µ } on the cut where
i
demonstrate, in our fermionic representation, the any- α = −1 are also flipped and the resultant configura-
i
onic nature of these vortex excitations. Let us start tion is a domain wall structure at p. The energy penalty
in the limit where J = J = 0. In this limit, the associatedwiththis domainwallisJ2J2/8J3. Forconve-
x y x y z
fermionized Hamiltonian reads nience let us define
H =J α (2d†d −1). (44)
z r r r
r I˜(i,j+1/2)=µz µz . (48)
X i−1,j+1/2 i+1,j+1/2
The ground state is thus 2N/2 degenerate, where N is
In terms original spin operators,I˜is, similar to Eq.(9),
number of sites. This degeneracy can also be under-
stood in the original spin language. In this limit, all I˜=σxσzσyσxσzσy, (49)
verticalbonds are disconnected. Let us consider a single 1 2 3 4 5 6
bond, (Jzσ1zσ2z). Without loss of generality, let us con- An illustration is provided in the lower corner plaquette
sider Jz > 0. There are two degenerate ground states of Fig.6.
|↑↓iand|↓↑i. ThisdegeneracyisrelatedtothelocalZ2 We shall now demonstrate that the anyonic nature of
symmetry which present for JxJy = 0. Without losing thevortexexcitation. Avortexontheplaquettecentered
generality,letussetJx =0. Inthiscase,wecandefine a at (i,j+1/2) is characterized by two kinks on the same
localunitarytransformationwhichisjustσ1xσ2x ononeof bond,oneinthechannelofαandanotherinthe channel
the x-bonds. The effect of this operator is to transform of µz. There are four equivalent ways to create a vortex
Jz to −Jz on the two vertical bonds connected to the on the plaquette centered at (i,j+1/2):
x-bond. If J J 6= 0, such transformation is impossible
x y
andweexpectthedegeneracytobelifted. Thelowestor- PR(i,j+1/2) = iA B (i′,j+1) (50)
w w w
dercontributionfromJx andJy termisthusexpectedto i′>i
be in the second order of J J , i.e., J2J2/J3. This cor- Y
x y x y z PL(i,j+1/2) = iA B (i′,j+1) (51)
responds to a fourth-order perturbation. Indeed, Kitaev w w w
hasshownthisistrueinthelanguageofspinoperators[5]. iY′<i
The non-constanteffective Hamiltonian up to 4-th order PR(i,j+1/2) = iA B (i′,j) (52)
b b b
isthusproportionaltotheproductofthefourhorizontal i′>i
bonds of a given plaquette, (σxσx)(σyσy)(σyσy)(σxσx). Y
1 2 2 3 6 5 5 4 PL(i,j+1/2) = iA B (i′,j). (53)
In the language of our Majorana fermions A, this is b b b
i′<i
Y
He(f4)f ∝ (A1A2)(A2A3)(A6A5)(A5A4) The operators PR (PR) are related to PL (PL) by
w b w b
= (iA A )(iA A ). (45) the gauge transformations encapsulated by Uw (Ub) of
1 6 3 4 i i
9
Σ Σ Σ row, j = 2n. L can be separated into two parts. The
z z z C
first part is the product of I˜that covers the plaquettes
Σ Σ on the odd rows enclosed by the loop C. The second
Σx Σy part is the product of σzσz on the vertical bonds whose
y x b w
black site is enclosed inside the loop C. In the projected
Σ Σ subspace on which the effective Hamiltonian lives, the
Σx Σy
y x vertical bond part is the identity 1. Therefore, the loop
operator is just the product of I˜over the plaquettes on
ΣΣxΣ Σ Σ ΣΣy the odd rows enclosed by C, LC = (−1)nC where nC is
y z z z x Σ Σ the number of vortices that are located on the odd rows
- z z and inside the loop C. In another words, if we bring a
vortexonoddrowaroundanothervortexonanevenrow,
Σ Σ Σ an extra minus sign is generated, while no such minus
ΣxΣzΣy sign if we bring a vortex on an even (odd) row around
y z x
another vortex also on an even (odd) row. Putting all
of the pieces together, we fleshed out, by building on
FIG. 6: Loop operator to movea vortex along a loop.
the fermionic representation, the anyonic nature of the
vortex excitations.
2
Eq.(13). We also notice that PR,L = 1. There-
b,w
fore, PR,L can either create a (cid:16)vortex(cid:17)or annihilate an
b,w VII. STRING AND BRANE TYPE
existing vortex at (i,j+1/2). A naive way to move an CORRELATORS DICTATED BY LOCAL
anyonverticallyfrom(i,j+1/2)to(i,j+5/2)istoapply SYMMETRY
PL(i,j+5/2)PL(i,j+1/2). It turns out that it is more
b w
convenient to multiply two extra phase terms, the first
In this section, we shall apply symmetry considera-
one contains the multiplication of z−bonds centered at
tions [8] [13] and review condensed remarks from [13]
(i′,j+3/2) with i′ <i
concerningthesymmetriesofthismodelandgeneralcon-
siderations regarding non-vanishing correlators in simi-
lar matter coupled theories which are reviewed in the
[(iA B )(iA B )] ,
b b w w z−bonds
i′<i ! Appendix (B). We will show directly from symmetry
Y considerations that the Kitaev model on the hexagonal
and the second one contains the multiplication of I˜cen- lattice [5] does not exhibit any two point correlations of
teredat(i′,j+3/2)withi′ ≤i, I˜(i′,j+3/2). After length larger than one and that any non-vanishing cor-
i′≤i
includingthesetwophases,theshiftingoperatorT from relation function generally amounts to a string operator
y
(i,j+1/2)to (i,j+5/2)takes aQsimple form interms of (either closed or open). The fermionization that we em-
spin operators ploy here [9] [10] gives rise to precisely such string like
correlators whose form is dictated by symmetry. In the
T =σx(i,j+1)σy(i,j+2). (54) matter coupled gauge analogy of Section(B), all possi-
y
ble correlators correspond to open meson or closed pho-
We are now ready to construct the loop operator that ton lines. We will show that Kitaev’s model supports
moves vortex around and study the statistic properties finite valued brane type correlators. By brane correla-
of vortex excitations. Let us first create a vortex on the tors, we allude to correlators which span portions of the
plaquette centered at (i,j + 1/2) with PR(i,j + 1/2), twodimensionalplane. This result- as it applies for only
w
which applies Sz on all site-3 (white site at top-right the two point correlations- was also recently derived by
corner)oftheplaquettes(i′,j+1/2)withi′ ≥iasshown a Majorana fermion construction [12]. In what follows,
in Fig.6. To move this vortex along a closed loop c, we we show how gauge symmetry considerations effortlessly
can construct a a loop operator LC by multiplying Ty mandate this result.
along vertical lines and σz along horizontal parts of the
loop, as illustrated in Fig.6. The loop operator can be
expressed by
A. Two point correlators
LC = I˜(i′,j′+1/2) [σwzσbz]l . (55) theWsepicnl-asipmintchoartrebloatthionatbfietnwiteeenanadnyattwzoersopitnesmwpheircahtuarree
j(i′′+,Yjj′=)∈oCdd, lYb∈C ! separatedbymorethanonelatticeconstantmustvanish.
To see this, we note that this system displays the local
An explanation of the notation is due. Without losing symmetries of Eq.(9). We now fuse these symmetries
generality,let us assume the anyonis locatedon aneven together and see that we have a symmetry associated
10
with any closed contour C, by with|ψianygroundstatewhichisasimultaneouseigen-
state of H and Oˆ . The eigenvalues of Oˆ can only be
OˆC = σpγpC. (56) ±1. We can now sChow that unless p and qCare linked by
p∈C a single step along a direction a each of the expectation
Y
values in Eq. (58) vanishes. For example, for the first
In Eq. (56), the polarization index γpC is chosen to cor- term in the final expression, we have
respond to the direction (γ = x,y,z) of the single bond
emanating from site p which does not lie in the contour hψ|σpaσqb|ψi=(hψ|OˆC)σpaσqb(OˆC|ψi)
C. Any given site p forms the endpoint of three bonds; =hψ|(Oˆ σaσbOˆ )|ψi=−hψ|σaσb|ψi, (59)
two ofthese bonds lie in the contourC andonly a single C p q C p q
bond ending in p does not lie in C. unless p and q which are linked by a single step along
We now briefly relate these symmetries to degenera- a direction a. Similarly, the second term in Eq. (58)
cies. IftwocurvesC andC′ shareanoddnumberofsites is seen to vanish. Here, the commutator [H,σb] leads
q
in commonthen OC andOC′ anticommute. Thus if, e.g, to sums of bilinears which involve the lattice site j and
we choose to simultaneously diagonalize H and OC and a nearest neighbor site k. When multiplied by σpa this
determine a ground state |g1i in that common eigenba- leads an expression which is cubic in the spin operators.
sis, then OC′|g1i is a new degenerate ground state. This Ifallofthethreelatticesitesaredifferent,wecanchoose
degeneracy applies not only to the ground state sector a path C such that this expression changes sign under
but to all energy levels. a unitary transformation corresponding to Oˆ . Similar
C
higher order terms in Eq. (58) unless p and q which are
linked by a single step along a direction a. This result
1. T >0 reaffirms (from a purely symmetry point of view) the
more detailed derivation of [12].
By employing these symmetries along with Elitzur’s
theorem [19] at temperatures T > 0, we find that the
finite temperature correlator hσaσbi vanishes unless (1) B. Higher order string and brane type correlators
p q
p and q are nearest neighbor sites and that (2) a = b
corresponds to the direction (a = x,y,z) between the We now turn to higher order spin correlations. A mo-
two lattice sites i and j in question. This is so as unless ment’s reflection reveals that invariance under the gen-
conditions (1) and (2) are satisfied, there is at least one erallocal symmetries of Eq.(9) allows only for the string
loopC for which[σaσb] is notinvariantunder Oˆ ofEq. or brane operators (either closed or open) to be finite:
p q C
(56) and for which in other words, all spins must form continuous clusters
along lines (strings) or lie in a fragment of the two di-
Oˆ [σaσb]Oˆ =−[σaσb]. (57) mensional plane.
C p q C p q
String and brane type correlators may involve, in the
thermodynamiclimit,aninfinitenumberoffieldsandal-
low different topologies than that of closed loops alone.
2. T=0
The fields may be ordered along an open string (or col-
lection of such strings) or in higher dimensions may in-
TheT =0relation(evenfordifferenttimes[12])issim-
volve sophisticated combinations of fields at all lattice
ilarlyderived. AlloftheoperatorsOˆ commutewithone
c sites which we term as brane type correlators.
another. [This is seen as for the minimal loop C which
The best known example of such a string correlator is
contains a single hexagon, two neighboring hexagons
that in the spin S =1 AKLT chain [22] [23] in which
leading to operators Oˆ and Oˆ which contain two com-
1 2
mon sites which lead to two anti-commutation relations k−1
4
between the different Pauli matrices. [5]. The product hSz eiπSj Szi= . (60)
i k 9
of operators each acting of a single hexagon leads to the (cid:16)j=Yi+1 (cid:17)
most generalsymmetry operator of Eq. (56) with an ar-
Eq.(60) holds for arbitrarily far separated sites i and
bitrary path C.] Consequently, we may simultaneously
diagonalize H of Eq. (1) with all of the operators Oˆ . k. As this correlator is, asymptotically, far larger than
c theusualspin-spincorrelator,itisoftensaidtocapturea
The dynamical T =0 two spin correlation function
hidden order. We may construct string operators which
hψ|σa(0)σb(t)|ψi are more sophisticated variants of the operator appear-
p q inginEq.(60). Apartfromspinchains,[22][23][24]such
= hψ|σa(0)exp(iHt)σb(0)exp(−iHt)|ψi correlators also appear, amongst others, in doped Hub-
p q
bard chains and related systems, [25] [26] [27] cold atom
= hψ|σaσb|ψi+ithψ|σa[H,σb]|ψi
p q p q chains [28], in spin leg ladders [29], and may be related
(cid:16) (it)2 to non-local constructs in Quantum Hall systems [30].
+ 2! hψ|σpa[H,[H,σjb]]|ψi+... (58) In doped Hubbard chains, the string correlator decays
(cid:17)
