Table Of ContentExactly solvable Kitaev model in three dimensions
Saptarshi Mandal1 and Naveen Surendran2
1 The Institute of Mathematical Sciences, C.I.T Campus, Chennai 600113, India∗
2The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy†
(Dated: January 29, 2009)
We introduce a spin-1 model in three dimensions which is a generalization of the well-known
2
Kitaevmodelonahoneycomblattice. FollowingKitaev,wesolvethemodelexactlybymappingitto
9 atheoryofnon-interactingfermionsinthebackgroundofastaticZ2gaugefield. Thephasediagram
0 consists of a gapped phase and a gapless one, similar to the two-dimensional case. Interestingly,
0 unlike in the two-dimensional model, in the gapless phase the gap vanishes on a contour in the
2 k space. Furthermore, we show that the flux excitations of the gauge field, due to some local
constraints, form loop like structures; such loops exist on a lattice formed by the plaquettes in
n
the original lattice and is topologically equivalent to the pyrochlore lattice. Finally, we derive a
a
low-energy effective Hamiltonian that can be used to study the properties of the excitations in the
J
gapped phase.
9
2
PACSnumbers: 75.10.Jm,03.67.Pp,71.10.Pm
]
l
l I. INTRODUCTION a relatively simple platform—the Hamiltonian involves
a
h only two-body interactions—to study concepts such as
- topological order and fractional excitations. Thus,
s The study oftopologicalphaseshasbeenactivelypur-
e sued in condensed-matter systems for some years. This not surprisingly, the various many-body aspects of the
m model have been thoroughly investigated.19,20,21,22,23,24
has resulted in the emergence of a new paradigm in the
There have also been some generalizations to other two-
. theoryofquantumphasetransitions: certainphasetran-
t dimensional (2D) lattices.25,26 It is then worthwhile to
a sitions cannot be described in terms of local order pa-
m rameters associated with spontaneously broken symme- findgeneralizationsofthemodelinhigherdimensions. In
this paper, we introduce and study a three-dimensional
- tries; instead, the phases in such transitions are char-
d acterized by topological order.1 The most famous ex- (3D) version of the Kitaev model. Three-dimensional
n models exhibiting topological order have been studied
ample of topological order in a quantum system is in
o the phenomenon of fractional quantum Hall effect.2,3,4 previously.27,28,29,30
c
[ Other examples—experimental and theoretical—include
quantumspinliquids,5,6,7,8 quantumdimermodels,9,10,11
2 An exact solution of the Kitaev model is possible due
quantum loop models,12 etc.
v to the existence of a macroscopic number of locally con-
9 Recently, ithasbeen proposedthattopologicalphases served quantities: this facilitates the mapping of the
2 can be used to do quantum computation.13,14 The main model to a quadratic Hamiltonian of Majorana fermions
2 obstacle in the realization of quantum memory—the ba- hopping in the background of a static Z gauge field.
0 sic ingredient of a quantum computer—is decoherence: 2
The three-dimensional model we construct also has the
.
1 it is difficult to prepare states that are robust to exter- above feature, which renders it exactly solvable. It has
0 nal noise. Kitaev15 suggested that topologically ordered
a gapped phase and a gapless one, just as in 2D, and
8 states can be used to overcome this problem. He illus-
although the phase boundaries are identical to the lat-
0 trated these ideas in a spin-1 model on a hexagonal lat-
: 2 ter, the nature of certain excitations is quite different.
v tice which, quite remarkably,can be solved exactly. The In the three-dimensional model, the excitations of the
i groundstate ofthe Kitaev model has two phases: in one
X gauge field are localized on “loops”. This gives rise to
phase the elementary excitations have a gapin the spec-
the possibility that, as yet unverified in our model, such
r trumandareAbeliananyons;thesecondphaseisgapless
a excitations can obey nontrivial statistics, since double
intheabsenceofanexternalmagneticfield,butdevelops
exchange in loops—unlike points—is topologically non-
a gap when the field is switched on, and then the exci-
trivial in three dimensions (for example, see Ref. 29).
tations are non-Abelian anyons. In topological quantum
computation,braidingofnon-Abeliananyonsisessential
fortherealizationofuniversalquantumgates.16 Regard- The paper is organizedas follows: in Sec. II we define
ing the feasibility of a physical realization of the Kitaev the spin Hamiltonian and then rewrite it using a Majo-
model, there has been a proposal to realize it on an op- rana fermion representation of spin 1. In Sec. III the
2
tical lattice.17,18 low-energy spectrum for fermionic excitations is derived
Apart from its potential application in quantum com- whileSec. IVdiscussestheexcitationsofthegaugefield.
putation, the Kitaev model is an interesting many-body We then derive the low-energy effective Hamiltonian in
systembyitself. First,exactsolutionsarerareindimen- the large J limit in Sec. V and end with a discussionof
z
sions higher than one, and second, the model provides our results in the last section.
2
of disconnected rows or disconnected columns. (ii) As
one goes along a particular row (column), at each site
thereisalinkwhosedirectionalternatesbetweenpositive
and negative z axes. That is, there is a criss-crossing
structurebetweenadjacentplaneswhichensuresthatthe
z
latticeistrulythreedimensional—despiteacoordination
y
numberof3—andnotasetofmutuallydisconnectedtwo-
x
dimensional surfaces.
FIG. 1: The honeycomb lattice: the three types of links are
labeled x, y,and z.
z
II. HAMILTONIAN
y
a
3
Before presenting the three dimensional model, let us 4
5 x
briefly describe the Kitaev model in 2D. It is a system
of spin-1 degrees of freedom located at the vertices of a a 6
2 3 2
honeycomb lattice. In the honeycomb lattice, there are 2
three types of links which, distinguished by their orien- p 7
1
tations, are labeled x, y and z (see Fig. 1). The Kitaev a1 8
Hamiltonian is
10
H =−Jx σixσjx−Jy σiyσjy−Jz σizσjz, 9 z
<Xi,j>x <Xi,j>y <Xi,j>z y
(1) x
whereσa’sarethePaulimatrices,and<i,j > indicates
a
that i and j belong to a link of a-type.
FIG.2: The3Dlattice: thefoursitesinsidetheloop(marked
For our purposes, the key points to note about the
1-4)constituteaunitcell;a1,a2,anda3arethebasisvectors.
Hamiltonian in Eq. (1) are that for any spin, only one Plaquette p consists of sites marked 1-10.
of the components couples to a particular neighboring
spin,andthecomponentwithnon-zerocouplingstrength
is different for each of the three neighbors. As we will
see later, this leads to the existence of a set of mutually To parametrize the lattice sites, we first note that the
commutingconservedplaquette operators,whichinturn unitcellcontainsfoursites. Thepositionvectorofaunit
makes the problem exactly solvable. cell is given by
Wenotetwofeaturesofthehoneycomblatticethatare
pertinent to the construction of the 3D lattice: (1) The R = ma1+na2+pa3, m,n,p Z, (2)
∈
coordination number of the lattice is 3. (2) The three a = 2xˆ, a =2yˆ, a =xˆ+yˆ+2ˆz, (3)
1 2 3
types of links x,y and z are distributed in such a way
that two links of the same type do not touch each other. wherexˆ,yˆ, andzˆ areunit vectorsalongx,y andz direc-
tions, respectively. The four sites within a unit cell are
at
A. Lattice
yˆ yˆ
r =R ˆz, r =R ,
1 2
− 2 − − 2
To facilitate visualization, we will first describe how
yˆ yˆ
to obtain the 3D lattice starting from the familiar cubic r3 =R+ , r4 =R+ +zˆ. (4)
lattice. Let i,j,k Z be the x,y and z coordinates 2 2
∈
of the latter. The new lattice is obtained by removing To define a Kitaev-type Hamiltonian, we need one more
those sites that satisfy one of the following conditions: ingredient, viz, the labeling of links. To this end, we al-
(1) k = 0mod4 and i = 0mod2, (2) k = 1mod4 and ternatelyassignxandy labels tothe links ineachofthe
j = 0mod2, (3) k = 2mod4 and i = 1mod2, and (4) rows and columns that lie on the x-y plane; the remain-
k =3mod4 and j =1mod2. ing links, the ones along the z axis, are labeled z. [The
Thisamountstodepletingthecubiclatticebyhalf,and ambiguityintheassignmentofxandylabelswithineach
the resultant lattice has coordinationnumber 3 (see Fig. row(column)isresolvedbydemandingperiodicity.] This
2). We note that: (i) The x-y planes alternately consist way of labeling ensures that the three links emanating
3
from each site have different labels. Now the definition dimensional lattice we have constructed. Explicitly,
oftheHamiltonianinEq. (1)canbeappliedtothethree-
H = J σx(R)σx(R a ) J σy(R)σy(R+a a ) J σx(R)σx(R)
− x 1 4 − 3 − y 1 4 1− 3 − x 2 3
R
Xh
J σy(R)σy(R+a ) J σz(R)σz(R) J σz(R)σz(R) . (5)
− y 3 2 2 − z 1 2 − z 3 4
i
Forsimplicityofnotation,wewillcontinuetousethefor- fermions. At each site j, we introduce four Majorana
mal expressionfor H in Eq. (1), where sites are referred operators bx, by, bz and c , which satisfy the following
j j j j
to by asingle lower-caseindex,andrevertto the explicit relations.
form in Eq. (5) only when the calculation demands it.
bα† =bα, c† =c ,
j j j j
bα,bβ =2δ δ , c ,c =2δ ,
B. Conserved quantities { j k} jk αβ { j k} jk
bα,c = 0. (9)
{ j l}
Let l = (j ,j ,...,j ) be a sequence of lattice sites
1 2 n The Majorana fermions act on a Fock space which is
such that j and j are neighbors for m = 1,2,...n
m m+1 fourdimensional,whereasthespinHilbertspacehastwo
with j j . Then l represents a loop. Equivalently,
n+1 1 dimensions. We define the physical space as consisting
≡
the links formed among j also uniquely determine the
m of those states which satisfy the constraint
loop. Let
D ξ = ξ , j where D =bxbybzc . (10)
W = σajm, (6) j| i | i ∀ j j j j j
l j
jYm∈l Dj2 = 1; thus, (1,Dj) form the elements of a Z2 gauge
group. The spin operators are defined as
where a is the label of the link that connects j to its
jm m
neighbor which is not in l. It is easy to see that σα =ibαc . (11)
j j j
[Wl,H]=0, [Wl,Wl′]=0, l,l′. (7) When restricted to the physical space, the above opera-
∀
tors satisfy the standard spin-1 algebra. The projector
However, not all such operators are independent. To 2
to the physical space is given by
see this, we define a geometrical operation of combining
two loops as follows: the combination of loops l1 and l2 1+D
is the loop l12 formed by all the links in l1 and l2 ex- Pj = 2 j. (12)
cept those that are common to both. Then, it is easy
to see that W W = W . This means that for a All the states related by a gauge transformation project
l1 l2 ± l12
lattice with open boundary conditions, any loop opera- on to the same physical state.
tor can be written as a product of those defined on the Starting from a generic spin model, H σα , we can
{ j}
plaquettes—the elementary loops which cannot be ob- obtain a fermionic Hamiltonian, H bα,c , by using Eq.
tained by combining smaller loops. For example, in Fig. { j j}
(11). Since [H,P ] = 0 j, the eigenstates of H can be
2,Wp correspondingto the plaquettep, consistingofthe j ∀ e
obtained from those of H by projecting the latter to the
sites marked 1,2,...,10, is
physical spacee.
Wp =σ1xσ2yσ3yσ4yσ5zσ6xσ7yσ8yσ9yσ1z0. (8) Substituting Eq. (11)ein Eq. (1), we obtain
There are further constraints among Wp corresponding H = i Aˆ c c , (13)
jk j k
to different plaquettes, and we will discuss them in Sec. 2
j,k
IV, where the excitations are studied. X
e J uˆ if j and k are linked,
Aˆ = αjk jk (14)
jk 0 otherwise,
(cid:26)
C. Hamiltonian in Majorana fermion uˆ = ibαjkbαjk, (15)
representation jk j k
where α is the type of the link between j and k. We
jk
To solve the Hamiltonian, following Kitaev,15 we use note that uˆ = uˆ , and in the sum the links are
jk kj
−
a representation of Pauli matrices in terms of Majorana treated as directed and therefore counted twice. We use
4
a hat to emphasize that uˆ is an operator; u is the of free fermions hopping on a d-dimensional hypercubic
jk jk
corresponding eigenvalue and takes values 1. Further- latticewithhoppingamplitude tij eiθij betweennearest-
more, ± neighbor sites i and j was stud|ied| by Lieb.31 Moreover
it was shown that, if t is reflection symmetric about
ij
| |
[H,uˆ ]=0, [uˆ ,uˆ ]=0 <j,k >,<l,m>. (16) certain planes that does not contain any sites, then the
jk jk lm
∀
ground-state energy is the lowest when the flux of the
Theerefore, the Hilbert space breaks up into various sec- phase along the plaquettes, Φ θij, equals π if
tors, each corresponding to a particular set ujk ; the thelengthoftheloopis0mod4≡,and<zije>roifthelengthis
{ } P
matrix elements of H betweenstates belonging to differ- 2 mod 4. Unfortunately, Lieb’s result cannot be directly
ent sectors are zero. The Hamiltonian in a given sector appliedto our casebecause ourlattice does nothavethe
isobtainedbyreplaceingtheoperatorsuˆjk withtheircor- required reflection symmetry.
responding eigenvalues, u :
jk
i
Hu = Ajkcjck, (17) Wenumericallystudiedthe ground-stateenergyofthe
2
j,k HamiltonianinEq. (17)forlatticescontainingupto864
X
e sites using periodic boundary conditions. The ground-
where Ajk is obtained from Eq. (14) by substituting state energy in the sector with uniform flux (Wp =
uˆjk with ujk. The gauge-invariant and hence physical +1, p)was comparedwith that of other selectively cho-
conserved quantities are sen ∀flux configurations. As will be discussed in Sec. IV,
duetosomelocalconstraints,theplaquettesthatareex-
10
citedthemselvesformloopsinanembeddedlattice. The
W = uˆ , (18)
p jmjm+1 smallest such loop consists of six plaquettes. We consid-
m=1
Y eredexcitations ofloops of varyinglength andfound the
f
energyincreasingwithincreasinglength. We alsolooked
where p = (j ,j ,...,j ), as before, is a plaquette. W
1 2 10 l
atexcitationsofmultipleloops. Ineverycaseconsidered,
is related to W in Eq. (6) as follows:
l
we found that the energy of the lowest energy state is
f
greater than that of the W =+1 uniform flux state. In
W = W , = P . (19) p
l Pl lPl Pl j Sec. V, we will getfurther confirmationofthis result, at
Yj∈l least in the large Jz limit, througha perturbative analy-
f
sis. Thereforeit is reasonableto assumethat the ground
From now on we will simplify the notation, following
Kitaev,15 by not making distinction between operators state belongs to this sector.
in the physical and extended Hilbert spaces, i.e., we will
drop tilde from all the operators acting in the latter.
u = 1 for all links is the obvious choice among the
jk
configurations of link variables that gives W = +1, p.
p
III. GROUND STATE AND SPECTRUM Of course, any configuration related to this one by∀a
gauge transformation will also satisfy the above condi-
Next question to be addressed is: Which sector of tion on W ’s. With this choice, the Hamiltonian in its
p
u does the ground state belong to? The problem explicit form becomes
ij
{ }
H = i J c (R)c (R a )+J c (R)c (R+a a )+J c (R)c (R)
x 1 4 3 y 1 4 1 3 x 2 3
− −
R
Xh
+J c (R)c (R+a )+J c (R)c (R)+J c (R)c (R) (20)
y 3 2 2 z 1 2 z 3 4
i
where r and a are given in Eqs. (2) and (3). To diago- and
i
nalize the Hamiltonian, we next do a Fourier transform.
π dk π dk π dk (2xˆ ˆz) (2yˆ ˆz) ˆz
cµ(r)= 2π1 2π2 2π3e−ik·rcµα(k), (21) b1 = 4− , b2 = 4− , b3 = 2.
Z−π Z−π Z−π
with µ=1,2,3,4 and where
Using the property, c ( k) = c†(k), the Hamiltonian
µ − µ
k = k b +k b +k b , becomes,
1 1 2 2 3 3
5
π dk π dk π dk i
H = 1 2 3 eik3δ c†(k)c (k)+δ c†(k)c (k)+J c†(k)c (k) +h.c. , (22)
2π 2π 2π 2 k1 1 4 k2 3 2 z 1 2
Z−π Z−π Z−π " (cid:26) (cid:27) #
where δki = Jx + e−ikiJy, for i = 1,2. Furthermore, we define ∆k = |δk1|2+|δk1|2+2Jz2 , and φk such that
e−ik3δk1δk2 ≡|δk1||δk2|eiφk. The above Hamiltonian is easily diagonalized a(cid:0)nd we obtain the spe(cid:1)ctrum to be
1
E(k)=±2√12"∆k±(cid:20)∆2k−n(cid:0)Jz2−|δk1||δk2|(cid:1)2+2Jz2(1−cosφk)|δk1||δk2|o(cid:21)21#2. (23)
C
3
2
1 1.9
k20 1.2 1.98
−1
1.6
0.5
−2
0.2
−3
−3 −2 −1 0 1 2 3
k1 A B
FIG.3: Inthekspace,thecontouronwhichthegapvanishes
FIG. 4: The phase diagram: it shows the plane defined
binseextpwtroetejoencttzheeedrocooannntkdo3utr=ws.o0. Cploarnree.spJoxnd=inJgyv=alu1e,saonfdJzJzariessvhaoriwedn bJyy =JxJ+z =Jy0+, BJzc=orr1e.spoPnodinsttoAJyco=rre1spaonnddJszt=o JJxx==01, aanndd
C correspondstoJz =1andJx =Jy =0. Theshadedregion
is thegapless phase.
In the ground state, all the negative-energy states are
filled. The system is gapless if solution exists for E =0.
Figure 4 depicts the phase diagram on a section of
Since the two termsinside the curlybracketsinEq. (23)
the parameterphase. Interestingly,the phase diagramis
are positive definite, for E = 0, both of them have to
symmetricinthethreecouplingconstantsthoughtheydo
vanish, i.e., notappearsymmetricallyintheHamiltonian(thez-links
cannot be transformed to x or y links by any symmetry
1
J2 = J2+J2+2cosk J J 2 transformationofthelattice). Noristhespectrum,given
z x y 1 x y
by Eq. (23), symmetric in the coupling constants.
h i 1
J2+J2+2cosk J J 2, (24)
× x y 2 x y
cosφk = 1.h i (25) We end the discussion on the ground state by noting
that spin correlations can be calculated in exactly the
The values of k1 and k2 for which the gap vanishes are same way as in 2D.19,20 The only non-zero correlations
determined by Eq. (24); k3 is then given by Eq. (25). are those constructed out of the interaction terms that
Solutions of Eq. (24) exist only when Jz Jx+Jy, appear in the Hamiltonian.
≤
J J +J and J J +J ; these conditions are
x y z y z x
≤ ≤
same as that for the 2D Kitaev model. Figure 3 shows
the plot of contours satisfying Eq. (24) projected on to IV. FLUX EXCITATIONS
k =0 plane, where we have set J =J =1 and varied
3 x y
Jz from zero to two. The contour shrinks to the point The Kitaev model in 2D has anyonic excitations, i.e.,
(0,0) as Jz approaches Jx +Jy = 2, i.e., when the gap there are particle-like excitations which obey nontrivial
opens up. statistics. Anyons are very specific to 2D and cannot
existinhigherdimensions;thefundamentalreasonbeing
that in D > 2 there are no nontrivial paths which take
6
a particle around another. However, in 3D, excitations
localized on loops may obey nontrivial statistics. In this
section, we show that there are excitations in our model 18
8 7
whicharelocalizedonloops. However,wedonotaddress 9 6
the issue of their statistics in this paper. 17 5
We have seen that in the ground state W = 1 for 14 18
p 8 16
all plaquettes p. It then follows that the excitations are 10
4
of two types: (1) Flux configurations which violate the 1 17
condition W = 1, i.e., W = 1 for some of the loops. 13 14
p p − 11 12 15
It can be looked upon as creating a π flux over those
loops. (2) Fermionic excitations in the background of (a) (b)
static configurations of W .
p
7
Wewillnextshowthattheexcitationsofthefirsttype 16
have the structure of loops. Earlier we mentioned that 8 6 5 3 4
9
not all Wp are independent; now we will find the con- 1 14 15
2
straints among them. There are four types of plaque-
4
ttes which are not related by translation. Let a,b,c, and 10 3 12 13
d be the labels for the different types [see Figs. 5(a)- 11
1 2
5(d)]. Consider a part of the lattice, shown in Fig. 5(e),
which consists of four adjacent plaquettes—each a dif- (c) (d)
ferent type. Here the links are labeled 1-20. Let the
correspondingloopoperatorsbeW ,W ,W ,andW ,re-
a b c d
spectively. In terms of the spin variablesthese operators 7
are 6
18
Wa = σ1x1σ1z2σ1y3σ1x4σ1x7σ1x8σ8zσ9yσ1x0σ1x, 8 5 x−link
W = σz σy σx σxσxσzσyσxσx σx , 9 b y−link
Wcb = σ11z4σ2y1σ53yσ164yσ45xσ56zσ67yσ78yσ89yσ11x80,17 c 17 4 16 z−link
Wd = σ1x1σ1z2σ1y3σ1y4σ1y5σ1x6σ4zσ3yσ2yσ1y. (26) 3 14
a
10
Using the relations σxσyσz =i and σa2 =1, d 15
j j j j 1
2
13
W W W W =1. (27)
a b c d
11 12
The above constraint has a graphical interpretation.
Note that a loop operatorcan also be written as a prod-
(e)
uct of the interaction terms on the links contained in
the loop. In Fig. 5, the object (e), which is obtained
FIG. 5: [(a)-(d)] The four types of plaquettes. (e) Part of
byputtingtogetherthe fourloops(a)-(d), representsthe
the lattice involving four such adjacent plaquettes; the cor-
left-hand side of Eq. (27). Evidently, each link in (e)
responding operators give rise to a constraint. The ellipses,
is shared by two of the Wp’s. Since the square of any labeleda-d,respectively,representeachoftheloopsandform
interaction term is one, Eq. (27) immediately follows. a “tetrahedron”.
the new lattice thus obtained; topologically, it is the py-
Foropenboundaryconditions,relationssuchasEq.(27) rochlore lattice, an arrangement of corner-sharing tetra-
exhaust all the constraints. To find the configurations hedra (see Fig. 6). (The edges of the geometrical object
of W which are consistent with the constraints, it is formed by the centers of four adjacent loops are not of
p
{ }
instructive to consider a lattice obtainedby representing equal length, and hence do not form an exact tetrahe-
each plaquette by a single site. A plaquette has a step dron but a stretched one. However, the connectivity of
like structure, consisting of tworectangles perpendicular L is same as that of the pyrochlore lattice, and we will
to the x-y plane connected by another rectangle on the continue to refer to the basic objects as tetrahedron.)
x-y plane. Each loop can be uniquely represented by a Inthisdescription,thefourplaquettesthatgiveriseto
point at the center of the rectangle on the x-y plane [in the constraint in Eq. (27) are the four sites of a tetrahe-
Fig. 5(e)such points are marked by ellipses]. Let L be dron, and each tetrahedron corresponds to an indepen-
7
τz τz
4 4
τx τy
5 3
τy τx τx
τy 3 2 5
6 τy
6
τx τz
τz 2 1
1
x−link
(1a) y−link (1b)
z−link
τz τz
4 4
τy τx
5 3
τx τx τy
3 6 5
τ2y τy
FIG. 6: The lattice L formed by the plaquettes—the py- τ1z τ6x 2
rochlore lattice. Loops such as the dashed one, which goes
τz
through six sites, are the shortest (apart from the basic tri- 1
angles).
(2a) (2b)
dent constraint. Therefore, any configuration satisfying
FIG.7: RepresentationofthefluxoperatorsinHeff associated
alltheconstraintswillhaveanevennumberofW taking
p with thefour typesof plaquettes.
value of 1 in each tetrahedron, where p is now the site
index in−L. Now it is clear how to obtain such config-
urations: draw a loop which does not cross itself and The ground state of H is highly degenerate: the two
C 0
which lies entirely within the tetrahedra, and let spins on each of the z-links can be either or
| ↑↑i | ↓↓
. This degeneracy is lifted only at the sixth order of
1, if p , i
Wp = −1, othe∈rwCise. perturbation theory. We now calculate the effective low-
energy Hamiltonian to this order.
(cid:26)
The groundstate subspace of H can be thought of as
0
Anyclosedself-avoidingloopcontainsanevennumberof the Hilbert space of effective spin 1’s located at each z
sites(0,2,or4)belongingtoanyparticulartetrahedron; link. Let τx,τy and τz be Pauli m2atrices acting on the
m m m
hencealltheconstraintsaresatisfied. Inotherwords,the
effective spin at the z link denoted by m, such that,
flux excitations have the structure of loops in the lattice
L. τz = ,
m|↑↑im |↑↑im
τz = . (31)
m|↓↓im −|↓↓im
V. EFFECTIVE HAMILTONIAN IN THE A plaquette consists of four z-links and six x or y-
LARGE J LIMIT
z links. Consider a plaquette by also including the two
z-links that are directed out of the loop (see Fig. 7).
Let us label the z-links in such an object with an index
i=1,2,...,6. The labeling scheme for the four different
To study the excitations in the gapped phase, one can types of loops is shown in the figure. We associate with
followKitaev’s15approachtothe2Dmodel—aperturba- each plaquette p a flux operator B defined as follows.
p
tive analysis in the limit J J ,J . The unperturbed
z x y
Hamiltonian H and the per≫turbation H are B =τz τa2τa3τz τa5τa6, (32)
0 1 p p1 p2 p3 p4 p5 p6
H0 = −Jz σizσjz, (28) wlohoeprea,ttfaocrhied=to2,th3,e5i,t6h,zailin=kxarieftthheestawmoe;liannkds ain=thye
<Xij>z otherwise. The flux operatorassociatedwith the vairious
H = J σxσx J σyσy, (29)
1 − z i j − z i j types of loops are also shown in Fig. 7. Then, to sixth
<Xij>x <Xij>y order—which is the leading order—the effective Hamil-
H = H +H . (30) tonian, ignoring the constant terms arising at the lower
0 1
8
orders, is fermionic spectrum by mapping the spin model to one
of free fermions in the background of a static Z gauge
2
H = 7Jx4Jy2 B 7Jy4Jx2 ′B , (33) field; the system has a gapped phase and a gapless
eff −256J5 p− 256J5 p one. Quite interestingly, the gap vanishes on a con-
z p z p
X X tour in the k space; this could be related to some ac-
wheretheunprimedsumisovertheloopsoftypes1aand cidentaldegeneracies—unrelatedtoanysymmetryofthe
2a in Fig. 7, which contain four x links and two y links, Hamiltonian—of the ground state in the classical limit.
while the primed sum is over loops of types 1b and 2b, The two-dimensional Kitaev model has been shown to
whichcontainfoury links andtwoxlinks. Furthermore, havesuchadegeneracyintheclassicallimit,whichgrows
exponentially with the system size.32 This result can be
[Bp,Bp′] = 0, (34) readilygeneralizedtoourmodel. Itwillbeinterestingto
see how quantum fluctuations lifts this degeneracy.
B2 = 1, p,p′. (35)
p ∀ We have further shown that the excitations of the
gaugefield,duetosomelocalconstraints,havethetopol-
Thus, the operators B can be simultaneously diagonal-
p
ogy of loops. As a first step towards understanding
ized for all values of p and their eigenstates will also be
the nature of the excitations in the gapped phase, we
the eigenstatesofH . FromEq. (35)itfollowsthatthe
eff have derived an effective Z gauge theory in the limit
eigenvalues of B are 1. Therefore, the ground state 2
p
± J J ,J . The ground state of the effective Hamilto-
GS will be such that z x y
eff ≫
| i nian thus obtained is trivially solved and it is consistent
B GS = GS , p. (36) with the assumption we made in the calculation of the
p eff eff
| i | i ∀ fermionic spectrum that the ground state is vortex free.
It is straightforward to see that B is the projection Further study of the effective theory should tell us more
p
of W —the conserved quantities of the spin model de- about the properties of the excitations, such as whether
p
fined in Eq. (8)—on the ground-state subspace of H . they obey fractional statistics.
0
Therefore, in the large J limit, Eq. (36) is a confirma- YaoandKivelson26haveintroducedatwo-dimensional
z
tion of our numerically verified assumption earlier that Kitaev model in which one type of plaquettes is a tri-
the ground state of H is vortex free. angle; the existence of loops with odd number of links
The excitations are the states which violate Eq. (36). resultsinthespontaneousbreakingoftime-reversalsym-
i.e.,B = 1forpinsomesubsetofplaquettes. However, metry. Wementionthatinourmodelalsothetimerever-
p
−
not all B are independent—there exist constraints sim- sal symmetry can be similarly broken by replacing each
p
ilar to Eq. (27)—therefore the plaquettes with B = 1 vertex in the lattice with a triangle.
p
cannot be chosen arbitrarily. To understand the prop−er- Note added : Towardthe completion of this paper, we
ties of the excitations, such as their statistics, a detailed came acrossthe workby Si andYu in Ref. 33 which dis-
analysis is required. This is currently being done. cussesavarietyofexactlysolvableKitaevmodelsinthree
dimensions, none of which are identical to the model in-
troduced here.
VI. SUMMARY AND DISCUSSION
We have constructed and solved a three-dimensional Acknowledgments
spin-1 model which is a generalization of the Kitaev
2
model on a honeycomb lattice. Based on the methods We thank G. Baskaranand R. Shankar for very useful
used by Kitaev,15 we calculated the exact low-energy discussions.
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