PHYSICAL REVIEW B 79, 134427 共2009兲
⌫-matrix generalization of the Kitaev model
Congjun Wu, Daniel Arovas, and Hsiang-Hsuan Hung
Department of Physics, University of California, San Diego, California 92093, USA
共Received 19 February 2009; published 22 April 2009兲
We extend the Kitaev model defined for the Pauli matrices to the Clifford algebra of ⌫ matrices, taking the
4 ⫻ 4 representation as an example. On a decorated square lattice, the ground state spontaneously breaks
time-reversal symmetry and exhibits a topological phase transition. The topologically nontrivial phase carries
gapless chiral edge modes along the sample boundary. On the three-dimensional 共3D兲 diamond lattice, the
ground states can exhibit gapless 3D Dirac-cone-like excitations and gapped topological insulating states.
Generalizations to even higher rank ⌫ matrices are also discussed.
DOI: 10.1103/PhysRevB.79.134427
PACS number共s兲: 75.10.Jm, 73.43.⫺f, 75.50.Mm
I. INTRODUCTION
Topological phases of condensed matter have attracted increasing attention in recent years. Examples include spin liquids, fractional quantum Hall 共FQH兲 states, and topological
insulators, which exhibit fractionalized excitations and
statistics.1–13 A particularly novel application has been the
proposal of utilizing non-Abelian elementary excitations of
topological phases to achieve fault-tolerant topological quantum computation.7,14 To this end, Kitaev,7 in a seminal paper,
introduced a model of interacting spins on a honeycomb lattice which reduces to the problem of Majorana fermions
coupled to a static Z2 gauge field. The ground state is topologically nontrivial and breaks time-reversal 共TR兲 symmetry,
and the elementary excitations are anionic, with non-Abelian
statistics. Much progress has been made toward understanding the nature of this phase9,15–23 and in generalizing the
model to other lattices8,24,25 and to three dimensions.26 Very
recently, a generalization to a decorated honeycomb lattice
exhibiting a chiral spin-liquid ground state was made by Yao
and Kivelson.8 On the experimental side, the ␯ = 25 FQH state
is expected to realize non-Abelian anyons.27–29 Time-reversal
invariant topological states have also been realized in HgTe
semiconductor quantum wells30 and in Bi1−xSbx alloys.31
The solvability of the Kitaev model depends crucially on
the property of the Pauli matrices, i.e., 兵␴i , ␴ j其 = 2␦ij and
␴x␴y␴z = i, which are the simplest example of a Clifford algebra. This gives rise to the constraint that all the above
models are defined in lattices with coordination number 3,
and thus most of them are on lattices of dimension two.
Extending the Kitaev model to more general lattices, three
dimensions, and large spin systems enriches this class of
solvable topological models. These extensions naturally involve
higher
ranked
Clifford
algebras,
with
2n ⫻ 2n共n ⱖ 2兲-dimensional matrices, which can be interpreted as high spin-multipole operators. Some early work on
exactly solvable models in the ⌫-matrix representation of the
Clifford algebra has been done in Refs. 32–34.
In this article, we generalize the Kitaev model from the
Pauli matrices to the Clifford algebra of ⌫ matrices. For the
4 ⫻ 4 representation, we construct a model in a decorated
square lattice with coordination number 5, which can be interpreted as a spin- 23 magnetic model with anisotropic interactions involving spin-quadrapole operators only. It is inter1098-0121/2009/79共13兲/134427共11兲
esting that although each spin-quadrapole operator is TR
invariant, the ground state spontaneously breaks TR symmetry. Such a state is a topologically nontrivial chiral spinliquid state with extremely short-ranged spin correlation
functions. The topological excitations are expected to be
non-Abelian. The ⌫-matrix formalism is also convenient to
define a three-dimensional 共3D兲 counterpart of the Kitaev
model on the diamond lattice. By breaking the TR symmetry
explicitly, a gapless spin liquid with a 3D Dirac-cone-like
spectrum is found. Topological insulating states with TR
symmetry also may be elicited on the diamond lattice. A
generalization to even higher rank ⌫ matrices is also discussed, wherein a topologically nontrivial spin-liquid state
appears along which manifests a suitable defined “timereversal-like” symmetry.
II. 2D CHIRAL SPIN LIQUID WITH ⌫ MATRICES
A. Remarks on the nature of Kitaev’s model
Before elucidating the details of our model, it will prove
useful to reflect on why the Kitaev model is equivalent to
noninteracting fermions in a static Z2 gauge field. The ⌫
matrices obeying the Clifford algebra 兵⌫a , ⌫b其 = 2␦ab may be
represented in terms of 2n Majorana fermions ␩ and 兵␰a其,
where a = 1 , . . . , 2n − 1. Then define the 2n − 1 ⌫ matrices
⌫a = i␩␰a. The product,
⌳ = ⌫1⌫2 ¯ ⌫2n−1 = i␩␰1␰2 ¯ ␰2n−1 ,
共1兲
then commutes with each of the ⌫a and furthermore satisfies
⌳2 = 共−1兲n−1; hence we can choose ⌳ = ␭ = i共n−1兲, which is to
say that all states in each local Hilbert space satisfy ⌳兩⌿典
= ␭兩⌿典. Now consider a lattice L of coordination number z
= 2n − 1 in which the link lattice is itself z partite. That is to
say that each link can be assigned one of z colors, and no two
links of the same color terminate in a common site. On each
link 具ij典 of the lattice, then, we can write interaction terms
⌫ai ⌫aj = −uij · i␩i␩ j, where i and j are the termini of the link
and uij = i␰ai ␰aj , where a is the color label of the link. Note that
u2ij = 1, furthermore 关uij , ukl兴 = 0 for all 具ij典 and 具kl典, and in
particular even if these links share a common terminus.
Thus, the set 兵uij其 defines a configuration for a classical Z2
gauge field. Note that uij = −u ji. The model is then
134427-1
©2009 The American Physical Society
PHYSICAL REVIEW B 79, 134427 共2009兲
WU, AROVAS, AND HUNG
H = − 兺 Jij⌫ai ⌫aj ,
共2兲
具ij典
= 兺 Jijuij · i␩i␩ j ,
共3兲
具ij典
where the “color” index a is associated with each particular
link. The honeycomb lattice satisfies the above list of desiderata, with z = 3. Accordingly, the Kitaev model has interactions ␴xi ␴xj on 0° links, ␴iy␴yj on 120° links, and ␴zi ␴zj on 240°
links.
For n = 2, the ⌫ matrices are of course the Pauli matrices,
and the set 兵1 , ␴x , ␴y , ␴z其 forms a basis for rank-2 Hermitian
matrices. For n = 3, in addition to the five ⌫a matrices, we can
define 共 2n−1
2 兲 = 10 additional matrices,
⌫ab ⬅ − 2i 关⌫a,⌫b兴 = − i␰a␰b ,
共4兲
resulting in a total of 1 + 5 + 10= 16, which of course forms a
basis for rank-4 Hermitian matrices. For n = 4, we add the
abc
共 2n−1
⬅ ␩␰a␰b␰c to the ⌫ab 共21 total兲, ⌫a 共7
3 兲 = 36 matrices ⌫
total兲, and 1, resulting in the 64 element basis of rank-8
Hermitian matrices, etc.
For each closed loop C on the lattice, one can define a Z2
flux, which is a product,
FC =
兿
具ij典苸C
共5兲
uij ,
Fi1i2¯iN = − ⌫iaNa1⌫ia1a2 ¯ ⌫iaN−1aN .
2
A second point regarding the fluxes FC is that if n共C兲 is the
number of links contained in C, then traversing C clockwise
rather than counterclockwise results in a flux of FC−1 = 共
−1兲n共C兲FC. Thus, the Z2 flux reverses sign for odd length
loops if the loop is traversed in the opposite direction. For
loops of even length, the flux is invariant under the direction
of traversal.
B. Model definition
where the product is taken counterclockwise over all links in
C. If C is a self-avoiding loop of N sites 共and hence N links兲,
then
1
FIG. 1. 共Color online兲 The decorated square lattice 共right兲 with
linked diagonal bonds for the Hamiltonian of Eq. 共8兲. Each unit cell
共left兲 consists four sites 共A, B, C, and D兲, ten links, and six
plaquettes including two squares and four triangles. The bonds are
classified into five types as marked with labels a which run from 1
to 5 and 1⬘ to 5⬘.
N
共6兲
These fluxes are all conserved in that they commute with
each other and with the Hamiltonian. Under a local gauge
transformation, the Majoranas transform as ␩i → −␩i, which
is equivalent to taking uij → −uij for each link emanating
from site i. The gauge-invariant content of the theory thus
consists of the couplings 兵Jij其 and the fluxes 兵FP其 associated
with the elementary plaquettes P. For a given set of fluxes,
there are many 共gauge-equivalent兲 choices for the uij. By
choosing a particular such gauge configuration, the Hamiltonian of Eq. 共3兲 may be diagonalized in a particular sector
specified by the Z2 fluxes.
It is worth emphasizing the following features of the FC as
defined in Eq. 共5兲. First, a retraced link contributes a factor of
−1 to the flux because uij · u ji = −1. This has consequences for
combining paths. If two loops C and C⬘ share k links in
common, then FCC⬘ = 共−1兲kFCFC⬘, where CC⬘ is the concatenation of C and C⬘, with the shared links removed. Consider, for
example, the triangles ABD and BCD in the left panel of Fig.
1. The combination of these triangles yields the square
ABCD. One then has
FABD · FBCD = uABuBDuDA · uBCuCDuDB ,
=uABuBCuCDuDA共− 1兲 = − FABCD
共7兲
since the single link BD is held in common but is traversed
in opposite directions.
Our generalization of Kitaev’s model involves an n = 3
system on the decorated square lattice depicted in Fig. 1. The
link lattice is five-partite: each lattice node lies at the confluence of five differently colored links. We will take the side
length of each square to be a. The lattice constant for the
underlying Bravais lattice, i.e., the distance between two
closest A sublattice sites, is then a⬘ = 2a. The model is that of
Eq. 共2兲,
H = − J1
兺
⌫1i ⌫1j − J2
兺
⌫4i ⌫4j − J5
1 links
− J4
4 links
兺
⌫2i ⌫2j − J3
兺
⌫5i ⌫5j .
2 links
兺
⌫3i ⌫3j
3 links
共8兲
5 links
While the structural unit cell shown in the left panel of Fig.
1 contains distinct bonds, labeled with a among 兵1 , . . . , 5其 or
兵1⬘ , . . . , 5⬘其, and is solvable with ten distinct couplings Ja, we
shall assume J1 = J1⬘, etc.
The 4 ⫻ 4 ⌫ matrices may be explicitly taken as
⌫1 = i
冉 冊
0 −I
I
0
,
⌫2,3,4 =
冉
␴ជ
0
ជ
0 −␴
冊
,
⌫5 =
冉 冊
0 I
I 0
,
共9兲
ជ are the 2 ⫻ 2 unit and Pauli matrices, respecwhere I and ␴
tively. These five ⌫ matrices are in fact spin-quadrupole operators for the spin- 23 system.35 The other ten ⌫ matrices,
defined above in Eq. 共4兲, contains both spin and spin octupole operators.36,37 For later convenience, the ten ⌫ab matrices are also written explicitly here as
134427-2
⌫-MATRIX GENERALIZATION OF THE KITAEV MODEL
⌫12,13,14 =
⌫34,42,23 =
冉 冊
冉 冊
ជ
0 ␴
␴ជ 0
␴ជ 0
ជ
0 ␴
⌫25,35,45 = i
,
⌫51 =
,
冉
ជ
0 −␴
␴ជ
冉 冊
I
0
0 −I
0
冊
PHYSICAL REVIEW B 79, 134427 共2009兲
冦 冧 冦 冧
␩
␩
1,3
C ␰
C = ␰1,3
− ␰2,4,5
␰2,4,5
共10兲
.
From the six Majorana fermions, we may fashion three
Dirac fermions, viz.,
c04 = 21 共␩ + i␰4兲,
共11兲
c15 = 21 共␰1 − i␰5兲,
共12兲
c23 = 21 共␰2 − i␰3兲.
共13兲
With these definitions, the ⌫a operators are even under TR
while the ⌫ab operators are odd. Note that R␩R = ␩. Thus, the
effect of the time-reversal operation on the fluxes FC is the
same as that of reversing the direction in which the loop is
traversed, i.e., RFCR = 共−1兲n共C兲FC, where n共C兲 is the number
of links in C.
D. Projection onto the physical sector
As we have seen, in terms of the Majorana fermions, the
⌫ matrices are represented as
The three diagonal ⌫ matrices can be represented as
⌫ =
⌫ a = i ␩ ␰ a,
1,
共14兲
†
⌫15 = 2c15
c15 − 1,
共15兲
†
⌫23 = 2c23
c23 − 1.
共16兲
4
†
c04 −
2c04
The basis vectors of the physical space in terms of the Sz
eigenstates are as follows:
† † †
兩+ 23 典 = c23
c15c04兩⍀典,
共17兲
†
兩+ 21 典 = ic15
兩⍀典,
共18兲
†
兩− 21 典 = c23
兩⍀典,
共19兲
†
兩− 23 典 = − ic04
兩⍀典,
共20兲
where 兩⍀典 is the reference vacuum state. In other words, the
physical states have odd fermion occupation number n
†
†
†
c04 + c15
c15 + c23
c23. The R matrix defined in Sec. II C can
= c04
†
†
兲共c23 − c23
兲.
be represented as R = −i共c15 + c15
For each of the six plaquettes per unit cell, one defines a
Z2 flux as in Eq. 共5兲. For example, for the ABD⬘ triangle
shown in Fig. 1 enclosed by the a = 2, a = 3, and a = 5 bonds,
we have
5
2 2
FABD⬘ = 共i␰A3 ␰B3 兲共i␰B5 ␰D
兲共i␰D
␰A兲 = − ⌫A23⌫B35⌫D52 .
⬘
⬘
⬘
共21兲
These fluxes are all conserved in that they commute with
each other and with the Hamiltonian. As pointed in Refs. 7
and 8, the flux configuration for the ground state on the triangular lattice is odd under TR symmetry; thus the ground
state breaks TR symmetry and is at least doubly degenerate.
共22兲
⌫ab = − i␰a␰b .
共23兲
We further demand that any physical state 兩⌿典 must satisfy
⌳i兩⌿典 = −兩⌿典, where ⌳i = ⌫1i ⌫2i ⌫3i ⌫4i ⌫5i on each lattice site i
关see Eq. 共1兲兴. That is, each state in the eigenspectrum is also
1
an eigenstate of the projector P = 兿i 2 共1 − ⌳i兲, so the local
Hilbert space at each site is four dimensional rather than
eight dimensional. Since 关H , P兴 = 0, the two operators can be
simultaneously diagonalized. For any eigenstate 兩⌿典 of H in
the extended Hilbert space 共with local dimension eight兲, we
have that P兩⌿典 is also an eigenstate of H, and with the same
eigenvalue. Thus, we are free to solve the problem in the
extended Hilbert space, paying no heed to the local constraints, and subsequently apply the projector P to each
eigenstate of H if we desire the actual wave functions or
correlation functions. Note also that while the projector P
does not commute with the link Z2 gauge fields uij, nevertheless 关P , FC兴 = 0 for any closed loop C, so the projector does
commute with all the Z2 fluxes, which, aside from the couplings, constitute the gauge-invariant content of the Hamiltonian.
E. General Majorana Hamiltonian
The general noninteracting lattice Majorana Hamiltonian
is written as H = 21 兺i,jHij␩i␩ j, with H = H† = −Hⴱ and
兵␩i , ␩ j其 = 2␦ij. Let R denote a Bravais lattice site, i.e., a site
on the A sublattice, and the index l 苸 兵1 , . . . , 2r其 labels a
basis element; our model has r = 2. The Majorana fermions
satisfy 兵␩l共R兲 , ␩l⬘共R⬘兲其 = 2␦RR⬘␦ll⬘, and Fourier transforming
to ␩l共k兲 = N−1/2兺R␩l共R兲e−ik·共R+dl兲, where N is the number of
unit cells and dl is the location in the unit cell of the lth basis
site, we arrive at the Hamiltonian,
H=
1
兺 Hll⬘共k兲␩l共− k兲␩l⬘共k兲,
2 k
共24兲
where Hll⬘共k兲 = 兺RHll⬘共R − R⬘兲eik·共R⬘−R+dl⬘−dl兲 satisfies
C. Time reversal
The TR transformation is defined as the product T = RC,
where C is the complex-conjugation operator and R is the
charge conjugation matrix, satisfying R2 = −1 and R† = R−1
= Rt = −R. Explicitly, we can take R = ⌫1 ⌫3 = ␰1␰3. Note that
R⌫aR = −共⌫a兲t and R⌫abR = 共⌫ab兲t. Acting on the Majoranas,
the complex-conjugation operator C is defined so that
ⴱ
ⴱ
Hll⬘共k兲 = Hl⬘l共k兲 = − Hl⬘l共− k兲 = − Hll⬘共− k兲,
共25兲
and ␩l共−k兲 = ␩†l 共k兲. The eigenvalues of H共k兲 occur in pairs
兵E j共k兲 , −E j共−k兲其, where j 苸 兵1 , . . . , r其 where 2r is the number
of basis elements. Written in terms of Dirac fermions, the
diagonalized Hamiltonian takes the form
134427-3
PHYSICAL REVIEW B 79, 134427 共2009兲
WU, AROVAS, AND HUNG
r
H = 兺 兺 共2␥†j,k␥ j,k − 1兲E j共k兲,
共26兲
k j=1
and therefore the ground-state energy is
r
E0 = − 兺 兺 兩E j共k兲兩
共27兲
k j=1
and the excitation energies are ␻ j共k兲 = 2兩E j共k兲兩.
F. Bulk and edge spectra
Following the procedure outlined by Kitaev,7 we represent
the Hamiltonian H in the extended Hilbert space in terms of
free Majorana fermions hopping in the presence of a static Z2
gauge field, as in Eq. 共3兲. The spectra can be solved in each
gauge sector with a specified distribution of the Z2 phases
uij = ⫾ 1.
In the extended Hilbert space, all the link phases uij mutually commute with each other and with the Hamiltonian.
The product of the link phases around a given plaquette
gives the Z2 flux associated with that plaquette, as in Eq. 共5兲.
It is the Z2 fluxes of all the triangular and square plaquettes
which define the gauge-invariant content of the model.
The structural unit cell, as shown in the left panel of Fig.
1, consists of four sites, ten links, and six plaquettes. Suppose that the arrangement of fluxes FC has the same period as
this structural cell. What of the link phases uij? It is easy to
see that if the total Z2 flux of the structural unit cell is +1;
i.e., if the product of the FC over the six plaquettes in the
structural unit cell is +1, then the uij may be chosen so as to
be periodic in this unit cell. In such a case the magnetic unit
cell coincides with the structural unit cell.38 If, however, the
net Z2 flux per structural unit cell is −1, then the smallest
magnetic unit cell 共i.e., periodic arrangement of the link
phases uij兲 necessarily comprises two structural cells. Here,
we assume that the net flux per structural cell is +1, so the
magnetic and structural unit cells coincide. Thus, there are
five 共and not six兲 Z2 degrees of freedom per unit cell, which,
using the labels of Fig. 2, can be taken to be the link phases,
uAD ⬅ ␴1,
uAC⬘ = ␴2,
uAD⬘ = ␴4,
uAB⬘ = ␴3 ,
uBD⬘ = ␴5 .
共29兲
共30兲
The fluxes of the triangular and square plaquettes are then
given by
FABCD = ␴1
FBC⵮D⬘ = ␴5 ,
FAD⬘C⬙B⬘ = ␴3␴4
FC⬘DA = − ␴1␴2 ,
FABD⬘ = − ␴4␴5
FAB⬘C⬘ = ␴2␴3 .
uij phases consistent with zero total flux per cell is therefore
identical or gauge equivalent to one of these 32 configurations.
We set the coupling on the horizontal links to JH, on the
vertical links to JV, and on the diagonal links to JD. The
independent nonzero elements of Hll⬘共k兲 are then
共28兲
The remaining five values of uij can then be fixed, and we
take
uAB = uCB = uC⬙B⬘ = uCD = uC⬙D⬘ = 1.
FIG. 2. 共Color online兲 When the magnetic and structural unit
cells coincide, there are five Z2 gauge degrees of freedom, which
we take to be ␴1–5 as shown, with uAD = ␴1, etc. The other uij are
positive for links where the black arrow points from i to j.
HAB共k兲 = iJH共ei␪1/2 + ␴3e−i␪1/2兲,
共31兲
HAC共k兲 = iJD␴2e−i共␪1+␪2兲/2 ,
共32兲
HAD共k兲 = iJV共␴4ei␪2/2 + ␴1e−i␪2/2兲,
共33兲
HBC共k兲 = − 2iJV cos共␪2/2兲,
共34兲
HBD共k兲 = iJV␴5ei共␪1−␪2兲/2 ,
共35兲
HCD共k兲 = 2iJH cos共␪1/2兲,
共36兲
where we define the angles 共␪1 , ␪2兲 = 共kxa⬘ , kya⬘兲.
We have found that the ground-state energy is minimized
when the Z2 flux through each of the four squares in the unit
cell is F䊐 = −1, and the triangles all are the same value, i.e.,
F䉭 = ⫾ 1. Under time reversal F䊐 → F䊐 and F䉭 → −F䉭. Note
that the Z2 flux through either square which contains a diag2
= −1 if the triangles contain the same
onal bond is F䊐 = −F䉭
flux. The flux pattern with F䊐 = F䉭 = −1 is achieved with the
choice
共␴1, ␴2, ␴3, ␴4, ␴5兲 = 共− 1,− 1, + 1,− 1,− 1兲.
Thus, there are 25 = 32 possible distinct flux configurations
which are periodic in this unit cell. Any arrangement of the
共37兲
With this link flux assignment, the Hamiltonian matrix H共k兲
takes the form
134427-4
⌫-MATRIX GENERALIZATION OF THE KITAEV MODEL
H共k兲 =
冢
PHYSICAL REVIEW B 79, 134427 共2009兲
0
2iJH cos共␪1/2兲
− iJDe−i共␪1+␪2兲/2
− 2iJV cos共␪2/2兲
− 2iJH cos共␪1/2兲
0
− 2iJV cos共␪2/2兲
− iJDei共␪2−␪1兲/2
iJDei共␪1+␪2兲/2
2iJV cos共␪2/2兲
0
2iJH cos共␪1/2兲
2iJV cos共␪2/2兲
iJDei共␪1−␪2兲/2
− 2iJH cos共␪1/2兲
0
The eigenvalues of H共k兲 are found to be
E1,2共k兲 = − 2
冑
J2D
4
2
2
+ 共JH
+ JV2 兲f共k兲 ⫾ JD冑JH
+ JV2 g共k兲,
共39兲
E3,4共k兲 = + 2
冑
J2D
4
2
2
+ 共JH
+ JV2 兲f共k兲 ⫿ JD冑JH
+ JV2 g共k兲,
共40兲
where
2
cos2
JH
f共k兲 =
冉 冊
冉 冊
1
1
␪1 + JV2 cos2 ␪2
2
2
2
JH
+ JV2
g共k兲 = cos共 21 ␪1兲cos共 21 ␪2兲 .
,
共41兲
共42兲
In Fig. 3 we plot the total energy per site for our model for
all possible flux configurations for our model 共32 in total兲,
where we have taken JH = JV = J, which we here and henceforth assume. We explore the properties of our model as a
function of the dimensionless parameter JD / J. Since time
reversal has the effect of sending F → −F or all odd-
.
共38兲
membered loops, every state must have an even-fold degeneracy. We find that the flux configurations with the two
lowest-lying total energies are each twofold degenerate, and
the third-lowest-lying total energy flux configuration is eightfold degenerate. In Fig. 4 we show the spectra E j共k兲 for the
lowest energy flux configuration.
Our model exhibits a topological phase transition as increasing JD / J. At JD / J = 0, the dispersion is gapless, with a
Dirac cone at 共␪1 , ␪2兲 = 共␲ , ␲兲. As JD / J ⫽ 0, this spectrum
becomes massive with a gap ⌬共␲ , ␲兲 = 2JD. This is a topologically nontrivial phase, characterized by nonvanishing
Chern numbers, with concomitant gapless edge modes inside
the gap between bands 2 and 3 in a sample with an open
boundary, as depicted in Fig. 5共a兲. As JD / J increases, the
second and third bands are pushed toward each other at the
Brillouin-zone center. These bands eventually touch when
JD / J = 2冑2, forming there a new gapless Dirac cone. For
JD / J ⬎ 2冑2, the system is in a topologically trivial phase in
which the edge states no longer exhibit a spectral flow, and
(a)
FIG. 3. 共Color online兲 Total energies 共per site兲 versus ␣
= tan−1共JD / J兲 for our decorated square lattice model. Curves for all
possible flux configurations are shown. The flux configurations for
the two lowest-lying total energy states are twofold degenerate owing to time reversal, which reverses the flux in the odd-membered
loops. The third-lowest-lying total energy state has an eightfold
1
degenerate flux configuration. At ␣ = 2 ␲ the horizontal and vertical
bond strengths vanish and the system becomes a set of disconnected
dimers.
冣
(b)
(c)
FIG. 4. 共Color online兲 The band structure of Eq. 共38兲 at JD / J
⬎ 0. 共a兲 JD / J = 1 with a massive Dirac spectrum at 共␲ , ␲兲 with the
gap value 2JD; 共b兲 JD / J = 2冑2 where a gapless Dirac cone appears;
and 共c兲 JD / J = 3.5. The gap value at 共0,0兲 is 2兩JD − 2冑2J兩. A topological phase transition occurs from a topological nontrivial phase
at JD / J ⬍ 2冑2 to a trivial phase at JD / J ⬎ 2冑2.
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WU, AROVAS, AND HUNG
the ␰’s can be expressed as a product of the Z2 gauge phases
defined on different links; i.e.,
␰ai ␰aj = 兿 共− iuaij兲.
兿
具ij典
具ij典
共43兲
Similarly to the work of Ref. 8, it is natural to expect that
each topological excitation of a Z2 vortex traps an unpaired
Majorana mode. These vortex excitations will exhibit nonAbelian statistics. We are now performing numerical calculations to confirm this prediction.
III. EXTENSION TO 8 Ã 8 ⌫ MATRICES
The above procedure is readily extended to even higher
rank Clifford algebras, such as the n = 4 case with eight Majorana fermions and seven anticommuting 8 ⫻ 8 ⌫ matrices.
We choose the Majoranas to be 兵␩r , ␰a其 where r = 1 , 2 , 3 and
a = 1 , 2 , 3 , 4 , 5. We write ⌫a = i␩1␰a as before, and we define
⌫6a = i␩2␰a. The Hamiltonian is
(a)
5
H = − 兺 Ja
a=1
6a
共⌫ai ⌫aj − ⌫6a
兺
i ⌫ j 兲,
a links
= 兺 Jijuij · i共␩1i ␩1j − ␩2i ␩2j 兲.
具ij典
(b)
FIG. 5. 共Color online兲 共a兲 The gapless edge modes of ␩ fermions across the band gap appear in open boundary samples in the
topologically nontrivial phase at JD / J ⬍ 2冑2. 共b兲 The edge modes
become nontopological at JD / J ⬎ 2冑2.
they remain confined within the band gap, as shown in Fig.
5共b兲. In this case, the edge states are no longer topologically
protected, and they are sensitive to local perturbations, and
indeed can be eliminated by sufficiently strong such perturbations. The gap located at 共␪1 , ␪2兲 = 共0 , 0兲 is found to be
⌬共0 , 0兲 = 2兩JD − 2冑2J兩.
G. Spin correlation functions
As is the case with Kitaev’s model,18 the spin correlation
function in our ground state is short ranged due to the conserved flux of each plaquette. We take the ground-state wave
function 兩⌿G典 = P兩⌿典 where 兩⌿典 is the unprojected state, with
each uaij taking a value ⫾1. Then 具⌿G兩⌫ai ⌫bj 兩⌿G典
= 具⌿兩P␩i␩ j␰ai ␰bj 兩⌿典. Unless sites i and j are linked, and unless a = b appropriate to the bond 具ij典, it is always possible to
find a loop whose flux is flipped by ␰ai or ␰bj . In that case,
since ␩i␩ j and P do not change the loop flux, the above
expectation value vanishes. A similar reasoning shows that
any two-point correlation of the type of 具⌿G兩⌫ai ⌫cd
j 兩⌿G典 or
cd
具⌿G兩⌫ab
i ⌫ j 兩⌿G典 vanishes as well.
Generally speaking, a nonlocal spin correlation function is
finite only if, in the Majorana representation of its operators,
共44兲
共45兲
Physical states are projected, at each site, onto eigenstates of
the operator ⌳ = i␩1␩2␩3␰1␰2␰3␰4␰5 with eigenvalue +i.
We obtain the same behavior of ␩1,2 modes as before and
a flat band of zero energy mode of ␩3. Within a fixed gauge
choice of uij, H is invariant under a suitably defined TR-like
transformation of as T̃␩1T̃−1 = ␩2 and T̃␩2T̃−1 = −␩1, which is
not related to the physical TR transformation. In the topologically nontrivial phase, the ␩1,2 edge modes have opposite
chirality, which is robust in the absence of a perturbation of
the type ⌫6 ⬅ i␩1␩2, which breaks the symmetry under T̃.
This behavior is similar to that of the 3He-B phase in two
dimensions, which has been recently identified as a TRinvariant topological superconductor.10
IV. ⌫-MATRIX MODELS IN THE 3D DIAMOND LATTICE
The ⌫-matrix analogy of the Kitaev model is also extendable to the three-dimensional 共3D兲 systems. We will consider
here a diamond lattice, which is fourfold coordinated, as illustrated in Fig. 6. In the following, we will first describe a
model with a 3D Dirac cone and broken TR symmetry, and
then another one exhibiting 3D topological insulating states
with TR symmetry maintained.
A. 3D spin liquid with Dirac cone excitations
We first consider an n = 3 model 共4 ⫻ 4 ⌫ matrices兲 in the
diamond lattice with explicit time-reversal symmetry breaking. Recall that the diamond lattice is bipartite, consisting of
two fcc sublattices. Each A sublattice site is located in the
center of a tetrahedron of B sites, and vice versa, as shown in
Fig. 6. We define the unit vectors,
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PHYSICAL REVIEW B 79, 134427 共2009兲
4
a
H0 = − 兺 Ja 兺 ⌫Ra ˜⌫R+ê
,
a
R
a=1
共50兲
4
= 兺 兺 JauRa · i␩R˜␩R+êa .
共51兲
R a=1
We use the tilde to distinguish operators which reside on the
B sublattice from those which reside on the A sublattice.
Here we have defined
a
uRa = i␰Ra ˜␰R+ê
FIG. 6. 共Color online兲 The 3D diamond lattice with the nearest
separation a and two sublattices A 共filled circles兲 and B 共hollow
circles兲. The unit vectors ê1,2,3,4 are defined from each A site to its
four B neighbors 共see text兲.
ê1 =
ê2 =
1
冑3 共−
1
冑3 共1,−
1,1,1兲
1,1兲
ê3 =
ê4 =
1
冑3 共1,1,−
1
冑3 共−
1兲,
1,− 1,− 1兲,
共46兲
which point from a given A sublattice site to its four B sublattice neighbors. The basis vectors for the underlying fcc
Bravais lattice, which can be taken to be the A sublattice
itself, are then
冑3 共0,1,1兲,
a2 = ê2 − ê4 =
冑3 共1,0,1兲,
a3 = ê3 − ê4 =
2
冑3 共1,1,0兲.
ab
b
⌫R+ê
⌫Ra ˜⌫R+ê
a
b1 =
冑3
2 ␲共−
b2 =
冑3
2 ␲共1,−
b3 =
冑3
2 ␲共1,1,−
a−êb
b
= − uRa uR+ê
a−êb
· i␩R␩R+êa−êb
共53兲
and
2
共47兲
A general A sublattice site lies at R = n1a1 + n2a2 + n3a3 The
two-element diamond unit cell may be taken to consist of the
A site at R and the B site at R + ê4. It is also useful to define
the null vector a4 ⬅ ê4 − ê4 = 0.
The Brillouin zone of the fcc-Bravais lattice is a dodecahedron. The elementary reciprocal lattice vectors, which
form a basis for a bcc lattice, are
˜⌫a ⌫ab˜⌫b = − ua ub · i˜␩
␩R+êb .
R+êa˜
R R
R+ê R R+ê
a
b
共54兲
These operators both break time-reversal symmetry, owing to
the presence of the ⌫ab and ˜⌫ab factors. However, owing to
the commuting nature of the Z2 link variables uRa , we may
add terms such as these to H0 and still preserve the key
feature that the Hamiltonian describes Majorana fermions ␩
and ˜␩ hopping in the presence of a static Z2 gauge field.
To recover the point group symmetry of the underlying
lattice, we sum over contributions on each loop 具abc典. We
define the operators
1,1,1兲,
ab
b
⌫R+ê
VRab = ⌫Ra ˜⌫R+ê
a
a−êb
,
共55兲
1,1兲,
a
b
⌫Rab˜⌫R+ê
.
ṼRab = ˜⌫R+ê
1兲,
共48兲
and satisfy ai · b j = 2␲␦ij, where i , j 苸 兵1 , 2 , 3其. Any vector in
k space may be decomposed in to components in this basis,
viz.,
k=
as the Z2 gauge field on the link between R and R + êa.
Within the diamond lattice, one can identify four classes
of hexagonal loops. Starting at any A sublattice site, we can
move in a cyclical six-site path by traversing consecutively
the displacement vectors 兵êa , −êb , êc , −êa , êb , −êc其. We label
this hexagon by the indices 具abc典. Without loss of generality
we can assume a ⬍ b ⬍ c since any permutation of these indices results in the same loop, traversed in either the same or
the opposite sense, depending on whether the permutation is
even or odd, respectively. According to the theorem of
Lieb,39 the gauge flux in the ground state is −1 in each such
loop; thus we can set uRa = 1 on each link, for each a
苸 兵1 , 2 , 3 , 4其, because a circuit around a six-site loop involves three AB links on which the Z2 gauge field is uij =
+ 1 and three BA links on which uij = −1. Consider now
2
a1 = ê1 − ê4 =
共52兲
a
␪1
␪2
␪3
b1 +
b2 +
b3 .
2␲
2␲
2␲
We then have k · aa = ␪a, with ␪4 = 0.
We begin with the Heisenberg-type Hamiltonian,
a
共56兲
The time-reversal violating term in our Hamiltonian is then
written as
H1 = 兺
R
共49兲
b
4
共habVRab + h̃abṼRab兲.
兺
a⬍b
共57兲
We reiterate that this model is also exactly solvable due to
the commutativity of the link fluxes uRa , and we assume uRa
= 1 for all a and R.
Transforming to k space, we have the Hamiltonian matrix,
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PHYSICAL REVIEW B 79, 134427 共2009兲
WU, AROVAS, AND HUNG
冉
Hll⬘共k兲 =
␻共k兲
⌬共k兲
ⴱ
˜ 共k兲
⌬ 共k兲 − ␻
冊
共58兲
,
where, after performing a unitary transformation to remove a
phase 21 e⫿i共␪1+␪2+␪3兲/4 from the diagonal terms, we have
4
␻共k兲 =
兺 hab sin共␪a − ␪b兲,
共59兲
a⬍b
4
˜ 共k兲 =
␻
兺 h̃ab sin共␪a − ␪b兲,
共60兲
a⬍b
FIG. 7. 共Color online兲 A noncoplanar hexagonal ring within the
diamond lattice.
4
⌬共k兲 = i 兺 Jaei␪a .
共61兲
a=1
The eigenvalues are E⫾共k兲 = ⫾ 冑␻2共k兲 + 兩⌬共k兲兩2. One finds
that E⫾共k兲 both vanish when k lies at one of the three inequivalent X points, which lie at the centers of the square
faces of the first Brillouin zone, at locations 21 共b2 + b3兲,
1
1
2 共b1 + b3兲, and 2 共b1 + b2兲. Expanding about the last of these,
we write ␪1 = ␲ + ␺1, ␪2 = ␲ + ␺2, and ␪3 = ␺3; and assuming
Ja = J and h̃ab = hab, we find to lowest order in ␺1,2,3 that
qy =
qz =
冑3
4
␺1 + ␺2 + ␺3兲,
共␺1 − ␺2 + ␺3兲,
冑3
4 共␺1
+ ␺2 − ␺3兲.
R+ê4
these individual Hamiltonians to form
a matrix H共k兲 of the form
5
H = H␩ − H␰
共66兲
B. Time-reversal symmetry properties
共63兲
˜ 共k兲 can be written as hk · ĝ⬜, where
Thus, the term ␻共k兲 = ␻
ĝ⬜ is a unit vector lying in the 共x , y兲 plane, and h is some
combination of the hab. The spectrum is therefore that of a
deformed Dirac cone, linear in the two directions ĝ⬜ and ẑ,
and quadratic in the third. These deformed Dirac cones can
be made gapped by introducing anisotropy in the J terms,
say, J4 ⫽ J1,2,3 ⬅ J. The system may then become a 3D topological insulator with TR symmetry breaking by developing
chiral surface states.
At this point we must ask whether the ␰5 are Majoranas.
Indeed, for the Hamiltonian H = H0 + H1, the ␰5 Majoranas
form a flat band at zero energy. To provide a dispersion for
5
the ␰5 branch, we define a new Hamiltonian H␰ which we
form from H␩ with the replacements ⌫a → ⌫5a. Equivalently,
we can skip to the final form of H in terms of ␩R and ˜␩R+ê4
and replace them with ␰5 and ˜␰5 , respectively. Combining
R
= ␻共k兲␥4 − Re ⌬共k兲␥14 + Im ⌬共k兲␥45 ,
共65兲
共62兲
If we write k = 21 共b1 + b2兲 + q, then we have
冑3
4 共−
冣
共64兲
+ 共h13 + h23 + h34兲␺3 ,
qx =
冢
where the row and column indices range over ␩, ␰ , ˜␩, and
˜␰5, consecutively. The ␥ matrices here are used as a basis for
4 ⫻ 4 Hermitian matrices, which are not to be confused with
the spin and spin-multipole operators denoted as ⌫ matrices.
Nevertheless we still choose ␥a and ␥ab taking the same values as the matrix forms of ⌫a and ⌫ab defined in Eqs. 共9兲 and
共10兲.
Again the spectra of Eq. 共66兲 exhibit gapless Dirac cones
at the three X points. This gapless excitation is robust if there
are no mixing terms between ␩ and ␰5 fermions, which
would appear as terms involving 兵␥2 , ␥3 , ␥24 , ␥34其, corresponding to same sublattice hopping, or 兵␥12 , ␥13 , ␥25 , ␥35其,
corresponding to alternate sublattice hopping. The introduction of such couplings can produce a gap in the spectrum and
give rise to spin-liquid states 共Fig. 7兲.
␻共k兲 = 共h12 − h13 − h14兲␺1 − 共h12 − h23 + h24兲␺2
⌬共k兲 = J共␺1 + ␺2 − ␺3兲.
H共k兲 =
0
0
␻共k兲
⌬共k兲
0
0
− ␻共k兲
− ⌬共k兲
ⴱ
0
0
− ␻共k兲
⌬ 共k兲
0
0
− ⌬ⴱ共k兲
␻共k兲
results in
5
The Hamiltonian matrix H共k兲 is Hermitian and can therefore be expanded as
5
H共k兲 = 兺 ␭a共k兲␥ +
a
a=1
5
兺 ␭ab共k兲␥ab ,
共67兲
a⬍b
where the couplings ␭a and ␭ab are real functions of their
arguments. Due to the conditions of Eq. 共25兲, we see that
these couplings come in two classes. The coefficients of
purely real, symmetric ␥ matrices must satisfy ␭共k兲 = −␭共
−k兲; we call this class odd or −. The coefficients of the
purely imaginary, antisymmetric ␥ matrices must satisfy
␭共k兲 = ␭共−k兲; we call this class even or +. Thus, we have
class − :I,␭2,␭4,␭5,␭12,␭14,␭23,␭34,␭15,␭35 ,
class + :␭1,␭3,␭13,␭24,␭25,␭45 .
共68兲
The periodicity under translations k → k + G through a reciprocal lattice vector then requires ␭共G / 2兲 = ␭共−G / 2兲 = 0 for
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PHYSICAL REVIEW B 79, 134427 共2009兲
TABLE I. Symmetry properties of the ␥ matrices.
␥
I
␥1
␥2
␥3
␥4
␥5
␥12
␥13
Class
T
P
PT
␥
Class
T
P
PT
−
+
−
+
−
−
−
+
+
−
−
−
−
+
−
−
+
+
+
+
−
−
+
+
+
−
−
−
+
−
−
−
␥14
␥15
␥23
␥24
␥25
␥34
␥35
␥45
−
−
−
+
+
−
−
+
−
+
−
−
+
−
+
+
−
−
+
−
−
−
−
+
+
−
−
+
−
+
−
+
P − even:I, ␥1, ␥2, ␥3, ␥12, ␥13, ␥23, ␥45 ,
FIG. 8. 共Color online兲 First Brillouin zone for the fcc
structure.
the odd class. For three-dimensional systems, there are eight
wave vectors within the first Brillouin zone which satisfy this
condition, i.e., k = 21 共n1b1 + n2b2 + n3b3兲 for ni = 0 or 1, which
are identified as the zone center ⌫, the four inequivalent L
points, and the three inequivalent X points 共see Fig. 8兲.
If we consider 共␩ , ␰5兲 as a Kramers doublet of pseudospin
up and down, we can define a TR-like antiunitary transformation T as
T = i␥24C,
T␰5T−1 = − ␩ .
␭2 = ␭12 = ␭24 = ␭25 = ␭34 = 0
T − even:I, ␥5, ␥15, ␥25, ␥35, ␥45 ,
共forbidden by P symmetry兲.
Note that the product PT does not reverse k and is given
by
PT = ␥25CI.
共71兲
C. 3D topological states with TR symmetry
Recently, topological insulating states in three dimensions
have attracted a great deal of attention.40–46 Below we will
show by adding the hybridization between ␩ and ␰5 fermions
that we can arrive at the gapped spin-liquid states which can
be topologically nontrivial.
We begin with the additional hybridization term,
共72兲
⌬H = 兺
R
再
4
m共⌫R5
+
4
b5
ab
⌫R+ê
兺 gab⌫Ra ˜⌫R+ê
−ê
a
冎
a
a⫽b
b5
a
⌫Rab˜⌫R+ê
兺 g̃ab˜⌫R+ê
a⫽b
b
a
b
共77兲
.
Note that
b5
ab
⌫Ra ˜⌫R+ê
⌫R+ê
a
共73兲
where I is the lattice inversion operator which inverts the
coordinates relative to the position r = 21 ê4. Under P we then
have the classification
5
+ ˜⌫R+ê
兲+
4
共forbidden by T symmetry兲.
We may also define a parity operator P as
P = ␥45I,
共76兲
A summary of the symmetry properties of the different ␥
matrices is provided in Table I.
Since T also reverses the direction of k, sending k → −k,
time-reversal symmetry requires that the coefficients of the
T-even ␥ matrices satisfy ␭共k兲 = ␭共−k兲, while the coefficients
of the T-odd ␥ matrices must satisfy ␭共k兲 = −␭共−k兲. Taking
into account the division into even and odd classes, we find
that time-reversal symmetry, i.e., THT−1 = H, requires the
vanishing of the following coefficients:
␭1 = ␭3 = ␭13 = ␭15 = ␭24 = ␭35 = 0
共75兲
共70兲
The ␥ matrices then divide into two classes under time reversal: even 共T␥T−1 = ␥兲 or odd 共T␥T−1 = −␥兲. We find
T − odd:␥1, ␥2, ␥3, ␥4, ␥12, ␥13, ␥14, ␥23, ␥24, ␥34 .
共74兲
Parity also acts on crystal momentum, sending k → −k. As a
result,
共69兲
where C is the complex-conjugation operation. Under this
operation, we have
T␩T−1 = ␰5,
P − odd:␥4, ␥5, ␥14, ␥15, ␥24, ␥25, ␥34, ␥35 .
a−êb
b
= uRa uR+ê
a−êb
5
· i␩R␰R+ê
a−êb
5
˜⌫a ⌫ab˜⌫b5 = ua ub · i␩
R+êa␰R+ê .
R R
R+ê R R+ê
a
b
b
,
共78兲
共79兲
Assuming once again that uRa = 1 for all R and a, and further
taking g̃ab = gba, this leads to the matrix Hamiltonian
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WU, AROVAS, AND HUNG
H共k兲 =
冢
0
␻共k兲
␤共k兲
⌬共k兲
ⴱ
0
− ⌬共k兲
␤ 共k兲 − ␻共k兲
ⴱ
0
− ␻共k兲 ␤共k兲
⌬ 共k兲
ⴱ
0
− ⌬ 共k兲 ␤ⴱ共k兲 ␻共k兲
冣
= ␻共k兲␥4 + Re ⌬共k兲␥14 − Im ⌬共k兲␥45 + Re ␤共k兲␥34
+ Im ␤共k兲␥24 ,
共80兲
where
␤共k兲 = im + i 兺 gabei共␪a−␪b兲 .
共81兲
a,b
The spectra for each momentum k is doubly degenerate, and
the two degenerate energy levels are found to be
E⫾共k兲 = ⫾ 冑␻2共k兲 + 兩⌬共k兲兩2 + 兩␤共k兲兩2 .
共82兲
In the presence of the T symmetry, Im ␤共k兲 = 0. This can be
achieved if we take m = 0 and the coupling matrix g to be
antisymmetric, i.e., gab = −gba. The system still possesses the
additional parity symmetry of P = ␥45I.
As a specific example, consider the case
h12 = h13 = h23 ⬅ h,
h14 = h24 = h34 ⬅ 0,
共83兲
and
g12 = g13 = g23 ⬅ 0,
g14 = g24 = g34 ⬅ g,
共84兲
where the elements of gab below the diagonal follow from
antisymmetry. Then
system remains gapless at three X points because 兩Im ⌬共k兲兩
= 0. By tuning J4 ⫽ J1,2,3 = J, the system can be made gapped,
where the gap at the X points is 兩Im ⌬共k兲兩 = 兩J4 − J兩. This situation is the same as the Hamiltonian of the 3D topological
insulator studied by Fu and Kane.41 Following their reasoning, whether or not the insulating states are topological can
be inferred from the parity eigenvalues of ␥45 of the occupied
states. The system is topologically nontrivial for J4 ⬎ J, in
which case it exhibit surface modes with an odd number of
Dirac cones in open surfaces.
On the other hand, if we relax the requirement of both T
and P symmetries and only keep the combined symmetry of
PT, then all five coefficients are allowed in Eq. 共80兲 and each
energy level remains doubly degenerate. However, the analysis in Ref. 41 no longer applies. We expect then a more
diverse set of topological insulators, the study of which will
be deferred to a future investigation.
V. CONCLUSION
In summary, we have generalized the Kitaev model from
the Pauli matrices to the Clifford algebra of ⌫ matrices. This
enriches the physics of topological states, including the twodimensional 共2D兲 chiral spin liquids with nontrivial topological structure, as well as that of topological spin liquids with
time-reversal-like symmetries. Possible topological insulating states on the 3D diamond lattice were also discussed.
Note Added. During the preparation of this work, we
learned of similar work on a ⌫-matrix extension of the Kitaev model on the square lattice in which a gapless algebraic
spin liquid was studied47 and also of work on threedimensional topological phases of the Kitaev model on the
diamond lattice.46
␻共k兲 = h关sin共␪1 − ␪2兲 + sin共␪1 − ␪3兲 + sin共␪2 − ␪3兲兴,
共85兲
␤共k兲 = g关sin ␪1 + sin ␪2 + sin ␪3兴.
共86兲
Note that ␻共k兲, Re ⌬共k兲, and Re ␤共k兲 all vanish at the eight
wave vectors k = 21 共n1b1 + n2b2 + n3b3兲 for ni = 0 or 1, which
includes the zone center ⌫, the four L points, and the three X
points 共which are the three Dirac points兲. If J1,2,3,4 = J, the
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