Translation-symmetry-protected topological orders in
quantum spin systems
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Kou, Su-Peng, and Xiao-Gang Wen. “Translation-SymmetryProtected Topological Orders in Quantum Spin Systems.” Phys.
Rev. B 80, no. 22 (December 2009). © 2009 The American
Physical Society
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http://dx.doi.org/10.1103/PhysRevB.80.224406
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PHYSICAL REVIEW B 80, 224406 共2009兲
Translation-symmetry-protected topological orders in quantum spin systems
Su-Peng Kou
Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China
Xiao-Gang Wen*
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
and Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2J 2Y5
共Received 10 August 2009; published 4 December 2009兲
In this paper we systematically study a simple class of translation-symmetry protected topological orders in
quantum spin systems using slave-particle approach. The lattice spin systems are translation invariant, but may
break any other symmetries. We consider topologically ordered ground states that do not spontaneously break
any symmetry. Those states can be described by Z2A or Z2B projective symmetry group. We find that the Z2A
translation symmetric topological orders can still be divided into 16 subclasses corresponding to 16 translationsymmetry protected topological orders. We introduced four Z2 topological indices ␨k = 0 , 1 at k = 共0 , 0兲, 共0 , ␲兲,
共␲ , 0兲, and 共␲ , ␲兲 to characterize those 16 topological orders on square lattice. We calculated the topological
degeneracies and crystal momenta for those 16 topological phases on even-by-even, even-by-odd, odd-byeven, and odd-by-odd lattices, which allows us to physically measure such topological orders. We predict the
appearance of gapless fermionic excitations at the quantum phase transitions between those symmetry protected topological orders. Our result can be generalized to any dimensions. We find 256 translation-symmetry
protected Z2A topological orders for a system on three-dimensional cubic lattice.
DOI: 10.1103/PhysRevB.80.224406
PACS number共s兲: 75.45.⫹j, 71.27.⫹a
I. INTRODUCTION AND MOTIVATION
For long time, people believe that Landau symmetry
breaking theory1 and the associated local order parameters2,3
describe all kinds of phases and phase transitions. However,
in last 20 years, it became more and more clear that Landau
theory cannot describe all quantum states of matter 共the
states of matter at zero temperature兲.4,5 A nontrivial example
of states of matter beyond Laudau theory is the fractional
quantum Hall 共FQH兲 states.6 FQH states do not break any
symmetry and hence cannot be described by broken symmetries. The subtle structures that distinguish different FQH
states are called topological order.4,5,7 Physically, topological
order describes the internal order 共or more precisely, the long
range entanglement兲 in a gapped quantum ground state. It
can be 共partially兲 characterized by robust ground-state
degeneracy.4,8 Recently, many different systems with topologically ordered ground states were found.8–16
Quantum spin liquid states in general contain nontrivial
topological orders. In the projective construction of spin liquid 共also known as slave-particle approach兲,17–23 there exist
many-spin liquids with low energy SU共2兲, U共1兲, or Z2 gauge
structures. Those spin liquids all have exactly the same symmetry. To distinguish those spin liquids, we note that although the spin liquids have the same symmetry, within the
projective construction, their ansatz are not directly invariant
under the translations. The ansatz are invariant under the
translations followed by different gauge transformations. So
the invariant groups of the ansatz are different. We can use
the invariant group of the ansatz to characterize the order in
the spin liquids. The invariant group of an ansatz is formed
by all the combined symmetry transformations and the gauge
transformations that leave the ansatz invariant. Such a group
is called the projective symmetry group 共PSG兲.24 Thus al1098-0121/2009/80共22兲/224406共11兲
though one cannot use symmetry and order parameter to describe different orders in the spin liquids, one can use the
PSG to characterize/distinguish the different quantum/
topological orders of spin liquid states.
The simplest kind of topological orders is the Z2 topological order where the slave-particle ansatz is invariant under a
Z2 gauge transformation. According to the PSG characterization within the projective construction, for system with only
lattice translation-symmetry, there can be two different
classes of Z2 topological orders labeled by Z2A and Z2B
共a Z2B ansatz has ␲ flux going through each plaquette兲.24 In
this paper, we will study the Z2A topological orders and ask
“are there distinct Z2A topological orders?” We find that
there are indeed distinct Z2A topological orders. They can be
labeled by four Z2 topological indices ␨k = 0 , 1 at k = 共0 , 0兲,
共0 , ␲兲, 共␲ , 0兲, and 共␲ , ␲兲. So the ␨k characterization is beyond the PSG characterization of quantum/topological order
and provides additional information for translation-symmetry
protected Z2 topological order.
II. GENERAL “MEAN-FIELD” FERMION HAMILTONIAN
OF Z2 TOPOLOGICAL ORDERS
We will use the projective construction 共the slave-particle
theory兲17,23 to systematically construct different translation
symmetric Z2 topological orders in spin-1/2 systems on
square lattice. In such a construction, we start with “meanfield” fermion Hamiltonian24
Hmean = 兺 ␺†i uij␺j + 兺 共␺†i ␩ij␺j† + H.c.兲 + 兺 ␺†i ai␺i ,
ij
ij
i
共1兲
where uij, ␩ij, and ai are 2 by 2 complex matrices. The ␩
term is included since our spin-1/2 systems in general do not
224406-1
©2009 The American Physical Society
PHYSICAL REVIEW B 80, 224406 共2009兲
SU-PENG KOU AND XIAO-GANG WEN
have any spin rotation symmetry. We like to mention that ai
are not free parameters. ai should be chosen such that
共uij,␩ij兲 † l
共uij,␩ij兲
兩␺i ␴ ␺i兩⌿mean
典 = 0,
具⌿mean
l = 1,2,3,
共2兲
where ␴l are the Pauli matrices. In this paper, we will only
consider translation invariant ansatz uij = ui+a,j+a and ␩ij
= ␩i+a,j+a. Those states are characterized by Z2A PSG and are
Z2A topological states.24
共uij,␩ij兲
Let 兩⌿mean
典 be the ground state of Hmean. Then a many共uij,␩ij兲
典
spin state can be obtained from the mean-field state 兩⌿mean
by projection
共uij,␩ij兲
共uij,␩ij兲
典 = P兩⌿mean
典,
兩⌿spin
共3兲
into the subspace with even numbers of fermion per site.
Here the projection operator is
P=兿
i
1 + 共− 1兲ni
,
2
and ni = ␺†i ␺i is fermion operator at site i.
We note that after the projection, each site can have either
no fermion or two fermions. If we associate the no fermion
state as the spin-down state and the two-fermion state as the
共uij,␩ij兲
典 can be
spin-up state, then the projected state 兩⌿spin
viewed as a quantum state for the spin system. This is how
we construct many-spin state from the mean-field Hamiltonian. For each choice of the ansatz 共uij , ␩ij , ai兲, this proce共uij,␩ij兲
典.
dure produces a physical many-spin wave function 兩⌿spin
So the ansatz 共uij , ␩ij兲 can also be viewed as a set of labels
共uij,␩ij兲
典 can be viewed as a
that label a many-spin state and 兩⌿spin
trial wave function for a spin-1/2 system, with uij and ␩ij
being variational parameters.
For the translation invariant ansatz, one can rewrite the
mean-field fermion Hamiltonian in momentum space by presenting
⌿k =
冢 冣
␺1,k
†
␺1,−k
␺2,k
†
␺2,−k
Hmean =
⌿k† ⌿k = 2,
Thus up to a constant in Hmean, we may assume M共k兲 to
satisfy
Tr M共k兲 = 0,
Thus, we may rewrite Eq. 共6兲 as
Hmean =
1
⌿k† U共k兲⌿k + 兺 ⌿k† U共k兲⌿k ,
兺
2 k=0
k⬎0
U共k兲 = M共k兲 − ⌫M T共− k兲⌫.
U共k兲 = − ⌫UT共− k兲⌫,
共8兲
冊
␴1 0
.
0 ␴1
†
⌿−k
= ⌿kT⌫.
In terms of ⌿k, Hmean can be written as
U共k兲 = U†共k兲.
Now we expand U共k兲 by 16 Hermitian matrices
M 兵␣␤其 ⬅ ␴␣ 丢 ␴␤,
␣, ␤ = 0,1,2,3,
共9兲
where ␴0 = 1. We have
U共k兲 =
兺 c兵␣␤其共k兲M 兵␣␤其 ,
兵␣,␤其
where c兵␣␤其共k兲 are real. The 16 4 ⫻ 4 matrices M 兵␣␤其 can be
divided into two classes: in one class, the matrices satisfy
共4兲
†
, ⌿−k兲 can be expressed in term of
We also note that 共⌿−k
†
共⌿k , ⌿k兲,
⌿−k = ⌫⌿kⴱ ,
共7兲
†
M共− k兲⌿−k = Tr M共k兲 − ⌿k† ⌫M T共− k兲⌫⌿k .
⌿−k
兵⌿Ik,⌿Jk⬘其 = ⌫IJ␦k+k⬘ ,
冉
Tr关M共k兲␴0 丢 ␴3兴 = 0.
Due to Eq. 共5兲,
,
Note that ⌿k satisfy the following algebra
⌫ = ␴1 丢 ␴0 =
⌿k† ␴0 丢 ␴3⌿k = 0.
Here k ⬎ 0 means that k ⫽ 0 and ky ⬎ 0 or ky = 0 and kx ⬎ 0.
Clearly U共k兲 satisfy
†
†
␺1,−k ␺2,k
␺2,−k 兲.
⌿k† = 共␺1,k
where
共6兲
where −␲ ⬍ kx, ky ⬍ +␲, and M共k兲 are 4 ⫻ 4 Hermitian matrices M共k兲 = M †共k兲. Here k = 0 means that 共kx , ky兲 = 共0 , 0兲,
共0 , ␲兲, 共␲ , 0兲, or 共␲ , ␲兲. Also kx and ky are quantized: kx
= 2L␲x ⫻ integer and ky = 2L␲y ⫻ integer, where Lx and Ly are size
of the square lattice in the x and y directions. Note that on an
even-by-even lattice 共i.e., Lx = even and Ly = even兲, 共kx , ky兲
= 共0 , 0兲, 共0 , ␲兲, 共␲ , 0兲, or 共␲ , ␲兲 all satisfy the quantization
conditions kx = 2L␲x ⫻ integer and ky = 2L␲y ⫻ integer. In this case,
兺k=0 sums over all the four points 共kx , ky兲 = 共0 , 0兲, 共0 , ␲兲,
共␲ , 0兲, and 共␲ , ␲兲. On other lattices, 兺k=0 sums over less
points. Say on an odd-by-odd lattice, only 共kx , ky兲 = 共0 , 0兲 satisfies the quantization conditions kx = 2L␲x ⫻ integer and ky
= 2L␲y ⫻ integer. In this case, 兺k=0 sums over only 共kx , ky兲
= 共0 , 0兲 point.
We note that
and
†
兵⌿Ik
,⌿Jk⬘其 = ␦IJ␦k−k⬘,
⌿k† M共k兲⌿k + 兺 ⌿k† M共k兲⌿k ,
兺
k⫽0
k=0
M = − ⌫M T⌫.
We call them “even matrices;” in the other class, the matrices
satisfy
共5兲
M = ⌫M T⌫.
We call them “odd matrices.”
224406-2
PHYSICAL REVIEW B 80, 224406 共2009兲
TRANSLATION-SYMMETRY-PROTECTED TOPOLOGICAL…
There are six even matrices,
M 兵30其 = ␴3 丢 ␴0,
M 兵12其 = ␴1 丢 ␴2,
M 兵22其 = ␴2 丢 ␴2 ,
For above six matrices, M 兵30其, M 兵12其, and M 兵22其 anticommute
with each other,
兵M 兵30其,M 兵12其其 = 0,
兵M 兵30其,M 兵22其其 = 0,
兵M 兵12其,M 兵22其其 = 0.
共10兲
M 兵33其, M 兵31其, and M 兵02其 anticommute with each other
the Z2A spin liquid兲 may close which indicates a quantum
phase transition. Thus if two gapped regions are always separated by a gapless region, then the two gapped regions will
correspond to two different quantum phases. We may say
that the two quantum phases carry different topological orders.
In the following, we introduce topological indices that can
be calculated for each gapped mean-field ansatz 共uij , ␩ij兲. We
will show that two gapped mean-field ansatz with different
topological indices cannot be smoothly deformed into each
other without closing the energy gap. Therefore, the topological indices characterize different Z2A topological orders
with translation-symmetry.
兵M 兵33其,M 兵31其其 = 0,
A. Topological indices
兵M 兵33其,M 兵02其其 = 0,
兵M 兵31其,M 兵02其其 = 0.
共11兲
In the Sec. II, we obtained the mean-field fermion Hamiltonian in momentum space 1, which has a form Hmean
= H共k ⬎ 0兲mean + H共k = 0兲mean. Let us diagonalize the meanfield fermion Hamiltonian at the points k ⬎ 0 as an example.
Presenting
While each of M 兵30其, M 兵12其, and M 兵22其 commutate with each
of M 兵33其, M 兵31其, and M 兵02其. For the coefficients of the even
matrices c兵␣␤其共k兲 in the mean-field fermion Hamiltonian,
兵␣␤其 = 兵30其 , 兵12其 , 兵22其 , 兵33其 , 兵31其 , 兵02其, we have
W共k兲⌿k =
c兵␣␤其共k兲 = c兵␣␤其共− k兲.
In addition, there exist ten odd matrices:
M 兵00其 = ␴0 丢 ␴0,
M 兵10其 = ␴1 丢 ␴0,
M 兵20其 = ␴2 丢 ␴0 ,
M 兵03其 = ␴0 丢 ␴3,
M 兵11其 = ␴1 丢 ␴1,
M 兵13其 = ␴1 丢 ␴3 ,
M 兵23其 = ␴2 丢 ␴3,
M 兵32其 = ␴3 丢 ␴2,
冢冣
␣k
†
␣−k
␤k
†
␤−k
,
where
W共k兲U共k兲W†共k兲 =
M 兵21其 = ␴2 丢 ␴1 ,
冢
␧1共k兲 0
0
0
0
− ␧1共k兲 0
0
0
0
␧2共k兲 0
0
0
0
− ␧2共k兲
,
共12兲
M 兵01其 = ␴0 丢 ␴1 .
For the coefficients of odd matrices c兵␣␤其共k兲 in the mean-field
fermion Hamiltonian, we have
冣
␧1共k兲 ⬎ 0, and ␧2共k兲 ⬎ 0, we find
H共k ⬎ 0兲mean
c兵␣␤其共k兲 = − c兵␣␤其共− k兲.
=
Thus for odd matrices, c兵␣␤其共k兲 are odd functions of kx , ky
and are fixed to be zero at momenta 共0,0兲, 共0 , ␲兲, 共␲ , 0兲,
共␲ , ␲兲
兵␧1共k兲␣k† ␣k + ␧2共k兲␤k† ␤k
兺
k⬎0
†
†
− ␧1共k兲␣−k␣−k
− ␧2共k兲␤−k␤−k
其.
We note that both ␣⫾k and ␤⫾k will annihilate the mean-field
ground state 兩⌿mean典,
c兵␣␤其共k = 0兲 = 0.
III. CLASSIFICATION OF Z2A TOPOLOGICAL
ORDERS
␣⫾k兩⌿mean典 = 0,
For a generic choice of uij and ␩ij, the corresponding
mean-field Hamiltonian 1 is gapped. Note that the energy
levels of the mean-field Hamiltonian 1 appear in 共E , −E兲
pairs. The mean-field state is obtained by filling all the negative energy levels. The mean-field Hamiltonian is gapped if
the minimal positive energy level is finite. The gapped meanfield Hamiltonian corresponds to a gapped Z2A spin liquid.
As we change the mean-field parameters uij and ␩ij, the
mean-field energy gap 共and the corresponding energy gap for
␤⫾k兩⌿mean典 = 0.
At the four k = 0 points, only the even M ␣,␤ appear and we
diagonalized the Hamiltonian differently. The four eigenvalues of U共k兲 are given by
2
2
2
共k兲 + c兵12其
共k兲 + c兵22其
共k兲
␧⫾⫾共k兲 = ⫾ 冑c兵30其
2
2
2
共k兲 + c兵31其
共k兲 + c兵02其
共k兲,
⫾ 冑c兵33其
and
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PHYSICAL REVIEW B 80, 224406 共2009兲
SU-PENG KOU AND XIAO-GANG WEN
W共c兵33其M 兵33其 + c兵31其M 兵31其 + c兵02其M 兵02其兲W†
H共k = 0兲mean
=
2
2
2
= 冑c兵33其
+ c兵31其
+ c兵02其
M 兵33其 .
1
兺 兵␧++共k兲␣k† ␣k + ␧+−共k兲␤k† ␤k + ␧−−共k兲␣k␣k†
2 k=0
So W共k兲 changes M 33 = ␴3 丢 ␴3 to
+ ␧−+共k兲␤k␤k† 其
W共k兲共␴3 丢 ␴3兲W†共k兲 = a共k兲␴3 丢 ␴3 + b共k兲␴0
= 兺 兵␧++共k兲␣k† ␣k + ␧+−共k兲␤k† ␤k其 + Const.,
丢
k=0
where
W共k兲U共k兲W†共k兲 =
冢
0
0
0
− ␧++共k兲 0
0
0
0
␧+−共k兲 0
0
0
0
− ␧+−共k兲
冣
2
共16兲
2
␨k = 1 − ⌰关␧+−共k兲兴,
2
2
2
2
共k兲
共k兲 + c兵12其
共k兲 + c兵22其
共k兲 − c兵33其
=1 − ⌰关c兵30其
.
共13兲
We note that W共k兲 diagonalizes the linear combination of
M 兵30其, M 兵12其, and M 兵22其 in Hmean:
2
2
共k兲 − c兵02其
共k兲兴,
− c兵31其
共17兲
where ⌰共x兲 = 1 if x ⬎ 0 and ⌰共x兲 = 0 if x ⬍ 0. If two topological ordered states have different sets of topological indices
共␨k=共0,0兲 , ␨k=共␲,0兲 , ␨k=共0,␲兲 , ␨k=共␲,␲兲兲, then as we deform one
state smoothly into the other, some ␨k must change. ␨k can
only change when
2
2
2
2
2
2
冑c兵30其
共k兲 + c兵12其
共k兲 + c兵22其
共k兲 = 冑c兵33其
共k兲 + c兵31其
共k兲 + c兵02其
共k兲.
W共c兵30其M 兵30其 + c兵12其M 兵12其 + c兵22其M 兵22其兲W†
2
2
2
= 冑c兵30其
+ c兵12其
+ c兵22其
M 兵30其 .
␴1 + c共k兲␴3 丢 ␴2 .
where a 共k兲 + b 共k兲 + c 共k兲 = 1.
The energy spectrum at k = 0 motivates us to present ␨k as
the four topological indices, one for each k = 0 point:
2
␧++共k兲 0
共15兲
共14兲
W共k兲 also diagonalizes the linear combination of M 兵33其,
M 兵31其, and M 兵02其 in Hmean:
At that point, the topological ordered state becomes gapless
indicating a quantum phase transition. Therefore, there are
16 different translation invariant Z2A spin liquids labeled by
␨k=共0,0兲, ␨k=共0,␲兲, ␨k=共␲,0兲,
␨k=共␲,␲兲 = 1111,1100,1010,1001,0101,0011,0110,0000,1000,0100,0010,0001,1110,1101,1011,0111.
B. Topological degeneracy
Let’s calculate topological degeneracies for different topological orders through the projective construction. Now
共m,n兲 共m,n兲
共uij ,␩ij 兲
典共兩m , n典 = 兩0 , 0典 , 兩0 , 1典 , 兩1 , 0典 ,
we use 兩m , n典 = 兩⌿mean
兩1 , 1典兲 to denote four degenerate ground states on a
is
defined
as
torus.
Here
共uij共m,n兲 , ␩ij共m,n兲兲
msx共ij兲
nsy共ij兲
msx共ij兲
nsy共ij兲
共−兲
uij , 共−兲
共−兲
␩ij兲. sx,y共ij兲 have values
共共−兲
0 or 1, with sx,y共ij兲 = 1 if the link ij crosses the x = Lx, or y
= Ly line共s兲 and sx,y共ij兲 = 0 otherwise.24,25 In fact, the four
共m,n兲 共m,n兲
共uij ,␩ij 兲
mean-field states 兩m , n典 = 兩⌿mean
典 are obtained by giving the fermion wave functions ␺共x , y兲 different boundary
conditions:
†
†
†
†
共− 兲␺1,k␺1,k+␺1,−k␺1,−k+␺2,k␺2,k+␺2,−k␺2,−k
†
†
†
†
†
†
†
†
= 共− 兲␺1,k␺1,k−␺1,−k␺1,−k+␺2,k␺2,k−␺2,−k␺2,−k
= 共− 兲␺1,k␺1,k+␺1,−k␺1,−k+␺2,k␺2,k+␺2,−k␺2,−k
†
†
†
†
†
†
= 共− 兲␣k␣k+␤k␤k+␣−k␣−k+␤−k␤−k
†
†
= 共− 兲␣k␣k+␤k␤k+␣−k␣−k+␤−k␤−k .
共19兲
Hence we have
␺共x,y兲 = 共− 1兲m␺共x,y + Ly兲,
␺共x,y兲 = 共− 1兲n␺共x + Lx,y兲.
projection to be nonzero. The total ␺ fermion number has a
†
␺−k兲 and
form N̂ = Nk⫽0 + Nk=0 where Nk⫽0ˇ = 兺k⬎ˇ 0共␺k† ␺k + ␺−k
†
Nk=0 = 兺k=0␺k␺k.
We note that, for k ⬎ 0,
共uij,␩ij兲
兺 共␺a,k␺a,k+␺a,−k␺a,−k兲
具兩⌿mean
典典
共− 兲a=1,2
†
共18兲
To obtain a physical state from the mean-field ansatz
共uij,␩ij兲
共uij,␩ij兲
典, one needs to project 兩⌿mean
典 into the subspace
兩⌿mean
with even numbers of ␺ fermion per site. So the mean-field
state must have even numbers of ␺ fermions in order for the
†
共uij,␩ij兲
典
= 共− 兲␣k␣k+␤k␤k+␣−k␣−k+␤−k␤−k兩⌿mean
†
†
共uij,␩ij兲
= 兩⌿mean
典,
†
†
共20兲
for k ⬎ 0. We see that the total number of the ␺ fermions on
all the k ⫽ 0 orbitals is always even.
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So to determine if the mean-field ground state contain
even or odd number of ␺ fermions, we only need to count the
number ␺ fermions at the four special points: 共0,0兲, 共0 , ␲兲,
共␲ , 0兲, and 共␲ , ␲兲. For k = 0,
共1兲 For ␧+−共k兲 ⬎ 0, the k-orbital will be filled by 0 particle: zero ␣ particle and zero ␤ particle.
共2兲 For ␧+−共k兲 ⬍ 0, the k-orbital will be filled by 1 particle: zero ␣ particle and one ␤ particle.
We find that the total number of ␺ fermion at the k = 0
points and at all k are given by
†
†
†
†
†
共− 兲␺1,k␺1,k+␺2,k␺2,k = 共− 兲␺1,k␺1,k−␺2,k␺2,k = 共− 兲共1/2兲⌿k␴3 丢 ␴3⌿k .
共21兲
Nk=0 mod 2 = N mod 2 = 兺 ␨k mod 2.
Using Eq. 共18兲, we find
1
+ b共k兲共␣k† ␤k − ␣k␤k† + ␤k† ␣k − ␤k␣k† 兲
2
1
+ i c共k兲共− ␣k† ␤k − ␣k␤k† + ␤k† ␣k + ␤k␣k† 兲
2
= a共k兲共␣k† ␣k − ␤k† ␤k兲 + b共k兲共␣k† ␤k + ␤k† ␣k兲
+ ic共k兲共− ␣k† ␤k + ␤k† ␣k兲.
共22兲
We see that
†
†
共− 兲␺1,k␺1,k+␺2,k␺2,k
†
†
†
†
†
†
= e␲i关a共k兲共␣k␣k−␤k␤k兲+b共k兲共␣k␤k+␤k␣k兲+ic共k兲共−␣k␤k+␤k␣k兲兴 .
共23兲
If we treat 共␣k , ␤k兲 as an isospin-1/2 doublet, then the above
operator generates a 2␲ rotation since a2共k兲 + b2共k兲 + c2共k兲
= 1. This is consistent with the following relation:
†
†
†
†
†
†
†
†
␣k共− 兲␺1,k␺1,k+␺2,k␺2,k = − 共− 兲␺1,k␺1,k+␺2,k␺2,k␣k ,
␤k共− 兲␺1,k␺1,k+␺2,k␺2,k = − 共− 兲␺1,k␺1,k+␺2,k␺2,k␤k .
†
共24兲
†
Since the operator 共−兲␣k␣k+␤k␤k has the same algebra as
†
†
共−兲␺1,k␺1,k+␺2,k␺2,k and the two operators are equal when
a共k兲 = 1, b共k兲 = c共k兲 = 0, we have
†
†
†
†
共− 兲␺1,k␺1,k+␺2,k␺2,k = 共− 兲␣k␣k+␤k␤k .
共25兲
Thus, the total fermion number at the four points, k = 共0 , 0兲,
共0 , ␲兲, 共␲ , 0兲, and 共␲ , ␲兲 satisfies
†
†
兺 共␣k␣k+␤k␤k兲
共− 兲Nk=0 = 共− 兲k=0
.
共26兲
For a given point of k = 0, the energies for the particles ␣ and
␤ are ␧++共k兲 and ␧+−共k兲. There are two situations:
共ee兲
共eo兲
共oe兲
共oo兲
共27兲
k=ˇ 0
1
1 †
⌿k␴3 丢 ␴3⌿k = a共k兲共␣k† ␣k − ␣k␣k† − ␤k† ␤k + ␤k␤k† 兲
2
2
Note that on even-by-even 共ee兲 lattice, all the four k = 0
points k = 共0 , 0兲, 共0 , ␲兲, 共␲ , 0兲, and 共␲ , ␲兲 are allowed. In this
case N mod 2 is the sum of all four ␨k=0 mod 2. On even-byodd 共eo兲 lattice, only two k = 0 points k = 共0 , 0兲, 共␲ , 0兲 are
allowed. In this case N mod 2 = ␨共0,0兲 + ␨共␲,0兲 mod 2.
Let us use the topological order 1000 as an example to
demonstrate a detailed calculation of the ground-state degeneracy. There are four degenerate mean-field ground states
共m,n兲 共m,n兲
共uij ,␩ij 兲
兩m , n典 = 兩⌿mean
典, m , n = 0 , 1. On an ee lattice, the state
兩0 , 0典 has periodic boundary conditions along both x and y
directions. Among the four k = 0 points, only k = 共0 , 0兲 point
has ␨k = 1 as indicated by the first 1 in the label 1000. As a
result, the ground state 兩0 , 0典 contains an odd number of
fermions and is unphysical: P兩0 , 0典 = 0. For the states 兩0 , 1典,
兩1 , 0典, and 兩1 , 1典, the k = 共0 , 0兲 point is not allowed. Consequently, N = 0 mod 2. Hence 兩0 , 1典, 兩1 , 0典, and 兩1 , 1典 are all
physical states. This gave rise to three-degenerate ground
states for the topological order 1000 on an 共ee兲 lattice.
Second, we calculate the ground-state degeneracy on an
eo 关or odd-by-even 共oe兲兴 lattice. For the state 兩0 , 0典, only two
k = 0 points are allowed: k = 共0 , 0兲 and k = 共␲ , 0兲. Due to
␨共0,0兲 = 1, 兩0 , 0典 is forbidden, P兩0 , 0典 = 0. For other states
兩0 , 1典, 兩1 , 0典, and 兩1 , 1典, k = 共0 , 0兲 is not allowed on an 共eo兲 关or
共oe兲兴 lattice. For the same reason, one obtains threedegenerate ground states 兩0 , 1典, 兩1 , 0典, 兩1 , 1典 on an 共eo兲 关or
共oe兲兴 lattice. Third, we calculate the ground-state degeneracy
on an odd-by-odd 共oo兲 lattice. For the state 兩0 , 0典, there is
only one k = 0 point: k = 共0 , 0兲. As a result, 兩0 , 0典 is not permitted by the projection operator, P兩0 , 0典 = 0. However, without the point k = 共0 , 0兲, other states 兩0 , 1典, 兩1 , 0典, and 兩1 , 1典 are
all physical. One also obtains three-degenerate ground states
兩0 , 1典, 兩1 , 0典, and 兩1 , 1典 on an 共oo兲 lattice.
By this method, we obtain topological degeneracies on
different lattices for the 16 topological orders. The results are
given in the following table:
1111
1110
1101
1011
0111
1100
0011
1001
0110
1010
0101
1000
0100
0010
0001
0000
4
4
4
–
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
1
4
4
2
2
4
4
2
2
4
2
2
2
4
2
2
2
4
2
4
2
4
2
4
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
C. Crystal momenta
Next, we calculate the crystal momenta for 16 topological
ordered states. Because the spin Hamiltonian is translation
invariance, the ground states carry definite crystal momenta.
To calculate the crystal momenta k, we note that the fermion
wave function satisfies the 共anti兲 periodic boundary condition.
The crystal momenta are given by
224406-5
PHYSICAL REVIEW B 80, 224406 共2009兲
SU-PENG KOU AND XIAO-GANG WEN
K̂兩⌿spin典 = 兺 k␺k† ␺k兩⌿spin典
K共1010兲
共ee兲
共eo兲
共00兲
共01兲
共10兲
共11兲
共␲ , 0兲
共0,0兲
共0,0兲
共0,0兲
共␲ , 0兲
共0,0兲
共0,0兲
共0,0兲
K共0110兲
共ee兲
共eo兲
共00兲
共01兲
共10兲
共11兲
共␲ , ␲兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
K共1001兲
共ee兲
共eo兲
共oe兲
共oo兲
共00兲
共01兲
共10兲
共11兲
共␲ , ␲兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
K共1000兲
k
=
兺
k␺k† ␺k兩⌿spin典
k⫽0
+兺
k␺k† ␺k兩⌿spin典.
共28兲
k=0
共m,n兲
共m,n兲
共uij ,␩ij 兲
典 at k ⫽ 0 has a form
The ground state 兩⌿mean
兿k⬎0␣k␣−k␤k␤−k兩0典␺ where 兩0典␺ is the state with no ␺ fermion. Thus, the total crystal momenta k are obtained as
K̂兩⌿spin典 = 兺 k␺k† ␺k兩⌿spin典 = 兺 k␨k兩⌿spin典,
k=0
共29兲
k=0
where we have used 共at k = 0兲
␺k† ␺k
mod 2 = ␨k .
Thus, to determine the crystal momenta, we only need to
focus on the cases at k = 0.
By this method, we obtain the crystal momenta on different lattices 共ee, eo, oe, and oo兲 for 16 topological orders. The
following tables show the crystal momenta of different
ground states:
K共0000兲
共ee兲
共eo兲
共oe兲
共oo兲
共00兲
共01兲
共10兲
共11兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
K共0011兲
共ee兲
共eo兲
共oe兲
共oo兲
共00兲
共01兲
共10兲
共11兲
共0 , ␲兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0 , ␲兲
共0,0兲
K共1100兲
共ee兲
共eo兲
共oe兲
共oo兲
共00兲
共01兲
共10兲
共11兲
共0 , ␲兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0 , ␲兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
K共1111兲
共ee兲
共eo兲
共oe兲
共oo兲
共00兲
共01兲
共10兲
共11兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共␲ , 0兲
共␲ , 0兲
共0,0兲
共0,0兲
共0 , ␲兲
共0,0兲
共0 , ␲兲
共0,0兲
K共0101兲
共ee兲
共eo兲
共00兲
共01兲
共10兲
共11兲
共␲ , 0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共␲ , 0兲
共0,0兲
共0,0兲
共oe兲
共oo兲
共oe兲
共oo兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共oe兲
共oo兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共ee兲
共eo兲
共oe兲
共oo兲
共00兲
共01兲
共10兲
共11兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
K共0100兲
共ee兲
共eo兲
共oe兲
共oo兲
共00兲
共01兲
共10兲
共11兲
共0,0兲
共0,0兲
共0,0兲
K共0010兲
共ee兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共eo兲
共oe兲
共oo兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共00兲
共01兲
共10兲
共11兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
K共0001兲
共ee兲
共eo兲
共oe兲
共oo兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共00兲
共01兲
共10兲
共11兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
K共0111兲
共ee兲
共eo兲
共oe兲
共00兲
共01兲
共10兲
共11兲
共0,0兲
共0,0兲
共0,0兲
共␲ , 0兲
共0,0兲
共0,0兲
共0,0兲
共0 , ␲兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
224406-6
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共0,0兲
PHYSICAL REVIEW B 80, 224406 共2009兲
TRANSLATION-SYMMETRY-PROTECTED TOPOLOGICAL…
K共1011兲
共ee兲
共eo兲
共oe兲
共oo兲
共␲ , 0兲
共00兲
共01兲
共10兲
共11兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
共0 , ␲兲
共0,0兲
共0,0兲
共0,0兲
共0,0兲
K共1101兲
共ee兲
共eo兲
共oe兲
共oo兲
␨k = 1 − ⌰关␧+−共k兲兴,
共0 , ␲兲
共0,0兲
共00兲
共01兲
共10兲
共11兲
共0,0兲
共0,0兲
共0,0兲
共␲ , 0兲
共0,0兲
共0,0兲
K共1110兲
共ee兲
共eo兲
共oe兲
共00兲
共01兲
共10兲
共11兲
共␲ , 0兲
共0,0兲
共0,0兲
共0,0兲
共0 , ␲兲
共0,0兲
共0,0兲
共0,0兲
共oo兲
共0,0兲
IV. HOW MANY DISTINCT Z2 TOPOLOGICAL ORDERS?
We would like to point out that the four topological indices ␨k at k = 0 really describe 16 classes of mean-field ansatz.
It is not clear if different mean-field ansatz give rise to different many-body spin wave functions. So it is possible that
the 16 sets of topological indices ␨k describe less than 16
classes of Z2 topological orders.
On the other hand, if two Z2 topological phases can be
separated through measurable physical quantities, such as
ground-state degeneracy and crystal momenta, then the two
topological phases will be really distinct.
Using the ground-state degeneracies on different types of
lattice we can group the 16 sets of topological indices ␨k into
7 groups: 兵0000其, 兵0001, 0010, 0100, 1000其, 兵0101, 1010其,
兵1100, 0011其, 兵1001, 0110其, 兵0111, 1011, 1101, 1110其, 兵1111其.
Each group has the same ground-state degeneracies. So we
have at least seven distinct Z2 topological phases. If we assume that the boundary condition labeling 共m , n兲 are not
physically observable, the crystal momenta distributions cannot further separate the above seven groups into smaller
groups. However, it is likely that the boundary condition
labels 共m , n兲 are physically observable by moving the unique
type of fermionic excitations 共the spinons兲 around the torus.
In this case all 16 sets of topological indices ␨k label distinct
Z2 topological phases.
To study translation-symmetry protected topological order
in three dimensions, we can still use the projective construction to construct translation symmetric Z2 topological orders
in spin-1/2 systems on cubic lattice, with mean-field fermion
Hamiltonian in Eq. 共1兲. The many-spin topological ordered
共uij,␩ij兲
共uij,␩ij兲
共uij,␩ij兲
state is given by 兩⌿spin
典 = P兩⌿mean
典 where 兩⌿mean
典 is
†
1+共−1兲␺i ␺i
2
2
2
2
共k兲 + c兵12其
共k兲 + c兵22其
共k兲 − c兵33其
共k兲
=1 − ⌰关c兵30其
2
2
共k兲 − c兵02其
共k兲兴.
− c兵31其
共0,0兲
共0,0兲
共0,0兲
ces ␨k = 0 , 1 for the Z2 topological orders in three dimensions
at k = 共0 , 0 , 0兲, 共0 , 0 , ␲兲, 共0 , ␲ , 0兲, 共␲ , 0 , 0兲, 共0 , ␲ , ␲兲,
共␲ , 0 , ␲兲, 共␲ , ␲ , 0兲, 共␲ , ␲ , ␲兲. The values of ␨k are determined from the signs of the energy spectrum at the eight k
= 0 points in momentum space:
is the projection
the ground state of Hmean and P = 兿i 2
operator. Here i = 共ix , iy , iz兲 denotes a site in a cubic lattice.
Furthermore, using the same method applied to twodimensional cases, we may define eight Z2 topological indi-
共30兲
where ⌰共x兲 = 1 if x ⬎ 0 and ⌰共x兲 = 0 if x ⬍ 0. Therefore, there
are totally 28 = 256 different translation invariant Z2A spin
liquids labeled by ␨k=共0,0,0兲, ␨k=共0,0,␲兲, ␨k=共0,␲,0兲, ␨k=共␲,0,0兲,
␨k=共0,␲,␲兲, ␨k=共␲,0,␲兲, ␨k=共␲,␲,0兲, and ␨k=共␲,␲,␲兲. Similar to the
cases in two dimensions, one cannot deform two states with
different 兵␨k其 into each other without a quantum phase transition that closes the energy gap.
V. TWO EXAMPLES OF Z2A TOPOLOGICAL ORDERS
Let us discuss two different translation symmetric Z2A
topological orders studied in Ref. 25 in more detail. The first
one is described by the following mean-field fermion
Hamiltonian24 within the projective construction:8
Hmean = 兺 ␺†i uij␺j + 兺 ␺†i ai␺i ,
ij
i
ui,i+x = ui,i+y = − ␹␴3 ,
ui,i+x+y = ␩␴1 + ␭␴2 ,
ui,i−x+y = ␩␴1 − ␭␴2 ,
a1i = ␷ ,
共31兲
where ␺ = 共␺1 , ␺2兲. The other Z2A spin liquid comes
from an exact soluble spin-1/2 model on square lattice—the
Its
Hamiltonian
is
H
Wen-plaquette
model.16
y x
y
x
. Using projective construction, one
= 16g兺iSi Si+xSi+x+ySi+y
can present a mean-field fermion Hamiltonian24 to describe
such a spin liquid:
T
† IJ
† IJ †
Hmean = 兺 共␺I,i
uij ␺J,j + ␺I,i
␩ij ␺J,j + H.c.兲,
具ij典
共32兲
where I , J = 1 , 2. It is known that the ground states for g ⬍ 0
and g ⬎ 0 have different symmetry protected topological orders. The ground state 共Z2A topological order兲 for g ⬍ 0 is
described by mean-field ansatz −␩i,i+x = ui,i+x = 1 + ␴3 and
−␩i,i+y = ui,i+y = 1 − ␴3. The ground state 共Z2B topological order兲 for g ⬎ 0 is described by mean-field ansatz −␩i,i+x
= ui,i+x = 共−兲iy共1 + ␴3兲 and −␩i,i+y = ui,i+y = 1 − ␴3.
We like to ask: whether the topological order for Z2A
gapped state in Eq. 共31兲 and the topological order for the
Wen-plaquette model in Eq. 共32兲 are the same one. The four
Z2 topological variables ␨k=0 can help us to answer the
question.
224406-7
PHYSICAL REVIEW B 80, 224406 共2009兲
SU-PENG KOU AND XIAO-GANG WEN
For the Z2A gapped state, a mean-field fermion Hamiltonian in momentum space becomes
c兵33其 = cos kx + cos ky ,
Hmean共k兲 = 兺 ⌿k† U共k兲⌿k + H.c.,
c兵31其 = ␩关cos共kx + ky兲 + cos共kx − ky兲兴 + ␷ ,
共33兲
k
c兵02其 = ␭关cos共kx + ky兲 − cos共kx − ky兲兴.
where
U共k兲 =
兺 c兵␣␤其共k兲M 兵␣␤其 ,
Because M 兵33其, M 兵31其, and M 兵02其 make up of an anticommuting basis, we have
兵␣,␤其
2
2
2
共k兲 + c兵02其
共k兲
␧+−共k兲 = 0 − 冑c兵33其
共k兲 + c兵31其
= − 冑共cos kx + cos ky兲2 + 兵␩关cos共kx + ky兲 + cos共kx − ky兲兴 + ␷其2 + 兵␭关cos共kx + ky兲 − cos共kx − ky兲兴其2 .
For the Z2A gapped state, one has
␨共0,0兲 = 1,
␨共0,␲兲 = 1,
␨共␲,0兲 = 1,
␨共␲,␲兲 = 1.
It belongs to the 1111 type of the topological order. The
topological degeneracy is 4, 4, and 4, for even-by-even,
even-by-odd, and odd-by-even, respectively. On odd-by-odd
lattices, the state has no energy gap.
Another example of topological order is the Wenplaquette model. The mean-field ansatz in momentum space
now becomes
U共k兲 =
兺 c兵␣␤其共k兲M 兵␣␤其 ,
兵␣,␤其
c兵30其 = cos kx + cos ky ,
c兵33其 = cos kx − cos ky ,
c兵20其 = sin kx + sin ky ,
c兵23其 = sin kx − sin ky .
There are two even matrices in the ansatz, M 兵30其 and M 兵33其.
We have
␧+−共k = 0兲 = 兩cos kx + cos ky兩 − 兩cos kx − cos ky兩.
For the topological order of the Wen-plaquette model, one
has
␨共0,0兲 = 0,
␨共0,␲兲 = 1,
␨共␲,0兲 = 1,
␨共␲,␲兲 = 0.
It belongs to the type of the topological order. The topological degeneracy is 4, 2, 2, and 2 for 共ee兲, 共eo兲, 共oe兲, and 共oo兲
lattices. Because topological order in the toric-code model
with translation invariance12 and that in the Wen-plaquette
model are equivalent, one can use the same wave function to
describe the toric-code model.
Then when one changes the Z2A gapped state denoted by
1111, into the topological order of the Wen-plaquette model
denoted by, quantum phase transition occurs with emergent
massless fermion at k = 共0 , 0兲 and k = 共␲ , ␲兲.
VI. NON-ABELIAN TOPOLOGICAL STATES
We like to point out that for the four Z2A topological
states described by 兵␨k其 = 1000, 0100, 0010, 0001, the corresponding mean-field Hamiltonian describes a superconducting state whose band structure has an odd winding
number.26,27 So those four Z2A topological states are closely
related to the topological spin liquid state obtained by the
projection of px + ipy SC states.27 As a result, the four Z2A
topological states have a topological order described by Ising
topological quantum field theory 共which is the same topological quantum field theory describing the non-Abelian
Paffien FQH state at ␯ = 1 / 2兲.28–32 This is consistent with the
fact that those four Z2A topological states all have threedegenerate ground states on torus and do not have time reversal symmetry. Therefore the four Z2A topological states
are non-Abelian states with excitations that carry nonAbelian statistics described by Ising topological quantum
field theory.
We believe that the four Z2A topological states described
by 兵␨k其 = 0111, 1011, 1101, 1110 are also non-Abelian states
described by Ising topological quantum field theory. However, 0111, 1011,1101, and 1110 states are different from the
1000, 0100, 0010, and 0001 states since they have different
ground-state degeneracy on 共oo兲 lattices. Some concrete constructions of those non-Abelian topological states and other
topological states discussed in this paper are given in Appendix B.
We also like to point out that the mean-field Hamiltonian
is a Hamiltonian for a superconductor. The Z2 topological
indices ␨k provide a classification of translation invariant
two-dimensional 共2D兲 topological superconductors as
discussed in Ref. 33.
VII. CONCLUSION
In this paper, we study topological phases that have the
translation-symmetry using the projective approach. We con-
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TRANSLATION-SYMMETRY-PROTECTED TOPOLOGICAL…
PHYSICAL REVIEW B 80, 224406 共2009兲
centrated on a class of topological phases described by Z2A
PSG. We find that 2D Z2A topological phases on square
lattice can be further divided into 16 classes, which can be
described by four Z2 topological variables ␨k = 0 , 1 at the four
points k = 共0 , 0兲, 共0 , ␲兲, 共␲ , 0兲, and 共␲ , ␲兲. Through the projected SC wave functions, we obtain the topological degeneracies and the crystal momenta for those 16 classes of topological states. This allows us to identify the topological
phase of the Wen’s plaquette model as the 0110 Z2A topological order. In addition, it is predicted that massless fermionic excitations appear at the quantum phase transition between different topological orders with translation
invariance.
What are the effective topological field theories for those
different symmetry protected Z2 topological orders on lattices? In Ref. 25, it is pointed out that the continuum effective theories for all the two-dimensional Z2 topological orders are same U共1兲 ⫻ U共1兲 mutual Chern-Simons 共MCS兲
theory. However, the lattice symmetry 共such as the
translation-symmetry兲 have different realizations in the
U共1兲 ⫻ U共1兲 MCS theory. In particular, the 1111 type and
0110 type Z2 topological orders and the corresponding realization of the lattice translation-symmetry in the MCS theory
are discussed in detail in Ref. 25. We believe that the symmetry protect Z2 topological ordered states discussed here are
also described U共1兲 ⫻ U共1兲 MCS theory with different realization of lattice translation-symmetry. The results obtained
in this paper apply to any lattices 共since they all have
translation-symmetry兲. Square lattice, triangle lattice, etc.,
have additional and different rotation and reflection symmetries. We can have richer symmetry protected topological orders protected by those additional symmetries. Many of
those symmetry protected topological orders can be described by the projective symmetry group.24 There are also
symmetry protected topological orders that cannot be described by projective symmetry group as we have seen in
this paper. Those additional topological orders may be described by different realizations of lattice rotation/reflection
symmetries in the U共1兲 ⫻ U共1兲 MCS theory.
If a matrix can be decomposed into an anticommuting basis,
U=
one has the determinant of the matrix U as
2
When one add another matrix U⬘ = 兺兵␣,␤其c̃兵␣␤其M 兵␣␤其 to U, we
have the determinant as
det共U + U⬘兲 = 关共c兵␣␤其 + c̃兵␣␤其兲2 + 共c兵␣⬘␤⬘其 + c̃兵␣⬘␤⬘其兲2
+ 共c兵␣⬙␤⬙其 + c̃兵␣⬙␤⬙其兲2 + ¯兴2 .
兵M 兵␣␤其,M 兵␣⬘␤⬘其其 = 0,
共A1兲
A famous example is Dirac/Clifford algebra, of which five ␥
matrices, M 兵33其, M 兵13其, M 兵11其, M 兵02其, and M 兵01其 make up an
anticommuting basis. For a matrix of the form
U=
兺 c兵␣␤其M 兵␣␤其,
兵␣,␤其
兵␣␤其 = 兵33其,兵13其,兵11其,兵02其,兵01其,
the eigenvalue of U are
2
2
2
2
2
+ c兵13其
+ c兵11其
+ c兵02其
+ c兵01其
,
⫾ 冑c兵33其
and the determinant of U is
2
2
2
2
2
+ c兵13其
+ c兵11其
+ c兵02其
+ c兵01其
兲2 .
det U = 共c兵33其
M 兵30其, M 兵12其, and M 兵22其 make up of an anticommuting basis.
For a matrix of the form
U=
兺 c兵␣␤其M 兵␣␤其,
兵␣,␤其
兵␣␤其 = 兵30其,兵12其,兵22其,
the eigenvalues of U are
2
2
2
+ c兵12其
+ c兵22其
.
⫾ 冑c兵30其
U can be diagonalized U = c兵30其M 兵30其 + c兵12其M 兵12其 + c兵22其M 兵22其
into a 4 ⫻ 4 matrix as
2
2
2
冑c兵30其
+ c兵12其
+ c兵22其
⫻
APPENDIX A: DEFINITION OF ANTICOMMUTING BASIS
In this appendix we define anticommuting basis. An anticommuting basis 共M 兵␣␤其 , M 兵␣⬘␤⬘其 , M 兵␣⬙␤⬙其 , . . .兲 is a maximum
set for several 4 ⫻ 4 matrices which anticommute each other
2
det U = 关c兵2␣␤其 + c兵␣⬘␤⬘其 + c兵␣⬙␤⬙其 + ¯兴2 .
ACKNOWLEDGMENTS
This research is supported by NSF Grant No. DMR0706078, NFSC Grant No. 10228408, and NFSC Grant No.
10874017.
兺 c兵␣␤其M 兵␣␤其 ,
兵␣,␤其
冢
0
0
0 −1 0
1
0
0
0
0
1
0
0
0
0 −1
冣
.
M 兵33其, M 兵31其, and M 兵02其 make up of another anticommuting
basis. One can diagonalize U = c兵33其M 兵33其 + c兵31其M 兵31其
+ c兵02其M 兵02其, into another 4 ⫻ 4 matrix as
兵M 兵␣⬘␤⬘其,M 兵␣⬙␤⬙其其 = 0,
2
2
2
冑c兵33其
+ c兵31其
+ c兵02其
⫻
兵M 兵␣␤其,M 兵␣⬙␤⬙其其 = 0,
¯.
224406-9
冢
1
0
0 −1
0
0
0
0
0
0
0
0
−1 0
0
1
冣
.
PHYSICAL REVIEW B 80, 224406 共2009兲
SU-PENG KOU AND XIAO-GANG WEN
APPENDIX B: THE EXAMPLES OF THE ANSATZ OF
DIFFERENT Z2A TOPOLOGICAL ORDERS
0001:
In this part we give one example for each type of topological orders. The ansatz U共k兲 in momentum space of the
16 classes topological orders are given by
1000:
冉 冊
冉 冊
冉 冊
冉 冊
冉 冊
冉 冊
0000:
0101:
1111:
C1
0
0
C1
,
␴3 0
,
0 By
C1
0
0
− C1
0110:
,
␴3 0
1010:
,
0 − By
␴3 0
1110:
,
0 − C1
0111:
␴3 0
,
0 − C3
C1 =
C2 =
C3 =
C4 =
冉
冉
冉
冉
冉
冉
0011:
1100:
␴
0
3
0
Bx
冊
冊
冊
冊
冊
冊
,
冉
␴3 0
,
0 C1
冉
␴3 0
,
0 C3
冊
冊
␴3 0
,
0 Bx+y
␴ =
3
␴3 0
,
0 − Bx
冢
冢
冉
␴3 0
.
0 C4
冊
0
1
1
− 1 − cos kx − cos ky
4
4
By =
冢
冊
0
␴3 0
1011:
,
0 − C2
冢
␴3 0
,
0 C2
1
1
1 + cos kx + cos ky
4
4
Bx =
冢
0010:
冉
The parameters above are defined as
0
␴3
1001:
,
0 − Bx+y
1101:
0100:
␴3 0
,
0 − C4
Bx+y =
冉
冉
冉
cos kx
− i sin kx − cos kx
cos ky
cos共kx + ky兲
冊
冊
,
,
i sin共kx + ky兲
− i sin共kx + ky兲 − cos共kx + ky兲
sin kx + i sin ky
sin kx − i sin ky
1
− cos kx + cos共kx + ky兲 − cos ky
2
1
cos kx − cos共kx + ky兲 − cos ky
2
sin kx + i sin ky
sin kx − i sin ky
1
− cos kx − cos共kx + ky兲 + cos ky
2
1
cos kx + cos共kx + ky兲 + cos ky
2
sin kx + i sin ky
sin kx − i sin ky
1
− cos kx − cos共kx + ky兲 − cos ky
2
1
cos kx + cos共kx + ky兲 − cos ky
2
sin kx + i sin ky
sin kx − i sin ky
1
− cos kx − cos共kx + ky兲 + cos ky
2
224406-10
i sin ky
− i sin ky − cos ky
1
cos kx − cos共kx + ky兲 + cos ky
2
One can check above results by the following table:
i sin kx
冣
,
冣
,
冣
,
冣
.
冊
,
冣
,
PHYSICAL REVIEW B 80, 224406 共2009兲
TRANSLATION-SYMMETRY-PROTECTED TOPOLOGICAL…
1 + 1 / 4 cos kx + 1 / 4 cos ky
cos kx
cos ky
cos共kx + ky兲
cos kx + cos ky − 1 / 2 cos共kx + ky兲
cos kx − cos ky − 1 / 2 cos共kx + ky兲
cos kx + cos共kx + ky兲 + 1 / 2 cos ky
cos kx + cos共kx + ky兲 − 1 / 2 cos ky
共0,0兲
共0 , ␲兲
共␲ , 0兲
共␲ , ␲兲
⬎0
⬎0
⬎0
⬎0
⬎0
⬍0
⬎0
⬎0
⬎0
⬎0
⬍0
⬍0
⬎0
⬎0
⬍0
⬎0
⬎0
⬍0
⬎0
⬍0
⬎0
⬍0
⬍0
⬍0
⬎0
⬍0
⬍0
⬎0
⬍0
⬍0
⬍0
⬎0
*http://dao.mit.edu/~wen
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18 G.
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