Table Of ContentConstruction and Field Theory of Bosonic Symmetry Protected Topological states
beyond Group Cohomology
Zhen Bi1 and Cenke Xu1
1Department of physics, University of California, Santa Barbara, CA 93106, USA
Weconstructaseriesofbosonicsymmetryprotectedtopological(BSPT)statesbeyondgroupco-
homology classification using“decorated defects” approach. Thisconstruction isbased on topolog-
icaldefectsofordinaryLandauorderparameters,decoratedwiththebosonicshortrangeentangled
(BSRE) states in (4k+3)d and (4k+5)d space-time (with k being nonnegative integers), which
5
do not need any symmetry. This approach not only gives these BSPT states an intuitive physical
1
picture,it also allows usto derivetheeffectivefield theory for all these BSPT states beyondgroup
0
cohomology.
2
r
p 1. INTRODUCTION whereBI isa2k+2formantisymmetricgaugefield,and
A
KIJ =iσy. This theory with k =0 (4d space) has been
4 Bosonic symmetry protected topological (BSPT) studiedcarefullyinRef.9,anditwasdemonstratedthat
2 states are bosonic analogues of fermionic quantum spin its 3d boundary is an “all fermion” 3d QED [10] which
Hallinsulatorandtopologicalinsulator,whichhavetriv- cannot be independently realized in 3d space, and it has
]
l ial bulk spectrum but nontrivial boundary spectrum, as a global gravitational anomaly [11].
e
- long as the system preserves certain symmetry. There As was pointed out by Ref. 12, 13, the state Eq. 2
r are roughly two types of BSPT states, their mathemat- can also have a time-reversal symmetry. For instance,
t
s ical difference is whether they can be classified and de- this action is invariant under ZT : i → −i,(B1,B2) →
. 2
t scribedbygroupcohomology[1,2]andsemiclassicalnon- (B2,B1). Butthisstateisalsostableifthetime-reversal
a
m linear sigma model field theory [3]. For example, the symmetryisbroken. Inthispaperwewillonlycountthis
well-known E bosonic short range entangled (BSRE) state as a BSRE state without any symmetry.
- 8
d state [18] [4, 5] in 2d space [19], and its higher dimen- All these BSRE states inevenspatialdimensions have
n sional generalizations [6] cannot be classified by group their descendant BSPT states in higher dimensions. All
o
cohomology. thesedescendantBSPTstatesarealsobeyondthegroup
c
[ Any nontrivial SPT state’s boundary state cannot ex- cohomology classification. Recently, a systematic math-
istbyitself,aslongasthesystempreservesthenecessary ematicalformalismfor BSRE andBSPT states has been
2
symmetry. ThismeansthattheboundaryofaSPTstate proposed in Ref. 14, which was based on cohomology of
v
1 must be “anomalous”. The relation between boundary G×SO(∞),whereGisthesymmetrygroup,andSO(∞)
7 anomaly and bulk SPT states has been studied system- is supposed to describe the gravitational anomaly. The
2 aticallyinRef.7. IfanontrivialSREstatedoesnotneed purposeofthecurrentworkistogiveaphysicalconstruc-
2 anysymmetrytoprotectitsboundary,thenitsboundary tion and field theory description of BSPT states beyond
0
musthavegravitationalanomaly. The2dp+iptopologi- the ordinary group cohomology classification. Our re-
.
1 calsuperconductor,andthe2dE statebothhavechiral sults are summarized in Table I.
8
0
edge states, which lead to gravitational anomaly. Ana-
5
logues of 2d E state can be found in all even spatial
1 8
: dimensions. In every (4k + 2)d space (or equivalently 2. GENERALITIES
v (4k+3)dspace-time),thereisaBSREstatewithZclas-
i
X sification described by action [6]
We will view the BSRE states without any symme-
r
try in even spatial dimensions (Eq. 1 and Eq. 2) as
a iKIJ
S = CI ∧dCJ, (1) base states. Our general strategy for constructing other
(4k+3)d Z 4π
beyond-Group-CohomologyBSPTstates,istofirstbreak
whereCI isa2k+1formantisymmetricgaugefield,and part or all of the symmetry by condensing an ordinary
KIJ is the Cartan matrix of the E group. These states Landau order parameter, then proliferating/condensing
8
the topological defects of the Landau order parameter.
have bosonic 2k−dimensional membrane excitations in
The nontrivial BSPT state and the trivial state are dis-
thebulk,andperturbativegravitationalanomaliesatthe
tinguished by the nature of the topologicaldefects: non-
boundary[6,8];Inevery(4k+4)dspace(orequivalently
(4k + 5)d space-time), there is a BSRE state with Z trivial BSPT states corresponds to the case where the
2
defects are decorated with the BSRE states in Eq. 1
classification described by action
or Eq. 2. The first example of such beyond-Group-
iKIJ Cohomology BSPT state, which is protected by Time
S = BI ∧dBJ, (2)
(4k+5)d Z 4π Reversal Symmetry T, was discovered in Ref. 15. This
2
state canbe constructedby proliferatingT-breakingdo- iΘ (iσy)IJ
L = ndn∧...∧dn∧ dBI ∧dBJ,
main walls decorated with the 2d E8 state. The topo- Dd,B ΩD−(4k+6) 8π2
logical term in the field theory Lagrangian density that D−(4k+6)
| {z } (6)
encodes the decoration reads:
where~n is a Landauorderparameterwith aunit length.
Ω = V × D!, V is the volume of the unit D-
D D D
LZ2T = i2πndn∧ KEIJ8CI ∧dCJ dimensional sphere. Here we assume all components of
3+1d 2π 8π2 orderparameter~ntransformnontriviallyunderthesym-
KIJ metry group.
= idθ∧ E8CI ∧dCJ
The equations above are also effectively equivalent to
8π2
KIJ the two equations in the follows:
= −iθ E8dCI ∧dCJ (3)
8π2 iΘ KIJ
L = ndn∧...∧dn∧ CI ∧dCJ. (7)
where the O(2) vector ~n is parametrized as ~n = Dd,A ΩD−(4k+3) 8π2
D−(4k+3)
(cosθ,sinθ). The T-symmetry transformationis | {z }
Z2T : (θn→1,n−2θ)→(n1,−n2), (4) LDd,B = ΩD−i(Θ4k+5)ndnD∧−(.4.k.+∧5d)n∧(iσ8πy)2IJBI ∧dBJ,
| {z } (8)
One can verify that the Eq. 3 is time-reversal invariant.
where the component n does not transform under any
Also, if we keep time-reversal invariance, then hn i = 0, 1
2 symmetry group, but the rest of the components all
namely hθi = 0 or π, which precisely corresponds to the
transformnontrivially. The equivalence betweenthe two
trivial and nontrivial BSPT state discussed in Ref. 15.
descriptions above can be made explicit by parametriz-
Meanwhile, across a T-breaking domain wall, θ continu-
ouslychangesfrom−π+0+ to π+0−. After integrating ing ~n as: ~n = (cosθ,sinθn2,sinθn3,···), then following
the derivation in Eq. 3, because the desired BSPT state
over the normal direction, the effective field theory left
is fully symmetric, hθi must be either 0 or π, which cor-
on the domain wall precisely describes a 2d E state.
8 responds to the trivial state and nontrivial BSPT state
The idea of “decorated domain wall” construction
respectively. And with hθi = π, Eq. 7,8 return to Eq. 5
of SPT states was further explored in Ref. 16. Do-
and 6.
main wall of Z or time-reversal symmetry is the sim-
2 Allthetermsaboveare“topological”inthesensethat
plest kind of topological defect. In our current work
theyareinvariantunderlocalcoordinatetransformation,
wewillconstructbeyond-group-cohomologyBSPTstates
because they do not involve the metric. We only wrote
using more general topological defects of other symme-
down the most important topological terms explicitly,
try groups. Here we want to clarify that in our current
but the readers should be reminded that there are other
worktheconcept“topologicaldefect”referstoatopolog-
terms that guarantee the system is in a fully gapped
ically stable configurationsof Landauorderparameter~n
and nondegenerate phase. For example, we need a term
in d−dimensionalspace Rd with a singularity I, and the
1/g(∂ ~n)2 in the field theory to control the dynamics of
singularity can be viewed as the boundary of Rd − I. µ
~n, and we must keep g large enough to disorder ~n; we
The Landau order parameter ~n has a soliton configura-
also need a BF theory term[15] ∼ (dB)2+ 1 B∧dC to
tion on Rd−I, which has no singularity any more. For 2π
gap out all the excitations of the CI field.
example, in 2d space a vortex core is a singularityat the
Naively,wecanalsowritedownthefollowingfieldthe-
origin (0,0), and it can be viewed as the boundary of
ory,withall~ncomponentstransformingnon-triviallyun-
R2−(0,0),whichistopologicallyequivalenttoaringS1.
der symmetry:
Avortexconfigurationcorrespondstoa1dsolitononS1,
basedonthesimplefactπ [S1]=Z. In3dspaceahedge- iΘ KIJ
1 L= ndn∧...∧dn∧ CI ∧dCJ.
hog monopole core is again a singularity at (0,0,0), and Ω 8π2
a hedgehog monopole corresponds to a soliton on space
Forexample,wecanwritedownsuchfieldtheoryin3+1d
R3−(0,0,0), based on the fact π2[S2]=Z. space-time, with ~n being an O(2) vector, and CI a one
Ingeneral,thefieldtheorieswewilldiscussinthiswork form vector gauge field. Then the physical meaning of
is a combination of the Θ-term of ~n discussed in Ref. 3
thisfieldtheoryisthat,thevortexcoreof~nwillhostthe
and Chern-Simons form of CI or BI in Eq. 1,2. The boundary state of the 2d E state, which must be gap-
8
explicitlyformofthetopologicalterminD−dimensional less. Then this means that we can never achieve a fully
space-time is:
gapped nondegenerate state by proliferating the vortex
iΘ KIJ loops. Thus this field theory will always be gapless, un-
LDd,A = ΩD−(4k+4)ndnD∧−(.4.k.+∧4d)n∧8π2 dCI ∧dCJ, lfeosrsewtheisefixeplldictithleyobryredaeksctrhibeeUs(t1h)esbyomunmdeatrryyooffa~n4.dTsphaercee-,
| {z } (5) rather than a 3d bulk state.
3
ForfieldtheoriesinEq. 5and6,ingeneralweconsider Symmetry 3+1d 4+1d 5+1d 6+1d
fixedpoints Θ=2πp withp∈Z. However,this doesnot U(1) 0 Z 0 Z×Z2
mean that we have a Z classified state. If we can show Z2 0 Z2 Z2 Z22
tqh∈atZtw,ocafinelbdethsemoorioetsh,lΘy c=on0naecntdedΘw=it2hπouqtwciltohsicnegrttahine U(1Z)⋊2T Z2T ZZ22 Z0 ZZ2222 Z32Z(2Z42)
bulk gap, then they must be in the same phase. In that
case, the classification will be reduced to Z . U(1)×Z2T Z2 0 Z32 Z22 (Z32)
Because our field theory is constructed wqith order pa- U(1)⋊Z2 0 Z2 Z22 Z×Z32
rameter~n andChern-SimonsformofCI orBI,the clas- U(1)×Z2 0 Z×Z2 Z2 Z×Z42
sification will depend on both sectors.
TABLE I: BSPT beyond Group Cohomology constructed
For pure C ∧ dC theory, the classification is Z, be-
fromdecoratedtopologicaldefects. Pleasenotethatthestates
cause its boundary state has perturbative gravitational
withingroupcohomologyclassificationisnotlistedhere. The
anomaly[6,8],whichhasZclassification. Thentheclas- case for U(1)×Z2 symmetry was not discussed in Ref. 14.
sification of the mixed field theory of ~n and CI only de- Ourresultslargely agree with Ref. 14. Theresultsin Ref.14
pends on the ~n sector. that do not fully agree with ours are highlighted in red.
For instance, we can take Eq. 3 as an example. Take
two copies of the field theories and couple them to each
3. EXAMPLES OF BSPT BEYOND GROUP
other:
COHOMOLOGY
i2π KIJ
L = n dn ∧ E8CI ∧dCJ +(1→2)
2π (1) (1) 8π2 (1) (1) In this section we study examples of beyond-group-
+ βn2,(1)·n2,(2)+λdC(I1)∧⋆dC(I2), (9) cohomology BSPT states with various symmetries up to
6+1 space-time dimensions. All these states are con-
where ⋆ is the Hodge star operator. Now we fix λ at
structed with Landau order parameters and the 2d E
a negative value, and tune β from negative to positive. 8
state or the 4d BSRE state in Eq. 2. Our results are
With negative β, effectively ~n and~n will align with
(1) (2) summarized in TABLE I. Our results are mostly consis-
eachother,thusn =n ,C =C ,thenthetwo
2,(1) 2,(2) (1) (2) tentwithresultsinRef.14,exceptionsarehighlightedin
theories will “constructively interfere” with each other,
red in the table.
and the final theory effectively has Θ = 4π; with pos-
itive β, effectively n = −n , C = C , thus
2,(1) 2,(2) (1) (2)
the two theories will “destructively interfere” with each
other,andthefinaltheoryeffectivelyhasΘ=0. Because
both theories are fully gapped and nondegenerate in the A. U(1) Symmetry
bulk, tuning the coupling between them does not close
the bulkgap(aslongasthecouplingisnottoostrongto
• In 4d space, there is a series of BSPT states with
overcome the bulk gap), thus the two effective coupled
U(1) symmetry that is beyond the group cohomology,
theorieswith Θ=0 andΘ=4π are smoothly connected
their field theory is given by:
withoutgoingthroughabulkphasetransition. Therefore
theByclacsosnifitrcaastti,onletfoursthcoensstiadteerEaqU. (31i)sBZS2P. T in 4d space LU4+(11)d = i22ππkndn∧ K8πEI2J8dCI ∧dCJ, (11)
with the following field theory:
where CI’s are rank-1 gauge field, and k can take arbi-
i2π KIJ
LU(1) = ndn∧ E8dCI ∧dCJ. (10) trary integer value. Physically this state can be viewed
4+1d 2π 8π2
as decorating the 2π vortex of U(1) order parameter
TheU(1)symmetryactsasU(1):(n +in )→eiφ(n + ~n=(n ,n ) (whichis a 2dmembrane in this dimension)
1 2 1 1 2
in ). Imagine we have two copies of the theory, the only withtheE state,andthenproliferatingthevortices. As
2 8
U(1) symmetry allowedcoupling between these two the- we have shown in the previous section, this phase has Z
ories would be β~n ·~n . Then for either sign of β, i.e. classification.
(1) (2)
for either ~n ∼ ~n or ~n ∼ −~n , the final effective The 3 + 1d boundary of this state can be a super-
(1) (2) (1) (2)
theory alwayshas Θ=4π (simply because (−1)2 =+1). fluidphasewithspontaneouslyU(1)symmetrybreaking,
Thus there is no symmetry allowed coupling that can whosevortexlinehoststheedgestatesofthe2dE state,
8
continuously connect Θ = 4π to Θ = 0. Therefore the i.e. a chiral conformal field theory with central charge
classification for this U(1) BSPT state is Z. c = 8. If we couple ~n to a U(1) gauge field, then after
For pure B ∧ dB theory, the classification is Z [9], we integrate out the gapped matter field ~n, the bound-
2
therefore the classification of the mixed state can only ary of the system will have a mixed U(1)-gravitational
be Z or trivial depending on the classification on the ~n anomaly, namely the stress tensor of the system is no
2
sector. longerconservedinsidetheU(1)fluxattheboundary. A
4
similar mixedgauge-gravityanomalywas alsostudied in Here we parametrize~n as~n=(cosθ,sinθ). The symme-
Ref. 17. try transformation is:
• In 6+1d space-time, there are two root states for
Z : (n ,n )→(n ,−n )
U(1) BSPT states beyond group cohomology, the first 2 1 2 1 2
state is described by the following field theory: (B1,B2)→(B2,B1)
i2πk KIJ θ →−θ. (19)
LU(1) = ndn∧dn∧dn∧ E8dCI ∧dCJ (12)
6+1d,A 12π2 8π2 Notice that BI must transform nontrivially under Z
2
with symmetry, in order to guarantee that the field theory
U(1): (n +in )→eiφ(n +in ), is Z2 invariant. We can also choose a different transfor-
1 2 1 2 mation for BI: Z : B → σzB, but this transformation
(n +in )→eiφ(n +in ). (13) 2
3 4 3 4 is equivalent to the previous after a basis change. In a
This state has Z classification. The state is constructed Z2 invariant state, hn2i = 0, i.e. hθi = 0 or π, which
by decorating the E states on the intersection of two corresponds to the trivial and BSPT state respectively.
8
U(1) vortices,and then proliferatethe vortices(the two- • In 6+1d space-time, there are two root states, both
vortex intersection is now a 2d brane in 6d space). ofwhichare descendantsofU(1) BSPTstates, andboth
The field theory of the second root phase is have Z2 classification:
LU(1) = i2πndn∧ (iσy)IJdBI ∧dBJ (14) LZ2 = i2π ndn∧dn∧dn∧ KEIJ8dCI ∧dCJ (20)
6+1d,B 2π 8π2 6+1d,A 12π2 8π2
where
with
U(1):(n +in )→eiφ(n +in ) (15)
1 2 1 2 Z :(n ,n ,n ,n )→−(n ,n ,n ,n ). (21)
2 1 2 3 4 1 2 3 4
andB’s are2-formfields. The state has Z classification
2
according to our rules. And physically this field theory
i2π (iσy)IJ
corresponds to decorating the U(1) vortex with the 4d LZ2 = ndn∧ dBI ∧dBJ (22)
6+1d,B 2π 8π2
BSRE state in Eq. 2.
with
B. Z2 Symmetry Z2 :(n1,n2)→−(n1,n2). (23)
• In 4+1d space-time, there is one nontrivialbeyond-
cohomology BSPT state with Z symmetry, and this C. Z2T Symmetry
2
state is the descendant of the U(1) beyond Group Co-
• In 3+1d space-time, it is well-knownthat there is a
homology state in the same dimension in the sense that
BSPT state beyond Group Cohomology [15]. The state
it canbe obtainedby breakingthe U(1)symmetry to its
can be understood by decorating ZT domain walls with
subgroup Z2 from Eq. 11: 2
the 2d E state:
8
i2π KIJ
LZ4+21d = 2π ndn∧ 8πE28dCI ∧dCJ (16) LZ2T = i2πndn∧ KEIJ8CI ∧dCJ
3+1d 2π 8π2
with
KIJ
Z :(n ,n )→−(n ,n ) (17) = −iθ E8dCI ∧dCJ (24)
2 1 2 1 2 8π2
while the classification of the state is now reduced to Z
2 with
because the n-sector is now Z classified.
2
• In 5 + 1d space-time, there is a Z2 classified new Z2T : (n1,n2)→(n1,−n2)
state which is not a descendant of any U(1) state dis-
θ →−θ. (25)
cussed in the previous subsection. Physically this state
is constructedby decoratingthe Z domainwallwith 4d θ is defined as before, hθi = 0 and π correspond to the
2
BSRE state: trivial and BSPT state respectively. This state has Z
2
i2π (iσy)IJ classification.
LZ5+21d = 2π ndn∧ 8π2 BI ∧dBJ • In 5+1d space-time, there are two root states, both
haveZ classification. The field theory for the firststate
(iσy)IJ 2
= idθ∧ BI ∧dBJ reads:
8π2
= −iθ(iσy)IJdBI ∧dBJ (18) LZ2T = i2πndn∧dn∧ KEIJ8dCI ∧dCJ (26)
8π2 5+1d,A 8π 8π2
5
with • In 5+1d space-time, there are two root states, both
areidenticalto the ZT state in the samedimension with
Z2T :(n1,n2,n3)→−(n1,n2,n3). (27) trivial U(1) symmetr2y transformation, and both are Z2
classified.
The physical meaning of this state is most transparent
• In 6+1d space-time, in Ref. 14 there are four root
if we start with a system with an enlarged SO(3)×Z2T states, all Z classified. However,we can only find three
symmetry, and ~n forms a vector under the SO(3) sym- Z classified2root states by our construction. The first
metry. Then Eq. 26 can be viewed as decoration of the 2
one is identical to the ZT state in 6+1d. The other two
hedgehog monopole of ~n with the 2d E state. Weakly 2
8 root states are given by:
breaking the SO(3) symmetry while preserving the ZT
2
steyrmnmateivtreyly,doweescnaontvchieawngtehethheedngaetuhroegomfotnhoispsotlaetae.s tAhle- LU(1)⋊Z2T = i2π ndn∧dn∧dn∧ KEIJ8dCI ∧dCJ (34)
6+1d,A 12π2 8π2
intersection of three ZT domain walls.
2
The field theory for the second root state is with
LZ2T = i2πndn∧ (iσy)IJBI ∧dBJ U(1): (n1+in2)→eiφ(n1+in2),
5+1d,B 2π 8π2 ZT : (n ,n ,n ,n )→(n ,−n ,−n ,−n ) (35)
(iσy)IJ 2 1 2 3 4 1 2 3 4
= −iθ dBI ∧dBJ (28)
8π2 and
with LU(1)⋊Z2T = i2πndn∧ (iσy)IJdBI ∧dBJ (36)
ZT : (n ,n )→(n ,−n ) 6+1d,B 2π 8π2
2 1 2 1 2
θ →−θ. (29) with
This state can be viewed as decoration of ZT domain U(1): (n +in )→eiφ(n +in ),
2 1 2 1 2
wall with the 4d BSRE state. ZT : (n ,n )→(n ,−n ) (37)
• In 6+1d space-time, there is one new state with Z 2 1 2 1 2
2
classification:
We suspectthestate wemissedhereis the mixedSPT
LZ2T = i2πndn∧ (iσy)IJdBI ∧dBJ (30) state described by Ed(G) in Ref. 14.
6+1d 2π 8π2
with E. U(1)×Z2T Symmetry
ZT : (n ,n )→−(n ,n )
2 1 2 1 2 • In 3+1d space-time,there is a state identical to the
(B1,B2)→(B2,B1). (31) pure ZT state with trivial U(1) symmetry transforma-
2
tion.
The state is constructed by decorating the vortex of ~n
• In 5+1d space-time,we find three Z classifiedroot
(or the intersectionof twoZT domainwalls)with the 4d 2
2 states. Two of them are identical to the ZT states in
BSRE state. 2
5+1d space-time, with trivial U(1) symmetry transfor-
mation. The third state is given by:
D. U(1)⋊Z2T Symmetry LU(1)×Z2T = i2πndn∧dn∧ KEIJ8dCI ∧dCJ (38)
5+1d 8π 8π2
• In 3+1d space-time, there is one nontrivialbeyond-
cohomologyBSPTstatewithU(1)⋊Z2T symmetry,butit with
is identical to the ZT state in the same dimension, U(1)
2
symmetry simply acts trivially. U(1): (n +in )→eiφ(n +in ),
1 2 1 2
• In 4+1d space-time, there is one root state with Z ZT : (n ,n ,n )→(n ,n ,−n ). (39)
2 1 2 3 1 2 3
classification:
LU(1)⋊Z2T = i2πkndn∧ KEIJ8dCI ∧dCJ. (32) TthheisinstteartseeccatinonbeovfiaewZeTdadsomdeacionrawtainllgatnhde2adUE(81)stvaotertoenx
4+1d 2π 8π2 2
(it can also be viewed as the hedgehog monopole of ~n),
with then proliferating both the domain walls and vortices.
• In 6+1d space-time, in Ref. 14 there are three Z
2
U(1): (n1+in2)→eiφ(n1+in2) classifiedrootstates. However,usingourmethodwe can
ZT : (n ,n )→(n ,−n ). (33) onlyconstructtwoZ classifiedrootstates. Thefirstone
2 1 2 1 2 2
6
is identical to the ZT state with trivial U(1) symmetry Thelastrootstatein6+1dspace-timeisdescribedby
2
transformation. The other one is:
LU(1)⋊Z2 = i2πndn∧ (iσy)IJdBI ∧dBJ (46)
LU(1)×Z2T = i2πndn∧ (iσy)IJdBI ∧dBJ (40) 6+1d,B 2π 8π2
6+1d 2π 8π2 with
with U(1): (n +in )→eiφ(n +in ),
1 2 1 2
Z : (n ,n )→(n ,−n ),
U(1): (n +in )→eiφ(n +in ), 2 1 2 1 2
1 2 1 2
(B1,B2)→(B2,B1). (47)
ZT : (n ,n )→−(n ,n ), B1(2) →B2(1). (41)
2 1 2 1 2
This state has Z classification.
2
One may ask whether field theory like Eq. 34 could
correspond to a new root state. However, there is no
consistent way to assign the U(1)×Z2T symmetry trans- G. U(1)×Z2 Symmetry
formationsonEq.34,namelyEq.34cannotbe invariant
underU(1)×ZT symmetry,althoughitisinvariantunder • In 4+1d space-time, we have two root states, both
2
U(1)⋊ZT symmetry. of which are descendants from pure U(1) state and pure
2
Z state respectively.
2
• In 5+1d space-time, there is only one root state,
F. U(1)⋊Z2 =O2 Symmetry which is the same as the state with Z symmetry only.
2
• In 6+1d space-time, there are five root states. The
• In 4+1d space-time, there is one root state iden- first three states can all be described by the same field
tical to the BSPT state with Z2 symmetry in the same theory:
dimension, the U(1) symmetry simply acts trivially.
• In 5+1d space-time, there are two root states, both LU(1)×Z2 = i2πkndn∧dn∧dn∧ KEIJ8dCI ∧dCJ (48)
Z classified. One is the same Z state with trivial U(1) 6+1d,A 12π2 8π2
2 2
action. The other one is given by: These three different states have the same form of La-
grangian, but they are distinguished from each other by
LU(1)⋊Z2 = i2πndn∧dn∧ KEIJ8dCI ∧dCJ (42) their symmetry transformations:
5+1d 8π 8π2
(1) U(1): trivial,
with
Z : (n ,n ,n ,n )→−(n ,n ,n ,n ).(49)
2 1 2 3 4 1 2 3 4
U(1): (n +in )→eiφ(n +in ),
1 2 1 2
Z2 : (n1,n2,n3)→(n1,−n2,−n3). (43) (2) U(1): (n1+in2)→eiφ(n1+in2),
Z : (n ,n ,n ,n )→−(n ,n ,n ,n ).(50)
2 1 2 3 4 1 2 3 4
This state can be viewed as decorating the 2d E state
8
on the intersection of U(1) vortex and Z domain wall.
2
Also,theO2symmetryisasubgroupofSO(3)symmetry, (3) U(1): (n1+in2)→eiφ(n1+in2),
thus the vortex-domain wall intersection is simply the (n +in )→eiφ(n +in ),
3 4 3 4
hedgehog monopole of the SO(3) vector~n.
Z : (n ,n ,n ,n )→−(n ,n ,n ,n ).(51)
2 1 2 3 4 1 2 3 4
•In6+1dspace-time,wefindfour rootstates,which
is more than the results in Ref. 14. Two of them are The classification of the three states are Z2, Z2 and Z
the same as the BSPT states with Z symmetry, both of respectively.
2
which areZ classified. The third rootstate is described The other two states are described by the following
2
by field theory:
LU(1)⋊Z2 = i2πkndn∧dn∧dn∧ KEIJ8dCI ∧dCJ (44) L6U+(11)d×,BZ2 = i22ππndn∧ (iσ8πy)2IJdBI ∧dBJ (52)
6+1d,A 12π2 8π2
again,thesetwostateshavedifferenttransformationsun-
with der symmetry groups:
U(1): (n +in )→eiφ(n +in ), (4) U(1): trivial,
1 2 1 2
(n3+in4)→eiφ(n3+in4), Z2 : (n1,n2)→−(n1,n2). (53)
Z : (n ,n ,n ,n )→(n ,−n ,n ,−n ) (45)
2 1 2 3 4 1 2 3 4
(5) U(1): (n +in )→eiφ(n +in ),
This state is Z classified. This state can be viewed as 1 2 1 2
Z : trivial. (54)
decorating the 2d E state on the intersection of two 2
8
vortices, then proliferate the vortices afterwards. The classification of the two states are both Z .
2
7
4. SUMMARY B 91, 134404 (2015).
[4] A. Y.Kitaev, Ann.Phys. 321, 2 (2006).
[5] A.Kitaev,http://online.kitp.ucsb.edu/online/topomat11/kitaev.
In this work, we construct field theories of beyond-
[6] C. Xu and Y.-Z.You, Phys.Rev.B 91, 054406 (2015).
Group-Cohomology BSPT states based on decorated
[7] X.-G. Wen,Phys. Rev.D 88, 045013 (2013).
topological defect picture. Our results are largely con- [8] L. Alvarez-Gaume and E. Witten, Nucl. Phys. B 234,
sistentwithRef.14,withafewexceptions. Welistedex- 269 (1983).
amples of BSPT states below six dimensional space, but [9] S. M. Kravec, J. McGreevy, and B. Swingle,
our construction can be straightforwardly generalized to arXiv:1409.8339 (2014).
all higher dimensions, as long as we use the generalized [10] C.Wang,A.C.Potter,andT.Senthil,Science343,6171
(2014).
base states in Eq. 1,2 for k ≥ 1. We also note that
[11] L. Kong and X.-G. Wen,arXiv:1405.5858 (2014).
Ref. 17 proposedthe SPT states beyondgroupcohomol-
[12] A. Kapustin,arXiv:1404.6659 (2014).
ogyshouldhave mixedgauge-gravityresponses,which is
[13] A. Kapustin,arXiv:1403.1467 (2014).
also consistent with the formalism used in our current [14] X.-G. Wen,arXiv:1410.8477 (2014).
work. [15] A. Vishwanath and T. Senthil, Phys. Rev. X 3, 011016
TheauthorsaresupportedbythetheDavidandLucile (2013).
PackardFoundationandNSFGrantNo. DMR-1151208. [16] X. Chen, Y.-M. Lu, and A. Vishwanath, Nature Com-
munications (2014).
The authors thank Xiao-Gang Wen for many inspiring
[17] J. Wang, Z.-C. Gu, and X.-G. Wen, Phys. Rev. Lett.
communications.
114, 031601 (2015).
[18] In this paper we define short range entangled state as
systems with gapped and nondegenerate bulk spectrum,
namelyithasnotopologicalentanglemententropy.Inour
definition, SREstates includeSPT states as a subset.
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B 87, 155114 (2013). therealphysicalspace.Thephrasespace-timewillalways
[2] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Science be written explicitly.
338, 1604 (2012).
[3] Z. Bi, A. Rasmussen, K. Slagle, and C. Xu, Phys. Rev.
