Topological Phases Protected By Reflection Symmetry and Cross-cap States

0.67 MB·English

by  Gil Young Cho

Topological Phases Protected By Reflection Symmetry and Cross-cap States
Gil Young Cho,1,2 Chang-Tse Hsieh,1 Takahiro Morimoto,3 and Shinsei Ryu1
1Institute for Condensed Matter Theory and Department of Physics,
University of Illinois at Urbana-Champaign, 1110 West Green St, Urbana IL 61801
2Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea
3Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama, 351-0198, Japan
(ΩDated: May 28, 2015)
Twisting symmetries provides an efficient method to diagnose symmetry-protected topological
(SPT) phases. In this paper, edge theories of (2+1)-dimensional topological phases protected by
reflectionaswellasothersymmetriesarestudiedbytwistingreflectionsymmetry,whicheffectively
5 puts the edge theories on an unoriented spacetime, such as the Klein bottle. A key technical step
1 takeninthispaperistheuseoftheso-calledcross-capstates,whichencodeentirelytheunoriented
0 natureofspacetime,andcanbeobtainedbyrearrangingthespacetimegeometryandexchangingthe
2 roleofspaceandtimecoordinates. Whenthesystemisinanon-trivialSPTphase,wefindthatthe
y correspondingcross-capstateisnon-invariantundertheactionofthesymmetriesoftheSPTphase,
a but acquires an anomalous phase. This anomalous phase, with a proper definition of a reference
M state, on which symmetry acts trivially, reproduces the known classification of (2+1)-dimensional
bosonic and fermionic SPT phases protected by reflection symmetry, including in particular the
6 Z8 classificationoftopologicalcrystallinesuperconductorsprotectedbyreflectionandtime-reversal
2 symmetries.

]
l I. INTRODUCTION other hand, gauging (non-spatial) symmetries effectively
e
- deconfinesasetofquasiparticles(anyons). Thefractional
r
statisticsofthebraidinginthegaugedtheorycanbeused
t Symmetry-protected topological (SPT) phases are
s to diagnose the original SPT phases.24,27
. gapped phases of matter which are not adiabatically
t
a deformable, under a given set of symmetry conditions, Twisting non-spatial unitary symmetries of SPT
m to a topologically trivial phase. While SPT phases do phases is by now reasonably well-understood. Toward
- not have an intrinsic topological order (i.e., do not sup- further developments of methodologies to SPT phases,
d port(deconfined)fractionalexcitations),theyaresharply it is necessary to extend the twisting or gauging pro-
n
(topologically) distinct from topologically trivial states, cedure to spatial and/or antiunitary symmetries, such
o
suchasanatomicinsulator,whichrespectthesamesetof as parity symmetry (P-symmetry) and time-reversal (T-
c
[ symmetries. Inotherwords,thedistinctionbetweenSPT symmetry).21–23,26,28 The purpose of this paper is to
and trivial phases cannot be made within Landau’s the- provide an efficient and intuitive method to diagnose
3
ory; SPT phases are beyond the classification of phases SPT phases protected by a spatial symmetry such as
v
5 of matter based on broken symmetries. A flurry of re- parity (reflection). We will study (1+1)-dimensional
8 centtheoreticalworksincludes,amongothers,thebreak- [(1+1)d] non-chiral gapless theories with spatial symme-
2 down (or collapse) of the non-interacting classifications tries, which are the edge theories of the corresponding
7 of fermionic SPT phases upon inclusion of interactions, (2+1)dbulkSPTphasesprotectedbyparity(P)symme-
0 non-trivial bosonic SPT phases and the possibility of try or parity combined with a unitary non-spatial sym-
1. symmetry-respecting surface topological order, classifi- metrysuchasCP-symmetry(paritysymmetrycombined
0 cation of interacting electronic topological insulators in with charge conjugation). In typical situations, in addi-
5 three dimensions, and proposals for a possible complete tion to P- or CP-symmetry, there are other non-spatial
1 classification of SPT phases. (For a partial and incom- symmetries such as U(1) symmetry or time-reversal (T)
:
v pletelistofrecentworksonSPTphases,seeRefs. 1–23.) symmetry.
i
X One of the most efficient and powerful methods to Our starting point is Ref. 26, where twisting parity
study SPT phases is to twist or gauge symmetries pro- within edge theories of SPT phases (protected by par-
r
a tecting SPT phases.9,24,25 Quite generically, global sym- ity) has been used to diagnose the edge theories. This
metries in quantum field theories can be twisted, i.e., twisting procedure leads to theories that are effectively
can be used to define twisted boundary conditions. It defined on an unoriented manifold, e.g., the Klein bot-
was proposed that the twisted theory can be used to di- tle, and thus suggests an interesting link between SPT
agnose the original SPT phases, i.e., to judge whether phases and so-called orientifold field theories - type of
or not the original theory is symmetry-protected, and theories discussed in unoriented superstring theory.29–38
distinct from topologically trivial phases. More specif- Wemakeonefurtherstepinthispaperbymakinguseof
ically, once twisted, the edge theory of an SPT phase the so-called cross-cap states in formulating orientifold
suffers from various kinds of quantum anomalies, such field theories. Cross-cap states are quantum states in
as a global anomaly under large U(1) gauge transfor- fieldtheories(conformalfieldtheories)obtainedbytwist-
mations, or a global gravitational anomaly.9,25,26 On the ingparitysymmetryandencode,quantummechanically,
2
the unoriented topology; the fact that the theory is put anomalies. We also draw some parallel between twist-
onanunorientedsurfaceisentirelyencodedinthecross- ingnon-spatialsymmetries(so-calledorbifoldprocedure)
capstates. Theyareakintoboundarystatesinboundary and twisting parity symmetries (orientifold). In Sec. III,
conformal field theories, which are obtained by exchang- we apply our strategy to the bosonic SPT phases pro-
ing the role of space and time coordinates. One promis- tected by P- or CP-symmetry together with other sym-
ing aspect of cross-cap states is that they are formulated metries,andfindthatthenon-invarianceofthecross-cap
andconstructedfullyintermsofmany-bodyphysics,and statesreproducesallthenon-trivialSPTphasesfoundin
hence are expected to capture the effects of interactions. Ref. 39. InSec.IV,wediscussrealfermionicSPTphases
We will identify quantum anomalies in the edge theo- (topological superconductors) protected by parity sym-
ries of SPT phases as the non-invariance of a cross-cap metry. By identifying quantum anomalies (anomalous
state under the action of the other symmetry than par- phases)inthecross-capstates,theresultsconsistentwith
ity. Correspondingly, a bulk theory which supports an knownmicroscopicanalysisofgappingpotentialsareob-
anomalousedgetheoryisdiagnosedasanon-trivialSPT tained.
phase. Our procedure to diagnose an edge theory with
parity symmetry can be summarized as follows:
II. CROSSCAP STATES AND SPT PHASES
(i): Theedgetheoryofagivenbulktheoryisputonan
unorientedspacetime,theKleinbottle. Inpractice,
In this section, some generalities of our approach to
this can be achieved by twisting boundary condi-
SPT phases are presented. We start by briefly reviewing
tions (orientifold procedure).
twisting and gauging non-spatial unitary symmetries of
(ii): The edge theory on the Klein bottle is then quan- SPTphases, inparticularwithin(1+1)dedgetheoriesof
tized. The cross-cap state corresponding to the (2+1)dSPTphases. Whentheedgetheoriesarerealized
twisting is constructed. at the boundary of non-trivial SPT phases, this twisting
procedure (orbifolding) reveals a conflict between differ-
(iii): In the quantized theory, the effects of other sym-
entsymmetriesatthequantumlevel,evenwhentheyare
metry, such as time-reversal, fermion number par-
mutually consistent at the classical level. This is an ex-
ity, etc. are studied. When these symmetries are
ample of quantum anomalies, signaling the impossibility
anomalous, i.e., when the cross-cap state is non-
of realizing the edge theory on its own once symmetry
invariant under these symmetries, the edge the-
conditions are imposed. Hence, it is also indicative of
orycannotbegappedwhilepreservingthesymme-
the presence of non-trivial bulk states. Twisting unitary
tries. The corresponding bulk phase is symmetry-
spatial symmetry, such as parity and CP symmetry, can
protected and topologically distinct from a trivial
be discussed in a similar way, resulting in an unoriented
phase.
spacetime manifold. Cross-cap states, which are quan-
Inthepreviouswork,26theedgetheoriesofSPTphases tum states fully encoding the unoriented nature of the
with P (cid:111)U(1) symmetry and those with CP (cid:111)U(1) theory,areintroduced. Apotentialconflictofothersym-
A V
metries with parity/CP symmetry can be studied in the
symmetry have been studied. The partition function
language of cross-cap states.
on the Klein bottle was shown to be anomalous (non-
invariant) upon threading a unit flux of the U(1) sym-
metry. Infact,thepartitionfunctionacquirestheanoma-
lous (−1)-sign, implying the Z classification of these A. Edge theories of SPT phases
2
SPT phases. The proposed formulation in this paper
in terms of cross-cap states reproduces these results for By definition, when going from a SPT phase to a triv-
SPTphasewithP(cid:111)U(1) andCP(cid:111)U(1) . Inaddition, ial phase in a phase diagram by changing parameters in
A V
this formulation in terms of cross-cap states offers a few the system’s Hamiltonian, one inevitably encounters a
technical advantages. First, the new formulation allows quantum phase transition, if the symmetry conditions
us to discuss SPT phases protected by a broader set of are strictly enforced. This in turn implies that if an
symmetries than P (cid:111)U(1) or CP (cid:111)U(1) . That is, SPTphaseisspatiallyproximatetoatrivialphase,there
A V
we do not need a continuous U(1) symmetry and a large should be a gapless state localized at the boundary be-
U(1) gauge transformation. Second, it is not necessary tweenthetwophases;thiscriticalstatecanbethoughtof
to compute the full (symmetry-twisted) partition func- as a phase transition occurring locally in space, instead
tion; all information necessary to diagnose edge theories of the parameter space of the Hamiltonian. As implied
are encoded in cross-cap states. by this construction, the edge state of a non-trivial SPT
The rest of the paper is organized as follows. In Sec. phase should never be removable (completely gapped)
II, some generalities on twisting/gauging parity symme- if the symmetries are strictly imposed. (It should be
tries and finding cross-cap states are presented. We fur- noted, however, that there are other interesting pos-
ther demonstrate that non-invariance of the cross-cap sibilities for symmetric surface states for (3+1)d SPT
states under symmetry operations is related to the non- phases.15,16,40,41) Hence, this critical boundary state sig-
invariance of the partition function, i.e., the quantum nals the topological distinction between the SPT and
3
trivial phases, and many properties of SPT phases can Observe that the boundary condition (??) is invariant
be extracted from their boundary physics. For exam- under global U(1) transformations
c
ple, by inspecting under which symmetry conditions a
givenedgetheoryisstable/unstable, onecanpredictun- 0=ψ (x)−e2πiαψ (x+(cid:96))
s s
der which symmetry conditions a given phase can be a =eiφQ(cid:2)ψ (x)−e2πiαψ (x+(cid:96))(cid:3)e−iφQ, (4)
s s
SPT phase.
Although studying the boundary instead of the bulk
where Q is the total charge operator associated to U(1)
c
reduces the dimensionality of the problem, it is still not
symmetry, and φ is an arbitrary real parameter. Simi-
straightforward to judge if a given state is topological
larly, the boundary condition (??) is invariant under the
or not. In principle, one could enumerate all possible global U(1)s transformation generated by eiφSz where
symmetry-allowed perturbations within the edge theory,
S is the total “charge” associated to U(1) symmetry.
z s
which can potentially gap out the edge. Without any
A crucial observation now is that, while the boundary
guiding principle, however, such brute force approach is
condition (??) is invariant under the global U(1) , the
s
quite cumbersome, and also, more fundamentally, does
corresponding ground state may carry an “anomalous”
not provide any intuition on the physics of SPT phases.
S quantum number,
z
Henceitisnecessarytohaveanefficientandilluminating
guiding principle for diagnosing topological properties of eiQφ|GS(cid:105) =ei×0×φ|GS(cid:105) =|GS(cid:105) ,
α α α
phases of matter with symmetries.
eiSzφ|GS(cid:105) =ei×2α×φ|GS(cid:105) =e2iαφ|GS(cid:105) . (5)
α α α
(The quantum numbers of the ground state can be com-
B. Symmetry twist and conflicting symmetries
puted explicitly, by using, for example, bosonization
and identifying an operator corresponding to the ground
Our approach to diagnose non-trivial SPT phases is state |GS(cid:105) . More precisely, the operator is given by
α
to identify quantum anomalies within edge theories. ∼ exp(iα(ϕ +ϕ )), where the left- and right-moving
L R
We illustrate our strategy by considering a fermionic electrons are identified as ψ ∼exp(±iϕ ). For de-
L/R L/R
SPT phase protected by Uc(1)×Us(1) symmetry, where tails, see Ref. 18.) The anomalous phase is nothing but
Uc/s(1)referstoU(1)symmetryassociatedwiththecon- chiral anomaly; in the presence of a U(1)c magnetic flux
servation of electromagnetic charge and z-component of twisting the boundary condition, the S quantum num-
z
spin Sz, respectively. This phase is akin to (2+1)d time- ber, which is conserved at the classical level, is not con-
reversal symmetric topological insulators (the quantum servedatthequantumlevel. Hence,onlywaytoreconcile
spinHalleffect),althoughbecauseoftheimposedconser- the U(1) ×U(1) symmetry and quantum mechanics is
c s
vationofSz,stateswithUc(1)×Us(1)symmetryareclas- to realize this (1+1)d theory as a boundary theory of
sifiedbytwointegraltopologicalinvariants(i.e.,“charge” a higher-dimensional system, a (2+1)d bulk SPT phase
and“spin”Chernnumbers). Wewillfocusonthecaseof respecting the U(1) ×U(1) symmetry.
c s
vanishing charge Chern number, and hence the allowed
Let us now be more general. Consider an edge theory,
phases are classified by the integer-valued spin Chern
whichiswrittenintermsofafundamentalquantumfield
number.
Φ(x)andisdefinedonaspatialcirclex∼x+(cid:96). Suppose
Consider the following Hamiltonian describing the there is a set of non-spatial unitary symmetry operators,
edge of the Sz-conserving quantum spin Hall phase with G, which leave the edge theory invariant. We consider a
Uc(1)×Us(1) symmetry defined on a circle of circumfer- twisting boundary condition by a group element g1 ∈G,
ence (cid:96)
Φ(x+(cid:96))=G Φ(x)G−1 =U ·Φ(x), (6)
(cid:90) (cid:96) (cid:104) (cid:105) 1 1 g1
H = dx ψ†(−vi∂ )ψ +ψ†(+vi∂ )ψ , (1)
↑ x ↑ ↓ x ↓ where G is the operator which implements a symmetry
0 1
operation g in the Hilbert space of the edge theory, and
1
rwehperreesexnt∈s[0th,(cid:96)e]ifsertmheiosnpactrieaaltciooonrdoipneartaetoofrthfoeredugpe/,dψo↑w/n↓ Uthge1 fiiseladuΦn(ixta)r,yi.me.,atarixunaicttairnygmonattrhixerienpterrenseanltiantdioenx ooff
spin, and v is the Fermi velocity. Any global symme- the symmetry. (The field Φ(x) can carry a set of in-
try in quantum field theories can be twisted (i.e., can dices representing internal degrees of freedom, which are
be used to twist boundary conditions). We choose Uc(1) suppressed in the equation above and in the following.)
symmetry to twist the boundary condition as With twisting, states in the Hilbert space, the ground
state |GS(cid:105) in particular, obey
ψ (x)=e2πiαψ (x+(cid:96)), s∈{↑,↓}. (2) g1
s s
[Φ(x+(cid:96))−U Φ(x)]|GS(cid:105) =0. (7)
Letthegroundstatewiththetwistedboundarycondition g1 g1
be |GS(cid:105)α, which satisfies Asanextstep,weconsidertheactionofanothersymme-
tryg ∈G. (g canbeequaltog , whichisthesituation
(cid:2)ψ (x)−e2πiαψ (x+(cid:96))(cid:3)|GS(cid:105) =0, s∈{↑,↓}. (3) 2 2 1
s s α relevant to the quantum Hall effect, but in our examples
4
below, g (cid:54)= g .) At the classical level, the g -twisted C. Spatial symmetry
2 1 1
boundarycondition(??)maybeinvariantunderg ∈G,
2
When considering SPT phases protected by spatial
0=Φ(x+(cid:96))−U Φ(x) symmetries, one can consider twisting the spatial sym-
g1
=G [Φ(x+(cid:96))−U Φ(x)]G−1. (8) metries. Let us consider a parity symmetry:
2 g1 2
PΦ(x)P−1 =U Φ((cid:96)−x), (11)
P
When this is the case, one may expect the twisted the-
ory after quantization is invariant under g as well. In
2 wherethespaceisdefinedonacirclex∼x+(cid:96). Twisting
particular, oneexpectsthepartitionfunctionand/orthe
by parity symmetry can be introduced in the following
ground state of the twisted edge theory is invariant un-
way.29,30,34,36–38
der g . When this expectation is betrayed, there is a
2 a. Loop channel We consider Euclidean spacetime
quantum anomaly.
[0,(cid:96)]×[0,β] parameterized by (x ,x ), where x is the
1 2 1
Before leaving this subsection, we comment on a con-
spatial coordinate and x is the imaginary time coordi-
2
nection between the twisting procedure and orbifold-
nate. In the path integral picture, the fields obey the
ing/gauging. Gauging and orbifolding have a similar
following twisted boundary conditions
(identical) effect in that we focus on a gauge singlet
(G-invariant) sector of the theory [although the gaug- Φ(x ,x +β)=PΦ(x ,x )P−1 =U Φ((cid:96)−x ,x ),
1 2 1 2 P 1 2
ing means in general imposing the singlet condition lo-
Φ((cid:96),x )=GΦ(0,x )G−1, (12)
cally (e.g., at each site of a lattice), while the projection 2 2
in orbifolding is enforced only globally]. Let us again
in which G may be implementing a non-spatial unitary
consider an edge theory with symmetry group G. By
symmetry g ∈ G, e.g., the fermion number parity.26 In
state-operator correspondence in quantum field theories,
the operator language, twisting by parity symmetry can
corresponding to the state |GS(cid:105) in the twisted theory,
g
beintroducedasaprojectionoperation. Thisamountsto
there is an operator, the twist operator σ , which, when
g
inserting the parity operator into the partition function,
acting on the ground state of the untwisted theory, cre-
ates |GS(cid:105) 42–44:
g (cid:104) (cid:105)
ZK =Trg Pe−βHloop((cid:96)) , (13)
|GS(cid:105) =σ (0)|0(cid:105). (9)
g g
whereH ≡H istheHamiltonianthatgeneratestime-
loop
translation in the x -direction. The trace is taken in the
(The location of the insertion of the operator σ is taken 2
g
Hilbert space defined for the quantum field obeying the
to be the origin in the radial quantization.) The twist
boundaryconditionΦ(x +(cid:96))=GΦ(x )G−1,asindicated
operator, when inserted in correlation functions, imple- 1 1
by the subscript g. (For later use, we also introduce
mentsthesymmetrytwistbyg,andsatisfythefollowing
another parity operator, P(cid:48), such that P(cid:48)·P−1 = G.)
algebraic relation with Φ(x) on the complex plane:
This representation of the partition function with the
choiceofx asadirectionoftime-evolutionwillbecalled
Φ(z,z¯)σ (w,w¯)=σ (w,w¯)U ·Φ(z,z¯). (10) 2
g g g the ”loop channel” picture.
b. Tree channel As we have seen in the case of non-
Starting from the original, untwisted, theory, one can spatialsymmetries,itisusefultodiscussthetwistedpar-
now consider including the ground states with twisted tition function in terms of the twist operator and the
boundaryconditionstodefineanextendedtheory. Inthe corresponding state. One may want to develop a similar
extended theory, the ground states with twisted bound- alternative picture for the case of twisting by parity. To
ary conditions (the ground states in the “twisted sec- thisend,oneneedstofindaconvenientandproper“time-
tors”), and hence the corresponding twist operators, are slice” that allows us to define a quantum state (|GS(cid:105) in
g
considered as an excitation. This procedure to generate the notation of the previous section for the non-spatial
anewtheoryfromtheuntwistedtheoryiscalledorbifold. symmetry), which implements the twisting condition.
Now, by further invoking bulk-boundary correspon- It is convenient to recall that any 2d compact unori-
dence, there is a corresponding bulk excitation (anyon). entedsurfacewithoutboundarycanbegeneratedfroma
Bulk statistical properties of the gauged theory can be sphere S2 by adding handles and cross-caps, which can
read off from the operator product expansions and fu- be thought of as a real projective plane RP . In partic-
2
sion rules obeyed by the twist operator(s). Now a given ular, the Klein bottle can be generated by first cutting
symmetrygroupGcanbeimplementedinvariousdiffer- out two discs from S2, and then identifying antipodal
ent ways in different SPT (and trivial) phases protected points of the resulting holes. In fact, one can rearrange
by G leading to different choices of U , and to differ- the path integral on the Klein bottle such that the fields
g
ent twist operators. By studying statistical (braiding) arenowdefinedon[0,(cid:96)/2]×[0,2β]. Theprecisestepsfor
properties of the twist operator(s), one can distinguish rearranging the spacetime is described in Fig.1.
different ungauged (original) theories.24 Weuse(σ ,σ )torepresentthisrearrangedspacetime.
1 2
5
(a) x2 (b) x2 C’ A’ twrhaenrselatHiotnreeinitshethtreeeHcahmanilnteolnpiaicntutrhea.tThgeenloeroaptecshatninmeel--
2𝛽𝛽 tree channel duality (also known as open-closed duality)
assertsthatthepartitionfunctionscomputedintheloop
𝛽𝛽B C A’ B C D’ B’ and tree channels agree.
𝛽𝛽
c. Interplay with non-spatial symmetries By con-
structing the cross-cap states, we have “gauged” parity
(c) A D B’x 1 A D x1 symmetry. As in the case of the SPT phases with non-
x2 (d) spatial symmetries, we now consider the effects of an-
2𝛽𝛽 A𝑙𝑙/’2 C𝑙𝑙’ 𝝈𝝈𝟐𝟐 A’ 𝑙𝑙/2C’ 𝑙𝑙 other non-spatial symmetry, represented by g ∈ G, on
2𝛽𝛽 the cross-cap states. We act with g on the cross-cap

B’ D’ B’ D’ boundary conditions,
B
𝛽𝛽 C B C
𝛽𝛽 [Φ(0,σ +β)−U Φ(0,σ )]|C (cid:105)=0
2 P 2 P
A D x1 A D ⇒G [Φ(0,σ2+β)−UPΦ(0,σ2)]G−1G|CP(cid:105)=0
⇒[U Φ(0,σ +β)−U U Φ(0,σ )]G|C (cid:105)=0
𝝈𝝈𝟏𝟏 g 2 P g 2 P
𝑙𝑙/2 𝑙𝑙 𝑙𝑙/2 ⇒(cid:2)Φ(0,σ +β)−U−1U U Φ(0,σ )(cid:3)G|C (cid:105)=0.
FIG. 1. Rearrangement of spacetime. (a) The original loop 2 g P g 2 P
channel (Euclidean) spacetime. Here the points (A,B) and (18)
(A(cid:48),B(cid:48)) in the spacetime are identified due to the boundary
Thus we deduce the following relation:
conditionsEq.(??). Thelinesegmentswiththesamesymbols
are also identified by the boundary conditions Eq. (??). (b)
G|C (cid:105)=|C (cid:105). (19)
WeshifttheboxofthespacetimeformedbyA(cid:48)B(cid:48)D(cid:48)C(cid:48) along P g·P·g−1
x by β. (c) After shifting the box, we flip the orientation of
2 If the cross-cap condition is invariant under g ∈ G,
theboxwithrespecttox =3(cid:96)/4. (d)Weslidetheboxalong
1 U = U−1U U , then we may expect that so is the
x by(cid:96)/2. Thenweobtainthecrosscapgeometry. Noticethat P g P g
1 cross-cap state, |C (cid:105) = G|C (cid:105), classically. However this
anyx ∈[0,β]isidentifiedwithitsantipodalpointattheslice P P
2
expected invariance may be broken down quantum me-
of“time”atx =0andx =(cid:96)/2. Byrenamingthevariables
1 1
chanically. This then signals that the theory is anoma-
x → σ and x → σ , we obtain the spacetime of the tree
1 1 2 2
channel. Here the σ direction is taken to be a direction lousandshoulddescribetheedgeofaSPTphasedefined
1
of (fictitious) time-evolution. At the boundary of spacetime in one higher dimension.
located at σ =0 and σ =(cid:96)/2, there are cross-caps.
1 1
III. BOSONIC SPT PHASES
The fields now obey the cross-cap boundary conditions
In this section, we apply our strategy above to edge
Φ(0,σ +β)=U Φ(0,σ ), theoriesofbosonicSPTphasesconsistingofasinglecom-
2 P(cid:48) 2
ponentnon-chiralboson(i.e.,atwo-componentchiralbo-
Φ((cid:96)/2,σ +β)=U Φ((cid:96)/2,σ ),
2 P 2
son with 2×2 K-matrix). We consider situations where
Φ(σ1,σ2+2β)=G(cid:48)Φ(σ1,σ2)G(cid:48)−1, (14) parity(P)oracombinationofparityandchargeconjuga-
tion(CP)isapartofthefullsymmetrygroup,whichcan
where
potentially protect the edge theory from gap-opening.
P2 =P(cid:48)2 =G(cid:48). (15) In Ref. 26, SPT phases protected by P or CP sym-
metry together with a continuous U(1) symmetry were
By further rotating (σ1,σ2) coordinates by 90◦, we can studiedbyusingageneralizationofLaughlin’sargument.
take σ1 as a time-coordinate. In this coordinate system, Itwasfoundthatnon-trivialZ2 bosonicSPTphasescan
the time-evolution is generated by a (fictitious) Hamil- exist in the presence of the following combinations of
tonian (denoted by Htree in the following). The fact symmetries:
that the system is defined on an unoriented surface is
• P (cid:111)U(1) (“spin U(1)”),
now encoded in boundary conditions at σ = 0 and A
1
σ1 =(cid:96)/2. Wethenintroducecross-capstateswhichobey • CP (cid:111)U(1) (“charge U(1)”),
V
these boundary conditions as:
wherethesubscriptA/V representsthe“axial/vectorial”
[Φ(0,σ2+β)−UP(cid:48)Φ(0,σ2)]|CP(cid:105)=0, nature of the U(1) symmetry.
[Φ((cid:96)/2,σ +β)−U Φ((cid:96)/2,σ )]|C (cid:105)=0. (16) In Ref. 39, following the spirit of Refs. 17 and 45, a
2 P 2 P(cid:48)
microscopicstabilityanalysisforthebosonicedgetheory
The partition function can be written in terms of the iscarriedoutbyenumeratingpossiblegappingpotentials
cross-cap states as in the presence of various discrete symmetries. It was
found that the bosonic edge theory can be ingappable
ZK =(cid:104)CP(cid:48)|e−2(cid:96)Htree(2β)|CP(cid:105), (17) (protected) in the presence of the following symmetries:
6
• P ×C,P ×T given by
• P ×TC, CP ×T, T ×C. 1 k 1 R
p= (p +p )= , p˜= (p −p )= w,
2 L R R 2 L R α(cid:48)
SPTs protected by these symmetries are all classified by k R k R
Z2. In addition, it was also found that there are SPT pL = R + α(cid:48)w, pR = R − α(cid:48)w, (25)
phases protected by
where k and w are an integer. In terms of these momen-
• T ×C×P tum eigenvalues, the compactification conditions on the
boson fields are
The classification of these SPT phases is Z4.
2
Inthefollowing,wewillstudytheseSPTphasesbyus- ϕ (x+(cid:96))−ϕ (x)=+πα(cid:48)p ,
L L L
ing cross-cap states and by identifying quantum anoma- ϕ (x+(cid:96))−ϕ (x)=−πα(cid:48)p ,
R R R
lies. Aswewillshow,thisanalysisreproducesexactlythe
φ(x+(cid:96))−φ(x)=πα(cid:48)(p −p )=2πRw,
same classification as in Ref. 39, and hence it gives us a L R
perspective complimentary to the generalized Laughlin’s θ(x+(cid:96))−θ(x)=πα(cid:48)(p +p )=2πα(cid:48)k. (26)
argument on the Klein bottle in the “loop channel” pic- L R R
ture, and to microscopic stability analysis.
The Hilbert space is constructed as a tensor product
of the bosonic oscillator Fock spaces, each of which gen-
erated by pairs of creation and annihilation operators
A. Free compactified boson
{α ,α } and {α˜ ,α˜ } , and the zero mode
m −m m>0 m −m m>0
sectorassociatedtox andp . Wewilldenotestates
L,R L,R
We start from the free boson theory on a spatial ring in the zero mode sector by specifying their momentum
of circumference (cid:96) defined by the partition function Z = eigenvalues as
(cid:82)
D[φ]exp(iS) with the action
|p,p˜(cid:105)=|k/R,Rw/α(cid:48)(cid:105), k,w ∈Z, (27)
1 (cid:90) (cid:90) (cid:96) (cid:20)1 (cid:21)
S = dt dx (∂ φ)2−v(∂ φ)2 , (20) ormoresimplyas|k,w(cid:105). Alternatively,theFouriertrans-
4πα(cid:48) v t x
0 formation of the momentum eigenkets defines the “posi-
tion” eigenkets, which we denote by
where the spacetime coordinate of the edge theory is de-
noted by (t,x), v is the velocity, α(cid:48) is the coupling con- |φ ,θ (cid:105) 0<φ ≤2πR, 0<θ ≤2πα(cid:48)/R. (28)
0 0 0 0
stant, and the φ-field is compactified as
The two basis are related by
φ∼φ+2πR, (21)
(cid:90) 2πR (cid:90) 2πα(cid:48)/R
|p,p˜(cid:105)= dφ dθ e−ipφ0−ip˜θ0|φ ,θ (cid:105). (29)
with the compactification radius R. The canonical com- 0 0 0 0
0 0
mutation relation is
(cid:88)
[φ(x,t),∂tφ(x(cid:48),t)]=i2πα(cid:48)v δ(x−x(cid:48)−n(cid:96)). (22) B. Symmetries
n∈Z
Various symmetries of the single-component compact-
We use the chiral decomposition of the boson field, and
ified boson theory are listed below.
introduce the dual field θ as
d. U(1)×U(1) symmetry In the free boson theory,
when there is no perturbation, there are two conserved
φ=ϕ +ϕ , θ =ϕ −ϕ . (23)
L R L R U(1) charges, one for each left- and right-moving sector,
defined by
Themodeexpansionofthechiralbosonfieldsisgivenby
(x± =vt±x) (cid:90) (cid:96)
N = dx∂ ϕ =α(cid:48)πp , (30)
L,R x L,R L,R
ϕL(x+)=xL+πα(cid:48)pLx(cid:96)+ +i(cid:114)α2(cid:48) n(cid:88)(cid:54)=0αnne−2πin(cid:96)x+, They satisfy 0
n∈Z
ϕR(x−)=xR+πα(cid:48)pRx(cid:96)− +i(cid:114)α2(cid:48) n(cid:88)(cid:54)=0α˜nne−2πin(cid:96)x−, Corresponding[lϕyLt,oNtLh]e=se[ϕcoRn,sNerRv]e=d qαu(cid:48)πani.tities, the(f3r1ee)
n∈Z
bosontheoryisinvariantunderthefollowingU(1)×U(1)
(24)
symmetry
where [α ,α ] = [α˜ ,α˜ ] = mδ and [x ,p ] =
m −n m −n mn L L U :φ→φ+δφ, θ →θ+δθ,
[x ,p ] = i. The compactification condition on the bo- δφ,δθ
R R
:ϕ →ϕ +δϕ , ϕ →ϕ +δϕ . (32)
sonfieldsimpliestheallowedmomentumeiganvaluesare L L L R R R
7
In terms of the conserved charges, the generators of the C. Crosscap States
U(1)×U(1) transformations are given by
UL =eiδϕLNL/(α(cid:48)π) =eiδϕLpL, We now move on to the tree channel picture and con-
δϕL struct cross-cap states by twisting P- and CP- symme-
UR =eiδϕRNR/(α(cid:48)π) =eiδϕRpR, tries. After rearranging spacetime and exchanging the
δϕR
role of space and time, the spacetime is, space×time=
U =UL UR =ei(δφp+δθp˜). (33)
δφ,δθ δϕL δϕR 2β×(cid:96)/2. We parameterize this Euclidean spacetime by
Note also that U acts on the momentum eigenkets (σ ,σ ). Theoriginalspacetimeof[0,(cid:96)]×[0,β]isparam-
δφ,δθ 1 2
as eterized by (x ,x ) (for the mapping between the two
1 2
spacetime see Fig. 1).
U |p,p˜(cid:105)=ei(pδφ+p˜δθ)|p,p˜(cid:105). (34)
δφ,δθ
e. C-symmetry Particle-hole symmetry or charge
conjugation (C-symmetry) is unitary and acts on the 1. P-symmetry
bosonic fields as
m πα(cid:48) Wefirstconsiderthecross-capstateobtainedbytwist-
C :φ→−φ+ncπR, θ →−θ+ cR ing parity symmetry, defined in Eq. (??). The corre-
sponding cross-cap states are defined by
:(x ,x )→(x ,x ), (35)
1 2 1 2
(cid:20) (cid:21)
where (n ,m )∈{0,1}. From these transformation laws φ(σ )−φ(σ +β)−n πR |C (n ,m )(cid:105)=0,
c c 2 2 p p p p
ofthebosonfields,wereadofftheactionofC-symmetry
on the position basis as (cid:20)θ(σ )+θ(σ +β)− mpπα(cid:48)(cid:21)|C (n ,m )(cid:105)=0. (41)
2 2 R p p p
C|φ ,θ (cid:105)=eiδ|−φ +n πR,−θ +m πα(cid:48)/R(cid:105), (36)
0 0 0 c 0 c
By mode-expansion, the cross-cap condition (??) trans-
where eiδ is an unknown phase factor. In order to have
latesintothecorrespondingconditionforeachmode. To
the relation C|p,p˜(cid:105)∝|−p,−p˜(cid:105), expected from the com-
findanexplicitformofthecross-capstates,wefirstfocus
mutationrelationbetweenC andp,p˜,thephaseδ hasto
on the zero mode sector of the boson fields:
be a constant (independent of φ and θ ). The action of
0 0 πα(cid:48)p˜σ
C-symmetry on the momentum eigenstates is given by φ(σ )=x +x + 2 +··· ,
2 L R β
C|p,p˜(cid:105)=eiδe−ipncπR−ip˜mcRπα(cid:48)|−p,−p˜(cid:105) θ(σ )=x −x + πα(cid:48)pσ2 +··· . (42)
=eiδe−iπknc−iπwmc|−p,−p˜(cid:105), (37) 2 L R β
Withinthezeromodesector, wesolvethecross-capcon-
where p = k/R and p˜ = wR/α(cid:48). Since δ is constant,
ditions (??). The first condition (??) can be reduced to
the phase ambiguity is fixed once we specify the action
ofC onareference state,e.g.,|p,p˜(cid:105)=|0,0(cid:105). Inouranal- πα(cid:48)p˜−n πR=0 mod 2πR
p
ysis presented below, the reference state and its charge
R
conjugation parity eiδ plays an important role. ⇒p˜= (2N +n ), N ∈Z. (43)
α(cid:48) p
f. T-symmetry Antiunitary time-reversal operator
T acts on the boson fields as Similarly, the second condition can be solved as:
T :φ→φ+nTπR, θ →−θ+ mTRπα(cid:48) 2(xL−xR)+ 2παβ(cid:48)pσ2 +πα(cid:48)p= mpRπα(cid:48) mod 2πRα(cid:48)
:(x ,x )→(x ,−x ), (38) πα(cid:48)
1 2 1 2 ⇒p=0, (x −x )=(2N˜ +m ) , N˜ ∈Z. (44)
L R p 2R
where n ,m ∈{0,1}.
T T
g. P-symmetry Parity P is defined by Solving the conditions (??) and (??), we find the cross-
cap state |C (n ,m )(cid:105) in terms of the momentum eigen-
m πα(cid:48) p p p
P :φ→φ+n πR, θ →−θ+ p ket {|p,p˜(cid:105)} as
p R
(cid:88)
:(x1,x2)→((cid:96)−x1,x2), (39) |Cp(np,mp)(cid:105)= (−1)mpN|0,2N +np(cid:105). (45)
N∈Z
where n ,m ∈{0,1}.
p p
Thefullcross-capstateisobtainedbyincludingtheparts
h. CP-symmetry The above symmetries can be
related to the oscillatory modes:
combined. For example, CP-symmetry is a non-local
unitary symmetry, and defined by (cid:113) √ (cid:34) (cid:88)∞ (−1)n (cid:35)
R 2exp − α α˜ |C (n ,m )(cid:105). (46)
CP :φ→−φ+ncpπR, θ →θ+ mcRpπα(cid:48) n=1 n −n −n p p p
For our purpose of diagnosing SPT phases, however, it
:(x ,x )→((cid:96)−x ,x ), (40)
1 2 1 2 turns out that it is enough to focus on the zero-mode
where n ,m ∈{0,1}. sector.
cp cp
8
2. CP-symmetry which is invariant when
2πα(cid:48)
Next we consider CP-symmetry. The corresponding 2δθ =0 mod . (52)
R
cross-cap conditions are given by
Thus, at least classically, we expect the theory to be in-
(cid:20) (cid:21)
variantwhenδθ =πα(cid:48)/R. Ontheotherhand,thisinvari-
φ(σ )+φ(σ +β)−n πR |C (n ,m )(cid:105)=0,
2 2 cp cp cp cp ance may not be maintained at the quantum level. The
(cid:20)θ(σ2)−θ(σ2+β)− mcRpπα(cid:48)(cid:21)|Ccp(ncp,mcp)(cid:105)=0. carnodssc-acnappicstkautepamnaaynonmotalboeusinpvhaarsiea;notnuencdaenreaUsδiAlθy=cπhαe(cid:48)/cRk
(47) UA |C (n ,0)(cid:105)=(−1)np|C (n ,0)(cid:105). (53)
πα(cid:48)/R p p p p
By solving these conditions within the zero mode sector, Thus the edge theory, when enforced parity symme-
we obtain, as the cross-cap state, try with n = 1, is anomalous for U(1) symme-
p A
try. This invariance/non-invariance is equivalent to the
(cid:88)
|C (n ,m )(cid:105)= (−1)ncpN|2N +m ,0(cid:105). (48) invariance/non-invariance of the Klein bottle partition
cp cp cp cp
N∈Z function under large gauge transformations discussed in
Ref. 26. Observe also that the anomalous phase here is
Z -valued (i.e., a sign), and this implies that the classifi-
2
D. Diagnosis of SPT phases cation is Z since the two copies of the theory is trivial.
2
This agrees with the loop channel calculation.26,39
1. P (cid:111)U(1) and CP (cid:111)U(1) symmetries SPT phases protected by CP (cid:111) U(1) can be dis-
A V V
cussed in the same manner as P (cid:111)U(1) . The cross-cap
A
We start with the case of the symmetry group P (cid:111) state obtained by twisting CP is given by |Ccp(0,mcp)(cid:105)
U(1)A, which has been studied in Ref. 26 by using a in Eq. (??). The U(1)V symmetry is generated by
generalization of Laughln’s gauge argument. In the gen- UV =exp(ipδφ) as
δφ
eralized Laughlin’s argument, the edge theory is put on
an unoriented spacetime, such as the Klein bottle, and UδVφ :θ →θ, φ→φ+δφ. (54)
then the invariance under flux threading (a large gauge
Underthissymmetryaction,thecross-capcondition(??)
transformationofU(1) symmetry)ofthepartitionfunc-
A with n =0 is invariant when
tion of the edge theory is investigated. More specifically, cp
the analysis in Ref. 26 is performed in the loop-channel
2δφ=0 mod 2πR. (55)
channel picture. In the following, we will reproduce this
result in terms of the tree-channel calculations, i.e., by On the other hand, we find
using the cross-cap state.
Therelevantcross-capstateis|Cp(np,0)(cid:105)presentedin UδVφ=πR|Ccp(0,mcp)(cid:105)=(−1)mcp|Ccp(0,mcp)(cid:105). (56)
Eq. (??). We set m = 0 since n ∈ {0,1},m = 0
p p p
is enough to classify the P (cid:111) U(1) -symmetric SPT Thustheedgetheorywithparitysymmetrywithmcp =1
A and U(1) is anomalous.26,39 Here the anomalous phase
phases according to Ref. 39. However, it is straightfor- V
is the Z sign, and thus classification is Z as in the
ward to generalize the analysis below to more general 2 2
previous case.
sets of {n ,m }. The U(1) symmetry is generated by
p p A
UA =exp(ip˜δθ): Theaboveanalysisintermsofthecross-capstatesisa
δθ reformulation of Ref. 26; while in Ref. 26 the Klein bot-
UA :θ →θ+δθ, φ→φ. (49) tle partition functions are computed in the loop channel
δθ picture, here we have considered and studied the cross-
LetusactwithUA onthecross-capcondition(??). The cap states in the tree channel. A few technical remarks
cross-capconditioδnθforφistriviallyinvariantunderUA. are in order. (i) We have considered adiabatic trans-
δθ formations of the cross-cap states by acting with UA
For the condition written in terms of the dual field θ δθ
or UV and by continuously changing δθ or δφ, respec-
(with mp =0), δφ
tively. In terms of the original, loop channel picture,
UA{θ(σ )+θ(σ +β)}(UA)−1UA|C (n ,m )(cid:105)=0 these twisting parameters should appear as a twisting
δθ 2 2 δθ δθ p p p
angle in twisting boundary conditions in the spatial di-
⇒{θ(σ )+θ(σ +β)+2δθ}UA|C (n ,m )(cid:105)=0.
2 2 δθ p p p rection. This can be seen explicitly as follows. By using
(50) the formula G|C (cid:105)=|C (cid:105), the partition function in
P gPg−1
the tree channel can be written as,
Thus, the cross-cap condition is transformed into
θ(σ )+θ(σ +β)=0 ZK =(cid:104)CP(cid:48)|e−2(cid:96)HtreeUδAθ|CP(cid:105)
2 2
⇒θ(σ2)+θ(σ2+β)+2δθ =0, (51) =(cid:104)CP(cid:48)|e−2(cid:96)Htree|CUδAθ·P·U−Aδθ(cid:105). (57)
9
This can be written in the loop channel as toformthecross-capstate|C ,(n ,m )(cid:105)andthenstudy
cp p p
the action of C on the cross-cap:
ZK =Tr e−βHloop (58)
P(cid:48)UAP−1UA
δθ −δθ C|C (n ,m )(cid:105)=eiδ(cid:48)(−1)npmp|C (n ,m )(cid:105), (61)
cp p p cp p p
BynotingUAP−1UA =P−1UA ,andtakingP(cid:48) =P,
δθ −δθ −2δθ
where eiδ(cid:48) is an overall phase factor, which, as in Eq.
ZK =TrUA e−βHloop, (59) (??), cannot be determined. This again generates the
−2δθ
same classification as Eq. (??).
i.e.,2δθ (notδθ)appears,intheloopchannel,asatwist-
Thissituationshouldbecontrastedwiththecaseswith
ing angle for a spatial boundary condition. When 2δθ
a continuous U(1) symmetry; in the latter case, one can
is an integer multiple of 2π, the twisted system is large
compare the phase acquired by the cross-cap state when
gauge equivalent to the system without twisting. acting with UV and UV ; One can follow the evolu-
(ii) The current analysis in the tree-channel picture δθ δθ+πR
tion of UV|C (n ,m )(cid:105) adiabatically. In the present
has a few advantages over the loop-channel calculations. δθ cp cp cp
case, becauseofthe discretenatureofC-transformation,
First of all, in Ref. 26, we rely heavily on the exis-
such comparison is not possible.
tence of a continuous U(1) symmetry (either U(1) or
A
U(1) )asinLaughlin’sthoughtexperimentinthequan-
V
tum Hall effect. Therefore, it is not entirely obvious how
3. P ×T symmetry
the methodology in Ref. 26 can be generalized to SPT
phases which lack continuous (U(1)) symmetries. The
We now consider the cases where we have both P/CP
reformulation in the tree channel and in term of cross-
and T/CT symmetries. As in the case of P ×C, we can
capstates, however, indicatesanaturalwaytodealwith
first twist P/CP to obtain the corresponding cross-cap
SPTswithoutU(1)symmetry. Infact,theprocedurede-
state. The action of the remaining symmetry, T/CT, on
scribed above can be generalized to cases without U(1)
the cross-cap state can be studied. However, unlike C-
symmetry. (See the following sections.) Second, in the
symmetry,whichisanon-spatialunitarysymmetry,how
currentreformulationinthetreechannel,thereisnoneed
T/CT symmetry acts in the tree-channel is non-trivial.
tocomputethefullpartitionfunction,althoughitispos-
We illustrate this point for the case of P ×T symmetry.
sibletocomputethepartitionfunctionbyusingcross-cap
Let us consider the bosonic edge theory in the pres-
states.
ence of P-symmetry with m =0 and T-symmetry with
p
n = 0. We twist P-symmetry and consider the cor-
T
responding cross-cap state |C (n ,0)(cid:105) in Eq. (??). One
2. P ×C symmetry p p
can check the cross-cap condition is left invariant under
T-symmetry: the fields obey, in the loop channel, the
Todemonstratethatourmethodologyworksintheab-
following boundary conditions:
sence of a continuous U(1) symmetry, we now consider
SPTphasesprotectedbyP×C. Heretherelevantcross- φ(x ,x +β)≡φ((cid:96)−x ,x )+n πR,
1 2 1 2 p
cap state is |C (n ,m )(cid:105) in Eq. (??). We consider C-
p p p φ(x +(cid:96),x )≡φ(x ,x ),
1 2 1 2
symmetry (??) with n =m =0. One can check easily
c c
θ(x ,x +β)≡−θ((cid:96)−x ,x ),
that under the action of C-symmetry, the cross-cap con- 1 2 1 2
dition(??)isinvariant. Ontheotherhand,thecross-cap θ(x1+(cid:96),x2)≡θ(x1,x2). (62)
state may not, as one can check explicitly as
Under time-reversal, these conditions are transformed
C|C (n ,m )(cid:105)=eiδ(−1)npmp|C (n ,m )(cid:105), (60) into:
p p p p p p
by using Eq. (??). To judge if the cross-cap state is φ(x1,−x2)≡φ((cid:96)−x1,β−x2)+npπR,
anomalous or not, we need to know the overall phase φ(x +(cid:96),β−x )≡φ(x ,β−x ),
1 2 1 2
eiδ, which originates from our ignorance on the charge
πα(cid:48) πα(cid:48)
conjugation parity of the zero mode sector. Depending −θ(x ,−x )+m ≡θ((cid:96)−x ,β−x )−m ,
1 2 T R 1 2 T R
on our choice of δ, either one of phases with (n ,m ) =
p p πα(cid:48) πα(cid:48)
n(0a,t0u)r,a(l0,c1h)o,i(c1e,0w)ouolrd(nbpe,emiδp)==1,(1a,s1)thisisamnoemanaslowuse.aAs- −θ(x1+(cid:96),β−x2)+mT R ≡−θ(x1,β−x2)+mT R .
(63)
sign the charge conjugation parity +1 to the reference
state |p,p˜(cid:105) = |0,0(cid:105). With this choice, the edge the-
For example, we can derive the first line of Eq.(??) from
ory with the symmetry group P ×C is anomalous when
the first line of Eq.(??) by following steps.
(n ,m )=(1,1): thisresultisconsistentwiththemicro-
p p
scopic analysis given in Ref. 39. The phase acquired by φ(x1,x2+β)≡φ((cid:96)−x1,x2)+npπR,
thecross-capstateundertheactionofC-symmetryisZ2 ↔φ(x1,y2)≡φ((cid:96)−x1,y2−β)+npπR,
(i.e., sign), and hence the classification is Z . Further-
2 →φ(x ,−y )≡φ((cid:96)−x ,−y +β)+n πR,
more, one may twist CP symmetry, which can be gener- 1 2 1 2 p
↔φ(x ,−x )≡φ((cid:96)−x ,β−x )+n πR, (64)
atedbythemultiplicationofC andP,i.e.,CP =C×P, 1 2 1 2 p
10
where we renamed the variable y2 = x2 +β inbetween Thus C(cid:103)P or the original time-reversal symmetry is
the first and the second lines, and we acted T-symmetry anomalous. This result is consistent with the gapping
inbetweenthesecondandthethirdlines. Withthecom- potential analysis in Ref. 39.
pactification conditions, these are equivalent to In Appendix A, we present identification of quantum
anomalies using cross-cap states for other bosonic SPT
φ(x1,−x2)≡φ((cid:96)−x1,β−x2)+npπR, phases studied in Ref. 39.
φ(x +(cid:96),β−x )≡φ(x ,β−x ),
1 2 1 2
θ(x ,−x )≡−θ((cid:96)−x ,β−x ),
1 2 1 2 IV. REAL FERMIONIC SPTS
θ(x +(cid:96),β−x )≡θ(x ,β−x ), (65)
1 2 1 2
Inthissection,wediscussedgetheoriesof(2+1)dtopo-
which, with mere relabeling β−x →x , are equivalent
2 2 logical superconductors in the presence of discrete sym-
to the original boundary conditions.
metries. In particular, we consider topological supercon-
SinceT-transformationleavestheP-twistedboundary
ductors with reflection symmetry in symmetry classes
condition invariant, we expect the cross-cap state is also
D+R and BDI + R , and their CPT partners dis-
invariant under T-transformation, at least up to a phase + ++
cussed in Refs. 46–48, At quadratic level, i.e., in the ab-
factor. (Recall that we are after this possible anomalous
sence of interactions, topological classification for these
phase of the cross-cap state.) To compute this phase,
symmetryclassesarefoundtobeZ andZ, respectively.
we need to know how T-transformation looks like in 2
For the latter, it was found in Ref. 11, the Z classifi-
the tree channel. In the tree channel, T-transformation
cation collapses into Z in the presence of interactions.
should look like a parity transformation (in fact, CP- 8
Ourdiscussioninthissectioncantriviallybeextendedto
transformation, since T flips the sign of φ):
other fermionic SPT phases supporting complex (Dirac)
fermion edge modes.
φ(σ ,σ )→−φ(σ ,2β−σ ),
1 2 1 2
πα(cid:48)
θ(σ ,σ )→θ(σ ,2β−σ )+m . (66)
1 2 1 2 T R A. Majorana fermion edge states
This CP-transformation in the tree-channel picture, ob-
Consider an edge theory of a (topological) supercon-
tained from T-transformation in the loop-channel pic-
ductor consisting of N flavors of non-chiral real (Majo-
ture, is denoted by C(cid:103)P in the following. If the phase f
rana) fermions described by the Hamiltonian
field φ and its dual θ obey the standard commutation
relation, this C(cid:103)P-transformation must be unitary, i.e., Nf (cid:90) (cid:96)
(cid:88)
it must preserve, in particular, the Heisenberg algebra H = dx [ψa(−vi∂ )ψa +ψa(+vi∂ )ψa]. (70)
L x L R x R
obeyed by the zero modes, [xL,pL] = [xR,pR] = i. If a=1 0
it were defined in the original coordinates (x ,x ), this
1 2 The fermi velocity v is set to be identical for all fermion
unitary C(cid:103)P-transformation is CPT-dual of T-symmetry. flavors for simplicity. The fermion fields obey the canon-
Since T-transformation in the loop channel preserves ical anticommutation relations
theboundarycondition,thisshouldbesointreechannel
(cid:88)
as well. The cross-cap conditions {ψa(x),ψb(x(cid:48))}=2πδab δ(x−x(cid:48)+(cid:96)m),
L L
m∈Z
[φ(σ )−φ(σ +β)−n πR]|C (n ,0)(cid:105)=0, (cid:88)
2 2 p p p {ψa(x),ψb(x(cid:48))}=2πδab δ(x−x(cid:48)+(cid:96)m). (71)
R R
[θ(σ )+θ(σ +β)]|C (n ,0)(cid:105)=0, (67)
2 2 p p m∈Z
The(1+1)dnon-chiralfermionicedgetheory(??)canbe
are transformed into, by the C(cid:103)P transformation,
realized at the edge of (2+1)d topological superconduc-
tors in symmetry classes DIII, D+R and BDI + R .
[−φ(2β−σ2)+φ(β−σ2)−npπR]C(cid:103)P|Cp(np,0)(cid:105)=0, The fermionic edge theory (??) is i+nvariant under +th+e
(cid:104) πα(cid:48) following three symmetries:
θ(2β−σ )+m
2 T R (i)Thefermionnumberparityconservation,wherethe
πα(cid:48)(cid:105) fermion parity operator is given by
+θ(β−σ2)+mT R C(cid:103)P|Cp(np,0)(cid:105)=0. (68)
Nf
(cid:88)
Due to the compactification condition, these cross-cap Gf =(−1)F, F = Fa, (72)
conditions are equivalent to the original conditions. a=1
While the crosscap conditions are preserved by C(cid:103)P, the where F is the total fermion number operator for the
a
cross-cap state |C (n ,0)(cid:105) (??) is not invariant when a-th flavor,
p p
n =m =1,
p T 1 (cid:90) (cid:96)
F = dxiψaψa. (73)
C(cid:103)P|Cp(np,0)(cid:105)=(−1)mTnp|Cp(np,0)(cid:105). (69) a 2π 0 L R