A Bosonic Analog of a Topological Dirac Semi-Metal: Effective Theory, Neighboring Phases, and Wire Construction

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by  Matthew F. Lapa

ABosonicAnalog ofaTopologicalDiracSemi-Metal: Effective Theory, Neighboring Phases, and
WireConstruction
Matthew F. Lapa,1 Gil Young Cho,1,2 and Taylor L. Hughes1
1Department of Physics and Institute for Condensed Matter Theory,
University of Illinois at Urbana-Champaign, 61801-3080
2Department ofPhysics, KoreaAdvanced InstituteofScienceandTechnology, Daejeon305-701, Korea
(Dated:December6,2016)
Weconstructabosonicanalogofatwo-dimensionaltopologicalDiracSemi-Metal(DSM).Thelow-energy
6 description of the most basic 2D DSM model consists of two Dirac cones at positions k0 in momentum
1 space. ThelocalstabilityoftheDiracconesisguaranteedbyacompositesymmetryZ2TI±,where istime-
T
0 reversaland isinversion.Thismodelalsoexhibitsinterestingtime-reversalandinversionsymmetrybreaking
I
2 electromagneticresponses.InthisworkweconstructabosonicversionbyreplacingeachDiracconewithacopy
oftheO(4)NonlinearSigmaModel(NLSM)withtopologicalthetatermandthetaangleθ = π. Onecopy
c ±
ofthisNLSMalsodescribesthegaplesssurfaceterminationofthe3DBosonicTopologicalInsulator(BTI).We
e
D computethetime-reversalandinversionsymmetrybreakingelectromagneticresponsesforourmodelandshow
thattheyaretwicethevalueonegetsintheDSMcasematchingwhatonemightexpectfrom,forexample,a
5 bosonicCherninsulator. WealsoinvestigatethestabilityoftheBSMmodelandfindthatthecompositeZ2TI
symmetryagainplaysanimportantrole. Alongthewayweclarifymanyaspectsofthesurfacetheoryofthe

] BTIincludingtheelectromagneticresponse,thechargesandstatisticsofvortexexcitations,andthestabilityto
l
e symmetry-allowedperturbations. Webrieflycommentontherelationbetweenthevariousdescriptionsofthe
- O(4)NLSMwithθ=πusedinthispaper(adualvortexdescriptionandadescriptionintermsoffourmassless
r
t fermions)andtherecentlyproposeddualdescriptionoftheBTIsurfaceintermsof2+1dimensionalQuantum
.s Electrodynamicswithtwoflavorsoffermion(N = 2QED3). InasetoffourAppendixeswereviewsomeof
t thetoolsusedinthepaper,andalsoderivesomeofthemoretechnicalresults.
a
m
- I. INTRODUCTION systemdoesnotformasuperconductor(weneedtopreserve
d
U(1) ). Withtheseconditionsthe2+1-dDSMformsarobust
n c
topologicalsemi-metalphase. Interestingly,if we relax con-
o Massless 2+1-d Dirac fermions are one of the most well-
c studiedsystemsincondensedmatterphysics. Suchfermions dition (ii) then the system will form a gapped insulator, but
[ willtypicallyhaveanunusualelectromagneticresponse(e.g.,
often appear in relativistic field theories1, but more impor-
aquantumanomalousHalleffect6orachargepolarization7).
3 tantlyareknowntobethelow-energydescriptionoftheelec-
v tronicstructuresome2Dmaterials,e.g.,graphene2,andasthe We can find examples of systems with an odd number of
2 effectivetheoryofthesurfacestatesoftime-reversalinvariant massless Dirac cones as well. If we do not require time-
3
3+1-d topologicalinsulators3. In fact, in the latter two con- reversalsymmetrythenthereexist2+1-dlatticemodelswhich
6
texts alone, there have been thousandsof articles in the past have an odd number of Dirac cones, e.g., a Chern insula-
2
0 decadethatdiscussthepropertiesofthisfermionsystem. tor model tuned to the topological critical point represents
. Theimpetusfortheintensefocuson2+1-dDiracfermions such a system6. On the other hand, there is another way to
1
0 wastheexperimentaldiscoveryofgraphene2. Yearsearlier4,5 avoidtheFermiondoublingtheoremwhilemaintainingtime-
reversal( ). However,thisrequiressomethingmoredrastic,
6 it had been theoretically predicted that the electronic band
T
1 structure of graphene near the Fermi-level would be linear i.e., we can produce an odd number of 2+1-d Dirac cones,
: dispersing, gapless cones, i.e., massless Dirac fermions. In- and maintain , by considering the surface of a 3+1-d -
v T T
invariant(electron)topologicalinsulator(TI).Thenon-trivial
i deed,theuniquesignatureoftheDiracfermionswasquickly
X confirmed in quantum Hall measurements on graphene2. Z2 3+1-d topological phase is known to have an odd num-
r GrapheneitselfhasfourDiraccones,twomorethanthemin- berofmasslessDiracconesonitssurfacewithacharacteris-
a ticspin-momentumlockingfeatureofthestatesontheFermi
imum of two required to satisfy the Fermion doubling theo-
surface. Additionally, there must be at least one massless
remin2+1-dsystemswithtime-reversalinvariance.Thisthe-
Diracconelocatedata time-reversalinvariantmomentumin
orem implies that a 2+1-d material with time-reversal sym-
theBrillouinzone. Thisisunlikethegeneric2+1-dDSMfor
metry cannot harbor an odd number of gapless Dirac cones.
whichtheDiracconescanexistatarbitrarypointsintheBril-
Hence, the system will have a semi-metallic nature with an
louin zone8 It is well-knownthat theories with an odd num-
evennumberofpoint-likeFermisurfaces,andisoftenreferred
berof2+1-dmasslessDiracconestypicallyexhibittheparity
toasatopologicalDiracsemi-metal(DSM).Remarkably,this
anomaly1, and there are usually subtle features that must be
2+1-d (semi-)metal is relatively stable upon the requirement
carefully examined when considering the properties of such
ofsomeadditionalconstraints:(i)inter-conescatteringacross
systems.
theBrillouinzoneissuppressed(translationsymmetryissuf-
ficient for non-interacting fermions), (ii) intra-cone gapping More recently there have been rapid developments in un-
termsareforbidden(minimallywe needthe compositesym- derstanding symmetry-protected topological (SPT) phases
metryoftime-reversalcombinedwithinversion),and(iii)the with interactions9–26. One development in which we are
2
particularly interested is the prediction that there could be agaplesssemi-metalphasefromthevalueofitspolarization
bosonicanalogsoftheelectrontopologicalinsulators. Some response.Finally,inSec.VIIweprovidethedetailsoftheap-
examples are the Bosonic Integer Quantum Hall Effect propriatewirebundlesandcouplingstogeneratethebosonic
(BIQHE)15,16,27–33 andthe3D -invariantBosonicTopologi- semi-metalusingacoupled-wirearray.Followingtheconclu-
cal Insulator(BTI) 17,18,22,34. TThe formeris characterizedby sionswehaveasetofdetailedAppendixesthatreviewsome
its quantized Hall conductance, which must come in integer of the technicaltools used in the paper, and also containex-
multiplesof2e2/h,whilethelatterischaracterizedbyaquan- plicitderivationsofsomeofourmoretechnicalresults.
tizedmagneto-electricpolarizabilitywithaΘ-angleof2πin-
stead of the usual value of π for the non-trivialphase of the
electrontopologicalinsulator35. Thesebosonicphasesarenot II. MOTIVATIONANDOVERVIEWOFRESULTS
topologicallyordered,buttheyareSPTsthatrequireinterac-
tionstoexist;atzerothordertheinteractionsservetoprevent In this section we provide additional background motiva-
thesystemofbosonsfromformingatrivialBosecondensate. tion, describe the logic behindour constructionof a bosonic
Consider, fora moment, the 3+1-dBTI. In analogyto the analog of a topological DSM, and give an overview of our
electron TI we expect the surface states to exhibit unusual results. Henceforth, we call such a system a Bosonic Semi-
properties. Indeed, for one example, the surface theory can Metal (BSM). Readers interested in the technical details of
exhibit an effectively 2+1-d -breaking phase with a Hall thepapercanrefertothespecificsectionsformoreinforma-
conductanceof e2/h whichTis forbiddenfor a purely 2+1- tion. As mentioned above, the main goal of this paper is to
±
d BIQHE phase. To understandthe propertiesof this exotic construct a model of gapless bosons in 2+1-d which shares
surfacestateseveralequivalentrepresentationsofthesurface manyofthepropertiesoftheminimaltwo-nodeDSMoffree
theoryhavebeengivenintheliterature:(i)anetworkmodelof fermionsstudied,forexample,inRef.7. Themainproperties
quasi-1Dstripsthatarearrayedtoformasurfaceandcoupled, wewillbeinterestedinare: (1)theelectromagneticresponse
(ii) a dualdescription of the surface bosonic theoryin terms of the system to perturbations which break time-reversal or
ofdualvortices,and(iii)aneffectivefield-theorydescription inversion symmetry, and (2) the perturbative stability of the
in terms of the O(4) Nonlinear Sigma Model (NLSM) with gapless, low-energyeffective theory. As for any topological
a topologicaltheta termwith coefficientθ = π. Allthreeof semi-metal, translation symmetry is an important ingredient
theserepresentationsofthesurfacewerediscussedinRef.17. as it prevents any scattering processes between the different
ThedescriptionintermsoftheO(4)NLSMwithθ = π was Dirac or bosonic “cones” (which are generically located at
alsodiscussedinRef.36. Veryrecently,inspiredbynewde- differentpointsinmomentumspace).Indeed,inourBSMef-
velopmentsinthe descriptionoftheelectronTIsurface37–39, fectivetheory,translationsymmetrywillforbidperturbations
anewdualdescriptionoftheBTIsurfaceintermsof2+1-d whichcoulddrivethesystemintoagappedstate withonlya
QuantumElectrodynamicswith twofermionflavors(N = 2 trivialelectromagneticresponse.
QED ) was proposed40. This new dualdescriptionwas then Beforewebeginletusmakeanoteaboutunits. Inthispa-
3
derivedinacoupledwiresconstructioninRef.41. Whenany perweconsidersystemsconstructedfromfermionsorbosons
oneofthesetheoriesistunedtocriticalityitrepresentsasur- whichallcarryasingleunitofelectricchargee. Formostof
facestateinasymmetry-preservinggaplessphase. thepaperwe workin unitswheree = 1, butwillrestorethe
Inthisarticleourgoalistodevelopathoroughunderstand- chargee in allfinalresponseformulas. We also take ~ = 1,
ingofthesurfaceofthe3+1dBTI,andthentosubsequently which means that the conductancequantum e2 = 1 in our
h 2π
combine multiple copies of the theory to form a symmetry units. WealwaysexpressHallconductancesinunitsof e2.
h
preservingbosonicsemi-metalstatethatcanexistintrinsically WestartoutinSec.IIIbyreviewingthecontinuumdescrip-
in 2+1-d without breaking some requisite symmetries. This tionofthetwo-nodeDSM.Wefocusourreviewonthetime-
type of semi-metal represents the bosonic analog of a 2+1- reversalandinversionsymmetrybreakingelectromagneticre-
d DSM. We will present an effective theory for the bosonic sponsesoftheDSM,andalsothelocal(inmomentumspace)
semi-metalandexplorein detailtherequirementsforitssta- stability of the Dirac cones in the DSM. If the DSM is per-
bility, the resultingelectromagneticresponses, andpossibili- turbed by gap-inducingterms that break or then the re-
tiesforneighboringgappedphaseswithandwithoutintrinsic spectiveelectromagneticresponsesoftheDTSMItaketheforms
topologicalorder. We thenprovideanexplicitcoupledwires
constructionofthissemi-metalmodel.
e2
Our article is organized as follows: in Sec. II we give an = ǫµνλA ∂ A (2.1a)
T µ ν λ
overview of our main results, and in Sec. III we review the L 4π
e
properties of the 2+1-d fermion Dirac semi-metal. Next, in = ǫµνλB ∂ A , (2.1b)
I µ ν λ
L 2π
Sec.IV,wereviewsomepropertiesofthesurfacetheoryofthe
3+1-d -invariantBTIandprovidenewresultsandasynthe- where A is the potential for the external electromagnetic
µ
T
sisofpreviouswork.InSec.Vwediscussoureffectivetheory field, and B is another 3-vector field whose meaning is as
µ
forthe2+1-dbosonicsemi-metalbuiltfrommultiplecopiesof follows. ThetwoDiracconesoftheDSMarelocatedatdif-
thebosonicTIsurfacestates, includingthequasi-topological ferentpointsin theBrillouinzonewith a wave-vectordiffer-
electromagneticresponse,andthestability/instabilitiesofthis enceof2B (forsimplicitywechoosetheirlocationstobeat
i
criticalstate. InSec. VIwe deriveacriterionforidentifying k = (B ,B )),andthetwoconesareseparatedinenergy
± x y
±
3
by an amount2B . The effective Lagrangian represents tionforthismodelincludestheconventionalNLSM“kinetic
t T
L
a 2+1-d quantum Hall response with a Hall conductance of energy”term,andthetopologicalthetaterm,
1,while representsachargepolarizationandorbitalmag-
LI 1
netizationresponsewhose precise meaningwas discussed in S [U]= d3xǫµνλtr[(U†∂ U)(U†∂ U)(U†∂ U)],
Ref.7,andwhichwereviewinSec.VII.Wenotethatwehave θ 24π2 µ ν λ
Z (2.3)
suppresseda sign in these termsthattracksthe natureof the
where tr[ ] is the usual trace operation. In the action, the
inversionortime-reversalbreaking. ···
thetatermismultipliedbyaparameterθ,whichisanangular
Typicallypoint-nodesemi-metalsareunstablein2+1-dun-
variabledefinedmodulo2π. ForthesurfacetheoryoftheBTI
lessextrasymmetriesareimposed. ThestabilityoftheDSM
wehaveθ = π17. InSec.IVwereviewthecalculationofthe
isdueinparttothetranslationsymmetryofthesystem. This
time-reversal breaking electromagnetic response of this the-
symmetrypreventsscatteringprocessesbetweenDiraccones
oryviaitsdualvortexdescription(developedinRefs.17and
at differentlocations in the Brillouin zone. The local stabil-
42) and also discuss an alternate method for calculating this
ity (inmomentumspace) ofeachDirac conein the DSM (at
responsethatconfirmsthisresult.Wealsocommentonthere-
theleveloffreefermions)isthenguaranteedbyU(1) charge
c lationbetweenthedescriptionsoftheBTIsurfaceusedinthis
conservationsymmetryandacompositesymmetryZTI,con-
2 paper and the recently proposed dual description of the BTI
sistingofatime-reversaltransformationcombinedwithanin-
surfaceintermsofN = 2QED 40,41. Wethengoontogive
versiontransformation.Thesetwosymmetriesforbidtransla- 3
a carefuldiscussion of the effects that perturbationsallowed
tioninvarianttermswhichcouldgapouta singleDirac cone
bytheU(1) andZT symmetrieshaveonthesurfacetheory.
independentlyofanyoftheothers. c 2
TheseperturbationswereonlydiscussedbrieflyinRef.17.Fi-
HavingreviewedtheDSM,wecanmakethefollowingob-
nally,wereviewtheconstructionofthesymmetry-preserving
servationaboutaminimal,two-coneDSMwhichdirectlyin-
Z topologicallyorderedsurfacephaseoftheBTIwhichwas
formsourconstructionofaBSMmodel. SinceasingleDirac 2
firstderivedinRef.17.
cone is the surface theory for the 3+1-d Electron Topologi-
After all of this setup we are ready to introduce an effec-
cal Insulator (ETI)3 the degrees of freedom in the two-node
tivetheoryoftheBSM.InSec.Vweintroduceasystemwith
DSMcanbeviewedasbeingconstructedfromtwocopiesof
two copies of the O(4) NLSM with theta term. One copy
the ETI surface theory, but with the two copies separated in
has θ = π and the other copy has θ = π, and just as in
momentumspace. We arethereforemotivatedto constructa −
the case of the fermionic DSM, the two copies of the O(4)
model for a BSM from two copies of the surface theory for
NLSMarelocatedatpositionsk = (B ,B )inmomen-
the 3D Bosonic Topological Insulator (BTI), but with those ± ± x y
tum space. In ourdescriptionof the effectivetheorywe dis-
twocopiesalsoseparatedinmomentumspace. Accordingto
cernhowchargeconservation,translation,time-reversal,and
Ref. 17, one representation of the surface theory of the BTI
inversionsymmetriesactonthefieldsinthemodel,andthen
is the O(4) NLSM with theta term and θ = π, and it is this
computethetime-reversalandinversionbreakingelectromag-
theorywhichwediscussnext.
neticresponsesanalogoustothosefoundinthefermionDSM.
InSec.IVwegivealengthyreviewofthepropertiesofthe
We findthattheseresponsesalsoexistintheBSM case, and
O(4) NLSM with θ = π as it appears on the surface of the
have exactly twice the value of the responses in Eq. (2.1)
BTI.Therearetworeasonsforgivinganextendeddiscussion
forthefreefermionDSM. Thisdoublingofthe responsefor
oftheBTIsurfacetheory: (1)understandingjustonecopyof
bosonicvs. fermionicsystemsissimilartowhathappensfor
thistheoryisaprerequisiteforunderstandingourBSMeffec-
thecaseoftheETIandBTIin3D,andalsotheintegerquan-
tive theory, which consists of two copies of the surface the-
tumHalleffectsforfermionsandbosonsin2D15,17.
ory,and(2)weprovidealternatederivations(andalsoproofs
We then go on to give a partial discussion of the stability
in Appendixes B and C) for some of the properties of this
of our theory. We argue that the translation symmetry pre-
model. These discussions, and some additional new results,
vents us from coupling one copy of the theory to the other
lend further support to many of the claims about this model
copyinordertodrivethesystemintoatrivialinsulatingstate,
that havealready appearedin the literature. In particular we
and that the combined ZTI symmetry ensures the stability
provideanextendeddiscussiononthestabilityofthegapless 2
ofeachindividualO(4)NLSM.Finally,wediscusssome2D
natureoftheO(4)surfacetheorythatwewillrequireforour
topologicallyorderedphaseswhichcanbeaccessedfromour
discussionoftheBSMtheory.
BSM model by condensing suitable boundstates of the vor-
Tobegin,werecallthattheO(4)NLSMcanbeequivalently
ticesinthetheory.Inparticular,wefindaphasewithZ Z
formulatedintermsofanSU(2)matrixfield 2× 2
topologicalorderwhichbreakseitherthetime-reversalorthe
b b∗ inversionsymmetryoftheoriginalBSMmodel.Thisphaseis
U =(cid:18)b21 −b∗12(cid:19) , (2.2) efossuenndtiianllyRetwf.o1c7o,pbiuetsionfwthheicZh2thtoeptoimloeg-irceavlleyrsoarldaenreddthpehainse-
wherethecomponentsb andb areinterpretedasrepresent- versionsymmetryofthe BSM exchangethe two copies. We
1 2
ingphysicalbosonsonthe surfaceoftheBTI. Assuch, they alsodiscussphaseswithZ topologicalorderwhichbreakei-
2
transformunderthephysicalU(1) chargeconservationsym- thertheinversionorthe time-reversalsymmetryofthe BSM
c
metryas b eiχb forI = 1,2 (in unitswhere the boson model.
I I
→
chargee=1). Thistheoryalsohasatime-reversalsymmetry InSec.VIwegiveadifferentperspectiveonthestabilityof
ZT under which b and b are separately invariant. The ac- anysemi-metalphasebyrelatingthegaplessnessofthesemi-
2 1 2
4
metalto its polarizationresponse. In particular,we consider We constructinter-wire tunnelingtermswhich not onlygive
threebroadclassesof2Dgappedphaseswithtranslationsym- the desired O(4) NLSM’s with theta angles π and π, but
−
metry which can have a polarization response, and we show alsoshiftthetwocopiesoftheO(4)NLSM’stothelocations
thatfortheseclassesofgappedphasesthepolarizationinthe k = (B ,B )inmomentumspace(ourspecificconstruc-
± x y
±
x or y direction (in the presence of inversion symmetry) is tiongivesthecasewithB =0).Wethenshowhowtoassign
x
always of the form r e , where r Q is a rational num- transformationsunder time-reversal and inversion symmetry
2a0 ∈
ber and a is the lattice spacing. In addition, for each class to the fields in the coupled wires modelso that the transfor-
0
ofgappedphaseweareabletorelatethenumberr tosimple mationsof the fields in the BSM modelare recoveredin the
measurable properties of that phase. A crucial point is that continuumlimit. We concludeSec. VII with a discussion of
thethreeclassesofgappedphasesthatweconsiderarerepre- thedifferentphysicalinterpretationsofthecoupledwirecon-
sentative ofall gapped2D phaseswith translationsymmetry structionsoftheDSMandBSMmodels,andwealsoindicate
which could be expected to exhibit a polarization response. howinversionandtime-reversalbreakingperturbationsofthe
Ourresultthenimpliesthatageneric(i.e.,non-rational)value BSMmodelcanbeexploredwithinitsquasi-1Dcoupledwire
ofthepolarizationinasystemwithtranslationsymmetry,and description.
in particular a continuously tunable polarization, indicates a InAppendixAwereviewthecanonicalquantizationofthe
gaplesssemi-metalphase. Thisshowsthatthegaplessnessof O(4)NLSM,andalsoworkoutthecommutatorsforthisthe-
thesemi-metalisdirectlyrelatedtoitsphysicallymeasurable orywhenexpressedintermsoftheconstrainedbosonicvari-
polarization response. Since this response is expected to be ablesb andb . ThisinformationisusedinSec.IVtoinves-
1 2
reasonably robust, this also providesadditional evidence for tigate the effects of symmetry-allowed perturbations on the
thestabilityofthesemi-metalphaseitself. BTIsurfacetheory,andalsoinSec.Vtodiscussthestability
Finally,inSec.VIIwe giveanexplicitconstructionofthe oftheBSMmodeltosymmetry-allowedperturbations.InAp-
BSM model using an array of coupled 1D wires. This con- pendixBwestudyafamilyofexact,finiteenergyvortexsolu-
struction is motivated by the fact that 2+1-d fermion DSMs tionstotheNLSMequationsofmotion,andwecomputethe
canbe constructedoutofarraysof coupledwires7. A build- quantum numbers carried by global excitations in the back-
ingblockforthesimplesttwo-nodeDSMoffermionsisawire groundofasinglevortex. Inparticular,weareabletoprove
with a single 1+1-dmassless Dirac fermion. When coupled, the result, first argued for in Ref. 17, that the main effect of
arrays of these wires may exhibitthree related phases: (i) if thethetatermintheO(4) NLSMistoattacha charge θ of
2π
an array of these wires is dominated by an intra-wire topo- thebosonb1 tovorticesinthephaseofb2,andvice-versa. In
logicaltunnelingtermthenthesystembecomesa2+1-dweak AppendixCwediscusstheroleofthethetatermoftheO(4)
topologicalinsulatorthatexhibitsachargepolarizationparal- NLSMintheMinkowskispacetimepathintegralofthetheory.
leltothewires,(ii)ifthearrayisdominatedbyaninter-wire Finally,inAppendixDweresolveanapparentparadoxasso-
topological tunneling term then the array forms a Chern in- ciatedwithouralternativecalculation,viaauxiliaryfermions,
sulatorphasewith anintegerHallconductivityof e2/h, or ofthetime-reversalbreakingelectromagneticresponseofthe
(iii) if there is significant competition between an±intra-wire BTIsurfacetheory.
and inter-wire tunnelingthere can be a parentcritical phase,
i.e.,aDSM,whichisunstabletotheformationofphase(ii)if
time-reversalisbroken,andunstabletophase(i)iftheDirac III. REVIEWOFTHEFREE-FERMIONDIRAC
nodesmeetattheboundaryoftheBrillouinzoneandannihi- SEMI-METAL
late.
A key observation of this construction is that a wire with Before going into the details of our construction of a
a single 1+1-d Dirac fermion can be thought of as a narrow bosonic analog of a Dirac semi-metal (DSM), we first give
strip of a σxy = e2/h integerquantumHall system. Hence, areviewofthefreefermionDSMforthesimplecaseoftwo
by analogy, we can immediately propose a 1+1-d bosonic Dirac points in the Brillouin zone. Our review closely fol-
wire model to serve as the building block for a coupled- lows the discussion in Ref. 7, in which the electromagnetic
wire constructionof our2+1-dBSM state: a narrowstrip of responses of various topological semi-metals were derived.
the BIQHE, whichwill containthe degreesof freedomfrom Specifically,we discussa squarelattice modelofaDSM, its
both edges. An edge of the BIQHE can be described by an symmetryrequirements,itslow-energydescription,thetime-
SU(2)1Wess-Zumino-Witten(WZW)conformalfieldtheory reversalandinversionsymmetrybreakingelectromagneticre-
(CFT)15,17, thereforeour1+1-dbosonicwireswillconsistof sponses of the system, and finally the local stability of the
two (time-reversed)copiesof an SU(2)1 WZW theory. The Dirac nodes. A particularly importantpoint is that the local
fieldsineachwireconsistofbosonswhichcarrycharge1un- stability of the DSM, which means the stability against per-
dertheU(1)csymmetry. turbationsthatcangapoutindividualDiraccones,isguaran-
IthasbeenknownforsometimethatonecopyoftheO(4) teedbyenforcinga compositesymmetryZTI, whoseaction
2
NLSMwithθ = π canbeobtainedfromanarrayofcoupled consists of a time-reversaltransformationcomposedwith an
wires in which each wire contains a single SU(2) WZW inversiontransformation. Thiscompositesymmetrywillalso
1
theory17,42,43. After giving a brief review of this result, we playanimportantroleinourbosonicsemi-metalmodel.
then showhow ourBSM modelcan be derivedstarting with We also note here that the surface theory of the 3D Elec-
1+1-dbosonicwirescontainingtwoSU(2) WZW theories. tronTopologicalInsulator(ETI)isasingleDiracfermion.We
1
5
maythereforeviewthesimpletwoconeDSMasatheorycon- and t the bandstouch only at isolated pointsalong the line
y
structedfromsimilardegreesoffreedomastwocopiesofthe k = 0intheBrillouinzone. Thelocationofthesepointsis
x
surfacetheoryof the3D ETI(butwith thetwo copiesofthe determinedby theratioof m andt . We focusourattention
y
theoryhavingoppositehelicity).Thisobservationisthemoti- on the regime of m > 0 but m 2 (at m = 2 the band
≪
vationforourconstructionofabosonicsemi-metalfromtwo touching moves to k = π). In this regime the low-energy
x
copies of the surface theory of the 3D Bosonic Topological physicsofthissystemiscompletelydescribedbytwocontin-
Insulator(BTI),whichisanO(4)NLSMwiththetatermand uum Dirac Hamiltonians obtained by linearizing around the
thetaangleθ = π. Wereturntothispointinlatersections. twobandtouchings. Thesebandtouchingsarelocatedatthe
±
pointsk = (0, B )intheBrillouinzone,whereB isthe
± y y
±
positive solution, in the first Brillouin zone, to the equation
A. LatticemodelofaDSM m+t cos(B )=0.
y y
Performingak P expansionaroundk wefindthelow-
±
·
Wenowdescribealatticemodel,discussedinmoredetailin energyDiracHamiltonians ±(k)
H
Ref.7,whichrealizesaDSMphaseforacertainrangeofpa-
rameters. Themodelconsistsoftwospecies/orbitalsofspin- ±(k)=kxσx tysin(By)(ky By)σz . (3.6)
H ± ∓
lessfermionsathalf-fillingonthesquarelattice.Wetherefore
We emphasize that these two low-energyHamiltonianshave
have a two-componentcomplexfermion operator~c at each
n opposite signs on their k terms. This means that the two
site n = (n ,n ) of the square lattice. We take the lattice y
x y Dirac fermions which emerge at low-energy in this model
spacinga = 1. A numberofsymmetriesplayanimportant
0 have opposite helicity, i.e., the Berry phase for electrons on
role in this system. They are discrete translation symmetry,
the two Fermi surfaceswhen the chemicalpotentialis tuned
U(1) chargeconservationsymmetry,ZT time-reversalsym-
c 2 away from the Dirac points have opposite signs. This form
metry,andZIinversionsymmetry.Thefermionscarrycharge
2 of the low-energyHamiltonianfor each Dirac pointleadsus
1,sotheytransformunderU(1) as
c directlytothecontinuumdescriptionoftheDSM.
U(1) :~c eiχ~c , (3.1)
c n n
→
whereχ is a constantphase. The action of the time-reversal B. ContinuumdescriptionoftheDSM
andinversionsymmetriesonthefermionsisgivenintermsof
theanti-unitaryoperator andtheunitaryoperator ,respec- ForthecontinuumdescriptionoftheDSM,wetakeasour
T I
tively. We will specify the action of these operators on the starting point an effective Hamiltonian for two continuum
complexfermionsafterintroducingtheDSMmodel. Dirac fermions ψ and ψ with opposite helicity (which is
A B
TheBlochHamiltonianofthismodeltakestheform exactly what we found in the linearized Bloch Hamiltonian
fortheDSMmodel). Wethenshiftthezeroinmomentumin
H2D(k)=sin(kx)σx+(1−m−cos(kx)−tycos(ky))σz , theky directionbyBy. ThisleadstotheHamiltonian
(3.2)
where σa, a = x,y,z are the Pauli matrices acting on the DSM(k)=kxI σx+kyσz σz ByI σz , (3.7)
H ⊗ ⊗ − ⊗
orbitalspace.Thesin(k )termrepresentsacomplexhopping
x
forfermionsinthexdirection,whilethetermsmultiplyingσz where for simplicity we have taken tysin(By) = 1 to make
representa mass term aswell as realhoppingtermsin the x thedispersionoftheDiracconesisotropic. Inpositionspace
andydirections. Thissystemhastime-reversalandinversion theoriginallatticefermions~cnmaybewrittenatlowenergies
symmetrywhere and actonthefermionsas intermsofthetwocontinuumDiracfermionsψA andψB as
T I
~cn −1 =σz~cn, (3.3) ~cn ψ (x)eiByy +ψ (x)e−iByy , (3.8)
T T a ∼ A B
0
and
wherex = (x,y) = (n a ,n a ), andwehavetemporarily
x 0 y 0
~c −1 =σz~c . (3.4) restoredthelatticespacing.Nowwedefinethemulti-compo-
n −n
I I nentfermionoperatorΨ = (ψ ,ψ )T. Onecanshowthat
A B
I
We see that both time-reversal and inversion symmetry act and actonΨas
T
with opposite signs on the two species of fermion. We note
that the inversion symmetry also negates the spatial coordi- Ψ(x) −1 =σx σzΨ( x), (3.9)
I I ⊗ −
nate,and isanti-unitary.
T and
The energiesof the two bandsof this modelas a function
ofkaregivenby
Ψ −1 =σx σzΨ. (3.10)
T T ⊗
E(k) = sin2(k )+ 1 m cos(k ) t cos(k ) 2. Inparticular,theseoperatorsexchangeψ andψ , i.e.,they
± x x y y A B
± − − −
q (3.5) eachmapfermionsfromoneDiracconetotheother.
(cid:0) (cid:1)
When t = 0 and m = 0, this model has a band touching Atthispointwemaygoaheadandgenericallyallowforan
y
at k = 0 for any value of k . However, for non-zero m offsetB tothek locationoftheDiracpoints,aswellasan
x y x x
6
offsetB betweentheenergiesofthetwoDiracpoints. This bothoftheDiraccones. However,supposewetriedtoadda
t
leadstotheeffectiveHamiltonian linearcombinationofthesetwomassterms. Thetwopossible
linearcombinationsare
(k)=k I σx B σz σx+k σz σz
DSM x x y
H −ByI⊗⊗σz−+Btσz⊗⊗I. ⊗ (3.11) Σ± = 21(ΣI ±ΣT). (3.15)
To more clearly see the final structure we can pass to a La-
Addingjustoneofthesetermswouldgapouteitherψ (add
grangianformulationofthissystem. TheLagrangianhasthe A
Σ )orψ (addΣ ).However,bothofthesetermsareforbid-
form + B −
denbythecompositesymmetry . Therefore,thelocalsta-
TI
=Ψ¯ i∂/+A/+(σz I)B/ Ψ, (3.12) bilityoftheDiracconesisguaranteedbythecombinedtime-
L ⊗ reversal times inversion symmetry .44 If we enforce this
where we define the(cid:0)gamma matrices γ0(cid:1)= I σy,γ1 = symmetry, then it is impossible to gTaIp out one of the Dirac
iI σz, and γ2 = iσz σx, and where Ψ¯ =⊗Ψ†γ0. We cones independently of the other cone, and hence they can
−have⊗also employed the Fe⊗ynman slash notation ∂/ = γµ∂ , onlyberemovediftheyareperturbedenoughtocollidewith
µ
etc., and included minimal coupling to the external electro- eachotherinmomentumspace. Thismeansthatwithtransla-
magneticfieldAµ connectedtotheU(1)csymmetry. tion,TI,andU(1)c preservedtheDSMisa(perturbatively)
stable2+1-dsemi-metalphase.
C. ElectromagneticresponseoftheDSM
IV. THESURFACETHEORYOFTHEBOSONIC
TOPOLOGICALINSULATOR
Wenowbrieflyreviewtheelectromagneticresponseofthe
DSM to time-reversaland inversionsymmetry breakingper-
turbations.Thetwomassterms Inthissectionwereview,andalsoclarifysomeaspectsof,
the surface theory of the 3+1-d BTI. Since our BSM model
Σ =I σy (3.13) is constructed from two copies of the surface theory of the
I
⊗
Σ =σz σy , (3.14) BTI, it is essential that we discuss this theory in detail. The
T
⊗ surfacetheoryoftheBTIwasfirstderivedin Ref.17, where
are the only matrices that anti-commutewith the kinetic en- itwasobtainedfromanetworkmodelconstructedfromcou-
ergy terms of the DSM Hamiltonian in Eq. (3.11), and pre- plededgetheoriesof theBIQH state (we brieflydiscussthis
serve translation invariance (i.e., they do not couple ψ to networkmodelinSec.VII).TheauthorsofRef.17thenused
A
ψ ,whicharelocatedatdifferentpointsinmomentumspace). thistheory,aswellasadualvortexdescriptionofthetheory,
B
The first term Σ breaksinversionsymmetry, while the sec- toderivemanypossiblesurfacephasesfortheBTI.Thesedif-
I
ondtermΣ breakstime-reversalsymmetry. ferent possible surface phases were further investigated and
T
As shown in Ref. 7, perturbing the system with a term clarifiedinRef.18whichutilizedmonopoleconfigurationsof
mΣ leads to the 2D electromagnetic response from theexternalgaugefieldtoprobethepropertiesofthevarious
T T
− L
Eq. (2.1), i.e., it induces a quantum anomalous Hall effect phases.
withChernnumber/Hallconductance6σxy = e2/h. Onthe In this section we provide a detailed account of the sur-
±
other hand, perturbing the system with mΣI leads to the facetheoryoftheBTI,whichisequivalenttoanO(4)NLSM
−
quasi-1D electromagnetic response I from Eq. (2.1). This with theta term and theta angle θ = π. We first discuss the
L
response indicates that, when starting from the gapped, in- basic properties of this theory, and also the transformations
version breaking phase and taking m 0, the DSM limit of the O(4) field under the physical symmetry group of the
→
will have a chargepolarizationand/or orbitalmagnetization. BTI. We then give a summary of the dual description of the
Infact,theresponsedoesnotdependonthemagnitudeofm theory,butfroma differentpointof view thanthe one given
at all, andthe dependenceonm entersonly as a globalsign in Ref. 17. We then show how the time-reversal symmetry
sgn(m) multiplyingthe response formula. Interestingly,this breakingelectromagneticresponseoftheBTIsurfacecanbe
second response term depends crucially on the propertiesof obtainedfromthedualdescription.Wealsodescribeanalter-
theDiracnodes,i.e.,theirrelativepositionsinmomentumand nativemethodforcalculatingtheelectromagneticresponseof
energy. thetheory.Thismethodusesawell-knownformuladerivedby
AbanovandWiegmanninRef.45,whichallowsonetowrite
theoriginalO(4)NLSMasapathintegraloverasetofauxil-
D. Combined symmetryensureslocalstabilityoftheDirac iaryfermionswhichcoupletotheO(4)field. SincetheO(4)
TI
cones NLSM is such a difficultsystem to study, havingtwo differ-
entmethodsforcalculatingtheresponsewhichgivethesame
Weendthissectionwithaquickcommentaboutthestabil- answerisstrongcorroboratingevidence. Wenextdiscussthe
ityoftheDSM.We sawintheprevioussectionthattheonly stabilityofthegaplesstheory.Inparticular,wecarefullystudy
mass terms that we can add to Eq. (3.11) which are allowed theeffectsofsymmetry-allowedperturbations,someofwhich
bytranslationsymmetryarethetermsΣ andΣ . Ifweadd werebrieflydiscussedinRef.17. Finally,weendthesection
T I
onlyoneofthesemasstermstothesystemthenitwillgapout withabriefreviewofthesymmetry-preserving,topologically
7
orderedsurfacephasefortheBTIproposedinRef.17. After whereb andb aresubjecttotheconstraint 2 b 2 =1,
1 2 I=1| I|
allofthisiscompletewewillbereadytodiscusstheproper- which is equivalentto the original constraint N N = 1 of
P ·
tiesofthebosonicsemi-metalstate. theO(4)NLSM.Weshouldthinkofb andb asrepresenting
1 2
thephysicalbosonicdegreesoffreedomonthesurfaceofthe
BTI,andsowewillrefertob ,I = 1,2,as“bosonicfields”
I
A. TheO(4)NLSMwiththetaterm for the rest of the article. Using these fields we see that the
O(4) NLSM can be viewed as being essentially a theory of
Inthissubsectionwe reviewthedescriptionofthesurface twocomplexscalarfieldsb andb ,however,thesefieldsin-
1 2
of the Bosonic Topological Insulator (BTI) in terms of one teractwitheachotherdueto(i)theconstraint b 2 = 1,
I| I|
2+1-d O(4) Nonlinear Sigma Model (NLSM) with a topo- and(ii)thethetatermS [U].
θ
P
logical theta term having θ = π. The O(4) NLSM field TheBTI isa gappedbosonicphaseofmatterprotectedby
N=(N1,N2,N3,N4)isareal-valuedunitvectorfield(i.e., U(1) charge conservation symmetry and ZT time-reversal
c 2
N N = 1). The action for this theorywith a generaltheta symmetry. Under these symmetries the bosonic fields b
I
·
angletakestheform transformas
1
S = d3x g(∂µNa)(∂µNa)−θSθ[N], (4.1) U(1)c :bI →eiχbI (4.6)
wherewesuZmoverallrepeatedindices(µ = t,x,y anda = Z2T :bI(t,x)→bI(−t,x), (4.7)
1,2,3,4),andthethetatermis for I = 1,2, where x = (x,y) denotes the spatial coordi-
nates. These transformationsgive the total symmetry group
1
Sθ[N]= 12π2 d3xǫµνλǫabcdNa∂µNb∂νNc∂λNd. thestructureU(1)c⋊Z2T,wherethesemi-directproduct“⋊”
Z (4.2) indicatesthattheU(1)c andZ2T transformationsdonotcom-
mute with each other. As we explain in the next few para-
Thecoefficientg isapositivecouplingconstant. Smallg fa-
vorsanorderedphaseinwhichNa isconstanteverywherein graphs, the O(4) NLSM theory with a theta term only pos-
sessesthistime-reversalsymmetrywhenθ isanintegermul-
spacetime,whilelargeg favorsadisorderedphase. Thetheta
tipleofπ.
termonlyplaysaroleinthedisorderedphase,soweassume
thatweareworkinginthelargegregime. Toseewhytheonlytime-reversalsymmetricvaluesofθare
ForthedescriptionofthesurfaceoftheBTI,itismorecon- θ = nπ, n Z,wefirstmakeatransformationtoEuclidean
∈
venient to use a formulationof the O(4) NLSM in terms of spacetime. Euclideantimeisdefinedbyτ = it,andthetheta
terminEuclideanspacetimehastheform
anSU(2)matrixU whichisrelatedtotheunitvectorNvia
U =N4I+ 3 iNaσa. IntermsofU theactiontakesthe
a=1 i
form S [N]= d3x ǫµνλǫ Na∂ Nb∂ Nc∂ Nd,
P θ,E −12π2 E abcd µ ν λ
S = d3x 1 tr[∂µU†∂ U] θS [U], (4.3) Z (4.8)
2g µ − θ where d3x = dτd2x is the integration measure for Eu-
Z E
wherenow clideanspacetime,andnowµ,ν,λ = τ,x,y. Thethetaterm
is now imaginary,whichmeansthat e−θSθ,E[N] appearsas a
1
S [U]= d3xǫµνλtr[(U†∂ U)(U†∂ U)(U†∂ U)], phase factor in the Euclidean path integral. Under a time-
θ 24π2 µ ν λ
Z (4.4) reversaltransformationwe sendt → −t, i → −i(sincethis
symmetry is anti-unitary), and b (t,x) b ( t,x). Since
andtr[...] denotesthe usualtrace operationformatrices. In I → I −
τ =it,τ isinvariantunderthistransformation.Thereforewe
thisform,theO(4)NLSMisalsoknownastheSU(2)Prin-
findthatundertime-reversalS [N] S [N]48.
cipalChiralNonlinearSigmaModel(PCNLSM). θ,E →− θ,E
IfweimposeboundaryconditionsonNsuchthatNtends
The renormalizationgroup (RG) flows of generalSU(N)
toafixedconfigurationN atinfinityinalldirectionsofEu-
PCNLSM’s in the (g,θ) plane were studied qualitatively in 0
clideanspacetime,thenwemayidentifyEuclideanspacetime
Ref.46. Inthatpapertheauthorsarguedthatthetheorywith
with the sphere S3. The sphere S3 is also the configuration
θ = π could either be gapless or have a degenerate ground
spaceforthe O(4) NLSM, so inthissituation thetheta term
state. InthegaplesscasetheypredictedanRGfixedpointat
becomesquantized,
θ =πandg =g∗forsomefiniteg∗,whileforthedegenerate
case they predicted that g flows off to positive infinity. In
1
this paper we focus only on the first possibility of a gapless d3x ǫµνλǫ Na∂ Nb∂ Nc∂ Nd =n Z,
12π2 E abcd µ ν λ I ∈
theory. WemightalsosuspectthattheO(4)NLSMatθ = π Z (4.9)
is gapless on the grounds that its lower dimensional cousin,
wheren istheinstantonnumberofthefieldconfigurationN.
theO(3)NLSMwithθ =π,wasalsoshowntobegaplessin I
The quantizationof this integral followsfrom the homotopy
Ref. 47.
group π (S3) = Z. In fact, the theta term is just the pull-
For the description of the BTI surface, one writes U in 3
backtospacetimeofthevolumeformonS3. SinceEuclidean
termsoftwocomplexfieldsb andb as
1 2 spacetime(with theboundaryconditionsdiscussedabove)is
b b∗ justanothercopyofS3,theintegralisrequiredtobeaninte-
U = b1 −b∗2 , (4.5) ger,whichjustcountsthenumberoftimesthatthespacetime
(cid:18) 2 1 (cid:19)
8
S3wrapsaroundtheconfigurationspace(alsoS3)oftheO(4) stateisafermion,aswasdiscussedinRef.17.
NLSMfieldN.
IntheEuclideanpathintegral,thethetatermappearsinan
exponential, e−θSθ,E[N] = eiθnI, which shows that the pa- B. Time-reversalbreakingresponse
rameter θ is only defined modulo2π. We have already seen
thata time-reversaltransformationsendsθ θ. Itisthen In this section we discuss the calculation of the time-
→ −
immediatetoseethattheonlytime-reversalsymmetricvalues reversal breaking electromagnetic response of the O(4)
of θ are θ = nπ, n Z, since it is only these values of θ NLSM with θ = π. This response is in principle obtained
∈
whichsatisfyθ θ mod2π. ItwasshowninRef.17that by coupling the NLSM to the external electromagnetic field
≡ −
thegaplesssurfaceterminationoftheBTIisdescribedbythe A , turning on a small time-reversal breaking perturbation,
µ
O(4) NLSM with θ = π, and hence preservestime-reversal integrating out the matter fields b and b , and then setting
1 2
symmetry. the time-reversal breaking perturbation to zero. In practice,
Anothercommentcanbe madeaboutthe interpretationof however,itisverydifficulttointegrateouttheNLSMfielddi-
thethetaterminEuclideanspacetime.ItwasshowninRef.42 rectly,andsowemakeuseoftwoalternativeandcompletely
thattheone-instantonconfigurationoftheO(4)fieldNcanbe different methods for calculating the time-reversal breaking
re-interpretedintermsofvortexconfigurationsofthebosonic responseofthetheory. Thefactthatthese twomethodsgive
fieldsb1 andb2. RecallthatavortexofthefieldbI isapoint thesameanswerstronglysuggeststhattheansweristhecor-
inspacearoundwhichthephaseofbI windsby2π. In2+1-d rectone, eventhoughithasnot, as yet, beencheckedwith a
thespacetimetrajectory,orworldline,ofavortexisjustaline directcalculationintheO(4)NLSM.
(or curve) in spacetime. In Euclidean spacetime (compact-
ified to the sphere S3 via appropriate boundary conditions),
the worldlines of vortices become closed loops. In Ref. 42
1. Method1:DualVortexDescription
itwasshownthattheone-instantonconfigurationofthefield
Nisequivalenttoalinkingconfigurationinwhichtheworld-
Thefirstmethodforcalculatingthetime-reversalbreaking
line of a vortex in the phase of b links exactly once with a
1 responseoftheBTIsurfaceistousethedualvortexdescrip-
worldlineofavortexinthephaseofb . Sincethisconfigura-
2 tionoftheO(4)NLSMwhichbecomespossibleatthespecial
tioncontributesaphaseofeiθ totheEuclideanpathintegral,
value of θ = π. This dual vortex description was first ob-
the authors of Ref. 42 interpreted this to mean that a vortex
tainedinRef.42usingalatticeformulationofthetheory.The
in b and a vortex in b have a mutual statistical angle of θ.
1 2 continuum version of this dual vortex theory was then used
This means that a braiding process in which a vortex in b
1 extensivelyinRef.17tostudythepossiblesurfacephasesof
makesa completecircuitarounda vortexin b shouldresult
2 theBTI.Inthissectionwegiveareviewofthisdualdescrip-
in an overall phase factor eiθ for the wave functional of the
tionfromanalternativeperspectivethatiscomplementaryto
quantumfieldtheory. ThisresultwasthenusedinRef.17to
thatgiveninRefs.17and42.
deduceatopologicallyorderedsurfacephasefortheBTI.We
We have already explained how the O(4) NLSM can be
willreviewthistopologicallyorderedphaseattheendofthis
regarded as a theory of two complex scalar fields b and b
section. We remark in passing that similar arguments were 1 2
subjecttotheconstraint b 2 = 1. Thisconstrainthasa
alsousedinRef.49todeducethebraidingstatisticsofparti- I| I|
strongeffectonthephysicsofvorticesinthefieldsb andb .
cleandloop-likeexcitationsingaugedSPTphasesfromtheir 1 2
P
Recallthata vortexin the field b is a pointin space around
descriptionintermsofNLSM’swiththetaterm. 1
whichthephaseofb windsby2π. Atsuchapointthephase
Itisclear fromthe discussioninthe precedingparagraphs 1
ofb isundefined,andsotheamplitudeofb mustvanishat
thatthethetatermplaysanimportantroleinthephysicsofthe 1 1
thatpoint. However,since the fieldsb andb aresubjectto
O(4) NLSM. However, in this section we relied extensively 1 2
theconstraintdiscussedabove,thismeansthatinthecoreof
ontheinterpretationofthethetaterminEuclideanspacetime
avortexinb wehave b =1. Thisindicatesthatvorticesin
tounderstandits specialproperties. Tobetterunderstandthe 1 | 2|
b cantrapchargeofb andvice-versa.Infact,inMinkowski
quantummechanicsoftheO(4)NLSMwiththetaterm,itis 1 2
spacetimethemaineffectofthethetatermistoattachcharge
desirable to understand the role the theta term plays in the
θ of boson b to vortices in b and vice-versa. A heuristic
Minkowksi spacetime path integral. In Appendix C we ex- 2π 1 2
argumentforthiseffectwasgiveninRef.17. InAppendixB
plainthepreciseinterpretationofthethetaterminMinkowksi
we prove this result explicitly by computing the charges of
spacetime,andshowthatthethetatermdoesindeedcontribute
global excitations on the backgroundof certain exact vortex
aphaseeiθ tothepathintegralforconfigurationsoftheO(4)
solutionsoftheNLSMequationsofmotion.
fieldinwhichavortexinthebosonb makesacompletecir-
2 We first give a short review of the dualvortex description
cuitaroundavortexinb . Thisresultconfirmstheinterpreta-
1 ofthetheoryofanordinarychargedscalarfieldin2+1-d,and
tionofthethetatermgivenbySenthilandFisherin Ref.42,
referthe reader to Ref. 51 for a more detaileddescriptionof
which was based on an analysis of the theory in Euclidean
thistechnique.Considerfirstanordinarycomplexscalarfield
spacetime. Inaddition,followinganargumentfromRef. 50,
b,withaLagrangianoftheform
this result implies that a bound state of a vortex in b and a
1
vortexinb carriesintrinsicangularmomentumJ = θ . At
θ = π we2have J = 1, which means that the vortex2bπound = (∂µ iAµ)b2 µ b2 λ b4+... (4.10)
2 L | − | − 2| | − 4| |
9
Forlaterconveniencewewritebinadensityphaserepresen- andtheexternalfieldA . Thethetatermisentirelyresponsi-
µ
tationasb= ρeiϑ. Whenµ <0thissystemhasasymmetry- bleforthecouplingofφ toA . Finally,theconstantκ is
2,+ µ 2
brokengroundstate in whichρ = ρ¯= µ andthephase givenbyκ = g ,whereρ¯ istheabsolutevalueofb inthe
ofbislockedtoaparticularvalue(thusspo−ntλaneouslybreak- condensed2phas4eρ¯.22 2 2
ingtheoriginalU(1)c symmetryunderbp eiχb). Thelow- Interestingly,exactlyatthespecialvalueθ =π,itbecomes
→
energyexcitationsaboutthisgroundstatearethegaplessfluc- possible to replace our description of the theory in terms of
tuationsofthephaseϑofb(theGoldstonemodes),whichare b and φ with a much more symmetric dual description
1 2,+
describedby in terms of two types of vortices φ and φ , as we now
2,+ 2,−
explain. Atθ = π,thevortexφ carriescharge 1 ofboson
=ρ¯2(∂ ϑ A )2 +.... (4.11) 2,+ 2
L µ − µ b1. Inthiscasethecompositefield
The fluctuations ϑ consist of two parts, ϑ = ϑs +ϑv. The φ =φ b∗, (4.14)
smoothpartϑs consistsofsmallfluctuationsaroundthefixed 2,− 2,+ 1
vacuum value of ϑ. The second part ϑv consists of vortices carries charge 1 of boson b (note that we are using ∗ to
−2 1
inwhichthephasewindsbysomemultipleof2π aroundthe representanti-particles). Thefieldφ canbeunderstoodas
2,−
vacuummanifold(i.e.,thecircledefinedby b =ρ¯). a vortex-anti-bosonboundstate, and at θ = π it is a natural
| |
Intheusualboson-vortexdualityasequenceoftransforma- objectto considerbecauseofthe factthatitcarriesthe same
tionsisnowappliedto the LagrangianEq.(4.11)(morepre- magnitudeofchargeastheoriginalvortexφ (notethatwe
2,+
cisely, these transformationsare appliedto the pathintegral) can always define φ in this way for any value of θ, but
2,−
toobtainafinalLagrangianoftheform thisfieldonlytransformsnicelyunderthe symmetriesof the
theorywhenθ =π).
= (∂ iα )φ2 µ˜ φ2 λ˜ φ4+... Further justification for the introduction of the field φ2,−
µ µ
L | − | − 2| | − 4| | canbe obtainedbyrecallingthatat thespecialvalue θ = π,
1 1 2 1 thetime-reversalsymmetryoftheO(4)NLSMisrestored. It
− 4ρ¯2 2πǫµνλ∂ναλ − 2πǫµνλAµ∂ναλ.(4.12) followsthatthevortexφ2,+ shouldhaveawell-definedtrans-
(cid:18) (cid:19) formationundertime-reversalwhen θ = π. Vortices should
This expression features two new fields: the gauge field αµ transformintoanti-vorticesundertheactionoftime-reversal,
andthecomplexscalarfieldφ. Thefieldαµisanon-compact sincetime-reversalisananti-unitarysymmetry. Ontheother
gauge field which is introduced to represent the conserved hand,thetime-reversalpartnerofφ shouldhavethesame
2,+
number current Jµ of the original bosons b via the equa- U(1) chargeasφ inordertopreservethestructureofthe
c 2,+
tion Jµ = 21πǫµνλ∂ναλ. Non-compactnessof αµ is just the symmetry group of the BTI. It turns out that φ∗2,− has just
statement that ǫµνλ∂ ∂ α = 0, which guarantees the con- the rightpropertiesto bethe partnerof φ underthe time-
µ ν λ 2,+
servation of Jµ. The excitations of the new complex scalar reversaloperation.
field φ represent vortices in the phase of the original boson We see then that at the special value θ = π, the dual
b. Thevortexcurrentofb,definedbyKµ = 1 ǫµνλ∂ ∂ ϑv, description of the O(4) NLSM is given most naturally in
2π ν λ
is givenin this representationby the numbercurrentof φ as termsofthetwo-componentvortexfieldΦ =(φ ,φ )T,
2 2,+ 2,−
Kµ = i(φ∂µφ∗ φ∗∂µφ)(inotherwords,theU(1)charge which transforms under the U(1) and ZT symmetries ac-
− c 2
ofφisthevortexnumber). We havealso includeda number cordingto
ofpotentialenergytermswhichcouldappearintheactionfor
thevortexfieldφ. U(1)c :Φ2 eiχ2σzΦ2 (4.15)
→
NLWSeMnwowhilaeplpelayvitnhgisbteucnhtnriaqnusefotromtehde(baonseoanrlby2idinentthiceaOld(i4s)- Z2T :Φ2(t,x)→σxΦ∗2(−t,x). (4.16)
1
cussioncan be hadif onechoosesto dualizeb1 andleaveb2 In terms of the pair of vortex fields making up Φ2, the final
fixedinstead).Wethereforedefineanewcomplexscalarfield dualactiontakestheform
φ2,+ whichcreatesavortexinthephaseofb2. Fromthedis- = (∂ iα isA )φ 2+... (4.17)
µ 2,µ µ 2,s
cussionearlierinthissection,andtheresultsofAppendixB, L | − − 2 |
s=±
φ carries charge θ under the U(1) symmetry. We rep- X
re2s,e+nttheconserved2nπumbercurrentJ2µcofb2 usingthenon- 1 1 ǫµνλ∂να2,λ 2 1 ǫµνλAµ∂να2,λ,
compactgaugefieldα2,µ. − κ2 (cid:18)2π (cid:19) − 2π
Atthispoint,thedualvortexdescriptionoftheO(4)NLSM
where again the ellipses stand for possible potential energy
withgeneralangleθtakestheform
terms. Note that the two species of vortex carry the same
1 θ chargeofα2,µ,butoppositechargeofAµ.
L= g|(∂µ−iAµ)b1|2+|(∂µ−iα2,µ−i2πAµ)φ2,+|2+... AsstatedinRef.17,theoriginalbosonfieldb1 isnowrep-
resentedapproximatelyby
2
1 1 1
ǫµνλ∂ α ǫµνλA ∂ α , (4.13)
− κ2 (cid:18)2π ν 2,λ(cid:19) − 2π µ ν 2,λ b1 =φ2,+φ∗2,− , (4.18)
where the ellipses stand for possible potential energy terms. i.e., it is a bound state of a vortex (φ ) and an anti-vortex
2,+
Thefieldφ carrieschargeofboththedualgaugefieldα (φ∗ ). We should, however,take a momentto considerthis
2,+ 2,µ 2,−
10
equation carefully. Interestingly, the two sides of this equa- pathintegral. Therightmost2 2matrixinthetensorprod-
×
tiondonothavethesamedimensions. Thefieldb isdimen- uctsactsonthespinorindicesofψ ,whiletheleftandmiddle
1 a
sionless,whilethecomplexscalarfieldsφ andφ carry matricesinthetensorproductsactontheisospaceindices.
2,+ 2,−
dimensionsof(length)−12 (thisistruebecausethevortexcur- WedefinetwosetsofgammamatricesγµandΓa by
rentKµ hasdimensionsof(length)−2). Amoreprecisever- γ0 =I I σy (4.22a)
sionofthisequationwouldbetowrite ⊗ ⊗
γ1 = iI I σz (4.22b)
− ⊗ ⊗
b gφ φ∗ , (4.19) γ2 =iI I σx , (4.22c)
1 ∼ 2,+ 2,− ⊗ ⊗
and
wheregistheNLSMcouplingwhichhasunitsoflength,and
wherean arbitrarydimensionlessconstantcouldbe included
Γ1 =σx σx I (4.23a)
ontheright-handsideofthisequation. ⊗ ⊗
Γ2 =σy σx I (4.23b)
Thetime-reversalbreakingresponseoftheO(4)NLSMat ⊗ ⊗
θ = π can now be explored using the dual description in Γ3 =σz σx I (4.23c)
⊗ ⊗
Eq. (4.17). A gapped, time-reversal breaking phase is real- Γ4 =I σy I, (4.23d)
ized when, for example, φ condenses and φ becomes ⊗ ⊗
2,+ 2,−
gapped,orvice-versa.Inordertoinducethisphase,oneneeds whereIisthe2 2identitymatrix. Inthiscasewecanalso
×
toincludeinEq.(4.17)apotentialenergyoftheform defineafifthmatrixforthesecondset,
V(Φ )=µΦ†σzΦ+λ φ 4+λ φ 4, (4.20) Γ5 =I σz I. (4.24)
2 2 +| 2,+| −| 2,−| ⊗ ⊗
The first set of gamma matrices obey a Clifford Algebra in
whereλ arebothpositive. Thechoiceofwhichvortexcon-
± Lorentzsignature, γµ,γν =2ηµνI ,andareusedtocon-
denses and which is gappeddependson the sign of µ. Note 8×8
{ }
that the term Φ†σzΦ explicitly breakstime-reversalsymme- structthederivativeoperatorfortheDiracaction.Thesecond
2 setobeysaEuclideanCliffordAlgebra, Γa,Γb =2δabI ,
try. { } 8×8
andisusedtoconstructthemasstermswhichcoupleΨtothe
Whenφ condensesandφ isgapped,wemay(atlow
2,+ 2,− O(4)fieldN.
energies)setφ =0andφ =const.tofindthatthemin-
2,− 2,+ AccordingtoRef.45,afermionicactionoftheform
imum energyconfigurationis realized when α = 1A ,
2,µ −2 µ
whichyieldstheresponse
4
=Ψ¯ i∂/ cos(ν)MΓ5 sin(ν)M NaΓa Ψ,
= e2eµνλA ∂ A , (4.21) Lf − − a=1 !
Leff 4π µ ν λ X (4.25)
with large mass M > 0 will produce, after integration over
wherewe haverestoredthe chargee. Thisresponseyieldsa
thefermions,anO(4)NLSMoftheformofEq.(4.1)withthe
“half” bosonic quantumHall effect with σxy = 122he2. If we thetaanglegivenby
hadinstead condensedφ and gappedoutφ , we would
2,− 2,+
havefoundthesameresponsebutwiththeoppositesign. 9 1
θ =π 1 cos(ν)+ cos(3ν) . (4.26)
− 8 8
(cid:18) (cid:19)
2. Method2:Abanov-Wiegmannintegrationoverfermions Taking ν = π gives θ = π. The evaluationof this fermion
2
pathintegralisnotcompletelystraightforward,andsowere-
Thesecondmethodforcalculatingthetime-reversalbreak- ferthereadertoRef.45aswellasRef.53forexplanationsof
ingresponseoftheBTIsurfaceusesaformuladuetoAbanov thiscalculation.
andWiegmann45whichallowsonetoexpresstheO(4)NLSM Ifwe setν = π δ forsmallδ, thenthe actiontakesthe
2 −
with theta term as a path integral over a set of auxiliary form
fermions. The fermions in this construction must also carry
4
chargeunderthephysicalU(1)csymmetry,sowecandirectly =Ψ¯ i∂/ (Mδ)Γ5 M NaΓa Ψ. (4.27)
f
couplethefermionstotheU(1)cgaugefieldAµandtheninte- L − − a=1 !
grateoutthefermionstodeducetheelectromagneticresponse X
ofthesystem.AsimilarapproachwasusedrecentlyinRef.52 Sincetheonlytime-reversalinvariantvaluesofθaremultiples
tocalculatetheelectromagneticresponseofaBosonicInteger ofπ,thiscorrespondstoaddingasmalltime-reversalbreaking
QuantumHallstatein4+1-d. perturbationtotheaction(wewouldnowgetθ π(1 3δ)
≈ − 2
The starting point for this construction is a multi- afterintegratingoutthefermions). We now calculatethe re-
component fermionic field Ψ = (ψ ,ψ ,ψ ,ψ )T, where sponseofthetheoryinthepresenceofthisperturbation,and
1 2 3 4
eachofψ ,a = 1,2,3,4,isatwo-componentDiracfermion thentakethelimitδ 0.
a
→
in 2+1-d. In whatfollows we use tensor productnotation in Beforeweproceedwiththecalculation,wementionthefol-
ordertotreatspinorand“isospace”indicesonequalfooting. lowingpuzzle. ThecalculationinRef.45iscontrolledbyan
All indices are traced over in the evaluation of the fermion expansioninpowersofM−1,sowemusttakeM tobelarge