Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect

0.71 MB·English

by  Ashvin Vishwanath

Physics of three dimensional bosonic topological insulators: Surface Deconfined
Criticality and Quantized Magnetoelectric Effect.
Ashvin Vishwanath
Department of Physics, University of California, Berkeley, CA 94720, USA and
Materials Science Division, Lawrence Berkeley National Laboratories, Berkeley, CA 94720
T. Senthil
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
(Dated: October 19, 2012)
We discuss physical properties of ‘integer’ topological phases of bosons in D=3+1 dimensions,
protectedbyinternalsymmetriesliketimereversaland/orchargeconservation. Thesephasesinvoke
2 interactionsinafundamentalwaybutdonotpossesstopologicalorderandarebosonicanalogsoffree
1 fermiontopologicalinsulatorsandsuperconductors. Whileaformalcohomologybasedclassification
0 ofsuchstateswasrecentlydiscovered,theirphysicalpropertiesremainmysterious. Herewedevelop
2 a field theoretic description of several of these states and show that they possess unusual surface

t states,whichifgapped,musteitherbreaktheunderlyingsymmetry,ordeveloptopologicalorder. In
c thelattercase,symmetriesareimplementedinawaythatisforbiddeninastrictlytwodimensional
O
theory. While this is the usual fate of the surface states, exotic gapless states can also be realized.
For example, tuning parameters can naturally lead to a deconfined quantum critical point or, in
8
other situations, a fully symmetric vortex metal phase. We discuss cases where the topological
1
phases are characterized by quantized magnetoelectric response θ, which, somewhat surprisingly,

is an odd multiple of 2π. Two different surface theories are shown to capture these phenomena -
]
l the first is a nonlinear sigma model with a topological term. The second invokes vortices on the
e
surfacethattransformunderaprojectiverepresentationofthesymmetrygroup. Abulkfieldtheory
-
r consistent with these properties is identified, which is a multicomponent BF theory supplemented,
t
s crucially,withatopologicalterm. Apossibletopologicalphasecharacterizedbythethermalanalog
. of the magnetoelectric effect is also discussed.
t
a
m
- Contents VII. Candidate Phases with Half Quantized
d Surface Thermal Hall Effect 22
n
I. Introduction 1
o
VIII. Conclusions 23
c
[ II. Overview 4
IX. Acknowledgements 23

2
III. Transport Properties of 3D bosonic
v A. Derivation of 3D BF Surface States: 24
8 topological insulators: General
5 Constraints 6
B. Θ periodicity in multicomponent BF
0
theory 25
3 IV. Surface Theory of 3D Bosonic SPT
.
9 Phases 7 C. Other Symmetries: U(1)(cid:111) Z 25
0 A. Surface States and Projective Vortices 8 2
2
1. Surface States of a Bosonic topological D. Symmetry Transformation of Surface
1
: insulator: Symmetry U(1)(cid:111) Z2T 9 States with Topological Order 26
v B. Network Model Construction 9
i
X C. Synthesis of the Two Approaches 12 References 27
D. Analysis of the Surface Field Theory: Phase
r
a Diagram, Deconfined Criticality and
Fermionic Vortices 12 I. INTRODUCTION
V. 3D Topological Field Theories 14
Inthelastseveralyearswehavelearntofmanyphases
A. Two component BF theory of Bosonic and phase transitions of quantum many body systems
Topological Insulator 15 that go beyond conventional concepts of broken symme-
B. Bulk sigma model theory 17 try. Striking examples are gapped phases of matter with
topological order. These are characterized by emergent
VI. Other Symmetries: 19 excitations with unusual quantum statistics and ground
A. Symmetry ZT 19 statedegeneraciesthatdependonthetopologyoftheun-
2
B. Symmetry U(1)× ZT 20 derlying manifold1. Other examples are gapless phases
2
2
of matter where the gaplessness is protected but not by
a broken symmetry. The most familiar example of such
a phase is a Fermi liquid but gapless spin liquids and
various non-fermi liquid phases provide other examples.
Other examples of exotic gapless states include decon-
fined quantum critical points2, which nevertheless may
occur between conventional ordered phases. It has be-
comeclearthatallthese“exotic”phaseshavesomeemer-
gent non-local structure in their ground state wavefunc-
tion. Thisnon-localityislooselycapturedbysayingthat
the local degrees of freedom have long range entangle-
ment in the ground state wavefunction.
More recently it has become clear that some of the
interesting properties of such exotic phases are also
shared by phases with only short ranged entanglement.
The best example is provided by free fermion insula- FIG. 1: Schematic depiction of a 3D symmetry protected
tors/superconductors with topological band structure3. topological phase, with two conserved species of bosons
As is well known these insulators have non-trivial pro- (U(1)×U(1)) and time reversal symmetry (Z2T). The bulk
is insulating and corresponds to a condensate of vortices of
tected surface states but only have short range entangle-
both species (shown as black and white loops). In the topo-
ment.
logical phase, the vortex line of one species that ends on a
Let us now generalize the concept of a topological
surface carries half charge of the other species. Such a sur-
insulator. Consider a system of interacting bosons or face,itmaybeargued,doesnothaveatrivialgappedphase,
fermions in d dimensions, possibly including global sym- where the symmetries are preserved. Pictorially, the vortex
metries. Assume that there is a unique ground state linesmaybeviewedin1DHaldanephases,withhalfcharged
withgappedexcitationsonallclosedmanifoldsi.e. there end states.
is no topological order . We refer to such phases as
ShortRangeEntangled(SRE)states58. DodistinctSRE
phases exist that share the same symmetry, but differ at many of these results were obtained in a simpler way us-
theleveloftopology? Apossibledistinction,forexample, ingaChern-Simonsapproach12. Theclassificationshows
is presence of protected states at the boundaries. that interaction-dominated phases of bosons with a bulk
It is interesting to pose this question for interaction gap and a unique ground state on closed manifolds exist
dominated phases of matter, i.e phases that require in- thathavenon-trivialsurfacestates. Forinstanceind=2
teractions between the underlying particles to stabilize bosons with a conserved global U(1) but no other sym-
them. Aclassicexample(onewhichisthesubjectofthis metries have an “integer quantum Hall state” with non-
paper) are gapped phases of interacting bosons. When a trivialprotectededgestatesbutnofractionalbulkexcita-
global U(1) symmetry is present, associated with boson tions. In particular, the quantized Hall conductance was
number conservation, such phases correspond to bosonic predicted12 to be an even integer in units of q2/h, where
insulators. For bosons the non-interacting limit is a sim- q is the elementary charge of the bosons. Very recently
ple condensate so that interactions are necessary to sta- a simple physical realization of such a bosonic integer
bilize insulating phases. quantum Hall effect has been provided, and its proper-
Recently there has been much progress in identify- ties shown to agree with the formal classification13. In
ing such topological phases of bosons in diverse dimen- general, depending on the detailed symmetry group and
sions. In one spatial dimension (1D), these include dimensionality of the system, a variety of such “Sym-
states like the Haldane (or AKLT) state of gapped spin- metry Protected Topological” (SPT) phases have been
1 chains4. Using the matrix product representation of argued to exist. In d = 1 or in d = 2 we have a fair un-
gapped states5–8 they are argued to be classified by pro- derstanding of the universal physical properties of these
jective representations of the symmetry group (G) or SPT phases. However in d = 3, though the cohomology
equivalently by the second group cohomology of sym- classification allows for the existence of a number of dif-
metry group G. While in 2D and 3D such rigorous re- ferent SPT phases, including bosonic generalizations of
sultsarenotavailable,recently,Chen,Liu,GuandWen9 topological insulators, their physical properties have not
haveproposedthatSREtopologicalphasesofbosonsare thus far been elucidated. This will be our primary task.
captured by the higher dimensional cohomology groups. Potentially, the approach here could also serve to inde-
While Chen et al.9 restrict attention to the non-chiral pendently classify these phases, which however is left for
subset of these states (i.e. ones that do not have a net future work.
imbalance of left and right movers at the edge of ) pro- Surface States and Projective Vortices: Typically,
tected by symmetry, Kitaev10 has also considered chiral physicalmanifestationsoftopologicalphasesinvolvepro-
states. An explicit example of such a phase in d=2 was tected surface states and/or a quantized response. We
constructedbyLevinandGu11. Later,intwodimensions show that 3D SRE bosonic topological phases can also
3
be characterized in this way. First, we will see that such withbrokentimereversalonthesurfacethatremovesthe
phases only occur in the presence of symmetry. Then, surfacestates,haveaquantizedmagneto-electriceffect16
as with free fermion topological insulators, the surface captured by the topological theta term:
states are either gapless or, if gapped, must develop ei-
θ
ther conventional or topological order. Conventional or- L = E(cid:126).B(cid:126) (1)
θ 4π2
der involves breaking the underlying symmetry defining
thephase. Theotherpossiblegappedstateinvolvestopo- (we have set h=c=e=1 and E(cid:126), B(cid:126) are applied electric
logical order that develops just at the boundary. While
and magnetic fields). For fermion topological insulators
we discuss multiple routes that reach this conclusion, a
θ =π(mod2π),correspondingtoahalfintegerHalleffect
particularly intuitive picture is the following. Consider
onthesurface. The2π ambiguityinθ correspondstothe
for a moment a system with two independent species
factthatonemaydepositanintegerQuantumHalllayer
of bosons (species 1,2 whose numbers are independently on the surface17.
conserved by U(1)×U(1) symmetry) as well as time re-
One of our central observations is that for 3D bosonic
versal (ZT) symmetry. We describe a three dimensional
2 topologicalinsulators,withconservedchargeandtimere-
topological insulator of such a system where the surface
versal symmetry in the bulk, θ is only defined modulo
hasthefollowingunusualproperty. Ifbosonsofspecies1
4π, and the topological phase corresponds to θ = 2π.
is condensed on the surface, then its vortices carry half-
This implies, for example, the domain wall between op-
charge of species 2, and vice versa. This implies that
posite time reversal symmetry breaking regions on the
disordering the surface superfluid, which corresponds to
surface induces a protected mode, the edge state corre-
condensing vortices, also condenses species 2. That is, sponding to the σ = 2q2/h quantized Hall effect of
xy
thereisanobstructiontorealizingafullysymmetricand
bosons. The 4π ambiguity in θ corresponds to the fact
gapped surface. It is important that the vortices carry
that one may deposit an integer Quantum Hall layer of
fractional charge of species 2, which cannot be screened
bosons on the surface, which must have an even integer
by bosons on the surface. Time reversal plays an impor-
Hall conductance. The gapless surface state may there-
tant role in fixing this charge to be ±1/2. Some of the
fore be viewed as a Quantum Hall state that fluctuates
options then are (i) to break U(1) symmetry on the sur- between σ = ±q2/h. A powerful route to obtaining a
xy
face such as in the surface superfluid or (ii) to condense
theory of such a state is to start with the theory of the
pairs of vortices that carry no charge, which leads to
quantumphasetransitionbetweendistinctintegerquan-
Z topological order14, or (iii) both vortices and bosons
2 tum Hall states of bosons. Such an approach when ap-
are gapless but not condensed. For bosons, which are
plied to fermionic topological insulators correctly yields
typically either gapped or condensed, this corresponds
the single Dirac cone surface state. We therefore con-
to a (multi) critical point. In fact, it corresponds to a
struct a network model to describe the bosonic integer
deconfined critical point, analogous to that proposed in
quantum Hall transition in two dimensions and use that
thecontextofacontinuousquantumphasetransitionbe-
to derive a field theory of this transition. This then nat-
tweenaNeelandvalencebondsolid(VBS)onthesquare
urally leads us to the right effective field theory of the
lattice2. There it was pointed out that vortices of the
surface of the three dimensional bosonic topological in-
VBS order carry spin 1/2, i.e. ’half charge’ of the other
sulator. We show that that key properties, such as the
symmetry15. More exotic gapless Bose or vortex metal
presenceofprojectivevortices, emergefromthisstarting
phasesarealsopossible. Thekeypointhoweveristhata
point.
trivial gapped surface state preserving all symmetries is
Possible Phases with Half Quantized Surface Thermal
not possible. A gapped surface that preserves all sym-
Hall effect: Note, in the discussion above, the quantized
metries,necessarilyexhibitstopologicalorder. Moreover,
Halleffectofd=2SREbosonsimmediatelyconstrained
this topologically ordered state differs from a truly two
the physics in 3D. An analogous argument for thermal
dimensional phase with the same topological order and
Hall conductance can be made, which in d = 2 is quan-
symmetry. We discuss specific examples where the ac-
tized to 8 times the quantum of thermal conductance:
tisiodnisotifnscytmfrmometrtyhaotnotfhaensyutrwfaoceditmopenolsoiognicaallsetxatceit.ations κxy/T =8nπ23khB2 . Arealizationofn=1istheKitaevE8
state,with8chiralbosonsattheedge. Therefore,athree
Finally, a different symmetry breaking option is to
dimensional phase protected by time reversal symmetry
break time reversal, which can lead to a gapped surface.
canbeconceived,thathassurfacestatesatadomainwall
Giventhatthisisaninsulator, withgappedbosonsboth
betweenoppositeTbreakingregionsonthesurface,with
inthebulkandonthesurface,itcanbecharacterizedby
the 8 chiral boson modes of the E state. Indeed we will
8
aquantizedresponse-themagnetoelectricpolarizability
constructfieldtheoriesofsuchstates-whetherthesecan
θ. Thissurvivesonreducingtoasingleconservedcharge
berealizedinmicroscopicmodelsregularizedonalattice
(achargeU(1)alongwithtimereversalsymmetry)where
is an open question that we will briefly discuss. We note
this leads to θ =2π, which is a defining property of this
that if indeed this particular class of states exists then it
phase as elaborated below.
is likely to be outside the cohomology classification.
Magnetoelectric polarizability of bosonic insulators: It 3D ‘BF+FF’ Field Theory: The general arguments
is well known that free fermion topological insulators, above will be shown to be consistent with the following
4
d=3 field theory: II. OVERVIEW
2πL =(cid:88)(cid:15)µνλσBI ∂ aI +Θ(cid:88)KIJ(cid:15)µνλσ∂ aI∂ aJ AsthispaperislonganddiscussesbosonicSPTphases
3D µν λ σ 4π µ ν λ σ from several points of view it is helpful to provide an
I I,J overview. Let us first recall the defining properties of
(2)
such a phase. The first requirement is that it has a
where the index I refers to boson species, and bosons
bulk gap but no “intrinsic topological order” implying
four currents are represented by jµI = 1 (cid:15)µνλσ∂ BI
2π ν λσ a unique ground state on all closed manifolds (loosely
while the curl of aI represents the vortex lines. The first characterized as SRE). The second requirement is that
‘BF’ term18 just represents the 2π phase factor of tak- in the presence of a boundary the surface cannot be in
ing a particle around its vortex. The key topological a trivial gapped insulating phase that preserves all the
properties however are determined by the second term, symmetries. One of our goals in this paper is to seek
which attaches quantum numbers to vortices . To avoid an effective “Landau-Ginzburg” description of the sur-
topological order at the surface, and ensure bosonic ex- face theory which naturally incorporates this restriction.
citations, detK = 1 and diagonal entries of K are even Below we motivate our approaches to obtaining such an
integers. In most cases we will take two species with effective theory.
K = σx. Furthermore, time reversal symmetry con- It is worthwhile to emphasize a different point of view
strains Θ = 0,π, the latter being the topological phase. onSPTphases. OnedefinitionofSREisthattheground
This Θ for the internal gauge fields aI associated with state wavefunction can be smoothly deformed into a
vortex lines should be distinguished from the θ for the productoflocaldegreesoffreedomwithoutundergoinga
external electromagnetic field we discussed above. Cou- phasetransition. (Thisdefinition-usedbyChenetal.9 -
pling to an external electromagnetic field allows one to isabitmorerestrictivethananalternateoneusedbyKi-
obtain the quantized magneto electric effect discussed taev). An SPT phase may then be described as a phase
above. Related theories have appeared in the context withagroundstatewavefunctionwhichcanbesmoothly
of three dimensional topologically ordered phases and deformed into a product state if and only if the defining
superconductors19 whereonlythefirstBF terminEqn.2 global symmetry is broken. Let us now briefly discuss
appears with a different coefficient. the surface states in the presence of a boundary from
Recently,itwasproposedthatfreefermiontopological this point of view. If the surface cannot be in a trivial
insulators are captured by similar theories,20 again with gapped state then what else can it be? Clearly the sur-
onlythefirsttermofEqn. 2andcouplingtotheexternal facemayspontaneouslybreakthesymmetry. Theninthe
field. The surface states in such theories were argued absenceofthesymmetryinthebulkthereisnothingspe-
to be bosonized Dirac fermions20,21. While this is an cialaboutthesurfaceconsistentwithitbeingdeformable
intriguing idea, we point out certain problems with the into a product state. For a d = 3 system the surface is
specificimplementationinRef. 20. Inparticular,wefind two dimensional and there is the interesting possibility
noevidenceforsurfacefermions(seeAppendixA)inthe that the surface itself has intrinsic topological (or other
singlecomponentBFtheoryofRef. 20,21. Itispresently more exotic) order, for instance, a fractionalized insula-
unclearifthereisasuitablemodificationthatovercomes tordescribedbya2+1dimensionaldeconfinedZ gauge
2
these issues. Finding an effective field theory description theory. Can we envisage SPT phases where the surface
for 3D fermionic topological insulators remains an open necessarilyhassuchintrinsictopologicalorderthoughthe
problem. On the other hand the field theory discussed bulk does not? The answer is no. If we add a symmetry
in Ref. 22 retains only the second term, which leads to breaking perturbation to the bulk we should be able to
gaplessexcitationsinthebulk,asnotedinRef. 23,which get a trivial surface for an SPT phase. So the surface
differ from the gapped phases of interest here. Thus, it theory should be such that it allows for intrinsic surface
will be important to combine both these terms. Further topological order but does not require it. The effective
wewillneedtoallowformultiplespeciesofbosons. This field theories we obtain for the surface state will all have
isreminiscentoftheK-matrixdescriptionofseveralSPT these general features.
phases in 2D12, where a two component Chern-Simons To motivate our line of attack on 3D SPT phases we
theory, arising from two species of bosons, was required adopt in this paper, it is helpful to recall the physics in
to capture topological properties. lower dimensions, particularly in 2D. There, a number
of different symmetry protected topological phases can
We also display continuum field theoretic models in
be represented by the same 2 × 2K matrix12,24, K =
D =3+1 dimension that realize some of the topological
σ , and differ only in the way the edge fields transform
phases we describe. These are obtained as perturbations x
under different symmetries. The K matrix imposes the
of non-linear sigma models in the presence of a topologi-
following term in the edge Lagrangian:
calthetaterm. Thethetatermhastheeffectofendowing
topological defect configurations of the continuum fields
1
with non-trivial global quantum numbers. We show the Ledge1D = 2π∂tφ1∂xφ2−VIJ∂xφI∂xφJ
connection to the topological BF theory and to the the-
ory of protected surface states. the first term is determined by the bulk K matrix and
5
fixes the commutation relations between the edge fields boson phase variables. However, having multiple vortex
φ . The second term is non-universal and depends on fields demands a dual description like the one above, in
1,2
the detailed interactions. This is the Lagrangian for a termsofvorticescoupledtoagaugefield. Operatorsthat
1+1D Luttinger liquid of bosons, where the two phases insertbosonsactasmonopoleinsertionoperators. Inthe
φ may, for instance, be identified with the boson phase context of 2D deconfined quantum criticality, the action
I
(ϕ)andtwiceoftheconjugatephase(2θ)relatedtoden- above appeared where the density n ∼ Sz is the spin
sity wave order. However, in the edge theory above, the density of an easy plane antiferromagnet, and the gauge
transformation property of the fields is determined by invariant combination of the vortex fields ψ†ψ ∼ eiθ
+ −
the topology of the bulk. Thus, for example in the inte- is the valence bond solid order parameter. However, as
ger quantum Hall effect of bosons12,13 , both fields φ1,2 a theory of the surface on a 3D topological phase, the
are translated by the U(1) charge symmetry, which does fields will transform under internal symmetries, which
not occur in a purely 1D Luttinger liquid where only will dictate additional terms that can be added to this
one of the fields (the boson phase) is charged. Depend- Lagrangian. Nevertheless, this will prove a useful par-
ing on the transformation laws, cosine terms compatible ent theory of the 3D surface states, much like the 2D
with the symmetries may be added to the theory, which Luttinger liquid.
determines its ultimate state. In the previous example, An alternate derivation of the same surface theory is
with U(1) charge conservation symmetry which cannot arrived at by recalling that the surface state of the 3D
be spontaneously broken in 1D, the edge remains a gap- fermionic topological insulators, the single Dirac cone, is
less Luttinger liquid. In other cases, the edge may be also the dispersion at the critical point between integer
gapped after spontaneously breaking the symmetry11,12. quantum Hall plateaus25 in a clean two dimensional sys-
Nevertheless, it is useful to begin with the Luttinger liq- tem. The time reversal symmetry present in the former
uid edge theory above as a parent theory of the edge, caseautomaticallytunesthesystemtothecriticality. By
to which various anisotropies are added. Implicit in this analogy,hereweconsideranetworkmodel,poisedatthe
approach to 2d SPT phases is that it is a useful device transition point between two bosonic integer Quantum
to first consider the surface theory for a system with en- Hall states. The resulting theory is an O(4) non-linear
larged U(1)×U(1) (described by the 2×2 K-matrix al- sigma model with a topological term as in the Euclidean
ludedtoabove), andthenaddanisotropytermstobreak Lagrangian below:
the symmetry to the one under consideration.
This suggests that in three dimensions as well, we 1
L(cid:48) = Tr(∂ g†∂ g)
should first construct a model of the surface. We do edge2D 2κ µ µ
π
this in two different ways. First, we argue for a parent + i (cid:15)µνλTr[(g†∂ g)(g†∂ g)(g†∂ g)]
theory of the surface state, analogous to the Luttinger 24π2 µ ν λ
liquid in one lower dimension, and determine transfor-
where the O(4) vector has been written in terms of an
mation of various fields under the symmetry operations.
SU(2) matrix g. Reassuringly, this has been argued in
As in two dimensions we initially consider an enlarged
Ref. 26 to be equivalent to the deconfined quantum crit-
continuous global symmetry U(1)×U(1) and eventually
ical point surface theory once appropriate anisotropies
breakitdowntothesymmetryofinterest. Initsminimal
are introduced, as we also discuss below.
form, this leads to a deconfined quantum critical action:
Fermionic vortices and vortex metals: A natural case
toconsideriswhenthereisasingleU(1)symmetrylead-
L = (cid:88) |(∂ −iα )ψ |2+K(∂ α −∂ α )2 ing to a conserved ’charge’ and time reversal symmetry.
edge2D µ µ σ µ ν ν µ
Within our formalism, it is convenient to begin with two
σ=±
separately conserved species of bosons and then break
where the boson density is represented as the flux of the this symmetry down to a single physical U(1) by allow-
gaugefieldn=(∂ α −∂ α )/2π andthevorticesofthis ing for mixing between boson species. Assuming inde-
x y y x
boson field ψ are coupled minimally to the gauge field. pendentspecies,thetopologicalphaseischaracterizedby
σ
Theindexσreferstothemultiplecomponentsofthevor- vortices of one species carrying half charge of the other
texfieldwhichisforceduponusduetothesymmetriesof species (see figure 1). However, if the two species are
theproblem. Thekeyobservationisthatsincethevortex the same, then the resulting vortices are conveniently
isnotalocalobject,theycantransformbyanadditional viewed as fermionic (as argued in section IVD). Thus,
phasewhenactedbyasymmetry. Suchaprojectiverep- if the surface is superfluid (while the bulk remains in-
resentation of a symmetry group ensures that a trivial sulating), vortices may be thought of as point particles,
representationofthesymmetrydoesnotariseoncombin- and expected to be fermions. In the presence of a finite
ing with any number of bosons. For example, projective background density of charge, it may be operationally
representationscanleadtohalfchargedvorticesthatcan- difficult to define fermonic statistics27. However, if the
not be screened by integer charge bosons. Furthermore, conserved charge refers to S component of spin, then
z
projectiverepresentationsenforcedegeneracieswhichare timereversalforbidsbackgrounddensity,whichallowfor
crucial. If a single bosonic vortex species was present at statistics to be sharply defined. In this case the vortex
lowenergies,wecouldperformadualitybacktotheusual density is time reversal invariant and generically will be
6
non-zero in the ground state. Destroying the XY order of symmetry at the surface topologically ordered state
by proliferating the vortices will then lead to the forma- is distinct from what is allowed in a strictly two dimen-
tion of a Fermi surface of these vortices which as usual sional state with the same topological order and internal
will be coupled to a fluctuating U(1) gauge field. Thus symmetry. These issues are discussed in Appendix D.
in this case a rather exotic symmetry preserving state is Topological Surface States and Higher Dimensional
possible: a vortex Bose ’metal’28. The vortex metal is a Lieb-Schultz-Mattis Theorems: The feature that the two
compressible metallic state of bosons that preserves all dimensional surface state of a 3d boson topological in-
physical symmetries. Such a state was proposed, and its sulator cannot be in a trivial gapped insulating phase is
properties analysed, in Ref. 28, as a possible quantum reminiscent of the Lieb-Shultz-Mattis (LSM) theorem4
vortex liquid state in two space dimensions. Here we see and its generalization30,31 to states of bosonic systems
that this rather exotic state arises very naturally at the at a fractional filling on clean 2d lattices. Indeed in
surface of a 3d boson SPT phase when the surface su- both cases either a symmetry must be broken or there
perfluidity is destroyed. We note here that the general is topological order, or more exotic long range entan-
properties of doped Conformal Field Theories in 2+1 glement possibly with protected gapless excitations. It
dimensions and the possibility of obtaining Bose metal is thus perhaps no surprise that the field theoretic de-
phases has also been recently discussed29. scriptionswe obtainofthe surface statesofthe 3d boson
Surface topological order and symmetry: It will also topological insulator are closely related to those describ-
be extremely useful to consider the properties of sur- ing deconfined criticality of two dimensional bosons. For
face states which have intrinsic topological order even example, a possible direct transition between superfluid
though the bulk does not. A simple but powerful exam- and valence bond solid (VBS) order controlled by a de-
ple is a state with Z topological order. In cases with a confined quantum critical point was discussed in2 These
2
global U(1) symmetry these may be obtained by pairing field theories naturally incorporate the restriction that
and condensing vortices. A key feature of Z topolog- thereisnotrivialinsulatingphase. Thereishoweverone
2
ical order is the existence of three distinct topological important difference. For the surface states discussed in
quasiparticles (often denoted the electric, magnetic, and thispaper,thetrivialinsulatingphasedoesnotexisteven
their composite particles). These three all have mutual in the presence of disorder that breaks translation sym-
π statistics. Further in the common kind of Z topologi- metry. Correspondingly gapless surface critical points
2
cal order (which is what is pertinent for this paper) two aredescribedbyrandomversionsofdeconfinedquantum
of these are bosons while their bound state is a fermion. criticality. In contrast, the proofs of LSM and its gen-
Usually a trivial phase is obtained from such a topologi- eralizations assume translation symmetry since bosons
cal ordered phase by confinement. This occurs when one on two dimensional lattices can always form a localized
ofthebosonictopologicalquasiparticlescondense. When phaseinthepresenceofsufficientlystrongdisorder. This
the topologically ordered state is obtained at the surface is consistent with the fact that at least one of the order
of a boson SPT phase, we will show that both types of parameters that enter in 2D bosonic deconfined critical
bosonic topological quasiparticles transform nontrivially points involve spatial symmetry (like the VBS order),
under the defining symmetry. When such a quasiparti- while the unique property of the surface state of a 3D
clecondenses,thesurfacetopologicalorderisindeedlost topologicalphaseisthatitisabletoachievethesameby
buttheresultingnon-topologicalphasebreakstheglobal invoking purely internal symmetries.
symmetry.
Two dimensional bosons on clean lattices at commen-
III. TRANSPORT PROPERTIES OF 3D
surate filling also exhibit Z topological ordered phases.
2 BOSONIC TOPOLOGICAL INSULATORS:
Thereagaintherearetwobosonictopologicalquasiparti-
GENERAL CONSTRAINTS
cles - one (dubbed the chargon) which carries fractional
boson charge and the other (usually called the vison)
Webeginbyconsideringasystemofinteractingbosons
which transforms projectively under lattice symmetries.
ind=3spacedimensionsinthepresenceoftimereversal
In the presence of these lattice symmetries, again a triv-
and particle number conservation symmetries. Specifi-
ialgappedsymmetrypreservinginsulatorisforbiddenas
cally let us consider the situation where the boson field
condensation of either chargons or visons breaks global
b carries charge 1 under a global U(1) symmetry and
symmetry. Incontrastasoutlinedaboveatthesurfaceof
transforms as b → b under time reversal. The corre-
the boson SPT phase both bosonic quasiparticles trans-
sponding symmetry group is U(1)(cid:111)ZT. Assume it is in
form projectively under internal symmetries and these 2
a gapped insulating phase (at least in the absence of any
are therefore sufficient to preclude the trivial gapped in-
boundaries) and that there is a unique ground state on
sulator.
topologically non-trivial manifolds. For any such insula-
Strictly two dimensional time reversal invariant
tor in 3D, the effective Lagrangian for an external EM
abeliangappedphaseswererecentlyclassifiedinRef. 24.
fieldobtainedbyintegratingoutallthematterfieldswill
In the specific case of Z topological order, the realiza-
2 take the form
tionofsymmetryatthesurfaceofthe3dSPTphaseswe
discussisbeyondthisclassification. Thustherealization L =L +L (3)
eff Max θ
7
The first term is the usual Maxwell term and the second Hall conductivities change sign. The requirement above
is the ‘theta’ term in Eqn. 1: then means that (2σ ,2κ ) must correspond to the al-
xy xy
Several properties of the theta term are well known. lowed electrical/thermal Hall conductivity of a 2d boson
Firstundertimereversal, θ →−θ. Nextonclosedmani- IQHE state.
folds,theintegralof 1 E(cid:126).B(cid:126) isquantizedtobeaninteger For θ =2π, it follows that σ =1,κ =0. It is thus
4π2 xy xy
sothatthequantumtheoryisperiodicunderθ →θ+2π. “half” of the elementary 2d boson IQHE state.
These two facts together imply that time reversal sym- For θ =π, 2σ =1 and this is not allowed for the 2d
xy
metric insulators have θ =nπ with n an integer. Trivial bosonic IQHE. It follows therefore that 3d bosonic TIs
time-reversal symmetric insulators have θ =0 while free with no “intrinsic topological order” cannot have θ =π.
fermion topological insulators have θ =π. It is of course very easy to construct such states32–34
If we allow for a boundary to the vacuum and fur- (or other states with fractional θ) if we allow for frac-
ther assume that the boundary is gapped (if necessary tionalization of the boson but that violates our original
by breaking time reversal symmetry), then the θ term assumption.
leads to a surface Hall conductivity of θ . To see this, A 2d IQHE state with 2σ = 8,2κ = 8 is allowed
2π xy xy
assume a boundary (say at z = 0), θ = θ(z) is zero for and is discussed by Kitaev. Thus a 3d boson TI with
z < 0 and constant θ for z > 0. The action associated surface σ = 4,κ = 4 is allowed. Combining these
xy xy
with the θ term is two types of fundamental states generates the allowed
1 (cid:90) thermal and electrical Hall responses on the surface.
S = d3xdtθ(z)∂ Kµ (4)
θ 8π2 µ Later on in the paper we will discuss how these re-
1 (cid:90) dθ sultsfitinwiththeformalclassificationofSPTandother
= − d3xdt Kz (5) short ranged entangled phases in 3d. For now we reiter-
8π2 dz
ate the crucial observation of this section: A state with
θ (cid:90)
= d2xdt(cid:15)zνλκA ∂ A (6) EM response of θ = 2π necessarily describes a topologi-
8π2 ν λ κ cal insulator of T-reversal symmetric bosons while θ =π
∂B
Where A is the external electromagnetic potential and requires the presence of “intrinsic topological order”. In
Kµ = (cid:15)µνλκA ∂ A . This is a surface Chern-Simons the next Section we study the properties of this θ = 2π
ν λ κ
term and leads to a Hall conductivity θ/2π. boson topological insulator in detail.
For fermion topological insulators θ = π so that the
surfaceσ = 1. Ifweshiftθ →π+2nπ,thenthesurface
xy 2
σ = (n+ 1). This corresponds to simply depositing IV. SURFACE THEORY OF 3D BOSONIC SPT
anxyordinary 2integer quantum Hall state of fermions at PHASES
the surface of this insulator - hence this should not be
regarded as a distinct bulk state so that the only non- In this section we will derive the non-trivial surface
trivial possibility is θ =π. theory of one example of a 3D bosonic SPT phase. We
Now let us consider bosonic insulators. Again T- will soon specialize to the symmetries of the topological
reversal and periodicity imply θ = nπ and a surface insulator: charge conservation and time reversal sym-
σ = n/2. A crucial observation is that now θ = 2π metry (U(1) (cid:111) ZT), and exhibit a nontrivial topologi-
xy 2
must be regarded as distinct from θ = 0. At θ = 2π cal phase in three dimensions, built purely of bosons.
the surface σ = 1. But this cannot be obtained from To begin with we will assume that there are two species
xy
the surface of the θ = 0 insulator by depositing any 2d of bosons whose numbers are separately conserved, and
integer quantum Hall state of bosons. Recent work12,13 there is enlarged (U(1)×U(1)) (cid:111) ZT. Later we will
2
has shown (see Ref. 13 for a simple argument) that 2d breakthistojustU(1)(cid:111)ZT symmetrybyincludinginter-
2
IQHE states of bosons necessarily have σ even. Thus speciesbosonmixingtermsintheHamiltonian. Asimilar
xy
the surface state of the θ = 2π boson insulator is not a construction12 has proven to be very powerful in d = 2.
trivial2dstatebutratherrequiresthepresenceofthe3d We consider two approaches:
bulk. The first approach exploits the fact that a bosonic
Therefore θ = 2π necessarily corresponds to a non- SPT phase in d dimensions has surface states that cor-
trivial3dbosonicTI.θ =4πishowevertrivialasthenthe respond to a conventional theory of bosons in d − 1
surfacestatecanberegardedasa2dbosonicIQHEstate. dimensions except in the way symmetries are imple-
One may still obtain a 3D topological phase, but the mented. For example, the edges of SPT phases of
topologyisnotmanifestintheelectromagneticresponse. bosonsinD=2+1dimensionscorrespondtoconventional
We can sharpen and generalize this result. Under T- 1D Luttinger liquids, except for their unusual symme-
reversal as θ → −θ, nπ → −nπ. As the bulk state is try transformations12,13,37. We therefore consider a 2D
T-reversal invariant we require that the surface state at bosonic state to model the surface and assume the sur-
θ =−nπbeobtainablefromthesurfacestateatθ =+nπ face is a superfluid breaking one of the U(1) symmetries.
by depositing a 2d IQHE boson state. Let us charac- Then, vortices of this condensate may transform under
terize the surface state by both its electrical and ther- a projective representation of the remaining symmetry
malhallconductivities(σ ,κ ). UnderT-reversalboth group. In a projective representation, even the identity
xy xy
8
elementofthesymmetrygroupinducesaphaserotation. This precludes the possibility of obtaining a trivial insu-
Hence local operators, that can be physically measured, lating phase at the surface by condensing vortices.
must remain unchanged under the identity operation of
the symmetry group, since this corresponds to ‘doing
nothing’. However,vortices, whicharenon-localobjects,
A. Surface States and Projective Vortices
can transform projectively. One may attempt to restore
the U(1) symmetry by condensing vortices. However,
Consider a boson field at the surface with phase de-
theprojectivetransformationensuresthatwhenvortices
condense, they necessarily break another symmetry. In gree of freedom φ1, b†1 = eiφ1. We assume the bulk is
insulating and the surface is in the x, y plane.
this way both the boson and vortex condensates lead
to symmetry breaking, in line with our general expec- Thesurfacetheorycouldspontaneouslybreakaglobal
tationforthesurfaceofa3DSPTphase. Itisimportant U(1) symmetry of boson number conservation (a surface
that vortices transform projectively, so that they cannot superfluid)orstayinsulating. Morepreciselyasthebulk
be screened by bosons to obtain a trivial representation is always assumed insulating, the vortex line loops have
of the symmetry group. This is a generalization of the proliferatedinthebulk. Thesevortexlinespenetratethe
idea of quantum number fractionalization - for example surface at points, which may be viewed as point vortices
a particle with half charge changes sign under the 2π of the two dimensional surface theory, since there is no
phaserotationofbosons, implyingaprojectiverepresen- vortex line tension in the insulating bulk. These point
tation. Clearly, a half charge cannot be screened by any vortices are gapped when the surface is a superfluid. If
finite number of bosons. Projective representations were instead they are condensed the surface will be insulat-
alsorecentlyusedtoclassifySPTphasesinD=1+1where ing. Todescribevorticeswegotoadualdescription35,36,
they correspond to the ends of gapped one dimensional where we write the density and currents of the boson b1
topological phases. For example, the half-integer spin on the surface in terms of the field strengths of a gauge
edge states of spin-1 Haldane chains furnishes a projec- field j1µ = (cid:15)µνλ∂να2λ/2π. In particular the density of
tiverepresentationoftherotationgroup. Thissuggestsa bosons is: n1 = (∂xα2y −∂yα2x)/2π (the reason for the
physical picture of a 3D SPT phase, in which the vortex subscript 2 on α will soon be apparent). The boson in-
lineinthebulkissimilartoaHaldanechaintypegapped sertionoperatorseimφ1 correspondtomonopoleinsertion
phase, which necessitates low energy states on the sur- operators,sincetheyinsert2πmmagneticflux. Now,the
face where the vortex ends. In this section we specialize vortices Ψ2 are particles that couple minimally to the
tothesymmetriesofthetopologicalinsulator. Thenthis gauge field α2. In general there will be multiple vortex
procedure explicitly produces a topological phase char- speciesthattransformintoeachotherunderthesymme-
acterized by quantized magnetoelectric effect θ =2π. tryoperation. Letuslabelthembyi,so[Ψ2]i =ψ2i. All
these fields couple minimally to the gauge field.
Thesecondapproachwillbetodirectlyimplementthe
Thus we have for the dual surface theory:
propertydiscussedaboveinSectionIIIthatifthesurface
breaks T-reversal and is gapped then it has quantized (cid:88)
L = |(∂ −iα )ψ |2
Halltransport. IfT-reversalisnotbrokenapowerfulap- surf 1µ 2µ 2i
proach to obtain the surface theory is to start with the i,µ
theory of the quantum phase transition point between 1
+V(Ψ )+ f2 +... (7)
thetwobosonicquantumHallphasesthatcorrespondto 2i 2κ 2µν
the two T-broken surfaces. In the case of free fermion (8)
topological insulators, a similar reasoning leads to the
single Dirac cone surface state that describes the tran- where f =(∂ α −∂ α ).
2µν µ 2ν ν 2µ
sition between the σxy = ±21 states on the surface. For As argued above, one route to obtaining topological
free fermions the transition between these integer quan- surface states is if the surface vortices transform under
tum Hall states is described by a Chalker-Coddington a projective representation of the remaining symmetry.
network model38. For the bosonic problem of interest Vortices that transform projectively under a global sym-
hereweconstructananalogousnetworkmodelandshow metry are actually not at all unfamiliar. It describes the
it leads to a sigma model with a topological term. generic situation of two dimensional bosons on a lattice,
Theresultsofthesetwoapproachesarereadilyseento say, at some commensurate filling41,42. These projective
be connected. In both cases the field theories we obtain vortices play a crucial role in the theory of deconfined
for the surface have appeared previously in the context quantum criticality.
of deconfined quantum criticality. We discuss the phase We will only need to consider two component vortex
diagramofthesurfacestatesdescribedbythesefieldthe- fields, Ψ2 = (ψ2+, ψ2−) for a variety of cases consid-
ories. When inter-species tunneling is included the vor- ered in this paper. The gauge invariant combination
tices of the two species of bosons get confined to each Ψ†2σ+Ψ2 = ψ2∗+ψ2− = eiφ2 = b†2 then defines another
other. The resulting single vortex no longer transforms bosonic field . Now, Eqn. 7 closely resembles the action
projectively under the physical symmetries. However we for a deconfined quantum critical point (non-compact
argue that it is most conveniently viewed as a fermion. CP theory with easy plane anisotropy)2,40. We will
1
9
demonstrate how this emerges as the theory for the sur- Condensing single vortices will then break symmetry
facestates,andalso,inthenextsection,describeathree as described below. This is best analyzed by assuming
dimensional bulk theory that leads to this edge theory. separate number conservation of each species of boson,
n , n , in which case they can be coupled to an external
1 2
gauge potential A , A . Then the effective Lagrangian
1 2
1. Surface States of a Bosonic topological insulator: Eqn7 reads:
Symmetry U(1)(cid:111) ZT
2
a cTohnesseervaerde tUh(e1)sycmhamregteriaensdofZtThettimopeolroegviecraslailnssyumlamtoer-, L1 = (cid:88)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18)∂µ−iα2µ−isA22(cid:19)ψ2s(cid:12)(cid:12)(cid:12)(cid:12)2+....
2 s=±
try. The semi-direct product appears so that the charge
1 i
insertion operator eiφ is invariant under time reversal, + ((cid:15) ∂ α )2+ A (cid:15) ∂ α (12)
2κ µνλ ν 2λ 2π 1µ µνλ ν 2λ
which involves both φ → −φ and i → −i. Here we will 1
construct the surface theory of a 3D topological phase
Here s=±. The ψ are vortex fields of b which carry
2± 1
with these symmetries.
half charge of boson species 2. The flux of the gauge
Let us begin with an enlarged symmetry, two species
field α is precisely the conserved density of species 1;
1
of bosons that are separately conserved. Consider a con-
hence the last term in the above action where the exter-
densateofonespeciesb . Vorticesinthiscondensateare
1 nal probe gauge field A couples to this current. This
1
createdbythefieldΨ . Weneedtospecifytheprojective
2 action will need to be modified by including all symme-
representation for the vortices Ψ and the transforma-
2 try allowed perturbations. We will do this and analyze
tion of the bosons b†1. The remaining symmetry group thepossiblephasesbelow, butasapreview, considerthe
U(1)(cid:111) Z2T has a single projective representation (P1) effect of breaking time reversal symmetry by condensing
which acts as follows. Under a U(1) rotation by angle (cid:15), just one species of vortex say ψ . Anticipating a single
2−
thefieldsψ2± →e±i(cid:15)/2ψ2± andundertimereversal: Z2T: charge, consider an external field that couples equally
ψ2+ →ψ2∗− and ψ2− →ψ2∗+. Or more compactly: to the two charge densities A1 = A2 = A. The vor-
tex condensate forces α = 1A, which when substituted
2 2
intotheactionaboveyieldstheelectromagneticresponse
Ψ2 →ei2(cid:15)σzΨ2 :U(1) 1 A (cid:15) ∂ A ,whichyieldsσ =1onthesurface,indi-
Ψ2 →σxΨ∗2 :Z2T (9) c4aπtinµgµaνmλ aνgnλetoelectricrespoxnyseθ =2π,asadvertised.
Wenowchangetrackandobtainthesamesurfacetheory
Here the σ are the Pauli matrices in the standard repre- from a very different point of view.
sentation. Thusthevorticescarrycharge±1/2,ofbosons
oftheotherspecies. Thetimereversalsymmetrythatin-
terchanges the two vortex fields ensures that the vortex B. Network Model Construction
chargeisfixedexactlyathalf. Itisimpossibleto‘screen’
this charge with regular integer charged bosons.
ThegeneralconsiderationsofSectionIIIshowedthata
Thesetransformationlawsofcoursedeterminehowthe
timereversalsymmetricbosoninsulatorwithelectromag-
bosonoperatorb†2 =eiφ2 andtheirdensityn2transforms. neticresponsecharacterizedbyθ =2π isinatopological
We also need to specify how the bosons b†1 = eiφ1 and insulator phase. This key result relied on the observa-
density n1 transforms. The symmetry transformations tion that if T is broken at the surface to gap it out then
are: such a state has a quantized electrical Hall conductiv-
ity σ = ±1, and a thermal Hall conductivity κ = 0.
xy xy
φ → φ +(cid:15) U(1)
1,2 1,2 WhatifT-reversalisnotexplicitlybrokenatthesurface?
φ → −φ ZT (10) Thesurfacecanthenpotentiallybegapless. Whatisthe
1,2 1,2 2
nature of the resultant theory? To construct this theory
The conjugate boson numbers therefore transform as it is extremely instructive to learn from the example of
thefreefermionT-reversalsymmetrictopologicalinsula-
n1,2 → n1,2 U(1) tor. In that case if T is explicitly broken to gap out the
n1,2 → n1,2 Z2T (11) surface, then we get σxy = ±12. When T is unbroken, it
is possible to get a single massless Dirac cone which is
A necessary compatibility check is that Eqn. 7 is in- exactly the low energy theory of the transition between
variantunderthesymmetryoperation,whichcanbeveri- two integer quantum hall plateaus of fermions in d = 2.
fiedforthesetransformations. Forexample,timereversal Generically we can tune the chemical potential to move
symmetryisimplementedviaEqn. 9onthevortexfields away from the Dirac point to get a Fermi surface which
which is compatible with n (and hence α ) remaining encloses the Dirac point.
2 2
invariant while i → −i under time reversal. Moreover, The familiar free fermion example gives us a crucial
since the bosons carry charge, the monopole insertion clue to construct the theory of the T-reversal symmetric
operators are forbidden. surface state of the boson topological insulator. First
10
construct the low energy theory of the d = 2 integer 1 2 3 4 2N
quantumHallstateofbosonsasapotentialcandidatefor
t
the gapless surface state of the 3d topological insulator. 0
Then add perturbations allowed by symmetry to obtain
the generic surface theory. t
e
With this motivation we now study the IQHE plateau
transition of bosons in d=2.
IQHE quantum phase transition of bosons in
d=2
1 2 3 4 2N
We will study the phase transition using a “network”
t
model construction. The idea is to start with the theory 0
oftheedgestateandcoupletogetheroppositeedges. Let
uswarmupwiththefamiliarexampleoftheIQHEtran- t
e
sition of fermions from a state with σ =1 to one with
xy
σ =0. The model is defined by Fig. 2. It is described
xy
(cid:82)
by the Euclidean action S = dxdtL with
(cid:88) (cid:88)
L= c¯i(∂τ −isi∂x)ci− ti(c¯i+1ci+c¯ici+1) (1M3o)nday, August F27, I20G12.2: NetworkmodelforthefermionIQHEtransition. On
i the top all chiral edge modes are paired to yield an ordinary
insulator. On the bottom there is an unpaired edge mode to
with yield an integer quantum Hall insulator
t = t (14)
ieven e
full effective action is then
t = t (15)
iodd o
si = −(−1)i (16) S = S0+SW +St (18)
(cid:90) 1 (cid:88) (cid:16) (cid:17)
The first term is the sum of the actions of a single chiral S = dxdτ tr ∂ g†∂ g (19)
0 2λ µ i µ i
edge mode of the σxy = 1 fermion IQHE state taken i
to propagate in opposite directions for adjacent i. The (cid:88)
S = i s S [g ] (20)
W i WZW i
secondtermdescribeselectronhoppingbetweenopposite
i
moving edge channels. (cid:88) (cid:16) (cid:17)
When te < to all chiral edge channels are paired with St = − titr gi†gi+1+h.c (21)
partners and σ = 0. Conversely if t > t , then all i
xy e o
edgechannelsgetpairedexceptthetwoendonesandwe
with s and t as before. If t (cid:29) t we get the trivial
i i o e
getthefermionIQHEstatewithσ =1. Thetransition
xy insulator while if t (cid:29) t we get the boson IQHE state.
e o
occurs at t = t and is readily seen to yield a single
e o The transition occurs at t =t . A low energy theory of
e o
massless Dirac fermion in the continuum limit is taken
the transition is obtained by taking the continuum limit
in the i direction.
in the i direction. As the opposite moving edge channels
Let us now repeat this construction for the bosonic
have opposite WZW terms they nearly cancel, and it is
IQHE transition. The edge theory for the boson IQHE
necessary to carefully sum them. Fortunately precisely
state with σ = 2,κ = 0 has a pair of counterpropa-
xy xy thissumwasperformedinRef. 26wherethesamemodel
gating edge modes - one carries the charge and the other
arose in a different context. The result is the effective
is neutral12,13,37. It is convenient to write the effective
D =2+1 dimensional action
action of the edge as an SU(2) WZW theory:
1 S =(cid:90) d3x 1 tr(cid:0)∂ g†∂ g(cid:1)+iπL [g] (22)
S =(cid:90) dxdτ 1 tr(cid:0)∂ g†∂ g(cid:1)+iS [g] (17) eff 2κ µ µ θ
eff 2λ µ µ WZW
Thesecondtermisaθ termfortheSU(2)matrixvalued
(cid:18)b −b∗(cid:19) fieldg in2+1dimensionscorrespondingtoΠ3[SU(2)]=
Hereg isa2×2matrixwithentriesg = 1 2 . The Z. Inthepresentcontextourcalculationhasyieldedthis
b b∗
2 1 term at the value θ =π.
b ,b are the two physical boson fields which form the
1 2 We of course do not have full SU(2) symmetry rotat-
IQHE quantum Hall state.
ingbetweenb andb inthemicroscopicsystem. Forthe
A network model capable of describing the boson 1 2
time being let us assume that we have U(1)×U(1) sym-
IQHE state may now be written down and is defined
metrycorrespondingtoseparateconservationoftheb ,b
by Fig 3. Again we have an array of opposite edge 1 2
bosons. Further let us also assume there is a Z symme-
channels which are coupled together by boson hopping 2
−(cid:80)a=1,2(cid:16)b†iabi+1,a+h.c(cid:17) ∝ −tr(cid:16)gi†gi+1+h.c(cid:17). The atrsysuimntpetricohnasn.gTinhgebn1tahnedrebs2u.lLtsatoefrRweef.w2i6llsrheloawx athllatthtehsee