Zero temperature transition from d-wave superconductor to underdoped regime

12 Pages·0.25 MB·English

by  Jinwu Ye

Zero temperature transition from d-wave superconductor to underdoped regime
Jinwu Ye 1,2 and A. Millis 1,3
1Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106
2 Department of Physics, University of Houston, TX, 77204
3 Department of Physics, Rutgers university, Piscataway, New Jersey, 07974
1 (February 1, 2008)
0
0 By usingmutualflux-attachingsingulargauge transformations, wederiveaneffectiveaction de-
2 scribingthezerotemperaturequantumphasetransitionfromd-wavesuperconductortounderdoped
regime. Inthiseffectiveaction, quantumfluctuationgenerated vorticescoupletoquasi-particles by
n
a mutual statistical interaction with statistical angle θ, the vortices are also interacting with each
a
J other by long range interactions due to charge fluctuation. Neglecting the charge fluctuation, we
findafixedlinecharacterized byθ andcalculate theuniversalspinon, vortexandmutualHalldrag
3
conductivitieswhichcontinuouslydependonθ. ImplicationsfordoublelayerquantumHallsystems

] are given. When incorporating thecharge fluctuation,we emphasize theimportance of keepingthe
n periodicity of the mutual CS term at θ = ±1/2 to get the correct critical behaviors and point out
o possible future directions. The connection with Z2 gauge theory is discussed.
c
-
r I. INTRODUCTION hc/2e vortices.
p
u A lot of work has been done on a closely relatedprob-
s lemwherequasi-particlesarecoupledtothevorticesgen-
. In this paper, we are trying to study the nature of
t erated by external magnetic field inside the supercon-
a zerotemperature quantumphase transitionfromd-wave
ducting state [1–6]. In the following, we will first review
m superconductor at x > x to the underdoped regime at
c these work, then will extend the methods developed in
d- x<xc ofthehightemperaturesuperconductors(Fig.1). theseworktostudythisquantumphasetransitionwhere
n T the quasi-particles are coupled to the quantum fluctua-
o tion generated vortices.
c
Employing semi-classical approximation, Volovik
[
pointed out that the circulating supercurrents around

1
vorticesinduceDopplerenergyshifttothequasi-particle
v T*
spectrum, which leads to a finite density of states at the
2
3 nodes[1]. Thiseffect(Volovikeffect)hasbeenemployed
0 to attempt to explain the experimental observations [7]
1 of longitudinal thermal conductivity κ in [2].
xx
0 AF
Starting from BCS Hamiltonian, Anderson [3] em-
1 dSC
ployed the first single-valued singular gauge transforma-
t/0 underdoped regime xc d-wave superconductor x tiontostudy quasi-particledynamicsinthe mixedstate.
a Unfortunately, Anderson made an incorrect mean field
m Fig 1: The temperature ( T ) versus doping ( x ) diagram of approximationwhich violates the ”Time-Reversal” sym-
- highTc cuprates. metry[6],thereforeleadstotheincorrectconclusionthat
d
It is well known that inside the superconductor phase there is Landau levelquantizationof energy levels of the
n
o x > xc, there are low energy quasi-particle excitations quasi-particle. Franz and Tesanovic (FT) employed a
c near the four nodes of the Fermi surface, the quantum different single-valued singular gauge transformation to
v: phasefluctuationoftheCooperpaircondensatesaresup- map the quasi-particlein a square vortexlattice state to
i pressed,the positive and negative vortices are bound to- Dirac fermion moving in an effective periodic scalar and
X
gether. Asthe dopingdecreasesfromtherighttox ,the vectorpotentialwithzeroaverageandstudiedthequasi-
c
r phase fluctuation increases. At the underdoped regime particlespectrumnumerically[4,5]. Theydidnotseethe
a
x < x , the quantum phase fluctuations are so strong signatureofLandaulevelquantizationintheirnumerical
c
that they generate free hc/2e vortices, destroy the long calculations
range phase coherence of the d-wave superconductor, Unfortunately, in the vortex lattice state, the vector
therefore the superconducting ground state. However, potential only provides a periodic potential instead of
the local short-range pairing still exists, the low energy scattering quasi-particles, its effect on the energy spec-
quasi-particles at the four nodes remain. This quantum trum and other physical quantities is not evident. This
phase transition is driven by the condensation of hc/2e motivatedYetostudythequasi-particletransportinthe
vortices. Near the transition, there are strong coupling disorderedvortexstatewheretherandomABphasescat-
between the quasi-particles and the phase windings of tering may show its important effects. In Ref. [6], Ye
1
observedthat because infinite thin hc/2e vorticesdo not tual CS theory by the U(1) mutual CS theory, we find
break the T reversal symmetry, any correct mean field a fixed line characterized by the mutual statistical angle
theory should respect this T symmetry. He showed ex- θ and calculate the universal spinon, vortex and mutual
actly there is no Landau level quantization due to this Hall drag conductivities which continuously depend on
T symmetry. He applied the singular transformation in θ. This transition can also be viewed as a simplest con-
the FT gauge to disordered vortex state and found that finement and deconfinement transition. The properties
the long-range logarithmic interaction between vortices ofthe confinementanddeconfinement phasesonthe two
suppresses the fluctuation of superfluid velocity (scalar sides of the criticalpoint are also discussed. We stressed
potential), but does not affect the fluctuation of the in- explicitlythattheperiodicityofZ mutualCSgaugethe-
2
ternal gaugefield. Therefore the scalar field acquires a ” oryonthelatticeisnotpreservedbytheU(1)mutualCS
mass ” determined by the vortex density, but the gauge theoryinthe continuum. WhentheU(1)chargefluctua-
field remains ” massless ”. The quasi-particle scattering tionistakenintoaccount,wetentativelystillreplacethe
from the ”massless” internal gauge field dominates over Z mutualCStheorybythe U(1)CSmutualtheoryand
2
those from the well-known ”massive” Volovik effect and treat both U(1) fluctuations on the same footing. We
the non-magnetic scattering at sufficient high magnetic find the U(1) charge fluctuation turns the fixed line into
field. This dominant scattering mechanism is a purely a fixed point where charge ±e and spin 1/2 Dirac-like
quantummechanicaleffectswhichwascompletelymissed quasi-particle and charge 2e Cooper pairs are asymptot-
by all the previous semi-classical treatments [2]. In fact. ically decoupled. Although the result is pathological, it
it is responsible for the behaviorsofboth κ andκ in doessuggestthe spinonandholonareconfinedinto elec-
xx xy
high magnetic field observed in the experiments [7]. trons and Cooper pairs when condensing hc/2e vortices,
When the vortices are generated by quantum fluctu- in contrast to the condensing of double strength hc/e
ations or thermal fluctuations, themselves are moving vorticesdiscussedin[13,12]andreviewedinappendixC.
around, new physics may arise. From Newton’s third Although we are unable to solve the critical behaviors
law, intuitively, the vortices must also feel the counter- of a combined Z and U(1) theory in this paper, we do
2
actingABphasecomingfromthequasi-particles. There- stress that Z periodicity should be treated correctly to
2
fore there is a mutual AB phase scattering between vor- understand the critical behaviors of a theory with both
tices and quasi-particles. In this paper, we enlarge the discrete Z and continuous U(1) gauge fields.
2
one singular gauge transformation developed in [3,4,6] The paper is organized as the following. In the next
to two mutual flux-attaching singular gauge transfor- section, we derive the effective action with both the mu-
mations. We then apply them to study quasi-particles tual C-S interaction and the charge fluctuation, first in
coupled to moving vortices near the zero temperature thepathintegrallanguage,andthenincanonicalquanti-
quantum critical point. In canonical quantization, we zationrepresentation. In Sec. III, we concentrate on the
perform two singular transformations which are dual to effect of the mutual C-S interaction and calculate the
eachotherto quasi-particlesandmovingvorticesrespec- critical exponents and universal conductivities for gen-
tively. Just like conventional singular gauge transforma- eral statistical angle θ. Implications for double layers
tion leads to conventional Chern-Simon (CS) term, the Quantum Hall System are given. Some details are rele-
two mutual singular gauge transformations lead to mu- gated to appendix A. In appendix C, we concentrate on
tualCSterm. ThiselegantmutualCStermpreciselyde- the effect of charge fluctuation which is the only gauge
scribe the mutual AB phase scattering between vortices fluctuation in double strength hc/e vortices. In Sec IV,
and quasi-particles. Alternatively, in path-integral pre- wediscuss the combinedeffectofmutualC-Sinteraction
sentation,theeffectiveactiondescribesthequasi-particle and charge fluctuation, we also stress the importance to
moving in both vector and scalar potentials due to the keeptheperiodicity ofZ gaugefieldwhenconsideringa
2
phase fluctuations of quantum generated vortices. By a theory with both discrete Z and continuous U(1) gauge
2
duality transformation presented in Refs. [8,9] to a vor- fields. In Sec. VI, we discuss the connection of our ap-
tex representation, the quantum fluctuation generated proachto the currentfashion Z gauge theory and point
2
vortices couple to quasi-particles by a mutual CS term, out the possible open problems. In appendix B, we ap-
the vortices are also interacting with each other by long ply the method developed in the main text to quenched
range logarithmic interactions due to charge fluctuation random vortex array.
described by a Maxwell term. By carefully considering
the periodicity of this mutual CS term, it can be shown
that the effective action is essentially equivalent to the II. THE EFFECTIVE ACTION
dual vortexrepresentationderivedby Senthil and Fisher
(SF) from the elegant Z2 gauge theory [10] except the Following the notation in Refs. [4,6], we define d↑ =
Volovik effect missing in the Z gauge theory is also ex- c (x),d = c† [18]. Because there are equal number of
2 ↑ ↓ ↓
plicitly incorporatedinto the presentapproach. Neglect- positiveandnegativevortices,wedividethepositivevor-
ing the charge fluctuation first and replacing the Z mu- tices into two subsets φ ,φ and negative vortices into
2 p1 p2
2
two subsets φ ,φ . We define φ = φ +φ ,φ = is explicitly SU(2) spin invariant. Equivalently, we can
n1 n2 A p1 n1 B
φ +φ andintroduce the spinonby performingagen- start with the explicitly spin SU(2) invariant approach
p2 n2
eral singular unitary transformation d=Ud : advocated in Ref. [13] and perform the singular gauge
s
transformation in the p-h space.
Hs =U−1HU, U = ei0φA e−0iφB (1) facLt,qp ietnjoisys Ug(a1u)ge×syZmme),trythUe(1)fiurst× Ubesi(n1g) (uniin-
(cid:18) (cid:19) 2
form and second being staggered gauge symmetry:
where φ +φ =φis the phase ofthe Cooperpair. The
A B
Uniform (or external) U (1) gauge symmetry
originaltransformation[3,4]isdevisedforstaticvortices. u
Here we extend it to moving vortices whose phase φ is
c →c eiχ, d →d
alsofluctuating,therefore,dependsonboththespaceand α α α α
time. Thespinonischargeneutral. InAnderson’sgauge, φA →φA+χ, φB →φB +χ (6)
φ = 0,φ = φ or vice versa. In the former(latter), the
A B
spinon is electron-like ( hole-like). UnderthisuniformU(1)transformation,thecorrespond-
Expanding H around the node 1 where p~ = (p ,0), ing fields transform as:
s F
weobtainthelinearizedquasi-particleLagrangianL in
qp
φ→φ+2χ, A →A +∂ χ
the presence of the external gauge potential A : α α α
µ
v →v , a →a (7)
α α α α
Lu =ψ†[(∂ −ia )+v (p −a )τ3+v (p −a )τ1]ψ
qp 1 τ τ f x x 2 y y 1
d,v ,a all are invariant under this external U(1)
+ψ†ψ v v (~r)+iψ†τ3ψ v (~r)+(1→2,x→y) (2) α α
1 1 f x 1 1 τ transformation. Thereforethespinond ischargeneutral
α
where v = h¯∂ φ− eA is the gauge-invariant super- to the external magnetic field.
µ 2 µ c µ Staggered (or internal) U (1) gauge symmetry
fluid momentum, it acts as a scalar scattering potential, s
a = 1∂ (φ − φ ) is the AB gauge field due to the
pµhase w2inµdinAg of vBortices [14]. Note that the external cα →cα, dα →dαe−iχ
gauge potential only appear explicitly in the superfluid φ →φ +χ, φ →φ −χ (8)
A A B B
momentum v . Because there are equal number of pos-
µ
itive and negative vortices in both φ and φ , on the UnderthisinternalU(1)transformation,thecorrespond-
A B
average, the vanishing of v and a is automatically en- ing fields transform as:
µ µ
sured, in this case, the Anderson gauge turns out to be
moreconvenientthanFTgauge. Inthefollowing,weuse φ→φ, Aα →Aα
the electron-like Anderson gauge where aµ = 21∂µφ [17]. vα →vα, aα →aα+∂αχ (9)
We get the corresponding expression at node ¯1 and ¯2
by changing vf →−vf,v2 →−v2 in the above Eq. Although the spinon dα is charge neutral to the ex-
ternal magnetic field, it carries charge 1 to the internal
Llqp =ψ¯1†[(∂τ −iaτ)−vf(px−ax)τ3−v2(py−ay)τ1]ψ¯1 gauge field aα.
−ψ¯1†ψ¯1vfvx(~r)+iψ¯1†τ3ψ¯1vτ(~r)+(¯1→¯2,x→y) (3) secIttoirs,esianscyettoheresapliinzeonthiastchUaur(g1e)naecuttsraoln,lUy o(n1)thacetsboosnolny
s
Performing a P-H transformation ψ˜ = ǫ ψ† and onthefermionsector. Infact, Us(1)shouldbe adiscrete
1α αβ ¯1β localZ symmetry[10],becauseupanddownhc/2evor-
the corresponding expression at node ¯2, the above Eq. 2
tices are equivalent and do not break T symmetry [6].
becomes:
Thephase fluctuationis simply 2+1dimensionalX-Y
Ll =ψ˜†[(∂ +ia )+v (p +a )τ3+v (p +a )τ1]ψ˜ model:
qp 1 τ τ f x x 2 y y 1
+ψ˜1†ψ˜1vfvx(~r)+iψ˜1†τ3ψ˜1vτ(~r)+(1→2,x→y) (4) L = Kv2 = K(∂ φ−2A )2 (10)
ph 2 µ 2 µ µ
InordertomakethefinalexpressionsexplicitlySU(2)
invariant, we perform the singular gauge transformation After absorbing the scalar potential scattering part (
ψ12α = e−i(φA−φB)ψ˜1α and the corresponding transfor- Volovikterm)intoLph,wecanwritethetotalLagrangian
mation at node ¯2, then aµ → −aµ, Eq.4 takes the same L=Lqp+Lph as:
form as Eq.2 [15]. Adding the two equations leads to:
L=ψ† [(∂ −ia )+v (p −a )τ3+v (p −a )τ1]ψ
1a τ τ f x x 2 y y 1a
Lqp =ψ1†a[(∂τ −iaτ)+vf(px−ax)τ3+v2(py−ay)τ1]ψ1a +(1→2,x→y)+ K(∂ φ−Aeff)2 (11)
+ψ† ψ v v (~r)+iψ† τ3ψ v (~r)+(1→2,x→y) (5) 2 µ µ
1a 1a f x 1a 1a τ
where a = 1,2 is the spin indices. τ′s matrices are act- where Aeµff =2Aµ−K−1Jµ and the quasi-particle elec-
ing on particle-hole space. As intended, the above Eq. triccurrentis: J =ψ†τ3ψ ,J =v ψ†ψ ,J =v ψ†ψ .
0 j j x f 1 1 y f 2 2
3
1
From Eq.11, it is easy to identify the two conserved jv = ǫ ∂ ∂ φ
Noether currents: spinon current and electric current. µ 2π µνλ ν λ
The spinon current is given by: =ǫµνλ∂νavλ (17)
j0s =ψ1†(x)ψ1(x)+ψ2†(x)ψ2(x) where avµ =∂µφ/2π is the vortex gauge field.
Substituting the above expressions into Eq.15, we
js =ψ†(x)v τ3ψ (x)+ψ†(x)v τ1ψ (x)
x 1 F 1 2 2 2 reach:
js =ψ†(x)v τ1ψ (x)+ψ†(x)v τ3ψ (x) (12)
y 1 2 1 2 F 2 1
f2 +i∂ φjt −iAeffǫ ∂ ae
4K eµν µ µ µ µνλ ν λ
Obviously the spinon current only comes from quasi-
1
particle. Inprinciple,thespinoncurrentisnotconserved = f2 +iatjv−iAeffǫ ∂ ae (18)
4K eµν µ µ µ µνλ ν λ
due to scatterings between different nodes which lead
to anomalous terms not included in Eq.11. However, where at = ae − 1as is the total gauge field felt by the
theinter-nodescatteringsinvolvelargemomentatransfer µ µ 2 µ
moving vortices.
K~ −K~ fori6=j, sowe neglectthem due to momentum
i j Using Φ for the vortexoperatorand adding the quasi-
conservation in the long wave-length limit of the phase
particle part, we get the following effective action:
fluctuation [16]. The exact conserved spin current~jS is
µ
with ~σ/2 inserted in the above spinon currents. L=ψ†[(∂ −iaψ)+v (p −aψ)τ3+v (p −aψ)τ1]ψ
1 τ τ f x x 2 y y 1
The electric current is given by:
+(1→2,x→y)
∂L i
jµe =−∂Aeff =K(∂µφ−Aeµff)=K(∂µφ−2Aµ)+Jµ +|(∂µ−iaΦµ −iaµ)Φ|2+V(|Φ|)+ 2πθaψµǫµνλ∂νaΦλ
µ
1
+ f2 −iAeffǫ ∂ a −µǫ ∂ a (19)
(13) 4 µν µ µνλ ν λ ij i j
Where the first part coming from Cooper pair and the where V(|Φ|) = m2|Φ|2 + g |Φ|4 + ··· stands for the
Φ Φ
second from the quasi-particle. Although they are not short range interaction between the vortices. The last
separately conserved, their sum is. term is due to Berry phase in the boson representa-
The Noether current due to the the symmetry under tion [9] which can be absorbed into Aeff by redefining
µ
φ→φ+χ can be written as: Aeff → Aeff +iµδ , it acts like an external magnetic
µ µ µ0
field in the zˆ ( namely µ = 0 ) direction. There are one
∂L 1 1
jt = =K(∂ φ−Aeff)− js =je − js (14) species of vortex and N = 4 species of Dirac fermion(
µ ∂(∂ φ) µ µ 2 µ µ 2 µ
µ 2 spin components at 2 nodes at the upper plane), the
mutual statistical angle θ = ±1/2. We also changed the
It is a combination of electric and spinon currents,
notation by setting aψ =av,aΦ =as,ae =a.
therefore also conserved.
Although we derived the above equation in the
FollowingRefs.[8,9],weperformadualitytransforma-
electron-like Anderson gauge, in fact, it holds in any
tion to Eq.11
gauge. Eq.19 again enjoys the gauge symmetry U(1)×
K 1 U(1), the first only acting on the vortex sector is the
2 (∂µφ−Aeµff)2−i2∂µφjµs electric U(1) gauge field, the second on both the vortex
1 1 and fermion sectors is the U(1) mutual CS gauge field.
=ijeµ(∂µφ−Aeµff)+ 2Kje2µ−i2∂µφjµs The mutual Chern-Simon term enforces the constraints:
=ijµt∂µθ+ijµt∂µφ−ijµeAeµff + 21Kje2µ (15) 2mπeθajnµss =thaǫtµνwλh∂eνnaΦλa,2qπuθajsµvi-p=artǫicµlνeλ∂enνaciψλr.clesPhayvsiocratlelyx,, iitt
picksup aphase 2πθ. Equivalentlywhena vortexmoves
Where we have separated topological trivial spin-wave aroundaquasi-particle,italsopicksupaphase2πθ. Al-
part and topological non-trivial vortex parts. though the conventional C-S term breaks T symmetry
Integrating out the spin-wave part, we get the conser- and has periodicity under θ →θ+2 [19–23], the mutual
vation equation for the total current jt = je − 1js. In C-Stermdoesnot breakTsymmetryandhasperiodicity
µ µ 2 µ
fact, as shown in the previous paragraphs,je and js are under θ → θ+1. For example, θ = −1/2 is equivalent
µ µ
separately conserved. Therefore we can introduce spin to θ = 1/2. In fact, this mutual CS term at θ = 1/2
and electric gauge fields by: shouldbethesameastheZ mutualCStermonthelat-
2
ticediscussedinRef.[10]. Thereforestrictlyspeakingaψ
jµs =ǫµνλ∂νasλ andaΦ areZ2 gaugefields, onlyaµ is aU(1)gaugefield.
je =ǫ ∂ ae (16) The Doppler-shifted term ( Volovik effect) is encoded in
µ µνλ ν λ
the last term in Eq.19. This effect was not taken into
We can also define the vortex current: account in the Z gauge theory [10].
2
4
In canonical quantization language, it is very instruc- All the generated terms only renormalize the short
tive to compare the well-known singular gauge transfor- rangeinteractionsalreadyincludedinV(|Φ|). Themodel
mation leading to composite fermion in ν = 1/2 system is alsoclosely relatedto the Halldrag problemindouble
by Halperin, Lee and Read (HLR) [19] to the singular layer Quantum Hall system [26], therefore the model it-
gauge transformations performed in the spinon-vortex self is interesting on its own right and deserves detailed
systeminthispaper. Thecrucialdifferenceisthatinthe investigation.
former, we attach electron’s own θ flux to itself ( θ = 2
in ν =1/2 systemto keepfermion statistics intact), the
constraint∇×~a(~r)=2πθρ(~r)inCoulombgauge∂ a =0
i i A. Quantum Critical point
leads to composite fermion ψ coupled to conventional
c
CS term which breaks T symmetry, has periodicity un-
der θ →θ+2; however,in the latter, we attach vortex’s Inordertocalculateconductivities,weaddtwosource
θ =1/2 flux to quasi-particle or vise versa, the two con- fields Aψ and AΦ for the quasi-particles and vortex re-
µ µ
straints ∇ ×~aψ(~r) = 2πθjv(~r),∇ ×~aΦ(~r) = 2πθjs(~r) spectively:
0 0
in Coulomb gauges ∂ aψ = ∂ aΦ = 0 naturally leads to
quasi-particles and voiritices airei coupled by mutual CS L=ψa†γµ(∂µ−iaψµ −iAψµ)ψa
terminadditionaltothe electricalgaugefieldwhichme- i
+|(∂ −iaΦ−iAΦ)Φ|2+V(|Φ|)+ aψǫ ∂ aΦ (22)
diates the logarithmic interaction between vortices. The µ µ µ 2πθ µ µνλ ν λ
mutualCStermdoesnotbreakTsymmetry,hasperiod-
icity under θ →θ+1. where a = 1,2,3,4 stands for N = 4 species of Dirac
Itisimportanttopointoutthatwhenavortexismov- fermion. In fact, two Maxwell terms for aψ and aΦ can
ingaroundaclosedloop,itpickuptwophases,oneisZ be addedto the aboveequation,but they are subleading
2
phaseduetothespinoncurrent,describedbythemutual to the mutual CS term in the low energy limit. It is ex-
C-S term, another is U(1) phase due to the total electric pected that there is no periodicity under θ → θ +1 in
charge current described by the Maxwell term. In the the continuum limit.
following section,we will neglect the electric chargefluc- If the theory is properly regularized on the lattice, it
tuation and concentrate the mutual statistics term. In should has the periodicity under θ → θ +1, for exam-
Sec. IV, we will study the effect of charge fluctuation ple, θ = −1/2 should be equivalent to θ = 1/2. In fact,
described by the Maxwell term. For completeness, we θ =1/2mutualU(1)CStheorycouldbethesameasthe
relegate the review of the effect of charge fluctuation to Z2 mutual CS theory developed in [10].
appendix B. The RG calculation in Refs. [20–22] can be used to
For simplicity, we take the relativistic form for both show that θ is exactly marginal, therefore there is a line
fermionandboson,becausetheanisotropiesinEq.19are offixedpointsdeterminedbythemutualstatisticalangle
expectedtobeirrelevantnearthezerotemperatureQCP θ. In order to calculate the spin conductivity along this
[11]. fixedline, asourcefield couldbe introducedto coupleto
thespincurrent~jS =ψ†γ (~σ) ψ . Similarcalculations
µ α µ αβ β
follow.
III. THE EFFECT OF MUTUAL STATISTICS
Integrating out both fermion and boson leads to:
In this section, we neglect the charge fluctuation, 1 1
L=− aψ(−k)Πψ (k)aψ(k)− aψ(−k)ΠψΦ(k)aΦ(k)
namely setting aµ = 0 in Eq.19. The charge fluctuation 2 µ µν ν 2 µ µν ν
can be suppressed by condensing hc/e vortices reviewed 1 1
− aΦ(−k)ΠΦ (k)aΦ(k)− aΦ(−k)ΠΦψ(k)aψ(k)
in appendix C. We could add the kinetic and potential 2 µ µν ν 2 µ µν ν
terms for the hc/e vortex operator Φ to Eq.19: 1
2 − (aψ(−k)−Aψ(−k))ǫ k (aΦ(k)−AΦ(k))
2θ µ µ µνλ λ ν ν
L =|(∂ −i2a )Φ |2+V(|Φ |) (20)
Φ2 µ µ 2 2 1
− (aΦ(−k)−AΦ(−k))ǫ k (aψ(k)−Aψ(k)) (23)
AsshowninappendixC,thereisnomutualstatisticalin- 2θ µ µ µνλ λ ν ν
teractionbetweenspinonandhc/evorticesΦ . Thelong
2 where the exact forms of Π′s are dictated by gauge in-
range logarithmic interaction between hc/2e vortices Φ
variance and Furry’s theorem :
and hc/e vortices Φ is mediated by the electrical gauge
2
field a . Condensing <Φ >=Φ will generate a mass
µ 2 20 k k
term Φ2220(aµ)2t which dominates over the Maxwell term. Πψµν(k)=Π1(k)k(δµν − µk2ν)
Integrating out a leads to
µ ΠψΦ(k)=ΠΦψ(k)=Π (k)ǫ k
µν µν 2 µνλ λ
Φ1220[14(fµAνeff)2−iAeµffǫµνλ∂νjλv+(jµv)2t] (21) ΠΦµν(k)=Π3(k)k(δµν − kµkk2ν) (24)
5
Where Π ,Π ,Π are the polarizations for
1 2 3 = +
aψaψ,aψaΦ,aΦaΦ.
(a)
IfweareonlyinterestedinDC conductivities,forsim-
plicity, we can put ~k = 0, Eq.23 becomes ( for the most
=
general form, see appendix A):
(b)
L=−1aψ(−ω )Π |ω |aψ(ω )− 1aψ(−ω )Π ǫ ω aΦ(ω ) Fig2: Therenormalizedpropagatorsforaψ (thickwiggleline)
2 i n 1 n i n 2 i n 2 ij n j n andaΦ (thickdashedline). Thethinwigglelinestandsforaψ,the
− 1aΦ(−ω )Π |ω |aΦ(ω )− 1aΦ(−ω )Π ǫ ω aψ(ω ) dashedthinlineforaΦ,thethinsolidlineforfermionpropagators,
2 i n 3 n i n 2 i n 2 ij n j n thethicksolidlineforthebosonpropagators.
1
− (aψ(−ω )−Aψ(−ω ))ǫ ω (aΦ(ω )−AΦ(ω )) By using the bare propagators < aψaΦ >=
2θ i n i n ij n j n j n µ ν
−ǫ k /k2 andthe barefermionandbosonloopresults
1 µνλ λ
− 2θ(aΦi (−ωn)−AΦi (−ωn))ǫijωn(aψj(ωn)−Aψj(ωn)) (25)Πfµ0ν = Πbµ0ν = −1g62k(δµν −kµkν/k2), we can find easily
the renormalized aψ and aΦ propagators Gψ = GΦ =
µν µν
Ifwedefinea˜Φi (ωn)=ǫijaΦj(ωn),A˜Φi (ωn)=ǫijAΦj(ωn), −g2 1(δ −k k /k2). In contrast to the conventional
the above equation becomes diagonal in the spatial in- 16k µν µ ν
CStheorystudiedin [20,21],the propagatorsareeven in
dices i=1,2. Finally, integrating out aψ,a˜Φ leads to
i i k, this is because the theory respects T symmetry. On
the other hand, in contrast to the Maxwell propagators,
1 σψ σH Aψ
L=− (Aψ,A˜Φ) i (26) they behave as 1/k instead of 1/k2.
2 i i σH σΦ A˜Φ
(cid:18) (cid:19)(cid:18) i (cid:19) The three loop diagrams for Π1 are given by:
Where spinon, vortex and mutual Hall drag conduc-
tivities are: + +
1 Π
σψ =( )2 1
θ Π Π +(1/θ−Π )2
1 3 2
Fig 3: The three loop diagrams of Π1. The thick wiggle line
1 Π
σΦ =( )2 3 stands for the renormalized propagators of aψ, the thin solid line
θ Π Π +(1/θ−Π )2
1 3 2 stands for the fermion propagator. The one loop diagram is not
σH =(1)2Π2−θ(Π1Π3+Π22) (27) shown.
θ Π Π +(1/θ−Π )2
1 3 2 ByusingGψ andextractingthesymmetricpartofthe
µν
In fact, all the three conductivities can be written in gauge propagator in the large N results in Refs. [20,21],
the elegant connection formula: we are able to calculate the above three loop diagrams.
Furry’s theorem can be used to eliminate large number
ρij =(ρFB)ij −θǫij (28) of null diagrams. We get the following series:
with the conductivity tensor of fermion and boson given π 3 g4
Π =N (1+ +g8+···) (30)
by 1 8 16(2π)2
σ = Π1 −Π2 (29) where N = 4 is due to the sum over 4 species of Dirac
FB Π2 Π3 fermions.
(cid:18) (cid:19)
The three loop diagrams for Π are given by:
3
Although its form is similar to the conventional con-
nection formulas discussed in [19–21], the physical inter-
+ +
pretations of the conductivities are quite different (see
the following).
When the vorticesare generatedby externalmagnetic
field andpinned by impurities as discussedin [6],the to-
+ +
tal conductivity is the same as the fermion conductivity
becausethestaticvortexdonotcontribute. Asexplained
Fig 4: The three loop diagrams of Π3. The thick dashed line
in appendix B, the static vortices only feel the electric stands for the renormalized propagator of aΦ, the thick solid line
gauge field a , but not the statistical gauge field aΦ.
µ µ stands for the boson propagator. The one loop diagram is not
The above expressions are exact, but Π ,Π ,Π can
1 2 3 shown.
only be calculated perturbatively in the coupling con-
stant g2 = 2πθ. The renormalized propagators for aψ For bosons, only one loop result is known:
and aΦ can be found from the following Feymann dia- π
grams: Π3 =N (1+g4+g8+···) (31)
8
6
Note that both Π and Π are even functions of θ. g4
1 3 η =η − N (32)
From Furry’s theorem, one of the first non-vanishing Φ XY 12π2
diagram for Π2 is: where ηXY ∼ 0.038 is the anomalous dimension for the
3d XY model [24].
Thecorrelationlengthexponentcanalsobecalculated
to two loops by the insertion of the operator Φ†Φ:
Fig 5: OneofthefourloopdiagramsofΠ2. thethinsolidline
standsforthefermionpropagator,thethicksolidlinefortheboson X
propagator. All the other four loop diagrams can be obtained by
shufflingthepositionsofthethreebarepropagator lines. Fig 7: Thecrossstands fortheoperatorinsertionofΦ†Φ.
The series is Π = N(g6+g10+···) which is an odd The result is:
2
function of θ. g4
From Eq.27, σψ,σΦ are even functions of θ, but the ν =νXY − 12π2N (33)
mutual Hall drag conductivity is an odd function of θ.
whereν ∼0.672iscorrelationlengthexponentforthe
XY
These are expected from P-H transformation. Under
3d XY model [24].
the P-H transformation of the vortex operator Φ → Φ†
It is instructive to go to dual representation of Eq.22,
in Eq.22, it can be shown that AΦ,θ is equivalent to
µ namely go to the boson representation:
−AΦ,−θ. From Eq.26, we reach the same conclusions.
µ
Specifically, σH takes opposite values for θ = ±1/2, the L=ψa†γµ(∂µ−iaψµ −iAψµ)ψa+|(∂µ−iaφµ)φ|2+V(|φ|)
periodicity under θ → θ +1 is not preserved. Exper- + 1(fφ )2+iaΦǫ ∂ (aφ−aψ/θ)−iAΦǫ ∂ aφ (34)
imentally, the Hall drag conductivity can be detected 4 µν µ µνλ ν λ λ µ µνλ ν λ
by measuringthe transversevoltagedrop( or transverse where V(|φ|)=m2|φ|2+g |φ|4+···,
φ φ
temperature drop for thermal conductivity ) of spinons
IntegratingoutaΦ leadstothe constraintuptoapure
due to the longitudinal driving of vortices. The Hall µ
gauge:
dragconductivityindouble layerQuantumHallsystems
has been investigated by several authors [26]. In double aψµ =θaφµ (35)
layersystems,theelectronsindifferentlayersaretreated
Substituting theaboveconstrainttoEq.34andsetting
as two different species. There is a mutual CS interac- aφ =a , we find:
µ µ
tion between the two species (both are fermions) which
1
is directly responsible for this Hall drag conductivity, al- L=ψ†γ (∂ −iθa −iAψ)ψ + f2
thoughthe Coulombinteractionbetweenthe twospecies a µ µ µ µ a 4 µν
is responsible for the Coulomb( longitudinal ) drag [27]. +|(∂µ−iaµ)φ|2+V(|φ|)−iAΦµǫµνλ∂νaλ (36)
This example shows that no external magnetic field is
The above Eq. indicates that fermions and bosons are
needed to produce a Hall effect ! However,θ =±1/2are
coupled to the same gauge field whose dynamics is de-
very special points, U(1) mutual CS term reduces to Z
2 scribed by Maxwell term instead of C-S term. This Eq.
field, thereshouldbeno Halldragconductivity justlike
is simply 2+1 dimensional combination of spinor QED
α=±1/2 vortex leads to no Hall conductivity.
and scalar QED. The RG analysis at 4− ǫ by dimen-
Again, by using the large N result of [20,21], we find
sional regularization is possible, because the marginal
the anomalous dimensions of the fermion and vortex to
dimension of all the relevant couplings are 4. However
two loops:
a RG analysis directly at 2+1 dimension is formidable,
the physical meaning and the exact marginality of θ is
obscure in the boson representation. However they are
(a)
evident in the dual vortex representation Eq.22. It is
evident that there is no periodicity under θ → θ+1 in
+ Eq.36.
(b) If we perform duality transformation again on Eq.36
Fig6: Thetwoloopdiagramsfortheselfenergiesofspinon(a) togotothevortexrepresentation,thenwerecoverEq.19
andvortex(b). Therenormalizedoperatorsareused. upon neglecting the two Maxwell terms relative to the
mutual C-S term.
The results are:
In the next subsection, setting the two source terms
g4 vanishing, we look at the properties of the different
ηψ =−48π2 phases on the two sides of this quantum critical point.
7
B. Disorder and superconducting phases term in the continuum. In this section, we try to con-
sider the combined effects of Z mutual statistical gauge
2
In the disordered phase, the vortex condense <Φ>= fluctuation and U(1) electrical gauge fluctuation. After
Φ0 which generates a mass term for aΦµ in Eq.22 replacingthe Z2 discrete gaugetheoryby a U(1)mutual
CS theory, we treat U(1) mutual CS theory and U(1)
Φ2 electrical gauge theory on the equal footing. Although
0(aΦ)2 (37)
2 µ t the replacement leads to somewhat misleading conclu-
sions,the heuristics derivationsuggeststhatthe conden-
where the subscript t means transverse projection. sation of hc/2e vortex condensation indeed leads to the
IntegratingoutthemassiveaΦµ leadstoaMaxwellterm confinementof spinonand chargoninto Cooperpair and
for aψ: electron, in contrast to the condensation of hc/e vortex.
µ
We do not intend to understand the critical behaviorsof
1
L=ψ†γ (∂ −iaψ)ψ + (fψ )2 (38) acombinedZ2 andU(1)theoryinthispaper. Wesimply
a µ µ µ a 4Φ2 µν stress that Z periodicity should be treated correctly to
0 2
understand the critical behaviors of a theory with both
This is simply 2 +1 dimensional spinor QED which
discrete Z and continuous U(1) gauge fields.
2
was studied by large N expansion in [25]. In fact, we
reach the same description from the boson representa-
tion Eq.36. Because in the disordered phase, the boson
A. Quantum critical point
φismassive,thereforecanbeintegratedout,itgenerates
the Maxwell term 1 f2 which dominates over the ex-
isting non-critical M4maφxwµeνll term. We reach Eq.38 after Putting aΦµ →aΦµ −aµ in Eq.19, we get
identifying mφ ∼Φ20. L=ψa†γµ(∂µ−iaψµ)ψa+|(∂µ−iaΦµ)Φ|2+V(|Φ|)
In the superconductor phase, the vortex Φ is mas-
i 1
sive, therefore can be integrated out, it leads to the old + aψǫ ∂ aΦ−ia ǫ ∂ (aψ/θ+Aeff)+ f2 (40)
Maxwell term 1 (fΦ )2. Integrating out aΦ generates 2πθ µ µνλ ν λ µ µνλ ν λ λ 4 µν
4mΦ µν
a mass term: Integrating out the electric gauge field a leads to:
µ
m
L=ψa†γµ(∂µ−iaψµ)ψa+ 2Φ(aψµ)2t (39) L=ψa†γµ(∂µ−iaψµ)ψa+|(∂µ−iaΦµ)Φ|2+V(|Φ|)
i 1
where mΦ is the mass of the vortex. + 2πθaψµǫµνλ∂νaΦλ + 2(aψµ/θ+Aeµff)2t (41)
The Dirac fermions become free. In fact, we reachthe
same description from the boson representation Eq.36. Comparing to Eq.22, it is easy to see that the charge
Becausein superconductorphase,the bosonφ condense, fluctuation leads to a mass term for the gauge field aψ.
µ
therefore generates a mass term φ220(aµ)2t which renders Shifting aψµ/θ+Aeµff → aµ and adding the gauge fixing
the Maxwelltermineffective. We reachthe sameconclu- term 1 (∂ a )2,wecanintegrateoutthemassive gauge
2α µ µ
sion from both sides by identifying m ∼φ2. field a in Lorenz gauge α=0 and find:
Φ 0 µ
In short, in the disordered phase, the system is de-
scribedbyspinorQEDEq.38;inthesuperconductorside, L=ψa†γµ(∂µ−iθAeµff)ψa+|(∂µ−iaΦµ)Φ|2+V(|Φ|)
by free Dirac fermion. We can view the transition as 1
+ (fΦ )2−i(Aeff +θjs)ǫ ∂ aΦ+(js)2 (42)
the simplest confinement-deconfinement transition. In 4 µν µ µ µνλ ν λ µ t
the confined(disordered)phase, the bosonandfermion
are confined together by the fluctuating gauge field. In Note the Maxwell term for aΦ is generated by the in-
the deconfined ( superconductor) phase, the boson con- tegration over the massive aµ.
densed,thefermionbecomesfree. Thereisalineoffixed SettingAµ =0,integratingoutthefermionsonlyleads
pointgovernedby the mutualstatisticalangleθ separat- to higher derivative terms than the Maxwell term:
ing the two phases.
1
L=|(∂ −iaΦ)Φ|2+V(|Φ|)+ (fΦ)2+··· (43)
µ µ 4 µν
IV. THE EFFECT OF CHARGE FLUCTUATION where ··· means higher than second order derivatives.
Therefore the vortex and fermion are asymptotically de-
Inthissection,wetrytoinvestigatetheeffectofcharge coupled. It indicates that the charge fluctuation ne-
fluctuation on the fixed line characterized by the statis- glected in the last section destroy the fixed line char-
tical angle θ discussed in the last section. In the last acterized by θ. However, very different conclusions are
section, we pointed out the periodicity of Z gauge CS reached in Ref. [11]. We believe that the authors failed
2
term on the lattice is not preserved by U(1) gauge CS to treat the charge gauge field fluctuation correctly.
8
Justlike the lastsection, it is instructive to goto dual TheDiracfermionsbecomefreeandcarryspin1/2and
representation of Eq.41, namely go to the boson repre- charge 2θe. In fact, we reach the same conclusion from
sentation: the boson representation Eq.46.
In the superconductor phase, the vortex Φ is mas-
L=ψa†γµ(∂µ−iaψµ)ψa+|(∂µ−iaφµ)φ|2+V(|φ|) sive, therefore can be integrated out, it leads to the old
1 Maxwell term 1 (fΦ )2. Integrating out aΦ generates
+ (fφ )2+iaΦǫ ∂ (aφ−aψ/θ) 4mΦ µν
4 µν µ µνλ ν λ λ a mass term for aψ:
µ
1
+ (aψ/θ+Aeff)2 (44)
2 µ µ t L=ψ†γ (∂ −iaψ)ψ + mΦ(aψ)2+ 1(aψ/θ+Aeff)2
a µ µ µ a 2 µ t 2 µ µ t
Integrating out aΦ leads to the same constraint as
µ (49)
Eq.35 up to a pure gauge:
where m is the mass of the vortex.
aψ =θaφ (45) Φ
µ µ Diagonizingthe lasttwo massterms leads to a contin-
uously changing charge . In fact, we reach the same de-
Substituting theaboveconstrainttoEq.44andsetting
scription from the boson representation Eq.46. Because
aφ =a , we find:
µ µ insuperconductorphase,thebosoncondense<φ>=φ ,
0
therefore generates a mass term for a :
L=ψ†γ (∂ −iθa )ψ +|(∂ −ia )φ|2+V(|φ|) µ
a µ µ µ a µ µ
+ 14fµ2ν + 21(aµ+Aeµff)2t (46) L=ψa†γµ(∂µ−iθaµ)ψa+ φ220(aµ)2t + 12(aµ+Aeµff)2t
Comparing to Eq.36, the only difference is that the (50)
gaugefieldacquiresa mass due to the chargefluctuation
Which is essentially the same as Eq.49 after the identifi-
whichrenderstheMaxwelltermineffective. Uptoirrele-
vant couplings,we cansafely set aµ =Aeµff in the above cation mΦ ∼φ20.
Again, the pathological claims are due to the artifact
equation and find
of replacing the Z mutual statistical gauge theory by
2
L=ψa†γµ(∂µ−i2θAµ)ψa+|(∂µ−i2Aµ)φ|2+V(|φ|)+··· a U(1) mutual CS theory, then treating Z2 gauge field
and U(1) electrical gauge field on the same footing. Al-
(47)
though all these results are the artifacts of this replace-
ment,theydosuggestthatthemutualstatisticalinterac-
where ···meansthe irrelevantcouplingsbetweenbosons
tion between hc/2e vortex and spinons leads to the con-
and fermions.
finementofspinonandchargonintoelectronandCooper
The above Eq. leads to in-consistent conclusion that
pair.
the original charge neutral spinon ψ becomes charged:
carrying charge e for θ = 1/2 and −e for θ = −1/2.
Therefore we also reach ambiguous conclusions for θ =
V. DISCUSSIONS AND CONCLUSIONS
1/2 and θ =−1/2. The reason is that we replace the Z
2
CS gauge theory on the lattice which explicitly respects
The original singular gauge transformation [3,4] was
the equivalence between θ = 1/2 and θ = −1/2 by the
proposedforstaticvortices. Inthispaper,weextendthe
U(1) CS theory which does not respect this equivalence
singular gauge transformation to moving vortices gener-
inthecontinuumlimit. Identifyingaψ whichisadiscrete
µ ated by quantum fluctuation. By making a close anal-
Z gauge field with A which is a U(1) gauge field is a
2 µ ogytothe conventionalsingulargaugetransformationof
direct consequence of this problematic replacement.
FQHsystem,weperformsingulargaugetransformations
attachingfluxofmovingvorticestoquasi-particlesorvice
versa. Just like conventional singular gauge transforma-
B. Disordered and superconducting phases
tion leads to conventional CS term, the two mutual sin-
gular gauge transformations lead to mutual CS term. In
We follow the discussions in the previous section. In
thisway,weproposeaintuitiveandphysicaltransparent
the disordered phase, the vortex condense < Φ >= Φ
0 approach to bring out explicitly the underlying physics
whichgeneratesamasstermforaΦ inEq.41. Integrating
µ associated with the condensation of the hc/2e vortices.
out the massive aΦ leads to a Maxwell term for aψ.
µ µ BasedontheearlierworkbyBalents,FisherandNayak
[13],SenthilandFisherdevelopedZ gaugetheory[10,11]
1 1 2
L=ψ†γ (∂ −iaψ)ψ + (fψ )2+ (aψ/θ+Aeff)2 to study quasi-particlescoupled to vorticesgeneratedby
a µ µ µ a 4Φ2 µν 2 µ µ t
0 quantumfluctuations. BybreakingelectronsandCooper
(48) pairs into smaller constitutes: chargons with charge e,
9
spin 0 and spinons with charge 0,spin 1/2, SF intro- possible phases and to understand the critical behaviors
duced a local Z gauge degree of freedom to constrain of the phase transitions between these phases.
2
the Hilbert space to the original one. The effective ac- This work was supported by NSF Grant No. PHY99-
tion describes both chargons and spinons coupled to lo- 07949, university of Houston and DMR-97-07701. We
cal fluctuating Z gauge theory with a doping depen- thank M. P. A. Fisher, S. Girvin, S. Kivelson,P. A. Lee,
2
dent Berry phase term. By the combination of standard S, Sachdev, T. Senthil, Z. Tesanovic, Y. S. Wu and Kun
duality transformation of 3 dimensional XY model and Yang for helpful discussions.
that of Z gauge theory, the action is mapped into a
2
dual vortex representation where the hc/2e vortices and
spinonsarecoupledbyamutualZ CSgaugetheory. As APPENDIX A: THE MOST GENERAL FORM
2
usual vortices in XY model, the hc/2e vortices also cou- OF THE GAUGE PROPAGATORS
ple to a fluctuating U(1) gauge field which mediates the
long-range logarithmic interaction between the vortices. Adding the gauge fixing terms 1 ((∂ aψ)2+(∂ aΦ)2)
2α µ µ µ µ
Starting from the dual representation, the authors in to Eq.22, we can find the gauge field propagators by in-
Ref. [11] studied a transition from d-wave superconduc- verting the matrix
tor to confined Mott insulator driven by the condensing
ofhc/2evortices. Inordertostudythecriticalbehaviors Π (k)k(δ − kµkν)+ kµkν Π (k)ǫ k
1 µν k2 α 2 µνλ λ
osiftitohni,s pthaeryticruelpalracceodnfitnheemZentmauntduadle-CcoSnfithneeomryentontrtahne- Π2(k)ǫµνλkλ Π3(k)k(δµν − kµkk2ν)+ kµαkν !
2
lattice by U(1) mutual CS theory in the continuum and (A1)
performed renormalization Group (RG) analysis. Un-
fortunately, some of their RG analysis are incorrect as The results are:
demonstrated in Sec.IV.
αk k Π 1 k k
µ ν 3 µ ν
It was shown in the text that our resulting general ef- (G ) = + (δ − )
ψψ µν k4 Π Π +Π2k µν k2
fective action Eq.19 at the two special statistical angles 1 3 2
Π ǫ k
θ =±1/2isessentiallyequivalenttothedualvortexrep- (G ) =(G ) = 2 (− µνλ λ)
resentation of Z2 gauge theory developed by SF, except ψΦ µν Φψ µν Π1Π3+Π22 k2
it also bring out explicitly the Volovik effect which was αkµkν Π1 1 kµkν
(G ) = + (δ − ) (A2)
missed in SF’s Z2 gauge theory. ΦΦ µν k4 Π1Π3+Π22k µν k2
As explicitly stressed in this paper, the U(1) mutual
CS theory in the continuum does not have the required In the Landau gauge α = 0 and putting ~k = 0, we
periodicityunderθ →θ+1. Inordertopreservethispe- recover the results calculated in Sec. III.
riodicity when the mutual statistical angle takes special
values θ = ±1/2, a Z mutual CS theory must be en-
2
forced on the lattice. Again, two ways of regularization APPENDIX B: APPLICATION TO STATIC
mayleadtodifferentconclusions. Infact,theperiodicity DISORDERED VORTEX ARRAY
of the conventional CS theory under θ → θ +2 is also
a very intricate issue. On the one hand, the perturba- In Ref. [6], Ye pointed out that the long range Loga-
tive RG expansion in terms of the statistical angle θ in rithmic interaction between vortices suppress the super-
the continuum limit in Refs. [21–23] does not have this fluidvelocityfluctuation,butdoesnotaffecttheinternal
periodicity. On the other hand, properly regularized on gauge field fluctuation. He concluded that the quasi-
the lattice, the CS theorydoeshavethis periodicity [28]. particle scattering from the random gauge field domi-
The two different regularizationdo lead to different con- nate over that from the superfluid velocity ( the Volovik
clusions on the Quantum Hall transitions. It also leads effect). In this appendix, by using the formalism devel-
to notorious Hall conductivity difficulty at ν = 1/2 [29]. opedinthemaintext,weprovideadditionalevidencefor
Although we are unable to solve the critical behaviors this conclusion.
of this combined Z and U(1) gauge fields coupled to
2 For random pinned vortices, the static vortices do not
spinons and hc/2e vortices, we stressed the importance feel the statistical gauge field aΦ in Eq.19. This corre-
µ
ofputtingthetheoryonthelatticestokeeptheperiodic-
sponds to a mass term for this gauge field. Adding the
ityofZ2 mutualCStermandshedconsiderablelightson masstermtotheequationandintegratingoutaΦ,itgen-
µ
the physical picture and structure of the quantum tran- erates a Maxwell term for aψ [30]:
sitions driven by hc/2e vortices. We suggest that it may µ
be possible to trace out the Z2 gauge fluctuation on the 1
latticefirsttogetaneffectiveactionwithoutthediscrete L=ψa†γµ(∂µ−iaψµ)ψa+ 4(fµψν)2
gaugefields,thentakecontinuumlimitofitandperform 1
RGcalculation. Furtherworkisneededtosortoutallthe +|(∂µ−iaµ)Φ|2+V(|Φ|)+ 4fµ2ν −iAeµffǫµνλ∂νaλ (B1)
10