Competition between superconductivity and charge density waves

0.18 MB·English

by  Ki-Seok Kim

Competition between superconductivity and charge density waves
Ki-Seok Kim
School of Physics, Korea Institute for Advanced Study, Seoul 130-012, Korea
(Dated: February 2, 2008)
Wederiveaneffectivefieldtheoryforthecompetitionbetweensuperconductivity(SC)andcharge
density waves (CDWs) by employing the SO(3) pseudospin representation of the SC and CDW
order parameters. One important feature in the effective nonlinear σ model is the emergence of
7
Berry phase even at half filling, originating from the competition between SC and CDWs, i.e., the
0
pseudospinsymmetry. AwellknownconflictbetweenthepreviousstudiesofOshikawa[1]andD.H.
0
Lee et al.[2] is resolved by the appearance of Berry phase. The Berry phase contribution allows a
2
deconfinedquantumcriticalpointoffractionalizedchargeexcitationswitheinsteadof2eintheSC-

n CDWquantumtransitionathalffilling. Furthermore,weinvestigatethestabilityofthedeconfined
a quantumcriticalityagainstquenchedrandomnessbyperformingarenormalizationgroupanalysisof
J
aneffectivevortexaction. Wearguethatalthoughrandomnessresultsinaweakdisorderfixedpoint

3 differing from the original deconfined quantum critical point, deconfinement of the fractionalized
charge excitations still survives at the disorder fixed point owing to a nonzero fixed point value of

] a vortex charge.
l
e
- PACSnumbers: 71.30.+h,74.20.-z,74.20.Fg,71.10.Hf
r
t
s
.
t I. INTRODUCTION alizedchargeexcitationswitheinsteadof2easaresultof
a
thecompetitionbetweenSCandCDW.Furthermore,we
m
examinethe stabilityofthedeconfinedquantumcritical-
- Recently, it was proposed that when there exist
ityagainstquenchedrandomnessgeneratingtwokindsof
d two competing orders characterized by different pat-
n random potentials, a random mass term and a random
terns of symmetry breaking, the two order parame-
o fugacity one in the effective vortex action [Eq. (16)].
ters can acquire some topological Berry phases to al-
c Performingarenormalizationgroup(RG)analysisofthe
[ low a continuous quantum phase transition between the
two states, although forbidden in the Landau-Ginzburg- vortex action [Eq. (16)] in the London approximation
4 [Eq. (17)], we argue that deconfinement of the fraction-
Wilson (LGW) theoretical framework without fine-
v alized excitations still survives although the presence of
tuning of couplings admitting multi-critical points.[3, 4]
0 disorderleadstoanewquantumcriticalpointwithfinite
Especially, the quantum critical point in this quantum
2 disorderstrength. Wefindthatthestabilityofthedecon-
6 phase transition is quite exotic in the respect that ele-
finedquantumcriticalityoriginatesfromtheexistenceof
1 mentary excitations are fractionalized, thus called a de-
the charged critical point.
1 confined quantum critical point.[5, 6]
5 One deconfined quantum critical point was demon- Before going further, it is valuable to address sev-
0
strated in the competition between antiferromagnetic eral important differences between the present work and
/
at (AF) and valance bond solid (VBS) orders.[5, 6] Tanaka previous studies. Earlier studies[7] revealed that the
m andHuconsideredanSO(5)superspinrepresentationin- half-filled negative-U Hubbard model on a 2d square
cluding both the AF and VBS order parameters, and lattice is mathematically equivalent to the positive-U
-
d derived an effective nonlinear σ model for the SO(5) su- Hubbard model, using the particle-hole transformation.
n perspin variable from the spinon representation of the ThisequivalencemapstheXYorderedantiferromagnetic
o Heisenberg Hamiltonian.[3] One crucial feature in their phase of the spin system that results for positive-U to
c effective field theory is the presence of Berry phase for the superfluid phase of the negative-U problem. Like-
:
v the superspin field. They demonstrated that the com- wise, the Ising antiferromagnet (for positive U) maps to
Xi petition between AF and VBS is well described by the a CDW phase (for negative U). However,in these earlier
SO(5) nonlinear σ model with a topologicalBerry phase studies[7] the role of Berry phase was not investigated
r
a term. clearly, thus the LGW-forbidden continuous transition
and deconfined quantum critical points were not found
In the present paper we consider another concrete ex-
in the context of SC-CDW transitions.
ample, the competition between superconductivity (SC)
andchargedensitywaves(CDWs),asasimplifiedversion It is interesting to understand the origin of the Berry
of the competition between AF and VBS. Introducing phaseinthenegative-UHubbardmodelandthepositive-
an SO(3) pseudospin representation to include both the Uone. Thepositive-UHubbardmodelreducestothean-
SC and CDW order parameters, we derive an effective tiferromagneticHeisenbergmodelinthelarge-Ulimit. In
nonlinear σ model in terms of the O(3) pseudospin vari- the negative-U Hubbard model the low energy effective
able from the attractive Hubbard model. Interestingly, action can be mapped onto an effective model of hard-
a Berry phase term naturally appears in this σ model, core lattice bosons with a hopping amplitude of order
allowing a deconfined quantum criticalpoint of fraction- t2/U and repulsive nearest neighbor interaction of the
2
sameorderinthestrongcouplinglimitU →−∞.[8]One The local interaction term can be decomposed into
canshowthatthishard-corebosonmodelisequivalentto pairing and density channels in the following way
the antiferromagneticHeisenbergmodel, associatedwith
chargedegreesoffreedomtoformapseudospin.[7,9]The −3u c† c c† c =−u c† c† c c
Berryphaseinthenegative-UHubbardmodeloriginates 2 i↑ i↑ i↓ i↓ 2 i↑ i↓ i↓ i↑
i i
from the pseudospin (charge) SU(2) symmetry[7] while X X
u 2 u
it in the positive-U Hubbard model comes from the spin − c† c −1 − c† c −1 .
2 iσ iσ 2 iσ iσ
SU(2)symmetry. Itshouldbenotedthatthistopological Xi (cid:16)Xσ (cid:17) (cid:16)Xσ (cid:17)
phase appears even at half filling. On the other hand, it
was not allowed at half filling in recent studies.[10, 11] PerformingtheHubbard-Stratonovichtransformationfor
TheBerryphaseresultingfromthe chemicalpotentialin the pairing and density interaction channels, we find an
the boson Hubbard-type model[10, 11] is different from effective Lagrangianin the Nambu-spinor representation
the presentone because the presenceof the chemicalpo-
tential reduces the SU(2) pseudospin symmetry to the Z = D[ψ ,ψ†,ΦR,ΦI,ϕ ]e−RdτL,
i i i i i
U(1) one. This is the reason why there exists only the
Z
Berry phase coming from the chemical potential in the L= ψi†(∂τI−µτ3)ψi−t (ψi†τ3eiAijτ3ψj +H.c.)
bosonHubbard-typemodelwhileoureffectiveactionhas
both Berry phases resulting from the SU(2) pseudospin Xi Xhiji
symmetry and chemical potential. In other words, the − (ΦRi ψi†τ1ψi+ΦIiψi†τ2ψi+ϕiψi†τ3ψi)
competition between SC and CDWs results in a non- i
X
trivial Berry phase term even at half filling. Thus, the + 1 (ΦR2+ΦI2+ϕ2)− v (ψ†τ ψ +1). (2)
chemical potential plays the role of an additional Berry 2u i i i i i 3 i
i i
phase in the present effective theory. Furthermore, the X X
appearanceofBerryphaseathalffillingallowsotherpos-
c
sibledisorderedphasescorrespondingtovalancebondor- Hereψi is the Nambuspinor,givenbyψi = ci†↑ . ΦRi
ders in the pseudospin language. This resolves the well (cid:18) i↓ (cid:19)
andΦI aretherealandimaginarypartsofthesupercon-
knownconflictbetweenthetwopreviousstudies[1,2]that i
ducting orderparameterrespectively,andϕ aneffective
Ref. [2] does not admit a dimerised order while the pa- i
density potential. µ is an electron chemical potential
per [1] claims this phase is certainly possible. The emer-
which differs from its bare value µ as µ=µ +u/2.
gence of Berry phase at half filling clearly reveals how b b
the dimerised order appears. IntroducingapseudospinvectorΩ~i ≡(ΦRi ,ΦIi,ϕi),one
We would like to mention that the present quantum can express Eq. (2) in a compact form
transitionoccursbetween the XY orderedphase andthe
Ising antiferromagnetic one if one maps our negative-U Z = D[ψ ,ψ†,Ω~ ]e−RdτL,
i i i
problem to the positive-U one. This XY-Ising antiferro-
Z
magnetictransitionallowstheSO(3)pseudospindescrip- L= ψi†∂τψi−t (ψi†τ3eiAijτ3ψj +H.c.)
tion for the competition of SC and CDW fluctuations in
the context of the negative-U Hubbard model. On the Xi Xhiji
1
otherhand,theAF-VBSquantumtransitionrequiresthe − ψ†(Ω~ ·~τ)ψ + tr[Ω~ ·~τ −(µ+v )τ ]2
i i i 4u i i 3
SO(5) superspin description for the competition of AF
i i
X X
and VBS fluctuations.[3]
− v , (3)
i
i
X
II. EFFECTIVE FIELD THEORY
where we used the shift of ϕ → ϕ −µ−v . Integrat-
i i i
ing over the pseudospin field Ω~ , Eq. (3) recovers the
i
A. Derivation of the O(3) nonlinear σ model from
Hubbard model Eq. (1).
the attractive Hubbard model
In this paper we consider only phase fluctuations in
Ω~ , assuming amplitude fluctuations frozen thus setting
i
We consider the attractive Hubbard Hamiltonian
it as Ω~ = m~n with an amplitude m. Since our start-
i i
3u
H =−t c†iσeiAijcjσ − 2 c†i↑ci↑c†i↓ci↓ iwneguptoiilnizteisaastnroonnzgercoouapmlipnlgituadpeproofatchhedpesceoumdopsopsiinngfietlhde,
ijσ i
X X directional fluctuating field ~n into two complex boson
− vic†iσciσ. (1) fields, so called CP1 representiation[12]
iσ
X
Here t is a hopping integral of electrons, and u strength ~n ·~τ =U τ3U†,
i i i
ofon-siteCoulombrepulsions. A isanexternal(static)
ij z −z†
etilaecl.tromagnetic field, and vi a quenched random poten- Ui = z↓↑ z↑†↓ !, (4)
3
where U is an SU(2) matrix field in terms of a complex CDW is obtained to be the Heisenberg model in terms
i
bosonfieldz withpseudospinσ. Using the CP1 repre- of the O(3) pseudospin variable. One important mes-
iσ
sentation in Eq. (3), and performing the gauge transfor- sage in this effective action is that the Berry phase term
mation iS ω({S (τ)}) should be taken into account for the
i i
SC-CDW transition even at half filling. Furthermore,
Ψi =Ui†ψi, (5) thePchemical potential plays the same role as an exter-
Eq. (3) reads nal magnetic field, and the disorder potential a random
magnetic field.
Z = D[Ψi,Ψ†i,Ui]e−RdτL, If we consider half filling without disorder, i.e., µ =
Z v = 0, the XY order of hS±i =6 0 and hSzi = 0 is
L= Ψ†i(∂τI−mτ3+Ui†∂τUi)Ψi exipected in the case of J >>iV, identified witih SC. On
i theotherhand,theIsingorderofhSzi=6 0andhS±i=0
X i i
−t (Ψ†iUi†τ3eiAijτ3UjΨj +H.c.) arises in the case of V >> J, corresponding to CDW
because of the Berry phase, as will be discussed below.
Xhiji
One important question in this paper is how the SC-
1
+ tr[mτ −(µ+v )U†τ U ]2− v . (6) CDW transition appears in the presence of disorder.
4u 3 i i 3 i i
i i
X X ItiseasytoshowthattheHeisenbergmodelwithferro-
Since Eq. (6) is quadratic for the spinor field Ψ , one magneticXYcouplingsisthesameasthatwithantiferro-
i
can formally integrate out the spinor field to obtain magnetic ones. Performing the Haldane mapping of the
antiferromagneticHeisenberg model[14] with a magnetic
Seff =−trln ∂τI−mτ3+Ui†∂τUi−tijUi†τ3eiAijτ3Uj field in the z-direction, we obtain the O(3) nonlinear σ
m h i model
+ dτ − (µ+v )tr[U†τ U τ ]
2u i i 3 i 3
+Z vhi2+µ2Xi+m2+µvi −vi . (7) Sσ =iS (−1)iω({ni(τ)})+ g1 cβdx0 ddx (∂0nz)2
Xi (cid:16) 2u (cid:17)i Xi Z0 Z h
Expanding the logarithmic term for U†∂ U and +(∂0nx−i[µ+v]ny)2+(∂0ny +i[µ+v]nx)2+(∇xn)2 ,
i τ i
Ui†τ3eiAijτ3Uj, we obtain (1i0)
S ≈ tr[G (U†∂ U )]
eff 0 i τ i
where c is the velocity of spin waves, and g the coupling
i
X
1 strength between spin wave excitations. As Tanaka and
+2 trj[G0tijUi†τ3eiAijτ3UjG0tjiUj†τ3e−iAijτ3Ui] Hu derived an effective SO(5) nonlinear σ action of the
Xi superspin field for the AF-VBS transition, we derived
+ dτ −m (µ+v )tr[U†τ U τ ] an effective SO(3) nonlinear σ action of the pseudospin
2u i i 3 i 3 fieldfortheSC-CDWtransition. Furthermore,thiseffec-
Z h Xi tive σ action includes not only doping contributions but
v2+µ2+m2+µv
+ i i −v , (8) also disorder effects. On the other hand, in the SO(5)
i
2u
superspin σ model it is not clear how the doping effect
Xi (cid:16) (cid:17)i
modifiestheeffectiveactionbecauseachemicalpotential
where G = −(∂ I−mτ )−1 is the single particle prop-
0 τ 3 term breaks the relativistic invariance. In this case it is
agator. The first term leads to Berry phase while the
not clear even to obtain the topologicalterm. In the fol-
second results in an exchange interaction term. The re-
lowingwediscusshowthisσ actiondescribesthecompe-
sultingeffectiveactionisobtainedtobewithouttheelec-
titionbetweenSCandCDWinthepresenceofquenched
tromagnetic field A
ij disorder by focusing on the role of Berry phase.
β
Without loss of generality we use the parametrization
S =iS ω({S (τ)})+ dτH ,
eff i eff
i Z0
X
H =−J (SxSx+SySy)+V SzSz ~n =(sin(uϑ )cosϕ ,sin(uϑ )sinϕ ,cos(uϑ )),(11)
eff i j i j i j i i i i i i
ij ij
X X
− (µ+v )Sz, (9)
i i whereuisanadditionaltime-likeparameterfortheBerry
Xi phaseterm.[14]Wenotethatn+i =sinϑieiϕi corresponds
where the effective exchange coupling strength is given to the pairing potential Φ = ΦR +iΦI. Inserting Eq.
i i i
by J = V = 2t2/m.[13, 14] It is interesting that the ef- (11) into Eq. (10), and performing the integration over
fective Hamiltonianfor the competition between SC and u in the Berry phase term, we obtain the following ex-
4
pression for the nonlinear σ model Randomnessv playstheroleofadualrandommagnetic
i
field in vortices.
cβ
S =iS (−1)i dx (1−cosϑ )ϕ˙ Inthemeanfieldapproximationignoringvortex-gauge
eff 0 i i
i Z0 fluctuationscnm,onefindsthatthevortexproblemcoin-
X
cβ 1 cides with the well known Hofstadter one. If one consid-
+ dx ddx [sin2ϑ(∂ ϕ)2+(∂ ϑ)2] ersadualmagneticfluxf =p/qwithrelativelyprimein-
0 µ µ
g
Z0 Z tegersp,q (here,p=1andq =2),thedualvortexaction
cβ 1 has q-fold degenerate minima in the magnetic Brillouin
+ dx ddx [−(µ+v)2sin2ϑ+4i(µ+v)ϕ˙sin2ϑ]
0 g zone. Lowenergyfluctuationsneartheq-folddegenerate
Z0 Z
vacua are assigned to be ψ with l = 0,...,q −1. Ba-
+S , l
I
lents et al. constructed an effective LGW free energy
cβ
SI =I dx0 ddxcos2ϑ, (12) functional in terms of low energy vortex fields Ψl, given
Z0 Z by linear combinations of ψl.[10] Constraints for the ef-
where we introduced the action SI favoring the XY or- fective potential of Ψl are symmetry properties associ-
der. Thisprocedureisquiteparalleltothatinthe SO(5) ated with lattice translations and rotations in the pres-
σ model.[3] The chemical potential favors the XY or- ence of the dual magnetic field. In the present q = 2
der without the ”easy plane” anisotropy term. The easy case (corresponding to a π flux phase) there are two de-
plane anisotropy allows us to set ϑ = π/2. In this case generate vortex ground states at momentum (0,0) and
i
Eq. (12) reads (π,π). Introducing the linear-combined vortex fields of
Ψ =ψ +iψ andΨ =ψ −iψ whereψ andψ arethe
0 0 1 1 0 1 0 1
8
S =iπ [(−1)i+ (µ+v )]q lowenergyvortexfluctuationsaroundthetwodegenerate
XY i i
g groundstatesrespectively,andconsideringthesymmetry
i
X propertiesmentionedabove,onecanfindaneffectivelow
cβ 1 ρ
+ dx0 ddx ϕ˙2+ ϕ(∇xϕ)2 . (13) energy action. However, one important difference from
2u 2
Z0 Z h ϕ i thepreviousstudy[10]duetothecontributionofrandom
cβ Berry phase should be taken into account carefully. One
Here q =(1/2π) dx ϕ˙ is an integer representing an
i 0 0 i cautiouspersonmaydoubtifitismeaningfultoconsider
instanton number, here a vortex charge, and the pseu-
R the magnetic Brillouin zone in the presence of random-
dospin value S = 1/2 is used. Anisotropy in time and
ness. Actually, this is a correct question. In this paper
spatial fluctuations of the ϕ fields is introduced by u
ϕ we assume the existence of the magnetic Brillouin zone
andρ . The effective field theoryfor the SC-CDWtran-
ϕ since the limit of weak randomness is of our interest.
sition is given by the quantum XY model with Berry
Based on symmetry properties of the square lattice
phase in the easy plane limit of Eq. (10). It is clear
under π flux, we write down the effective action for low
that the topological phase appears even at half filling as
energy vortices with randomness
a result of the competition between SC and CDW. The
chemical potential plays the role of an additional Berry
phase in the phase field ϕ. Seff = dτd2r |(∂µ−icµ)Ψ0|2+|(∂µ−icµ)Ψ1|2
Z h
+m2(|Ψ |2+|Ψ |2)+u (|Ψ |2+|Ψ |2)2
0 1 4 0 1
B. Effective vortex action with both external and +v |Ψ |2|Ψ |2−v (Ψ∗Ψ +H.c.)
4 0 1 2 0 1
random dual magnetic flux
1
+ (∂×c)2 − dτd2rv(∂×c) . (15)
2e2 τ
To take into accountthe Berryphase contribution, we v i Z
resort to a duality transformation, and obtain the dual Intheeffectivevortexpotentialm2 isavortexmass,u a
4
vortex action local interaction, v a cubic anisotropy, and v breaking
4 2
Sv =−tv Φ†neic¯nm+icnmΦm+V(|Φn|) tthhee pUr(e1s)enpcheaosfertarnadnosmforBmeartriyonphΨa0s(e1)fo→r voeriϕti0c(1e)sΨ. 0T(1h)eirne
nm
X aretwoimportantdifferencesbetweenthecaseswithand
1 4
+ (∂×c)2 − v (∇×c) . (14) without disorder. In the absence of disorder the v term
2e2 µ ge2 i i 2
v Xµ v Xµ is given by −v8[(Ψ∗0Ψ1)4 +H.c.] owing to the four-fold
symmetry.[5, 10] However, the presence of weak disor-
Here Φ is a vortex field residing in the (2+1)D dual
n der implies that lattice translations and rotations are no
latticenoftheoriginallatticeµ=(τ,i),andc avortex
nm longersymmetries. This reduces the fourthpowerto the
gauge field. V(|Φ |) is an effective vortex potential. e
n v first one. Furthermore, we estimate that v is a random
is a coupling constant of the vortex field to the vortex 2
variabledepending on disorder. One canregardv as an
gauge field. c¯ is a background gauge potential for the 2
nm instanton fugacity.[5, 6] Thus, the estimation of the ran-
vortex field, resulting from the Berry phase contribution
domvariablev meansthatdisordermakestheinstanton
and satisfying at half filling 2
fugacity random. As another contribution of disorder v
(∇×c¯) =(−1)iπ. is a dual random magnetic field in the last term. This
i
5
term generates different kinds of random potentials, as III. ROLE OF DISORDER IN THE
will be seen later. DECONFINED QUANTUM CRITICAL POINT
Now we investigate the role of disorder in the decon-
Based on the effective vortex potential Eq. (15), one fined quantum critical point. In order to take into ac-
can perform a mean field analysis in the absence of dis- count the random potentials by disorder, we use the
order (v =0).[15] Condensation of vortices occurs in the replica trick to averageover disorder. The random mag-
case of m2 < 0 and u4 > 0. The signs of v4 and v8 de- netic field v and the random fugacity v2 in the vortex
termine the groundstate. For v <0,both vorticeshave action Eq. (15) would cause
4
anonzerovacuumexpectationvalue|hΨ i|=|hΨ i|=6 0,
0 1
N
and their relative phase is determined by the sign of ℑ
v8. In the case of v8 > 0 the resulting vortex state − dτdτ1 d2r2(∂×ck)τ(∂×ck′)τ1,
corresponds to a columnar dimer order, breaking both k,Xk′=1Z Z
the rotational and translational symmetries. In the case N ℜ
− dτdτ d2r (Ψ∗ Ψ +H.c.)
of v8 < 0 the resulting phase exhibits a plaquette pat- 1 2 0k 1k τ
tern, braking the rotational symmetries. On the other k,Xk′=1Z Z
hand, if v4 > 0, the ground states are given by either ×(Ψ∗0k′Ψ1k′ +H.c.)τ1
|hΨ i| =6 0,|hΨ i| = 0 or |hΨ i| = 0,|hΨ i| =6 0, and the
0 1 0 1
for Gaussian random potentials satisfying
sign of v is irrelevant. In this case an ordinary charge
8
density wave order at wave vector (π,π) is obtained,
hv(r)i=0, hv(r)v(r )i=ℑδ(r−r ),
breaking the translational symmetries. This mean field 1 1
hv (r)i=0, hv (r)v (r )i=ℜδ(r−r )
analysis coincides with that in Ref. [5]. 2 2 2 1 1
with the strength ℑ and ℜ of the random potentials, re-
spectively. Here k,k′ = 1,...,N denote replica indices,
At the critical point m2 = 0 the eighth-order term is and the limit N → 0 is done at the final stage of calcu-
certainlyirrelevantowingtoitshighorder. Furthermore, lations. However, inclusion of only this correlation term
the cubic anisotropy term (v4) is well known to be irrel- is argued to be not enough for disorder effects. Because
evant in the case of q < qc = 4, ignoring vortex gauge the gauge-field propagator has off-diagonal components
fluctuations.[16] As a result, the Heisenberg fixed point in replica indices, the vortex-gaugeinteractionof the or-
(v4∗ = 0 and u∗4 6= 0) appears in the limit of zero vor- derℑ2e4v generatesaquartictermincludingthecouplings
tex charge (ev → 0). Allowing the vortex gauge fields ofdifferent replicasofvorticesevenif this term is absent
atthe Heisenbergfixedpoint, the Heisenbergfixedpoint initially.[17]Theresultingdisorderedvortexactionisob-
becomes unstable, and a new fixed point with a nonzero tained to be
vortex charge appears as long as the cubic anisotropy v
4
is assumed to be irrelevant.[17, 18] This charged fixed ZR = DΨ0kDΨ1kDckµe−SR,
point seems to be qualitatively the same as that ob-
Z
tained in the absence of the dual magnetic field, i.e., the SR =Sv+Sd+Sf,
q = 1 case. However, one important difference is that N
the dual flux quantum (corresponding to an electromag- S = dτd2r |(∂ −ic )Ψ |2+|(∂ −ic )Ψ |2
v µ kµ 0k µ kµ 1k
neticchargeoftheoriginalboson)seenbythevortexfield
kX=1Z h
Ψ0(1) ishalvedduetothetwoflavorsofvortices.[10]This +m2(|Ψ |2+|Ψ |2)+u (|Ψ |2+|Ψ |2)2
0k 1k 4 0k 1k
implies that the boson excitations dual to the vortices
1
carry an electromagnetic charge e instead of 2e. These +v |Ψ |2|Ψ |2+ (∂×c )2 ,
4 0k 1k 2e2 k
fractionalizedexcitationsareconfinedtoappearasusual v
i
Cooper pair excitations with charge 2e away from the N
quantum critical point, resulting from the eighth-order Sd =− dτdτ1 d2r
term to break the U(1) gauge symmetry.[6] However, as k,k′=1Z Z
X
mentioned above, this v term becomes irrelevant at the ℜ
critical point, indicatin8g that the charge-fractionalized 2(Ψ∗0kΨ1k+H.c.)τ(Ψ∗0k′Ψ1k′ +H.c.)τ1
excitationsaredeconfinedtoappear. Thus,theSC-CDW N 1
W
transition at half filling occurs via the deconfined quan- − dτdτ1 d2r 2 |Ψqkτ|2|Ψq′k′τ1|2,
tumcriticalpointastheAF-VBStransition.[5]Thiscon- k,k′=1q,q′=0Z Z
X X
clusiondoesnotdependonwhetherthecubicanisotropy
N
ℑ
iissrreelleevvaannttoartntohteaitsotthreopchicarcgheadrgcerditificaxlepdopinoti.ntEtvoencaifuvs4e Sf =− dτdτ1 d2r2(∂×ck)τ(∂×ck′)τ1 (16)
k,k′=1Z Z
a new anisotropic charged fixed point, the eighth-order X
term associated with charge fractionalization would be withW >0. ThelastterminducedbydisorderinS has
d
irrelevant. the same form with the term resulting from a random
6
mass term. The correlation term S between random neutral (XY) fixed point of e∗2 = 0 and ρ∗ = 0 and the
f v
magnetic fluxes would be ignored in this paper. In the other,thecharged(IXY)fixedpointofe∗2 = 1 andρ∗ =
v λ
smallℑlimitthistermwasshowntobeexactlymarginal 0. The neutral fixed point is unstable against a nonzero
at one loop level.[17] charge e2 6=0, and the RG flows in the parameter space
v
The question is what happens on the deconfined of (ρ,e2) converge into the charged fixed point owing to
v
charged critical point when randomness is turned on. It 1−γe∗2 =1− γ <0.[6]
v λ
is not an easy task to take into account all of the terms Nextweexamine the roleofthe randomfugacityterm
on an equal footing in the RG analysis. To investigate ignoring vortex gauge fluctuations, i.e., e2 = 0. The
v
the role of the two disorder-induced terms of S in the random fugacity term can be rewritten in the following
d
deconfined charged critical point, one can consider two way
approximate ways. One is first to examine the random
ℜ
mdeacsosntfienremd,cdheanrogteeddcbryititchael pcoouinptl,inagndstrtehnegnthtoWse,eawththaet 2 cos(θ0k−θ1k)τcos(θ0k′ −θ1k′)τ1
happens if the random fugacity (ℜ) is turned on at a ℜ
weakdisorderfixedpoint. Theotherisfirsttoinvestigate = 4 cos[(θ0k−θ1k)τ +(θ0k′ −θ1k′)τ1]
the effect of the randomfugacity termonthe deconfined ℜ
charged critical point, and then to examine the random +4 cos[(θ0k−θ1k)τ −(θ0k′ −θ1k′)τ1]. (19)
mass term. In this paper we follow the second approach
In this expression we can find that the last term is the
because our main interest is to see the fate of the decon-
most relevant term owing to its sign. Thus, it is reason-
fined quantum criticality against randomness. It should
able to consider the following action for the RG analysis
be noted that the existence of the deconfined quantum
criticality is determined by the fugacity term.[6] N
ρ ρ
Toexaminetheroleoftherandomfugacityterminthe S ≈ dτd2r (∂ θ )2+ (∂ θ )2
R µ 0k µ 1k
2 2
charged critical point, we consider a phase-only action
kX=1Z h i
ignoring amplitude fluctuations of vortices.[19] This so- N
ℜ
c[5a,ll6e,d1L0]o.nTdohne eaffpepcrtoivxeimvaotritoenxwacatsioanlsios oubtitlaizinededintoRbeefs. −k,k′=1Z dτdτ1Z d2r4 cos[(θ0k−θ1k)τ −(θ0k′ −θ1k′)τ1].
X
N 1 ρ 1 This action was well studied in the context of Anderson
SR = dτd2r 2(∂µθqk−ckµ)2+ 2e2(∂×ck)2 localization in one dimensional systems when the flavor
Xk=1Z hXq=0 v i number of bosons is one.[21] In Ref. [6] we derived RG
N equations for the two-flavor sine-Gordon action. Simi-
ℜ
− dτdτ1 d2r2 cos(θ0k−θ1k)τ cos(θ0k′ −θ1k′)τ1,larly, one can easily obtain the following RG equations
k,k′=1Z Z for the stiffness ρ and the random parameter ℜ
X
(17)
dρ 2
=ρ+βℜ2 ,
where ρ is a stiffness parameter proportionalto the con- dl ρ
densation probability of vortices in the mean field level. dℜ 2
=(4−α )ℜ (20)
The parameter ℜ is also renormalized by the condensa- dl ρ
tion amplitude of vortices.
with positive numerical constants, β and α. In our con-
To see whether the randomcos termis relevantornot
sideration their precise values are not important. The
atthechargedfixedpoint,itisnecessarytochecktheex-
effect of two flavors appears as the factor 2 in the 1/ρ
istenceofthechargedcriticalpointwithoutthedisorder-
terms. One important difference between the present
induced term. Considering ℜ=0 in Eq. (17), we obtain
(2+1)D study andthe previous(1+1)D one[21] is that
theRGequationsforthestiffnessρandthevortexcharge
e2 the bare scaling dimensions of ρ and ℜ are given by 1
v
and4in(2+1)D while0and3in(1+1)D,respectively.
dρ This difference results inthe fact thatthere existno sta-
=ρ−γe2ρ,
dl v ble fixed points in (2+1)D while in (1+1)D there is
de2 a line of fixed points describing the Kosterliz-Thouless
dlv =e2v−2λe4v, (18) transition.[17, 21] Both the phase stiffness ρ and the pa-
rameter ℜ become larger and larger at low energy. This
whereγ andλarepositivenumericalconstants,[20]andl implies that depth of the random cos potential in Eq.
isausualscalingparameter. The lastterm−γe2ρinthe (17) becomes deeper and deeper, making the phase dif-
v
first equation originates from the self-energy correction ference θ −θ pinned at one ground position of the cos
0 1
of the vortex field owing to gauge fluctuations while the potential. Thisisthesignalofconfinementbetweenfrac-
term −λe4 in the second equation results from that of tionalized excitations, θ and θ .[6]
v 0 1
the gauge field due to screening of the vortex charge. In Combining Eq. (18) with Eq. (20), we obtain the RG
theseRGequationsthereexisttwofixedpoints;oneisthe equations for the stiffness ρ, the vortex charge e2, and
v
7
the random parameter ℜ vantatthechargedcriticalpoint. Then,wediscussedthe
effect of the random mass term on this fixed point, and
dρ 2
=ρ−γe2ρ+βℜ2 , found that the charged critical point becomes unstable,
dl v ρ and a weak disorder fixed point with a nonzero vortex
de2 chargeappears. We arguedthatsincethe randomfugac-
v =e2−2λe4,
dl v v ity term would still be irrelevant at this disorder fixed
point owing to the finite fixed point value of the vortex
dℜ 2
=(4−α )ℜ. (21) charge, deconfinement of fractionalized excitations sur-
dl ρ
vives in the weak disorder limit.
These RG equations tell us that the nonzero fixed point AcautiouspersonmayasktherelevanceofthisLGW-
value of the vortex charge (e2∗ = 1 ) in the second RG forbidden quantum transition because there has been
v 2λ
equationmakesthestiffness parameterρvanish(ρ∗ =0) no clear indication in actual physical systems so far.
in the first RG equation, causing the random parameter One way to justify this quantum transition is to find
to be irrelevant, i.e., ℜ∗ = 0 in the third RG equation. its one dimensional analogue. Considering spin fluctua-
Thissolutionisself-consistentwiththefirstRGequation. tions associated with the AF-VBS transition, its critical
Thisresultmeansthataslongasthestablechargedfixed field theory is well known to be an effective O(4) non-
point exists, the random fugacity term is irrelevant at linear σ model with a topological θ term as an SU(2)
the charged critical point. As a result, we find only one level-1 Wess-Zumino-Witten (WZW) theory.[4] This ef-
stable fixed point of e2∗ = 1 , ρ∗ = 0 and ℜ∗ = 0. fectivefieldtheorycanbederivedfromsomemicroscopic
v 2λ
The deconfined quantum criticality is stable against the models such as the bond-alternating spin chain[22] and
random fugacity term. thePeierls-Hubbardmodel[23]vianon-Abelianbosoniza-
Nowwe considerthe randommasstermatthis decon- tion. We believe that this procedure can be applied
fined charged critical point. At the tree level one can to charge fluctuations associated with competition be-
easily check that the random mass term is relevant at tween SC and CDWs. Actually, Carr and Tsvelik inves-
the charged critical point, indicating instability of the tigated the continuous SC-CDW transition in a quasi-
chargedfixed point againstdisorder. One-loopRG anal- one-dimensionalsystem.[24]Theyconsideredaneffective
ysis shows that a weak disorder fixed point appears if model of spin-gapped chains weakly coupled by Joseph-
the cubic anisotropy is irrelevant.[17, 18] One important son and Coulomb interactions. They obtained an effec-
point is that the fixed point value of a vortex charge tive field theory for SC and CDW fluctuations in the
is nonzero at the weak disorder fixed point, given by framework of the non-Abelian bosonization with weak
the value e2∗ = 1 of the charged critical point.[17, 18] interchain-interactions. They found its phase diagram
v 2λ
Furthermore, the fixed point value of the phase stiffness to show the SC and CDW phases, separated by line
would still be zero at the random charged critical point of critical points which exhibits an approximate SU(2)
because the vortex condensation should occur at ρ∗ =0. (charge) symmetry. They proposed that the critical line
Based on this discussion, we expect that the random fu- would shrink to a point in two dimensions, identified
gacitytermwouldstillbezeroattheweakdisorderfixed with the quantum critical point in the SC-CDW quan-
point. ThisimpliesthatalthoughthedimerisedorCDW tum transition. Furthermore, they discussed the rele-
phases may be unstable owing to disorder, turning into vance of their theory, considering the experimental sys-
glassyphases,deconfinement offractionalizedchargeex- temofSr2Ca12Cu24O41 built upfromalternatinglayers
citations is expected to survive at the disorder critical of weakly coupled CuO2 chains and Cu2O3 two-leg lad-
point. However,we admit that because we did not treat ders. One important difference is that the effective field
the two disorder-induced terms of S in Eq. (16) on an theory in Ref. [24] does not include a topologicalθ term
d
equal footing, the present result is not fully justified. while our field theory does allow the θ term. In this re-
spectthecorrespondencebetweenthepresentdescription
and the previous theory[24] is not complete. A further
IV. SUMMARY AND DISCUSSION investigationfortheone-dimensionalsystemisnecessary
near future.
In summary, we showed that the competition be- An important future work in this direction is to intro-
tween superconductivity (SC) and charge density waves ducespindegreesoffreedomassociatedwithanantiferro-
(CDWs) results in a non-trivial Berry phase for the SC magneticorder. Then,the resultingeffectivenonlinearσ
and CDW order parameters even at half filling, allow- modelwouldpossesanSO(4)∼=SU(2) SU(2)symmetry,
ing a deconfined quantum critical point of fractionalized where the former SU(2) is associated with spin, and the
N
charge excitations with e instead of 2e. We considered latter SU(2) pseudospin. A topological term would ap-
thestabilityofthedeconfinedquantumcriticalityagainst pearinthisSO(4)σmodel. Thecompetitionbetweenan-
quenched randomness generating two kinds of random tiferromagnetism, superconductivity, and density waves
potentials, a random mass term and a random fugacity remains to be solved.
oneinthe vortexaction. Within theLondonapproxima- K.-S. Kim would like to thank Dr. A. Tanaka for his
tion we showed that the random fugacity term is irrele- kind explanation of the conflict in Refs. [1, 2].
8
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[2] D. H. Lee and R. Shankar, Phys. Rev. Lett. 65, 1490 spinliquidintheweakdisorderlimit,Ki-SeokKim,Phys.
(1990). Rev. B 70, 140405(R) (2004), and Phys. Rev. B 72,
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(2005). [19] We note that a vortex action is given by a phase-only
[4] T.SenthilandM.P.A.Fisher,Phys.Rev.B74,064405 action in theduality transformation originally. SeeRefs.
(2006). [5, 10].
[5] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and [20] From the relation of ρR = |ΨR|2 = ZΨ−1|ΨB|2 = ZΨ−1ρB
M.P.A.Fisher,Science303,1490(2004); T.Senthil,L. itisnecessarytoknowthewavefunctionrenormalization
Balents,S.Sachdev,A.Vishwanath,andM.P.A.Fisher, constantZ .HereRandB representrenormalizedand
Ψ
Phys.Rev.B 70, 144407 (2004). bare, respectively. The renormalization factor Z can
Ψ
[6] Ki-Seok Kim, Phys.Rev.B 72, 035109 (2005). be easily obtained from the one-loop self-energy calcu-
[7] K. Borejsza and N. Dupuis, Phys. Rev. B 69, 085119 lation for the vortex field. The self-energy Σ(p) of the
(2004); C. N. Yang, Phys. Rev. Lett. 63, 2144 (1989); vortex field is given by Σ(p) = e2 d3k 1 (2p −
C. N. Yang and S. C. Zhang, Mod. Phys.Lett. B 4, 759 vR (2π)3|p−k|2
(1990); I. F. Herbut,Phys.Rev.B 60, 14503 (1999). k)µDµν(k)(2p−k)ν, where Dµν(k) = k12“δµν − kµk2kν”
[8] M. Keller, W. Metzner, and U. Schollwock, Phys. Rev. is the propagator of vortex gauge fields in the Landau
Lett.86, 4612 (2001), and references therein. gauge. We find Z−1 = 1−γe2, where γ is a positive
Ψ v
[9] E.Demler,W.Hanke,andS.-C.Zhang,Rev.Mod.Phys. numerical constant. In the same way we can obtain the
76, 909 (2004). RG equation for the vortex charge e2. From the rela-
v
[10] L. Balents, L. Bartosch, A. Burkov, S. Sachdev, and K. tion of e2R = Zce2B, we find the renormalization factor
Sengupta,Phys. Rev.B 71, 144508 (2005). Zc of the U(1) gauge field cµ. It can be derived from
[11] Z. Tesanovic, Phys. Rev.Lett. 93, 217004 (2004). the polarization function Πµν(q), given by Πµν(q) =
[[1123]] KX..-GS..KWimen,aPnhdyAs..RZeeve.,BPh7y2s.,R14e4v4.2L6et(t2.06015),.1025(1988). eλ2viRsa(2dpπ3ok)3si(t2iqv−|eqk−n)kµu|(2m2|qke−|r2kic)νal.cWonesotbatnati,naZndc =the1−pr2eλfaec2vt,owrh2eirne
[14] N. Nagaosa, Quantum Field Theory in Strongly Corre- the e2 term results from the two flavors.
v
lated Electronic Systems,Springer (1999). [21] T. Giamarchi and H. J. Schulz, Phys. Rev. B 37, 325
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