Table Of ContentEffective theory of fluctuating orbital currents in high-T cuprates
c
Kjetil Børkje1 and Asle Sudbø1
1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
(Dated: Received February 3, 2008)
We derive an effective dissipative quantum field theory for fluctuating orbital currents in clean
CuO sheets of high-T cuprates, based on a three-band model. The Coulomb repulsion term
2 c
between Cu- and O-sites is decoupled in terms of current operators representing horizontal and
vertical parts of circulating currents within each CuO unit cell of the lattice. The model has
2
ordering of currents at finite temperatures. The dissipative kernel in the model is of the form
8 |ω|/|q|, indicating Landau damping. Applications of the effective theory to other models are also
0 discussed.
0
2 PACSnumbers: 74.20.Rp,74.50.+r,74.20.-z
n
a
Constructing an effective description of the long-
J
wavelength and low-energy physics of high-T supercon-
0 c
3 ducting cuprates represents a profound and formidable
problem in physics. Such a description must be consis-
] tent with experimental observations of several anoma-
l
e lous normal state properties of these systems. Varma
-
r has recently proposed that quantum critical fluctuations
t associated with the breakup of a subtle order, involving
s
. circulating currents, could induce the observed anoma-
t
a lous normal state properties of high-T superconductors
m c
[1]. Essentially, the associated quantum critical fluctu-
- ations are suggested to produce a fluctuation spectrum
d FIG.1: (Color online) Thecirculating currentphaseΘ [1].
n resulting in a Marginal Fermi Liquid [2]. Recently, such Cu sites are grey circles, O sites are red. The unit cIeIll is
o a spectrum has been derived from a conjectured effec-
shown by the dashed square. A staggered magnetic moment
c tive field theory of circulating currents [3]. It should, pattern within each unit cell that repeats from unit cell to
[
however, be mentioned that a recent numerical evalua- unit cell (the curlof theblue directed circles) is indicated.
1 tion of the current-current correlations in a three-band
v t J-model with 24 sites, where doubly occupied sites
1 −
havebeenprojectedout,showsnoevidenceoftheorbital proposed by Varma [1], are the experiments reported in
1
6 current pattern proposed by Varma [4]. Ref. 5.
4 The particular form of proposed order involves circu- We derive an effective quantum field theory for fluc-
1. lating currents within a CuO2 unit cell where the cur- tuating orbital currents from a microscopic descrip-
0 rents run horizontally and vertically through a Cu site tion of clean CuO2 planes. We are primarily inter-
8 andcloseby direct hopping between O orbitals, as inFig. ested in investigating the intrinsic effects such fluc-
0
1. Three other equivalent patterns may be found by re- tuations have on the physics of the cuprates. We
:
v versingthe directionofthe currentthrougheachCu-site therefore neglect disorder, as was also done in Ref.
Xi in the horizontal and vertical directions. This results in 3. With ever improving sample quality, we expect
a pattern of staggered orbital magnetic moments within that the effective theory we derive should be use-
r
a each unit cell, such that the pattern repeats from unit ful. The starting point is the three-band model H =
cell to unit cell. A magnetic intensity of the type associ- ε d† d + K + K + H(1) + H(2), where
r,σ d r,σ r,σ pd pp int int
aritetipeedsorhwteaidtvhe[5tsh]o.efSaaibrnocbveeeennoorbroeibtpavoliromtuesadgtanhteetrtimhsmeodphysanesuardmeocigceansptiln-ylginubeleaeirnn- pPKyp,rd−=a2yˆt,pσd)P+r,σh[.dc†r.,]σ,(pKx,rp+pa2xˆ=,σ−−pxt,prp−a2xˆr,,σσ[−(pp†xy,r,r++a2a2xˆyˆ,σ,σ−+
the cuprates, it is important to investigate whether or p†x,r−a2xˆ,σ)(py,r+a2yˆ,σ − py,r−a2yˆ,σ) +Ph.c.] and Hi(n2t) =
not the proposed models for this novel type of order im- V r,σ,σ′nd,r,σ(npx,r+a2xˆ,σ′+npx,r−a2xˆ,σ′+npy,r+a2yˆ,σ′+
ply the presence or absence of prominent signals in such npPy,r−a2yˆ,σ′). We work with electron operators and the
quantities as specific heat or (indirectly) magnetization. vacuum is defined as empty dx2−y2, px and py orbitals.
Otherstaggeredorbitalmagneticpatternshavealsobeen The r-sum runs over the Cu-lattice. The Cu-O and O-
proposed, most notably the extension of the staggered O hopping is governed by the parameters t and t ,
pd pp
fluxphase[6]tofinitedoping[7]. Weemphasizethatour respectively, whereas ε is the difference in on-site en-
d
justification for focusing on the orbital current pattern ergy between the copper and oxygen orbitals. The term
2
H(1) represents on-site repulsion terms, for which we current from the left oxygen to the copper plus the cur-
int
make the crude assumption that their effect is to merely rentfromcoppertotherightoxygen. Wedefinejry,σ,σ′ in
renormalize the hopping parameters t t¯ = xt , the same way, but with a minus sign due to the d-wave
pd pd pd
→ | |
t t¯ = xt , where x is the deviation from symmetryoftheCu-orbital. Finiteexpectationvaluesof
pp pp pp
half-→filling [1]. |W|e also assu|me| the O-O repulsion to κx(y) would correspond to Landau-Pomeranchuk insta-
r,σ,σ′
be small. Hence, we only consider explicitly H(2), the bilites,believednottoberelevantinthecuprates. Thus,
int
Cu-O-repulsion. we retain only the decoupling fields that correspond
The interaction-term H(2) can be decoupled [8] in to spin diagonal expectation values of the operators
terms of bosonic fields couipntling to the bilinear fermion jrx,(σy,)σ′(τ), since hjrx,(σy,)σ(τ)i =6 0 in the current pattern
operators Aq(i,)σ,σ′ ≡ N−1/2 k a(xi,)k−qp†x,k−q,σ′ + depicted in Fig. 1. The fields retained, Jrx(y)(τ), are
a(yi,)k−qp†y,k−q,σ′ dk,σ with i = 1,P..,4(cid:0)[1]. Here, N is the real and hJrx(y)(τ)i = Vhjrx,(σy,)σ′(τ)iδσ,σ′, i.e. the fields
number of Cu lattice sites. We define a(1) = a(2) = represent charge currents on horizontal and vertical
(cid:1) x,k x,k O-Cu-O-links. The fields Jx(y)(τ) and the fermions
(3) (4) r
sin(k a/2) s , a = a = cos(k a/2) c and
x ≡ x,k x,k x,k x ≡ x,k are coupled by particle-hole excitations of the form
tah(y1e,k)C=u-−Cau(y2l,ka)tt=icesyc,okn,sata(y3,nk)t.=A−dais(y4c,k)us=siocny,okf,hwAh0(i,e)σr,eσ′iaaiss wiPhekr,eq,tσh(cid:16)eJt−ixmqcexd,ke−pqepn†xd,ekn−cqe,σwdaks,σo−m(itxte→d.yI)t−ishim.c.p(cid:17)o,rtant
translational invariant order parameters in the cuprates tokeepinmindthatthebosonicfieldsJx(y)(τ)transform
r
is found in [1]. While i = 2 transforms as the kinetic
as vectors under a change of coordinate system. Note
energy, i = 1 and i = 3,4 give rise to different current
that we could also have chosen the arguments of the
patterns. Since the observed magnetic signal [5] is con- a(i)’s to be k and not k q in A(i) , corresponding to
sistent with the current patterns of i = 3,4, we keep − q,σ,σ′
a decoupling in terms of currents defined on horizontal
only this in what follows. An effective model for the
and vertical Cu-O-Cu-links.
i = 1-part was considered in [9]. Observe the relation
N−1/2 kcx,k−qp†x,k−q,σ′dk,σ = 1/4 κxq,σ,σ′ +ijqx,σ,σ′ , Integrating out the fermion fields, we obtain
where, in real space, the partition function as Z = DJxDJy e−S,
P (cid:0) (cid:1)
where the effective action is given by S =
jrx,σ,σ′ ≡ 2i d†r,σ px,r+a2xˆ,σ′ +px,r−a2xˆ,σ′ −h.c. . (1) 21V q,ων Jqx(iων)J−xq(−iων)+Jqy(iωνR)J−yq(−iων) −
Trln −1+Σ . Using the gauge transformation
(cid:2) (cid:0) (cid:1) (cid:3) PG0 (cid:2) (cid:3)
In a unit cell centered on Cu, this is proportional to the p ip , p ip , we have
x,k,σ x,k,σ y,k,σ y,k,σ
(cid:2)→ (cid:3) →−
iω +ε µ 2t s 2t s
−1 (iω ,iω )=δ δ δ − 2nt1 s d− iωpd x,k1µ 4t spd y,sk1 , (2)
G0,k1k2,σ1σ2 n1 n2 k1,k2 n1,n2 σ1,σ2 pd x,k1 − n1 − pp x,k1 y,k1
2t s 4t s s iω µ
pd y,k1 pp x,k1 y,k1 − n1 −
0 c Jx (iω ) c Jy (iω )
δ x,k2 k12 12 y,k2 k12 12
Σ (iω ,iω )= σ1,σ2 c Jx (iω ) 0 0 , (3)
k1k2,σ1σ2 n1 n2 √βN cx,k1Jky12(iω12) 0 0
y,k1 k12 12
wherewehavedefinedk k k andω ω ω . ish,suchthatTrln −1+Σ =Trln −1 1Tr[ Σ]2+
For tpp =0,the non-inte1r2a≡ctin1g−pa2rtof th1e2p≡robnle1m−Gn0−21 O(J4),where−Trln(cid:2)GG00−1 give(cid:3)sthefreeGe0ne−rgy2 oftGh0enon-
mayeasilybediagonalizedintothreequasiparticlebands interacting system, and Σ involves the fluctuating fields
E(0) =0,E(±) =ε /2 (ε /2)2+4t2 (s2 +s2 ), of Jx and Jy. To second order in the fields Jx(y) and in
k k d ± d pd x,k y,k
space and imaginary time gradients, we have derived a
whichE(0),E(−) arefullaqndE(+) ispartiallyfilled. This
k k k quantumdissipativeeffectiveactionS =S +S , where
C Q
pictureisnotqualitativelyalteredbyt =0. Anonzero
pp
6
value of t is however vital for the realization of the
pp S = G−1 Ji(iω )Jj ( iω ),
current pattern. It is implicit that Jrx(y)(τ) 0 when C C,ij q ν −q − ν
t 0 [1]. h i → qX,ωνi,jX=x,y
pp
→ S = G−1 Ji(iω )Jj ( iω ), (4)
Q Q,ij q ν −q − ν
Expanding the last term [10], odd powers of J van-
qX,ωνi,jX=x,y
3
withG−1 =α +α q2+α q2,G−1 =α +α q2+α q2, energiesofthe eigenmodesofS aregivenby λ =α +
C,xx c l x t y C,yy c l y t x C ± c
G−1 = G−1 = α q q , G−1 = α ω2 +α |ων|qˆ2, (α +α )q2/2 (∆α)2q4+γq2q2, where ∆α = (α
C,xy C,yx xy x y Q,xx 0 ν d |q| y l t ± x y l −
G−Q,1yy = α0ων2 + αd|ω|qν||qˆx2 and G−Q,1xy = G−Q,1yx = αt)/2, and γ =αq2xy −(2∆α)2. Hence, for (αl,αt)>0, a
α |ων|qˆ qˆ . Here, qˆ = q /q. The dissipation ker- uniformly ordered state is stable in the classical domain
− d |q| x y x x | | belowsomecriticaltemperature,providedα2 <α2+α2.
nel is valid for ω /q 1. The limit ω /q 1 does xy l t
ν ν
| | | |≪ | | | |≫ The dissipation kernel essentially gives Landau damp-
not contribute to dissipation. The explicit expressions
ing,albeitanisotropicduetothedirectionalnatureofthe
for the coefficients α are unwieldy and of limited use.
i
fields. The dissipation is a result of coupling to the gap-
The equality ofthe diagonalandoff-diagonaldissipation
coefficients is only correct when tpp = 0. Changes when less particle-hole excitations in the band Ek(+). The sin-
tinppth6=e0foallroewisnmga.llNaontde uanlsiomtphoarttatnhti,satnhdeoaryremniegghltecnteodt igzuolnatrafloarmnd|ωveνr|/ti|cqa|liscucrorrernetcstaornelyunififtohremoradnedrinnotthmeohdour--
be applicable to the ordered phase, since the Fermi sur- lated at some nonzero reciprocal vector. It implies that
face is proposed to be gapped there [1]. However, it is the dynamical critical exponent z = 3 [12]. See however
thefluctuationspectruminthedisorderedphasewhichis Ref. 10.
importantinconnectionwiththe MarginalFermiLiquid Current amplitude fluctuations are expected to be
hypothesis [2]. high-energy excitations [1] and will therefore not deter-
We have divided the action into a classical (C) and mine the criticalpropertiesofthe model. Thus, we treat
a quantum (Q) part. At finite temperatures, only the the fields Jx(τ) and Jy(τ) as Ising variables. Reverting
r r
classicalpiece ofthe actionS needs to be consideredas to a real space Cu-lattice formulation and setting a=1,
C
far as critical properties are concerned. The excitation we obtain (up to constant terms)
β
SC = − dτ α˜xr,r′Jrx(τ)Jrx′(τ)+α˜yr,r′Jry(τ)Jry′(τ) + α˜xr,yr′ Jrx(τ)Jry′(τ)+Jry(τ)Jrx′(τ) , (5)
Z0 hXr,r′i(cid:16) (cid:17) hhXr,r′ii (cid:16) (cid:17)
β ∂Jx 2 ∂Jy 2 β
SQ = α˜0Z0 dτXr "(cid:18) ∂τr (cid:19) +(cid:18) ∂τr (cid:19) #+α˜dZ0 dτdτ′Xr,r′ Xi,j (cid:16)Jri(τ)−Jri′(τ′)(cid:17)Kirj−r′(τ −τ′)(cid:16)Jrj(τ)−Jrj′(τ′)(cid:17).
Here, r,r′ and r,r′ denote nearest-neighbor and bilinearcombinatoinofIsingvariables,andthesewillalso
h i hh ii
next-nearest-neighbor summations, respectively. For contributeto the coefficientsofthe quadraticterms even
r r′ = xˆ, α˜x = α˜ and α˜y = α˜ , whereas before a renormalization group analysis is carried out.
− ± r,r′ l r,r′ t
when r r′ = yˆ, α˜x = α˜ and α˜y = α˜ . The These terms are also taken into account by the tilde.
− ± r,r′ t r,r′ l
pαfi˜acxri,ryrea′nmt=etα˜edr−α˜α˜>xxry,yr0′w=ahnendα˜xrtyh−ewrhp′eon=sitriv±−e(rxˆs′e−m=iyˆd±)e.fi(xˆnitT+ehyˆem)acoatrneifdx- hanoIrdniztoogneJtnarxe±lrOayˆl-,tChwrueo-uOhg-ahlvineα˜ktα,˜.JlArx6=s,scαe˜oeutn.plfeArsomtcouFrJrirxeg±nuxˆtretlhi1vr,iontughgehorneα˜ilas,
Kr−r′(τ τ′) = Kr−r′(τ τ′)gˆr−r′ ˆgr−r′, where no reasonfor these couplings to be similar, and in fact a
gˆr−r′ =(−r r′)/r r′ and−Kr(τ)=1/(⊗r sin2(πτ/β)). detailed derivation shows that they are not [14].
− | − | | |
Fluctuations(Jx Jx,Jy Jy)correspondstogoing At finite temperature, we may ignore the inertial and
r →− r r → r
from the depicted current pattern (Fig. 1) to a new one dissipative terms, which reduces the model to a classical
whichisobtainedbyacounterclockwiserotationbyπ/2, model of two coupledIsing fields. Such a classicalmodel
(Jx Jx,Jy Jy) corresponds to clockwise rotation will suffice to study the breakup of the current pattern
r → r r →− r
of π/2, and (Jx Jx,Jy Jy) to a rotation of π. atfinite temperatures,while its quantumcriticalversion
r → − r r → − r
It is implied that in the dissipation kernel, we must use can only be accessed via the full dissipative field theory.
a short-distance cutoff in (τ,r)-space, since the expres- Note also that the dissipation kernel is non-local both in
sions are derived in the limit of low (ω,q). The tildes imaginary time and in space. The latter distinguishes
on the coefficients indicate that the model in Eq. 5 is this dissipation term from the Caldeira-Leggett type of
regularised on a lattice, and that the fields have been dissipation appropriate for an array of Josephson junc-
normalised to Ising-variables. Moreover, there will be tions [13, 15]. The non-locality in r-space is anisotropic
higher order (quartic) terms generated that simply in- forthe samereasonasforthe nearest-neighborcoupling.
volve local squares of Ising variables multiplied by som Eq. (5) may be rewritten on the form
4
β
SC = − dτ α¯ cos(θr,τ −θr′,τ)+(∆α˜)r,r′ sin(θr,τ +θr′,τ) +2 α˜xr,yr′ cos(θr,τ +θr′,τ), (6)
Z0 hXr,r′ih i hhXr,r′ii
β ∂θ 2 β
SQ = 2α˜0 dτ ∂rτ,τ +α˜d dτdτ′ Ji(θr,τ)−Ji(θr′,τ′) Kirj−r′(τ −τ′) Jj(θr,τ)−Jj(θr′,τ′) ,
Z0 Xr (cid:18) (cid:19) Z0 Xr,r′ Xi,j (cid:16) (cid:17) (cid:16) (cid:17)
where we have used the parametrization cos(θ ) = specific heat would be quenched. S in Eqs. (5,6) dif-
r,τ C
(Jx(τ) +Jy(τ))/2, sin(θ ) = (Jx(τ) Jy(τ))/2, and fers from the model of Ref. 3 in several respects. How-
r r r,τ r − r
θ (0,π/2,π,3π/2). We have defined α¯ = (α˜ +α˜ ), ever, a direct comparison is difficult, as it is not clear
r,τ l t
∈
(∆α˜)r,r′ = (α˜l α˜t) for r r′ = xˆ and (∆α˜)r,r′ = what physical quantitites the fields in Ref. 3 represent.
− − ±
(α˜ α˜ ) for r r′ = yˆ. Firstly, the Ising-exchange coupling terms in Eqs. (5,6)
l t
− − − ±
Eqs. (5) and (6) are the main results of this paper. are anisotropic, possibly highly anisotropic, due to the
These models describe a phase transition from a disor- bond-characteroftheIsingvariables. Moreover,theterm
dered bosonic state (a Fermi liquid), into a state with α˜xr,yr′JrxJry′ in Eqs. (5,6) is absent in Ref. 3. While
bosonic order in the form of ordered orbital currents. this term may be perturbatively irrelevant, it is far from
clear that α˜ is actually small. In addition, there also
We next proceed to discuss some qualitative aspects. xy
seemstobe adiscrepancybetweenthe dissipationkernel
Considerfirstthismodelatfinitetemperature,wherewe
ω /q derived here and the one employed in Ref. 3.
may use the approximation S SC. When (α˜l,α˜t) > 0 | ν| | |
≈
and α˜ = 0, the current pattern in Fig. 1 repeats uni- We expect our model to be generically useful in de-
xy
formly from unit cell to unit cell throughout the system scribing thermal and quantum critical fluctuations of di-
in the orderedstate. The specific heat has a logarithmic rected particle-hole bond variables in fermionic lattice
singularity at a critical temperature determined by the models.
condition sinh(2β α˜ )sinh(2β α˜ )=1, where β =1/T .
c l c t c c Acknowledgements. This work was supported by the
Anisotropy in the nearest-neighbor couplings suppresses
Research Council of Norway Grants No. 158518/431
thecriticaltemperatureandcriticalamplitudes,andnar-
and No. 158547/431 (NANOMAT), and Grant No.
rowsthe criticalregion,but does not alter the universal-
167498/V30(STORFORSK).ThehospitalityoftheCen-
ity class of the phase transition [16]. When (α˜ ,α˜ ) = 0
l t ter for Advanced Study at The Norwegian Academy of
and α˜ = 0, the ground state of the system features a
xy 6 Science and Letters is acknowledged, as well as useful
striped phase in the diagonal directions, irrespective of
discussions with C. M. Varma and Z. Tesanovic.
the sign of α˜ . Note also that when sign(α˜ )=sign(α˜ )
xy l t
6
and α˜ = 0, one obtains order with a period of twice
xy
the lattice constant.
Thedissipativeterminthismodelcomesfromthecou-
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k Abrahams,andA.E.Ruckenstein,Phys.Rev.Lett.,63,
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1996 (1989).
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sites. Wecouldalternativelyhavedefinedthecurrentson (2007).
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A.Mooketal,TalkatAspenCenterforPhysics,August
pect a different dissipation term in that case, due to the
2007.
finite modulation vector of the ordered currents. Note
[6] I. Affleck and J. B. Marston, Phys. Rev. B 37, 3774
also that the ω2-terms in S in Eq. (4), equivalently
ν Q (1988).
the inertial terms in Eqs. (5) and (6), are of multiband [7] S. Chakravarty, R. B. Laughlin, D. K. Morr, and C.
origin. Nayak, Phys.Rev.B 63, 094503 (2001).
[8] R. L. Stratonovich, Dokl. Akad. Nauk SSSR 2, 1097
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C (1957); J. Hubbard,Phys.Rev.Lett., 3, 77 (1959).
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r1r2r3r4 r1 r2 r3 r4 [9] H. C. Lee and H.-Y. Choi, Phys. Rev. B 64, 094508
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(2001).
cludeanAshkin-Tellertypeoffour-spininteraction,used [10] Note that this procedure [11] of integrating out the
in Ref. 3 to argue that the Ising type of singularity in fermions and expanding the logarithm in the case of a
5
|q|=0 order parameter and gapless fermions in general [13] A.O.CaldeiraandA.J.Leggett,Ann.Phys.(N.Y.)149,
leads tosingular coefficients and might not bevery suit- 374 (1984).
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al,Rev. Mod. Phys. 77, 579 (2005). site-variables, have directionality. See also A. Melikyan
[11] J. A. Hertz,Phys. Rev B 14, 1165 (1976). and Z. Tesanovic, Phys. Rev.B 74, 214511 (2005).
[12] The scaling function for the susceptibility is χ(q,ω) = [15] See also P. Werner, K. V¨olker, M. Troyer, and S.
(Z/T(2−η)/z)Φ±“(cTq)z,Tω”, where T is temperature, Z dCihssaikpraatvivaertpy,arPthoyfst.hReeavc.tiLoenttin.,t9h4is, (0s4p7a2t0ia1ll(y2e0x0t5e)n.dTehde)
is related to critical amplitudes, z is dynamical critical
transversefield Ising-chain isnon-local only intime, but
exponent, and η is the anomalous scaling dimension of
local in space, dueto the local character of the coupling
the relevant fields. For z = 3, the q-dependence of this
between theheat-bath oscillators and theIsing spins.
quantityisweakcomparedtotheω/T-dependenceatlow
[16] L. Onsager, Phys. Rev. 65, 117 (1944).
(q,ω) .
