No evidence for spontaneous orbital currents in finite size studies of three-band models for CuO planes

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by  Martin Greiter

No evidence for spontaneous orbital currents in finite size studies
of three-band models for CuO planes
Martin Greiter and Ronny Thomale
Institut fu¨r Theorie der Kondensierten Materie, Universit¨at Karlsruhe, D 76128 Karlsruhe
(Dated: February 6, 2008)
7
0
Wehavenumericallyevaluatedthecurrent-currentcorrelationsforthree-bandmodelsoftheCuO
0
2 planes in high-Tc superconductors at hole doping x = 1/8. The results show no evidence for
the orbital current patterns proposed by Varma. If such patterns exist, the associated energy is

n estimated to be smaller than 5 meV per link even if ǫp−ǫd = 0. Assuming that the three-band
a modelsareadequate,quantumcriticalfluctuationsofsuchpatternshencecannotberesponsiblefor
J phenomena occurring at significantly higher energies, such as superconductivity or the anomalous

1 properties of the pseudogap phase.
1
PACSnumbers: 74.20.Mn,74.72.-h,74.20.-z

]
n
o
The problem of high-T superconductivity has been doping level of x 0.19, an assumption consistent with
c c c ≈
somethinglikeaholygrailtothefieldofcondensedmat- a significant body of experimental data [11, 12, 13, 14,
-
r ter physics for the past two decades [1]. In the words 15]. Critical fluctuations around this point would then
p
u of R.B. Laughlin, it has not been a fight, but a war. be responsible for the anomalous properties of the pseu-
s It has been a traumatic experience for some of those dogap phase, and provide the pairing force responsible
.
t involved, but has also led to a plethora of new devel- for the superconducting phase which hides the QCP.
a
m opments extending far beyond the field. Many of the Interpreting the phase diagram in these terms, one is
experimental techniques used to study the systems, like immediately led to ask what the phase to the left of the
-
d angle resolved photo emission spectroscopy (ARPES) or QCP, i.e., for x < x , might be. The theory would
c
n scanning tunneling microscopy (STM), have undergone require a spontaneously broken symmetry beyond the
o revolutions with regard to resolution and data process- global U(1) symmetry broken through superconductiv-
c
[ ing. Thetheoryofsuperconductivityinhigh-Tc cuprates ity (which is often erroneously refered to as a broken
has been found as many times as victories must have gauge symmetry [16]). In addition, as the fluctuations
1 been proclaimed in civil wars, but while individuals be- are assumed to determine the phase diagramup to tem-
v
lieve to have the theory, there is no consensus what the peratures of several hundred Kelvin, the characteristic
5
4 theory should be. Many ideas, even though too general energy scale of the correlations inducing this symmetry
2 to qualify as complete theories of the cuprates, have in- violation must be at least of the same order of magni-
1 spired a vast amount of research in both high-Tc and tude. No definitive evidence of such a broken symmetry
0 other areas. Most prominently among them are the no- has been found up to now, even though several possi-
7
tions of a resonating valence bond (RVB) state [2], the bilities have been suggested. These include stripes [17],
0
/ gauge theories of antiferromagnetism [3], and the notion a d-density wave [18], and most recently a checkerboard
at of quantum criticality [4]. There have been, however, a charge density wave [19].
m few concise proposals which make falsifiable predictions.
The general consensus is that the low energy sec-
Intellectualmasterpiecesamongthemhavebeenthethe-
- tor of the three-band Hubbard model proposed for the
d ory of anyon superconductivity [5], the proposal of ki-
CuO planes (see (1) below) [20] reduces to a one-band
n
netic energysavingsthroughinterlayertunneling [6], the
o t–t′–J model, with parameters t 0.44, t′ 0.06,
SO(5)theoryofacommonorderparameterforsupercon- ≈ ≈ −
c and J 0.128 (energies throughout this article are in
v: ductivity andmagnetism[7],andamorerecentproposal eV) [21≈, 22, 23, 24, 25]. For the undoped CuO planes,
i that the anomalous properties of the cuprates may be the formal valances are Cu2+ and O2−. As the electron
X due to quantum critical fluctuations of current patterns configuration of Cu atoms is [Ar] 3d104s1, this implies
r formedspontaneouslyintheCuOplanes[8,9]. This last
one hole per unit cell, which will predominantly occupy
a
proposal is further investigated in this Letter.
the 3dx2−y2 orbital. As the onsite potential ǫp in the O
Theideaofaspontaneoussymmetrybreakingthrough 2px and 2py orbitals relative to the Cu 3dx2−y2 orbital
orbital currents was, as with so many major advances is generally assumed to be of the order of ǫ =3.6 (with
p
in physics, motivated by experiment. The normal state ǫ = 0), and hence smaller than the onsite Coulomb re-
d
of the cuprates at optimal doping shows a behavior pulsionUd 10.5forasecondholeinthe3dx2−y2 orbital,
≈
whichcanbeclassifiedasquantumcritical,andhasbeen itisclearthatadditionalholesdopedintotheplaneswill
rather adequately described by a phenomenological the- primarily reside on the Oxygens. The maximal gain in
ory called marginal Fermi liquid [10]. This phenomenol- hybridizationenergyisachievedbyplacingtheadditional
ogy suggests a quantum critical point (QCP) at a hole hole in a combination of the surrounding O 2p and 2p
x y
O -O
We hence undertook to estimate these couplings
@I (cid:0)(cid:9)
Cu
through numerical studies of finite clusters containing 8
O -O(cid:0)(cid:9) -@IO -O
@I (cid:0)(cid:9) @I (cid:0)(cid:9) unit cells, i.e., 8 Cu and 16 O sites, and periodic bound-
Cu Cu
O(cid:0)(cid:9) -@IO -O(cid:0)(cid:9) -@IO ary conditions (which do not frustrate but should en-
@I (cid:0)(cid:9) hance the correlations). The total number of holes on
Cu
O(cid:0)(cid:9) -@IO our cluster was taken N = 9 (5 up-spins and 4 down-
spins), corresponding to a hole doping of x = 1/8. We
FIG. 1: Orbital current pattern proposed byVarma. hadhopedthattheenergyassociatedwithadomainwall,
whichmaybeimplementedthroughatwistinthebound-
orbitals with the same symmetry as the original hole in ary conditions, would provide information regarding the
the Cu 3dx2−y2 orbital, which requires antisymmetry of couplingaligningthe orbitalcurrentsinneighboringpla-
the wave function in spin space, i.e., the two holes must quets, while the splitting between the lowestenergies for
formasinglet. Thispictureisstronglysupportedbydata a finite system would provide an estimate for the trans-
fromNMR[26]andevenmoredirectlyfromspin-resolved verse field. The result of our endeavors, however, is a
photoemission [27]. In the effective one-band t–J model daunting disappointment: The coupling is zero within
description of the CuO planes, these singlets constitute the error bars of our numerical experiments.
the “holes”movingina backgroundofspin1/2particles Let us now report our numerical studies in detail. To
localized at the Cu sites. begin with, we wish to study the three-band Hubbard
In contrast to this picture, Varma [8, 9] has proposed Hamiltonian H =H +H with
t U
that the additional holes doped in the CuO planes do
not hybridize into Zhang-Rice singlets, but give rise to H = ǫ np t d† p +p† d
circular currents on O-Cu-O triangles, which align into t Xi,σ p i,σ− pdhiX,ji,σ(cid:16) i,σ j,σ j,σ i,σ(cid:17)
a planar pattern as shown in Fig. 1. He assumes that
t p† p +p† p +V nd np ,
bthoethinttheer-haotopmpiincgCtouloamndbtphoetoenntsiiatel Vppodtenistialalrǫgerofthtahne − pphiX,ji,σ(cid:16) i,σ j,σ j,σ i,σ(cid:17) phdi,jXi,σ,σ′i,σ j,σ′
pd p
O 2p orbitals relative to the Cu 3dx2−y2 orbitals, an as- HU= Up npi,↑npi,↓+Ud ndi,↑ndi,↓, (1)
sumption which is not consistent with the values gen- Xi Xi
erally agreed on (see the list below (1)). Making addi-
tional assumptions, Varma has shown that the circular where h , i indicates that the sums extend over pairs of
currentpatternsarestabilizedinameanfieldsolutionof nearest neighbors, while di,σ and pj,σ annihilate holes in
thethree-bandHubbardmodel. Theorbitalcurrentpat- Cu 3dx2−y2 or O 2p orbitals, respectively. Hybertsen et
ternsbreaktime-reversalsymmetrybreaking(T)andthe al.[23]assumedtpd =1.5,tpp =0.65,Ud =10.5,Up =4,
discrete four-fold rotation symmetry on the lattice, but Vpd =1.2, and ǫp =3.6.
leave translational symmetry intact. The current pat- In order to be able to diagonalize (1) for 24 sites, we
tern is assumed to disappear at a doping level of about need to truncate the Hilbert space. A first step is to
x 0.19. ThephenomenologyofCuOsuperconductors, eliminatedoublyoccupiedsites,whichyieldstheeffective
c
≈
including the pseudogap and the marginal Fermi liquid three-band t–J Hamiltonian
phase, are assumed to result from critical fluctuations
H =P˜ H P˜ +H with (2)
around this QCP, as outlined above. eff G t G J
Motivated by this proposal, several experimental 1 1
groups have looked for signatures of orbital currents or HJ = Jpd (cid:18)Spi·Sdj − 4(cid:19)+Jpp (cid:18)Spi·Spj − 4(cid:19),
T violation in CuO superconductors. While there is no hXi,ji hXi,ji
agreement between different groups regarding the mani-
where
festation of T violation in ARPES studies [28, 29], a re-
cent neutron scattering experiment by Fauqu´eet al. [30] 1 1 4t2
indicatesmagneticorderwithintheunitcellsoftheCuO Jpd =2t2pd(cid:18)U ǫ + U +ǫ (cid:19), Jpp = Upp,
d p p p p
planes. Their results are consistent with Varma’s pro- −
posal, and call the validity of the one-band models into andthesumsinH arelimitedtopairswherebothneigh-
J
question. bors are occupied by holes. If P˜ only eliminates config-
G
In a recent article, Aji and Varma [31] have mapped urationswithmorethanoneholeonasite,thedimension
thefourpossibledirectionsofthecurrentpatternsineach oftheSz = 1 sub-sectoriswith164,745,504stillbeyond
tot 2
unit cell onto two Ising spins, and investigated the crit- ourcapabilities. Wehavehenceimplementedtwofurther
ical fluctuations. Within this framework, the coupling ways of truncating the Hilbert space: (a) We limit the
between and the transverse fields for these Ising spins maximal number of holes allowed in the O orbitals to
decide whether or under which circumstances the model Nmax. (b) We limit the maximal number of CuO links
ox
displays long-range order in the orbital currents. occupied with 2 holes to Nmax. For the values proposed
link
2
6 6
@I @I
+conventionforcurrents: @ - SCALE=100 +conventionforcurrents: @ - SCALE=100
(cid:0) (cid:0)
(cid:0)(cid:9) (cid:0)(cid:9)
~~ ~~ -1.8133 ~ 0.0354 ~ -1.8133 ~ ~~ ~~ -1.1076 ~ 0.3599 ~ -1.1076 ~
-2.9825120..70874698 ~y01.20.77688492.9825 0.027600..11423439 ~y00..11244393 -0.0276 -2.9825 -2.9813132..45115661 ~y21.35.14611562.9813 -0.041500..33334515 ~y00..33335451 0.0415 -2.9813
~ 0.0605 ~ -0.0812 ~ 0.0441 ~ -0.0812 ~ ~ 1.0751 ~ -0.1857 ~ 0.1832 ~ -0.1857 ~
-0.0812 0.081200..01633301 ~y-00..11932311-0.0479 0.0479-00..11393211 ~y00..10363310 -0.0812 -0.1857 0.1857-00..61819835 ~y00..31018835 -0.0158 0.015800..13108853 ~y-00..16188953 -0.1857
~ -0.0441 ~ 0.0479 ~ -0.0508 ~ 0.0479 ~ ~ -0.1832 ~ 0.0158 ~ -0.0676 ~ 0.0158 ~
0.0276--00..11244393 ~y--00..11423439-0.0276 0.1086-00..12366126 ~y-00..21631662-0.1086 0.0276 -0.0415--00..33335451 ~y--00..333345150.0415 0.3614-00..16742107 ~y-00..61471270-0.3614 -0.0415
~ -0.0354 ~ -0.0812 ~ 0.2467 ~ -0.0812 ~ ~ -0.3599 ~ -0.1857 ~ 0.9484 ~ -0.1857 ~
1.8133 -1.8133-00..58511578 ~y-00..53581976-0.0812 0.0812-00..35859167 ~y-00..851551871.8133 1.1076 -1.1076-00..10569315 ~y-01..16589117-0.1857 0.1857-10..61851971 ~y-00..016539511.1076
~~ ~~ -1.8133 ~ 0.0354 ~ -1.8133 ~ ~~ ~~ -1.1076 ~ 0.3599 ~ -1.1076 ~
FIG. 2: Current-current correlations hjk,k+xˆjl,mi multiplied FIG. 3: as in Fig. 2, butwith ǫp =0. Except for thevertical
by 102 for the ground state of (3) with ǫp = 3.6 on a 24 links,positivecorrelationsindicatealignmentwiththecurrent
site cluster (8 Cu = open circles, 16 O = filled circles) with pattern shown in Fig. 1.
PBCs. The reference link is indicated in thetop and (dueto
thePBCs) bottom left corner.
and parameters as assumed by Hybertsen, where no or-
bital currentpattern are expected. With the currentop-
by Hybertsen et al., either truncation should hardly af- erator for, e.g. an O-O link given by
fect the low energy sector. In Table I, we compare the
groundstateenergiesandthecurrent-currentcorrelation j =it p† p p† p , (3)
ondiagonallyandmaximallyseparatedlinksonthetorus k,l ppXσ (cid:16) l,σ k,σ − k,σ l,σ(cid:17)
for different values of Nmax and Nmax. This compari-
ox link the correlations function j j with an O-O link
k,k+xˆ l,m
son gives us confidence that it is not unreasonableto set h i
as reference link is depicted in Fig. 2. As expected, the
Nmax = 4, which reduces the Hilbert space dimension
link correlations fall off rapidly, and there is no indication of
to 93,595,824. Since the truncation(b) is predominantly
order.
projectingoutstates with highkinetic energy,we expect
The crucial result is that the correlations change con-
that the value of ǫ will not affect the validity of the
p tinuously and only quantitatively, but not really quali-
approximation. Excluding configurations with too many
tatively, as ǫ is lowered from 3.6 to 1.8, 0.9, 0.4, and
p
CuOlinksoccupiedbytwoholesshould,inanyoccasion,
finally to ǫ = 0, where the result is shown in Fig. 3
p
notweakenthetendencytoformorbitalcurrentpatterns.
(the ground state is now situated at the Γ point). This
Truncation (a), by contrast, is no longer reasonableif ǫ
p is the situation for which Varma has proposed that the
is small.
current pattern sketched in Fig. 1 would occur. Fig. 3,
Let us now turn to our results for the current-current
by contrast, shows no alignment of the currents.
correlationsin the groundstate (situatedat the Mpoint
The numerical experiments for the finite cluster can,
in the Brillouin zone) for our 24 site cluster with 9 holes
as a matter of principle, never rule out that a symmetry
is violated. We can use them, however, to put an up-
per bound on the size of the spontaneous currents, and
Nomxax E 102hJJidiag. Nlminakx E 102hJJidiag. hence the energy associated with these currents. If a
3 -0.705 -0.0085 2 -0.063 0.0004 current pattern as sketched in Fig. 1 were to exist, the
4 -0.835 -0.0304 3 -0.635 -0.0279 current correlation jk,k+xˆjl,l+xˆ for links far away from
h i
5 -0.877 -0.0488 4 -0.851 -0.0508 eachotherinarotationallyinvariantgroundstateshould
approach 1 xˆj xˆ 2, where xˆ denotes a state with
2h | k,k+xˆ| i | i
TABLE I: Ground state energies per unit cell and a current- a spontaneous current pointing in xˆ direction (the fac-
current correlation for various truncations of the Hilbert tor 1 arises because by choosing our reference link in x-
2
space. direction,weeffectively projectontotwoofthe four pos-
3
sibledirectionsforthe currentpattern). Fromthe values We further acknowledge the support of the computing
of 102 j j for the four horizontally connected facilitiesoftheINTattheForschungszentrumKarlsruhe.
k,k+xˆ l,l+xˆ
·h i
links in the center of Fig. 3, 0.1832,+0.0158, 0.0676,
− −
and +0.0158, which should all be positive if a current
pattern were present, we estimate 102 j 2 < 0.20
k,k+xˆ
·h i
andhence xˆj xˆ 2 <4 10−3 asanupperboundfor
h | k,k+xˆ| i · [1] J. Zaanen et al.,NaturePhysics 2, 138 (2006).
a current pattern we are unable to detect through the
[2] P. W. Anderson,Science 235, 1196 (1987).
error bars of our numerical experiment. We now denote
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xˆjk,k+xˆ xˆ by jpp. 78, 17 (2006).
h | | i
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pp
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pp p pp 2 p
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p [8] C. M. Varma, Phys. Rev.Lett. 83, 3538 (1999).
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j2
ε pp <5 10−3. [10] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E.
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[16] M. Greiter, Ann.Phys. 319, 217 (2005).
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[17] S.A.Kivelson,I.P.Bindloss,E.Fradkin,V.Oganesyan,
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184515 (2006).
quatelydescribedbythe three-bandHubbardmodel(1),
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p
presented here show that our conclusions remain intact Phys. Rev.B 39, 9028 (1989).
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p
Nonetheless, we should keep in mind that any analy- [27] L. H. Tjeng et al.,Phys. Rev.Lett. 78, 1126 (1997).
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We wish like to thank C.M. Varma, V. Aji, F. Evers [31] V. Aji and C. M. Varma, cond-mat/0610646.
andinparticularP.W¨olfleformanyilluminating discus-
sions of this subject. RT was supported by a PhD schol-
arship from the Studienstiftung des deutschen Volkes.
4