No evidence for spontaneous orbital currents in numerical studies of three-band models for the CuO planes of high temperature superconductors.
ABSTRACT We have numerically evaluated the current-current correlations for three-band models of the CuO planes in high-T(c) 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 estimated to be smaller than 5 meV per link even if [formula: see text]. Assuming that the three-band models are adequate, quantum critical fluctuations of such patterns hence cannot be responsible for phenomena occurring at significantly higher energies, such as superconductivity or the anomalous properties of the strange metal phase.
No Evidence for Spontaneous Orbital Currents in Numerical Studies of Three-Band Models
for the CuO Planes of High Temperature Superconductors
Martin Greiter and Ronny Thomale
Institut fu ¨r Theorie der Kondensierten Materie, Universita ¨t Karlsruhe, D 76128 Karlsruhe
(Received 13 December 2006; published 13 July 2007)
We have numerically evaluated the current-current correlations for three-band models of the CuO
planes in high-Tcsuperconductors 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 estimated to be smaller
than 5 meV per link even if ?p? ?d? 0. Assuming that the three-band models are adequate, quantum
critical fluctuations of such patterns hence cannot be responsible for phenomena occurring at significantly
higher energies, such as superconductivity or the anomalous properties of the strange metal phase.
DOI: 10.1103/PhysRevLett.99.027005PACS numbers: 74.20.Mn, 74.72.?h
The problem of understanding high-Tcsuperconductiv-
ity has been something like a holy grail to the field of
condensed matter physics for the past two decades . It
has turned out to be an exceedingly difficult problem, with
much of the effort invested just deepening the mysteries,
but it has also led to a plethora of new developments
extending far beyond the field. Many of the experimental
techniques used to study the systems, like angle resolved
photo emission spectroscopy (ARPES) or scanning tunnel-
ing microscopy (STM), have undergone revolutions with
regard to resolution and data processing. The theory of
superconductivity in high-Tc cuprates has been found
many times, but while individuals believe to have the
theory, there is no consensus what the theory should be.
Manyideas, eventhough toogeneral to qualify as complete
theories of the cuprates, have inspired a vast amount of
research in both high-Tcand other areas. Most prominently
among them are the notions of a resonating valence bond
(RVB) state  including the gauge theories of spin liquids
, and the notion of quantum criticality . There have
been, however, a few concise proposals which make falsi-
fiable predictions. Masterpieces among them have been the
theory of anyon superconductivity , the proposal of
kinetic energy savings through interlayer tunneling ,
the SO(5) theory of a common order parameter for super-
conductivity and magnetism , and a more recent pro-
posal that the anomalous properties of the cuprates may be
due to quantum critical fluctuations of current patterns
formed spontaneously in the CuO planes [8,9]. This last
proposal is further investigated in this Letter.
The idea of a spontaneous symmetry breaking through
orbital currents was, as with so many major advances in
physics, motivated by experiment. The normal state of the
cuprates at optimal doping shows a behavior which can be
classified as quantum critical and has been rather ade-
quately described by a phenomenological theory called
marginal Fermi liquid . This phenomenology suggests
a quantum critical point (QCP) at a hole doping level of
xc? 0:19, an assumption consistent with a significant
body of experimental data [11–15]. Critical fluctuations
around this point would then be responsible for the anoma-
lous properties of the strange metal phase and provide the
pairing force responsible for the superconducting phase
which hides the QCP.
Interpreting the phase diagram in these terms, one is
immediately led to ask what the phase to the left of the
QCP, i.e., for x < xc, might be. The theory would require a
spontaneously broken symmetry beyond the global U(1)
symmetry broken through superconductivity (which is
often erroneously referred to as a broken gauge symmetry
). In addition, as the fluctuations are assumed to de-
termine the phase diagram up to temperatures of several
hundred Kelvin, the characteristic energy scale of the
correlations inducing this symmetry violation must be at
least of the same order of magnitude. No definitive evi-
dence of such a broken symmetry has been found up to
now, even though several possibilities have been sug-
gested. These include stripes , a d-density wave ,
and most recently a checkerboard charge density wave
The general consensus is that the low energy sector of
the three-band Hubbard model proposed for the CuO
planes [see (1) below]  reduces to a one-band t ? t0?
J model, with parameters t ? 0:44, t0? ?0:06, and J ?
0:128 (energies throughout this Letter are in eV) [21–25].
For the undoped CuO planes, the formal valances are Cu2?
and O2?. As the electron configuration of Cu atoms is [Ar]
3d104s1, this implies one hole per unit cell, which will
predominantly occupy the 3dx2?y2 orbital. As the on-site
potential ?pin the O 2pxand 2pyorbitals relative to the Cu
3dx2?y2 orbital is generally assumed to be of the order of
?p? 3:6 (with ?d? 0), and hence smaller than the on-site
Coulomb repulsion Ud? 10:5 for a second hole in the
3dx2?y2 orbital, it is clear that additional holes doped into
the planes will primarily reside on the oxygens. The maxi-
mal gain in hybridization energy is achieved by placing the
additional hole in a combination of the surrounding O 2px
and 2pyorbitals with the same symmetry as the original
PRL 99, 027005 (2007)
13 JULY 2007
© 2007 The American Physical Society
hole in the Cu 3dx2?y2 orbital, which requires antisymme-
try of the wave function in spin space; i.e., the two holes
must form a singlet. This picture is strongly supported by
data from NMR  and even more directly from spin-
resolved photoemission . In the effective one-band t ?
t0? J model description of the CuO planes, these singlets
constitute the ‘‘holes’’ moving in a background of spin 1=2
particles localized at the Cu sites.
In contrast to this picture, Varma [8,9] has proposed that
the additional holes doped in the CuO planes do not
hybridize into Zhang-Rice singlets, but give rise to circu-
lating currents on O-Cu-O triangles, which align into a
planar pattern as shown in Fig. 1. He assumes that the
interatomic Coulomb potential Vpdis larger than the on-
site potential ?pof the O 2p orbitals relative to the Cu
3dx2?y2 orbitals, an assumption which is not consistent
with the values generally agreed on [see the list below
(1)]. Making additional assumptions, Varma has shown
that the circulating current patterns are stabilized in a
mean-field solution of the three-band Hubbard model.
The orbital current patterns break time-reversal symmetry
(T) and the discrete fourfold rotation symmetry on the
lattice, but leave translational symmetry intact. The current
pattern is assumed to disappear at a doping level of about
xc? 0:19. The phenomenology of CuO superconductors,
including the pseudogap and the marginal Fermi liquid
phase, are assumed to result from this symmetry breaking
and critical fluctuations around this QCP, as outlined
Motivated by this proposal, several experimental groups
have looked for signatures of orbital currents or T violation
in CuO superconductors. While there is no agreement
between different groups regarding the manifestation of
T violation in ARPES studies [28,29], a recent neutron
scattering experiment by Fauque ´ et al.  indicates mag-
netic order within the unit cells of the CuO planes. Their
results are consistent with Varma’s proposal and call the
validity of the one-band models into question.
In a recent article, Aji and Varma  have mapped the
four possible directions of the current patterns in each unit
cell onto two Ising spins and investigated the critical
fluctuations. Within this framework, the coupling between
and the transverse fields for these Ising spins decide
whether or under which circumstances the model displays
long-range order in the orbital currents.
We hence undertook to estimate these couplings through
numerical studies of finite clusters containing 8 unit cells,
i.e., 8 Cu and 16 O sites, and periodic boundary conditions
(which do not frustrate but should enhance the correla-
tions). The total number of holes on our cluster was taken
N ? 9 (5 up-spins and 4 down-spins), corresponding to a
hole doping of x ? 1=8. We had hoped that the energy
associated with a domain wall, which may be implemented
through a twist in the boundary conditions, would provide
information regarding the coupling aligning the orbital
currents in neighboring plaquettes, while the splitting be-
tween the lowest energies for a finite system would provide
an estimate for the transverse field. The result of our
endeavors, however, is a daunting disappointment: the
coupling is zero within the error bars of our numerical
Let us now report our numerical studies in detail. To
begin with, we wish to study the three-band Hubbard
Hamiltonian H ? Ht? HUwith
where h ;i indicates that the sums extend over pairs of
nearest neighbors, while di;?and pj;?annihilate holes in
Cu 3dx2?y2 or O 2p orbitals, respectively. Hybertsen et al.
 calculated tpd? 1:5, tpp? 0:65, Ud? 10:5, Up? 4,
Vpd? 1:2, and ?p? 3:6.
In order to be able to diagonalize (1) for 24 sites, we
need to truncate the Hilbert space. A first step is to elimi-
nate doubly occupied sites, which yields the effective
three-band t ? J Hamiltonian
Heff? PGHtPG? HJ
and the sums in HJare limited to pairs where both neigh-
bors are occupied by holes. PGeliminates configurations
with more than one hole on a site. The dimension of the
Let us now turn to our results for the current-current
correlations in the ground state (situated at the M point in
4? ? Jpp
2subsector is with 164745504 just within our
FIG. 1.Orbital current pattern proposed by Varma.
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13 JULY 2007
the Brillouin zone) for our 24 site cluster with 9 holes and
parameters as calculated by Hybertsen, where no orbital
current patterns are expected. With the current operator for,
e.g., an O-O link given by
the correlations function hjk;k?^ xjl;mi with an O-O link as
reference link is depicted in Fig. 2. As expected, the
correlations fall off rapidly, and there is no indication of
The crucial result is that the correlations change con-
tinuously and only quantitatively, but not really qualita-
tively, as ?pis lowered from 3.6 to 1.8, 0.9, 0.4, and finally
to ?p? 0, where the result is shown in Fig. 3 (the ground
state is nowtwofold degenerate andsituated at the ? point).
This is the situation for which Varma has proposed that the
current pattern sketched in Fig. 1 would occur. Figure 3, by
contrast, shows no alignment of the currents.
The numerical experiments for the finite cluster can, as a
matter of principle, never rule out that a symmetry is
violated spontaneously, as this would require an infinite
system. We can use our results, however, to put an upper
bound on the size of the spontaneous currents, and hence
the energy associated with these currents. If a current
pattern as sketched in Fig. 1 were to exist, the current
correlation hjk;k?^ xjl;l?^ xi for links far away from each other
in a rotationally invariant ground state should approach
2h^ xjjk;k?^ xj^ xi2, where j^ xi denotes a state with a spontaneous
current pointing in ^ x direction (the factor1
by choosing our reference link in x direction, we effec-
tively project onto two of the four possible directions for
the current pattern). From the values of 102hjk;k?^ xjl;l?^ xi for
the four horizontally connected links in the center of Fig.3,
?0:3726, ?0:1483, ?0:4123, and ?0:1483, which should
all be positive if a current pattern were present, weestimate
102hjk;k?^ xi2< 0:20 and hence h^ xjjk;k?^ xj^ xi2< 4 ? 10?3as
anupper boundfora current pattern weare unable to detect
through the error bars of our numerical experiment. We
now denote h^ xjjk;k?^ xj^ xi by jpp.
We roughly estimate the kinetic energy "ppper link
associated with a spontaneouscurrent jppof this magnitude
using jpp? npv and "pp?1
where npis the hole density on the oxygen sites (np?
0:33 for the state of Fig. 3), and obtain
A similar analysis for the CuO links, using data not shown
here, yields with 102hjk;k?^ x?^ yi2< 1:0 and hence j2
10?2(there is no factor1
2npmv2with m ? 1=2tpp,
< 5 ? 10?3:
2in this case) and nd? 0:47 an
< 4 ? 10?3:
102for the ground state of (2) with ?p? 3:6 on a 24 site cluster
(8 Cu ? open circles, 16 O ? filled circles) with periodic
boundary conditions (PBCs). The reference link is indicated in
the top and (due to the PBCs) bottom left corner.
Current-current correlations hjk;k?^ xjl;mi multiplied by
links, positive correlations indicate alignment with the current
pattern shown in Fig. 1.
As in Fig. 2, but with ?p? 0. Except for the vertical
PRL 99, 027005 (2007)
13 JULY 2007
We conclude that while we cannot rule out that orbital
current patterns exist, we can rule out that they are respon-
sible for the superconductivity, the properties of the pseu-
dogap phase, or the anomalous normal state properties
extending up to temperatures of several hundred Kelvin,
asthe energyassociated with the spontaneousloop currents
is less than 5 meV per link if such currents exist. Note that
this energy is not the condensation energy Ecper unit cell,
but a positive contribution to the energy of the current
carrying state, which would have to be (more than) offset
by other contributions (like the energy gain from aligning
the circulating currents according to the pattern Varma
proposed) if such a state were realized. We would expect
the transition temperature Tcof such a state to be of the
order of the effective coupling of the Ising spins introduced
byAjiandVarma andhence smaller than 5meV,while
we would expect Ec? Tc.
In this work, we have assumed that the CuO planes are
adequately described by the three-band Hubbard model
(1), but we have allowed ?pto be much smaller than
generally agreed upon, and based our estimates on the
extreme and to our purposes most unfavorable value ?p?
0. (Note that the ordered antiferromagnetic phase in un-
doped cuprates requires a finite ?p.) Numerical data not
presented here show that our conclusions remain intact if
we set Jpd? Jpp? 0 or/and double the value of the repul-
sion Vpd, which generates the orbital currents in Varma’s
mean-field calculation. They also hold for other lowenergy
states for the finite system (e.g., as situated at the M point
in the Brillouin zone for ?p? 0).
Nonetheless, we should keep in mind that any analysis
of a model can only reach a conclusionvalid forthis model.
The question of whether current patterns exist in CuO
superconductors can ultimately only be settled by ex-
periment. We consider it likely, however, that an even-
tual consensus among experiments will confirm our
Wewish like to thank C.M. Varma, V. Aji, F. Evers, and,
in particular, P. Wo ¨lfle for many illuminating discussions
of this subject. R.T. was supported by the Studienstiftung
des deutschen Volkes. We further acknowledge the support
of the computing facilities of the INT at the Forschungs-
 J. Zaanen et al., Nature Phys. 2, 138 (2006).
 P.W. Anderson, Science 235, 1196 (1987).
 P.A. Lee, N. Nagaosa, and X.G. Wen, Rev. Mod. Phys.
78, 17 (2006).
 S. Sachdev, Rev. Mod. Phys. 75, 913 (2003).
 R.B. Laughlin, Science 242, 525 (1988).
 P.W. Anderson, Science 268, 1154 (1995).
 E. Demler, W. Hanke, and S.-C. Zhang, Rev. Mod. Phys.
76, 909 (2004).
 C.M. Varma, Phys. Rev. Lett. 83, 3538 (1999).
 C.M. Varma, Phys. Rev. B 73, 155113 (2006).
 C.M.Varma,P.B. Littlewood,
E. Abrahams, and A.E. Ruckenstein, Phys. Rev. Lett.
63, 1996 (1989).
 J.L. Tallon and J.W. Loram, Physica (Amsterdam) 349C,
 L. Alff, Y. Krockenberger, B. Welter, M. Schonecke,
R. Gross, D. Manske, and M. Naito, Nature (London)
422, 698 (2003).
 D.v.d. Marel, H.J.A. Molegraaf, J. Zaanen, Z. Nussinov,
F. Carbone, A. Damascelli, H. Eisaki, M. Greven, P.H.
Kes, and M. Li, Nature (London) 425, 271 (2003).
 Y. Dagan, M.M. Qazilbash, C.P. Hill, V.N. Kulkarni, and
R.L. Greene, Phys. Rev. Lett. 92, 167001 (2004).
 S.H. Naqib, J.R. Cooper, J.L. Tallon, R.S. Islam, and
R.A. Chakalov, Phys. Rev. B 71, 054502 (2005).
 M. Greiter, Ann. Phys. (N.Y.) 319, 217 (2005).
 S.A. Kivelson, I.P. Bindloss, E. Fradkin, V. Oganesyan,
J.M. Tranquada, A. Kapitulnik, and C. Howald, Rev. Mod.
Phys. 75, 1201 (2003).
 S. Chakravarty, R.B. Laughlin, D.K. Morr, and C. Nayak,
Phys. Rev. B 63, 094503 (2001).
 J.X. Li, C.Q. Wu, and D.-H. Lee, Phys. Rev. B 74, 184515
 V.J. Emery, Phys. Rev. Lett. 58, 2794 (1987).
 F.C. Zhang and T.M. Rice, Phys. Rev. B 37, 3759
 H. Eskes and G.A. Sawatzky, Phys. Rev. Lett. 61, 1415
 M.S. Hybertsen, M. Schlu ¨ter, and N.E. Christensen, Phys.
Rev. B 39, 9028 (1989).
 M.S. Hybertsen, E.B. Stechel, M. Schlu ¨ter, and D.R.
Jennison, Phys. Rev. B 41, 11068 (1990).
 T.M. Rice, F. Mila, and F.C. Zhang, Phil. Trans. R. Soc. A
334, 459 (1991).
 R.E. Walstedt and W.W. Warren, Jr., Science 248, 1082
 L.H. Tjeng et al., Phys. Rev. Lett. 78, 1126 (1997).
 A. Kaminksi et al., Nature (London) 416, 610 (2002).
 S.V. Borisenko, A.A. Kordyuk, A. Koitzsch, T.K. Kim,
K.A. Nenkov, M. Knupfer, J. Fink, C. Grazioli,
S. Turchini, and H. Berger, Phys. Rev. Lett. 92, 207001
 B. Fauque ´, Y. Sidis, V. Hinkov, S. Pailhes, C.T. Lin,
X. Chaud, and P. Bourges, Phys. Rev. Lett. 96, 197001
 V. Aji and C.M. Varma, arXiv:cond-mat/0610646.
PRL 99, 027005 (2007)
13 JULY 2007