We provide, for the first time, in a doped strongly correlated system
(two-leg ladder), a controlled theoretical demonstration of the existence of a
state in which long-range ordered orbital currents are arranged in a staggered
pattern,coexisting with a charge density wave. The method used is the highly
accurate density matrix renormalization group technique.This brings us closer
to recent proposals that this order is realized in the enigmatic pseudogap
phase of the cuprate high temperature superconductors.Comment: The version accepted in Phys. Rev. Lett. 5 pages, 6 eps figures,
  RevTex

A. A. Nersesyan

B. I. Halperin

E. Dagotto

H. J. Schulz

J. B. Marston

J. O. Fjærestad

M. Troyer

P. W. Anderson

Sudip Chakravarty

U. Schollwöck

English

arXiv

Crossref

Broken Time-Reversal Symmetry in Strongly Correlated Ladder Structures

Schollwöck, U.

Chakravarty, S.

Fjærestad, J. O.

Marston, J. B.

Troyer, M.

Max Planck Society eDoc Server

Broken time-reversal symmetry in strongly correlated ladder structures

We provide, for the first time, in a doped strongly correlated system (two-leg ladder), a controlled theoretical demonstration of the existence of a state in which long-range ordered orbital currents are arranged in a staggered pattern, coexisting with a charge density wave. The method used is the highly accurate density-matrix renormalization group technique. This brings us closer to recent proposals that this order is realized in the enigmatic pseudogap phase of the cuprate high temperature superconductors

Schollwock, U.

Fjaerestad, J. O.

University of Queensland eSpace

P H Y S I C A L R E V I E W L E T T E R S week ending9 MAY 2003VOLUME 90, NUMBER 18Broken Time-Reversal Symmetry in Strongly Correlated Ladder Structures
U. Schollwo¨ck,1 Sudip Chakravarty,2 J. O. Fjærestad,3 J. B. Marston,3 and M. Troyer4
1Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
2Department of Physics and Astronomy, University of California Los Angeles, California 90095, USA
3Department of Physics, Brown University, Providence, Rhode Island 02912, USA
4Theoretische Physik, Eidgeno¨ssische Technische Hochschule, CH-8093 Zu¨rich, Switzerland
(Received 19 September 2002; published 5 May 2003)186401-1We provide, for the first time, in a doped strongly correlated system (two-leg ladder), a controlled
theoretical demonstration of the existence of a state in which long-range ordered orbital currents are
arranged in a staggered pattern, coexisting with a charge density wave. The method used is the highly
accurate density-matrix renormalization group technique. This brings us closer to recent proposals that
this order is realized in the enigmatic pseudogap phase of the cuprate high temperature superconductors.
DOI: 10.1103/PhysRevLett.90.186401 PACS numbers: 71.10.Fd, 71.10.Hf, 71.30.+h, 74.20.MnFIG. 1. Circulating plaquette currents on a ladder, character-
versial [11–14], and more precise and well-characterized istic of the DDW phase.The circulating current phases in correlated electron
systems, also called orbital antiferromagnets (OAF),
were first considered in the context of excitonic insulators
[1], but then discarded in favor of more conventional
order. After the discovery of the cuprate high temperature
superconductors, they were rediscovered [2,3] and called
staggered flux (SF) phases [2]. Many of their properties
were discussed, but were forgotten again in the absence of
experimental vindication. The discovery of an unusual
and robust regime called the pseudogap [4] in these super-
conductors has changed the picture once more. The
pseudogap mimics the momentum dependence of the
superconducting gap, but the state itself is not supercon-
ducting. In this context, two recent developments have
taken place: (i) Attempts have been made to explain this
regime in terms of fluctuations of SF order [5] and (ii) a
proposal has been made that it is not fluctuations, but a
true broken symmetry that is the origin of the pseudogap
[6,7]. This ordered state was called the singlet d-density
wave (DDW) following Ref. [8] and is the same as the
OAF and SF phases. In this Letter, we adopt the density
wave (DW) terminology, as it can describe large classes
of order parameters with orbital angular momentum. The
label d stands for angular momentum 2, as in atomic
physics. The conventional charge density wave (CDW)
in which charge is modulated in space is its s-wave
counterpart with angular momentum zero. The triplet
s-wave density wave is what is commonly called a spin
density wave. Another breakdown of time reversal sym-
metry in which the circulating currents obey translational
symmetry, as opposed to DDW, has also been pointed out
and has been argued to be responsible for the pseudogap
phase [9,10].
Although much indirect experimental evidence of
DDW can be argued to exist, a direct observation of
DDW would be Bragg reflection of neutrons carrying
magnetic moments from the staggered arrangement, on
the scale of a few A, of circulating currents. Recent neu-
tron scattering experiments have, however, been contro-0031-9007=03=90(18)=186401(4)$20.00 experiments are underway to establish this order. Thus,
theoretical exploration of microscopic models of corre-
lated electronic systems with controlled methods has
acquired urgency. We, therefore, study the simplest geo-
metrical structure in the form of a two-leg ladder [15] that
can support staggered orbital currents, as shown in Fig. 1.
Previous studies of DDW order in two-leg ladders have
used weak-coupling bosonization/renormalization group
(RG) analyses [16–23], density-matrix renormalization
group (DMRG) [24] analysis of the t-J model [23] and a
half-filled Hubbard-like model [25], or exact diagonal-
ization [26] of the t-t0-J model. At half filling, models
with long-range ordered currents have been found both
for spinless [16] and spinful [20–22,25] fermions. In
contrast, for doped ladders, power-law decay has been
found for spinless [17,19] and spinful [18–20] cases. For
the t-J ladder, only short-range order was found [23], and
the study of the t-t0-J model [26] yielded similar results.
The approach used in the present work is the accurate
DMRG method that can be used for arbitrary interaction
strength, unlike the weak-coupling bosonization/RG ap-
proaches. The results of our calculations are striking.
Although common t-J–type models do not exhibit
long-ranged DDW order, a separate class of repulsive
Hamiltonians show robust long-range DDW order even
in the presence of substantial doping. These have their
historical origin in a half-filled SO(5) invariant model on
a ladder [27]. At precise half filling, it was shown to
exhibit DDW in its phase diagram [21,25]. We shall
show that SO(5) invariance is irrelevant by considering
coupling constants very far from the ‘‘SO(5) parameters’’
and by substantially doping this model. The real reason
for success with this class of models is that it straddles2003 The American Physical Society 186401-1
FIG. 2. The correlation function Cr of the t-J-V-V0 model.
The parameters are J=t  0:4, V=t  3, V0=t  1, and   0:1.
The size of the ladder is 120 2. We kept 800 states and
performed 42 sweeps. The vertical scale is chosen in units
where t  1.
P H Y S I C A L R E V I E W L E T T E R S week ending9 MAY 2003VOLUME 90, NUMBER 18CDW and DSC-like (dx2y2-superconductor) states (more
precisely rung-singlet states), resulting in a local kinetic
exchange between them. This is much like the actual
situation in the cuprates in which the DDW phase is an
intermediate regime between a multiplicity of complex
charge ordered states and DSC [12,21,22,25].
We label the site of a ladder by i  r; l, where r 
1; . . . ; L is the rung index and l  1; 2 is the leg index. The
current operator between any two sites i and j, J i;j, is
J i;j  it
X

cyi;cj;  cyjci;; (1)
where cyi; is the creation operator of a fermion with spin
 at site i. We set the lattice spacing to unity.
There are at least two convenient ways of probing
DDW order. One approach is to measure the equal-time
rung-rung current correlation function in the ground
state,
Cr; R  hjrungR	 r=2jrungR r=2i; (2)
where jrungr  J r;1;r;2. In order to minimize the effect
of the boundaries of a finite ladder, we should choose R to
be the location of the central rung, and we shall denote
this correlation function as Cr
An alternative approach is to break the time reversal
symmetry explicitly by applying a source hjrung1 on
one end of the ladder and measuring the current induced
in the sample. The source term for DDW is necessarily
complex and a complex DMRG program is needed, which
is more demanding on memory and computer time.
Nonetheless, we have used both methods for cross
checks for every single case discussed in the present
Letter. For both methods, we use a finite size algorithm,
which is more reliable, performing sweeps to reach con-
vergence [24] .
To orient ourselves, we shall begin with the two-leg t-J
ladder, which is defined by the Hamiltonian:
HtJ  t
X
hi;ji;
Pcyi;cj;P 	 H:c:
	 J
X
hi;ji

Si  Sj 
ninj
4

; (3)
where ni is the total fermion occupation number at the
site i, and Si is the spin operator at the corresponding site.
P projects out doubly occupied sites and hi; ji denotes
pairs of nearest neighbor sites. The t-J model is the
simplest model that captures some aspects of the high
temperature superconductors. Removal of electrons from
the system is quantified by the doping parameter  
1 hnii. In actual experiments, DSC is observed in the
range   0:05–0:25; at small values of  lies the pseu-
dogap regime. In this model, the DDW correlations decay
exponentially with a correlation length   3–4 [23].
Additional next-nearest neighbor hopping, augmenting
the model, may be supposed to suppress the competing186401-2CDW and DSC phases and thus reveal DDW order, but
our calculations and those of Ref. [26] do not support
this idea.
A more interesting model is the t-J-V-V0 model, which
is a t-J model augmented by Coulomb repulsion terms V
and V0 between nearest and next-nearest neighbors:HtJVV0  HtJ 	 V
X
hi;ji
ninj 	 V 0
X
hhi;jii
ninj; (4)where hhi; jii denotes next-nearest neighbor pairs of sites.
In order to reduce the effect of open boundaries, a chemi-
cal potential term V 	 V 0ni is added to the boundary
sites. A typical parameter set, given by J=t  0:4, V=t 
3, V0=t  1, and   0:1, yields the current correlation
function Cr shown in Fig. 2.We observe what appears to
be a bubble of DDW extending up to 20 rungs, and then a
sinusoidally modulated exponential decay with   3.
Thus, although we observe DDW over a moderately long
range, there is no macroscopic order. We have probed this
model by the second method in which we induce a current
by a source at the edge. As shown in Fig. 3, the long-range
correlations that are observed at the first infinite-size step
disappear after a few sweeps and converge after six
sweeps to an exponential decay with a correlation length
  10. The moderately large bubblelike nature of Cr
and the different values of  in these two methods are
strong reasons for suspecting proximity to a first order
transition to the ordered DDW phase, as the boundary
effects seem to nucleate this phase.
Finally, we shall consider a different class of
Hamiltonians, in which a pair of electrons across a rung
is given an internal structure, much like a molecule. This
is an interesting way of generating a set of low energy
Hamiltonians [27] that are defined by186401-2
-0.0003
-0.0002
-0.0001
0
0.0001
0.0002
0.0003
0 20 100806040
-0.4
-0.2
0
0.2
0.4
0 10080604020
(a) (b)
10-6
10-5
0.0001
0.001
0.01
0.1
0 10 20 30 40 50 60
sweep 1
sweep 2
sweep 6
sweep 7
| <
 j pl
aq
(r)
 > 
|
r
FIG. 3. Absolute value of the current summed around a
plaquette, jhjplaqrij, as induced by an edge current, h 
0:01t, as a function of the distance r from the edge of a
60 2 ladder in the t-J-V-V0 model, with the parameters
described in the text. Note the convergence as a function of
sweeps. The number of states retained was 400 and the results
were found to scale with current at this level.
P H Y S I C A L R E V I E W L E T T E R S week ending9 MAY 2003VOLUME 90, NUMBER 18HtJ?UV?  t
X
hi;ji
cyi;cj; 	 H:c: 	
U
2
X
i
ni  12
	
X
r
J?Sr;1  Sr;2 	 V?nr;1  1nr;2  1:
(5)
Note that there are no longer any projection operators P ,
as in the previous examples, and the problem can be
treated for arbitrary interaction strength. In the half-filled
case, this Hamiltonian has a precise SO(5) symmetry
[27] when J?  4U	 V?. It also exhibits DDW order
[21,25]. As soon as the system is doped, or the parameters
are no longer finely tuned, there is no SO(5) symmetry.
The weak-coupling phase diagram at half filling obtained
from bosonization/RG, as shown in Fig. 4, gives us some
guidance as to where to look in our DMRG calculations.0
1
2
3
4
-1 -0.5 0 0.5 1
J ⊥
/V
⊥
U/V⊥
SO(5): J⊥ = 4(U + V⊥)
U + 3/4 J⊥ = V⊥
CDW
Rung singlet
DDW
Site
singlet
FIG. 4. The weak coupling phase diagram at half filling for
HtJ?UV? from a bosonization calculation. The open circles
correspond to the parameters in Fig. 5 and the open triangles
to those in Fig. 6.
186401-3This phase diagram is essentially identical to that of
Ref. [22], except that we also show the regime for U <
0. Other than the DDW and CDW, there are two relevant
states that can be adiabatically continued to resonating
valence bond states [28] of the short-range variety [29]—
rung singlet (jrsi) and site singlet states (jssi) [30],
jrsi /
Y
r

"
#



#
"

r
; jssi /
Y
r

"#


	


"#

r
:
The DDW lies between the CDW and the rung-singlet
phases.
We have studied the Hamiltonian in Eq. (4) for a range
of parameters and find long-range DDW order in the
doped model, which has nothing to do with SO(5) sym-
metry. Nonetheless, the DDW phase is situated between
the CDW and the rung-singlet phases, similar to the
weak-coupling bosonization results for half-filled lad-
ders. As a typical example, we have shown in Fig. 5 our
results for the rung current as induced by an edge current
of tiny magnitude 0:0001t. The parameters chosen were
U  0:25, t  V?  1, J?  0:8, 1.1, 1.5, 1.7, and  
0:04. As a response, we see robust long-ranged DDW
order in the middle of this range of J? with stripelike
features where pairs of holes reside; see, in particular,
Fig. 5(c), where we also plot the hole density, and the
coexistence with stripe order is especially evident from
the antiphase domain wall structure [31]. The induced
currents clearly alternate in sign and can be of the order-0.6
-0.3
0
0.3
0.6
0
0.02
0.04
0.06
0.08
0 20 40 60 80 100
-0.0004
-0.0002
0
0.0002
0.0004
0 20 40 60 80 100
(c) (d)
FIG. 5 (color online). Rung current jrungr as a function of
the location of the rung r in a t-J?-U-V? model at 4% doping
on a 100 2 ladder, with parameters U  0:25, t  V?  1,
and an edge current of 0.0001. The sequence of figures corre-
sponds to (a) J?  0:8, (b) J?  1:1, (c) J?  1:5, and (d)
J?  1:7. In (c), we show the profile of the hole density
depicted as solid dots corresponding to the scale on the right.
We kept up to 400 states and performed up to eight sweeps.
Note the vast differences in the scales of the current strengths.
186401-3
-0.2
0
0.2
-0.4
0.4
0 10 20 30 40 50
-0.5
-0.25
0
0.25
0.5
0 10 20 30 40 50
(a) (b)
FIG. 6 (color online). Rung current jrungr as a function of
the location of the rung r in a t-J?-U-V? model on a 50 2
ladder, with parameters t  V?  1, J?  1:2, and an edge
current of 0.0001. The sequence of figures corresponds to (a)
U  0:25 at 8% doping, and (b) U  0:5 at 4% doping. We kept
up to 400 states and performed up to eight sweeps.
P H Y S I C A L R E V I E W L E T T E R S week ending9 MAY 2003VOLUME 90, NUMBER 18of unity, in units of t, even though the source current is
infinitesimally small. For ladders of lengths 100, 150, and
200, and for parameters of Fig. 5(c), the current ampli-
tudes are, respectively, 0.56, 0.53, and 0.53, consistent
with long-range order, though in a numerical calculation
it is never possible to rule out a very slow decay. We have
studied the d-wave pairing correlations, and find only
extremely rapid decay in ladders that exhibit DDW
long-range order. This is as expected from previous nu-
merical work on the half-filled system [25], where it was
shown that the superconducting correlations decay expo-
nentially in the DDW phase. It is also in accord with the
phase diagram in Fig. 4, which shows that the region with
strong d-wave superconducting correlations, the ‘‘rung-
singlet’’ phase, is distinct from the DDW region. More
generally, bosonization leads to the prediction that the
phase with strong DDW correlations has only short-
ranged pairing correlations. For the case of Fig. 5(c),
we have also studied the low-lying excitations about the
ground state. We find a robust spin gap, again in accord
with general expectations. In Fig. 6, we show the results
for   0:08 and for U  0:5. For sufficiently strong
doping, roughly between 10% to 20%, DDW is suppressed
for these sets of couplings.
In summary, we have shown that there are repulsive
microscopic models that exhibit DDW order at finite dop-
ing, providing added support for the identification of the
pseudogap in the cuprates with this state. Our work also
raises the real possibility that an appreciation of the
complexity of many novel materials [32] may be impos-
sible without these remarkable broken symmetries, and it
is important to search for such complex quantum order in
an even wider class of Hamiltonians.
We thank Chetan Nayak and Asle Sudbø for stimulat-
ing discussions.We acknowledge support from a Gerhard-
Hess prize of the Deutsche Forschungsgemeinschaft
(U.S.), the U.S. National Science Foundation, Grants
No. NSF-DMR-9971138 (S. C.) and No. NSF-DMR-
9712391 (J. B. M.), the Research Council of Norway,
Grant No. 142915/432 (J. O. F.), and the Swiss National
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Broken time-reversal symmetry in strongly correlated ladder structures

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