Dynamical layer decoupling in a stripe-ordered, high T
superconductor
E. Berg,1 E. Fradkin,2 E.-A. Kim,1 S. A. Kivelson,1 V. Oganesyan,3 J. M. Tranquada,4 and S. C. Zhang1
Department of Physics, Stanford University, Stanford, California 94305-4060
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080
Department of Physics, Yale University, New Haven, Connecticut 06520-8120
Brookhaven National Laboratory, Upton, New York 11973-5000
(Dated: October 11, 2018)
In the stripe-ordered state of a strongly-correlated two-dimensional electronic system, under a
set of special circumstances, the superconducting condensate, like the magnetic order, can occur at
a non-zero wave-vector corresponding to a spatial period double that of the charge order. In this
case, the Josephson coupling between near neighbor planes, especially in a crystal with the special
structure of La2−xBaxCuO4, vanishes identically. We propose that this is the underlying cause of
the dynamical decoupling of the layers recently observed in transport measurements at x = 1/8.
High-temperature superconductivity (HTSC) was first
discovered [1] in La2−xBaxCuO4. A sharp anomaly [2] in
Tc(x) occurs at x = 1/8 which is now known to be indica-
tive [3, 4] of the existence of stripe order and of its strong
interplay with HTSC. Recently, a remarkable dynamical
layer decoupling has been observed [5] associated with
the superconducting (SC) fluctuations below the spin-
stripe ordering transition temperature, Tspin = 42K.
While Tc(x), as determined by the onset of a bulk
Meissner effect, reaches values up to Tc(x = 0.1) = 33 K
for x somewhat smaller and larger than x = 1/8, Tc(x)
drops to the range 2–4 K for x = 1/8. However, in other
respects, superconductivity appears to be optimized for
x = 1/8. The d-wave gap determined by ARPES has
recently been shown [6] to be largest for x = 1/8. More-
over, strong SC fluctuations produce an order of magni-
tude drop [5] in the in-plane resistivity, ρab, at T ≈ Tspin,
which is considerably higher than the highest bulk SC.
The fluctuation conductivity reveals heretofore un-
precedented characteristics (as described schematically
in Fig. 1): 1) ρab drops rapidly with decreasing tem-
perature from Tspin down to TKT ≈ 16K, at which
point it becomes unmeasurably small. In the range
Tspin > T > TKT , the temperature dependence of ρab
is qualitatively of the Kosterlitz-Thouless form, as if the
SC fluctuations were strictly confined to a single copper-
oxide plane. 2) By contrast, the resistivity perpendicular
to the copper-oxide planes, ρc, increases with decreasing
temperatures from T ⋆ >∼ 300 K, down to T
⋆⋆ ≈ 35 K.
For T < T ⋆⋆, ρc decreases with decreasing temperature,
but it only becomes vanishingly small below T3D ≈ 10 K.
Within experimental error, for TKT > T > T3D, the re-
sistivity ratio, ρc/ρab, is infinite! 3) The full set of usual
characteristics of the SC state, the Meissner effect and
perfect conductivity, ρab = ρc = 0, is only observed be-
low Tc = 4K. Thus, for T3D > T > Tc, a peculiar form
of fragile 3D superconductivity exists.
The above listed results are new, so an extrinsic
explanation of some aspects of the data is possible.
Here we assume that the measured properties do re-
flect the bulk behavior of La2−xBaxCuO4. We show
that there is a straightforward way in which stripe or-
35 K=
Meissner  State
~ 0 ~ 0
ab ~ 0 ρc ~
10 m Ω cm
~ 10 x  10 m Ω cm
}max { ~ 8 x 10 m Ω cm
~ 8 x 10
5 x 10
2 x 10 3
FIG. 1: Summary of the thermal phase transitions and trans-
port regimes in x = 1/8 doped La2−xBaxCuO4.
der can lead to an enormous dynamical suppression
of interplane Josephson coupling, particularly in the
charge ordered low-temperature tetragonal (LTT) phase
of La15/8Ba1/8CuO4, i.e. T ≤ Tco = 54 K.
The LTT structure has two planes per unit cell. In
alternating planes, the charge stripes run along the x or
y axes, as shown in Fig. 3. Moreover, the parallel stripes
in second neighbor planes are thought to be shifted over
by half a period (so as to minimize the Coulomb interac-
tions [7]) resulting in a further doubling of the number of
planes per unit cell, as seen in X-ray scattering studies.
Below Tspin, the spins lying between each charge stripe
have antiferromagnetic (AFM) order along the stripe di-
rection, which suffers a π phase shift across each charge
http://arxiv.org/abs/0704.1240v2
stripe, resulting in a doubling of the unit cell within the
plane, see Fig. 2c. Hence, the Bragg scattering from the
charge order in a given plane occurs at (2π/a)〈±1/4, 0〉
while the spin-ordering occurs at (2π/a)〈1/2± 1/8, 1/2〉.
SC order should occur most strongly within the charge
stripes. Since it is strongly associated with zero center-of-
mass momentum pairing, one usually expects, and typi-
cally finds in models, that the SC order on neighboring
stripes has the same phase. However, as we will discuss,
under special circumstances, the SC order, like the AFM
order, may suffer a π phase shift between neighboring
stripes if the effective Josephson coupling between stripes
is negative. Within a plane, so long as the stripe order
is defect free, the fact that the SC order occurs with
k = (2π/a)〈±1/8, 0〉 has only limited observable conse-
quences. However anti-phase SC order within a plane
results in an exact cancellation of the effective Josephson
coupling between first, second and third neighbor planes.
This observation can explain an enormous reduction of
the interplane SC correlations in a stripe-ordered phase.
Before proceeding, we remark that there is a preexist-
ing observation, concerning the spin order, which sup-
ports the idea that interplane decoupling is a bulk fea-
ture of a stripe-ordered phase. Specifically, although the
in-plane spin correlation length measured in neutron-
scattering studies in particularly well prepared crystals
of La2−xBaxCuO4 is ξspin ≥ 40a [8], there are essentially
no detectable magnetic correlations between neighboring
planes. In typical circumstances, 3D ordering would be
expected to onset when (ξspin/a)
2J1 ∼ T , where J1 is
the strength of the interplane exchange coupling. How-
ever, the same geometric frustration of the interplane
couplings that we have discussed in the context of the
SC order pertains to the magnetic case, as well. Thus,
we propose that the same dynamical decoupling of the
planes is the origin of both the extreme 2D character of
the AFM and SC ordering.
We begin with a caricature of a stripe ordered state,
consisting of alternating Hubbard or t–J ladders which
are weakly coupled to each other (Fig. 2). Such a car-
icature, which has been adopted in previous studies of
superconductivity in stripe ordered systems [9, 10, 11],
certainly overstates the extent to which stripe order pro-
duces quasi-1D electronic structure. However, we can
learn something about the possible electronic phases and
their microscopic origins, in the sense of adiabatic conti-
nuity, by analyzing the problem in this extreme limit. As
shown in the figure, distinct patterns of period 4 stripes
can be classified by their pattern of point group symme-
try breaking as being “bond centered” or “site centered.”
Numerical studies of t–J ladders [12] suggest that the dif-
ference in energy between bond and site centered stripes
is small, so the balance could easily be tipped one way or
another by material specific details, such as the specifics
of the electron-lattice coupling.
The simplest caricature of bond centered stripes is an
array of weakly coupled two-leg ladders with alternately
larger and smaller doping, as illustrated in Fig. 2a. This
a) Bond centered b) Site centered
c) Magnetic striped superconductor
Figure 2
(a) Bond centered
a) Bond centered b) Site centered
c) Magnetic striped superconductor
Figure 2
(b) Site centered
a) Bond centered b) Site centered
c) Magnetic striped superconductor
Figure 2
(c) Magnetic striped
FIG. 2: a) Pattern of a period 4 bond centered and b) site
centered stripe, with nearly undoped (solid lines) and more
heavily doped (hatched lines) regions. c) Sketch of the pair-
field (lines) and spin (arrows) order in a period 4 site centered
stripe in which both the SC and AFM order have period 8 due
to an assumed π phase shift across the intervening regions.
Solid (checked) lines denote a positive (negative) pair-field.
problem was studied in Ref. 10. Because a strongly inter-
acting electron fluid on a two-leg ladder readily develops
a spin-gap,[13] i.e. forms a LE liquid, this structure can
exhibit strong SC tendencies to high temperatures. Weak
electron hopping between neighboring ladders produces
Josephson coupling which can lead to a “d-wave like” SC
state.[14] However, the spin-gap precludes any form of
magnetic ordering, even when the ladders are weakly cou-
pled, and there is nothing about the SC order that would
prevent phase locking between neighboring planes in a 3D
material. For both these reasons, this is not an attractive
model for the stripe ordered state in La15/8Ba1/8CuO4.
(There is, however, evidence from STM studies on the
surface of BSCCO [15] of self-organized structures sug-
gestive of two-leg ladders.)
By contrast, a site-centered stripe is naturally related
to an alternating array of weakly coupled three and one
leg ladders, as shown in Fig. 2(b). Because the zero-point
kinetic energy of the doped holes is generally large com-
pared to the exchange energy, it is the three-leg ladder
that we take to be the more heavily doped. The three leg
ladder is known [9, 16] to develop a spin-gapped LE liquid
above a rather small [16] critical doping, xc (which de-
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FIG. 3: Stacking of stripe planes.
pends on the interactions). An undoped or lightly doped
one-leg ladder, by contrast, is better thought of as an
incipient spin density wave (SDW), and has no spin-gap.
Where the one-leg ladder is lightly doped it forms a Lut-
tinger liquid with a divergent SDW susceptibility at 2kF .
The phases of a system of alternating, weakly coupled
LE and Luttinger liquids were analyzed in [11]. How-
ever, the magnetic order in La15/8Ba1/8CuO4 produces a
Bragg peak at wave-vector (2π
〉 in a coordinate
system in which y is along the stripe direction. There-
fore, it is necessary to consider the case in which, in the
absence of inter-ladder coupling, the one-leg ladder is ini-
tially undoped, and the three leg ladder has x = 1
> xc.
Our model of the electronic structure of a single charge-
stripe-ordered Cu-O plane is thus an alternating array of
LE liquids, with a spin-gap but no charge gap, and spin-
chains, with a charge gap but no spin gap. None of the
obvious couplings between nearest-neighbor subsystems
is relevant in the renormalization group sense, because of
the distinct character of their ordering tendencies. How-
ever, certain induced second neighbor couplings, between
identical systems, are strongly relevant, and, at T = 0,
lead to a broken symmetry ground-state.
The induced exchange coupling between nearest-
neighbor spin-chains leads to a 2D magnetically ordered
state. The issue of the sign of this coupling has been
addressed previously [17, 18, 19] and found to be non-
universal, as it depends on the doping level in the inter-
vening three-leg ladder. For x = 0, the preferred AFM
order is in-phase on neighboring spin-chains, consistent
with a magnetic ordering vector of (2π/a)〈1/2, 1/2〉. For
large enough x (probably, x > xc), the ordering on
neighboring spin-chains is π phase shifted, resulting in
a doubling of the unit-cell size in the direction perpen-
dicular to the stripes, and a magnetic ordering vector
(2π/a)〈1/2± 1/8, 1/2〉. This ordering tendency has also
been found in studies of wide t–J ladders [12].
A question that has not been addressed systematically
until now is the sign of the effective Josephson coupling
between neighboring LE liquids. In the case of 2-leg lad-
ders, it was found [10, 12] that the effective Josephson
coupling is positive, favoring a SC state with a spatially
uniform phase. It is possible, in highly correlated sys-
tems, especially when tunneling through a magnetic im-
purity [20], to encounter situations in which the effective
Josephson coupling is negative, therefore producing a π-
junction. Zhang [21] has observed that, regardless the
microscopic origin of the anti-phase character of the mag-
netic ordering in the striped state, if there is an approx-
imate SO(5) symmetry relating the antiferromagnetism
to the superconductivity, one should expect an anti-phase
ordering of the superconductivity in a striped state. The
example of tunneling through decoupled magnetic impu-
rities [20] is a proof in principle that such behavior can
occur. However, interplane decoupling associated with
the onset of superconductivity is not seen in experiments
in other cuprates, and states with periodic π phase shifts
of the SC order parameter have not yet surfaced in nu-
merical studies of microscopic models [12]; this suggests
anti-phase striped SC order is rare.
The new proposal in the present paper is that, for
the reasons outlined above, the SC striped phase of
La15/8Ba1/8CuO4 has anti-phase SC and anti-phase
AFM order, whose consequences we now outline. We
can express the most important possible interplane
Josephson-like coupling terms compactly as
Hinter =
∆⋆j∆j+m
+ h.c.
where ∆j is the j-th plane SC order parameter. The
term proportional to the usual (lowest order) Josephson
coupling, J1,1, and indeed, J1,2 and J1,3 all vanish by sym-
metry. The most strongly relevant residual interaction is
the Josephson coupling between fourth-neighbor planes,
J1,4. Double-pair tunnelling between nearest-neighbor
planes, J2,1, is more weakly relevant, but it probably has
a larger bare value since it involves half as many pow-
ers of the single-particle interplane matrix elements than
J1,4. J1,4 and J2,1 have scaling dimensions 1/4 and 1
at the (KT) critical point of decoupled plains, so both
are relevant. Thus, they become important when the in-
plane SC correlation length ξ ∼ ξ1,4 ∼ [Jo/J1,4]
1/4 and
ξ2,1 ∼ [Jo/J2,1], where Jo is the in-plane SC stiffness.
We can make a crude estimate of the magnitude of
the residual interplane couplings by noting that the
same interplane matrix elements (although not neces-
sarily the same energy denominators) determine the in-
terplane exchange couplings between spins and the in-
terplane Josephson couplings. Defining Jm to be the
exchange couplings between spins m planes apart, this
estimate suggests that Jn,m/J0 ∼ [Jm/J0]
n. In undoped
La2CuO4, it has been determined [22] that J1/J0 ≈ 10
which is already remarkably small.
Although in-plane translation invariance forbids direct
Josephson coupling between adjacent planes, there is an
allowed biquadratic inter-plane coupling involvingM and
∆, the SDW and the SC order parameters,
δHinter = J1,s
∆∗j∆j+1Mj ·Mj+1 + h.c.
Even though M 6= 0 for T < Tspin, this term vanishes
because, not only the direction of the stripes, but also
the axis of quantization of the spins (due to spin-orbit
coupling) rotates [23] by 90◦ from plane to plane, i.e.
Mj · Mj+1 = 0. However, a magnetic field, H ∼ 6T ,
induces a 1st order spin-flop transition to a fully collinear
spin state [23] in which Mj ·Mj+1 6= 0.
Thus, for perfect stripe order, the anti-phase SC or-
der would depress, by many orders of magnitude, of the
interplane Josephson couplings, which explains the exis-
tence of a broad range of T in which 2D physics is ap-
parent. Accordingly, there still would be a transition
to a 3D superconductor at a temperature strictly greater
than TKT , when ξ(T ) ∼ ξ1,4 or ξ2,1, whichever is smaller.
The only evidence for the growth of ξ comes indirectly
from the measurement of ρab; by the time ρab is “un-
measurably small,” it has dropped by about 2 orders of
magnitude from its value just below Tspin, which implies
(since ρab ∼ ξ
−2) that ξ has grown by about 1 order
of magnitude. Thus, if some other physics cuts off the
growth of in-plane SC correlations at long scales, we may
be justified in neglecting the effects of Hinter.
Defects in the pattern of charge stripe order have con-
sequences for both magnetic and SC orders. A dislo-
cation introduces frustration into the in-plane ordering,
resulting in the formation of a half-SC vortex bound to it.
For the single-plane problem, this means that the long-
distance physics is that of an XY spin-glass. Since there
is no finite T glass transition in 2D, the growth of ξ will
be arrested at a large scale determined by the density
of dislocations. The same is true of the in-plane AFM
correlations. Both ξ and ξspin should be bounded above
by the charge stripe correlation length, ξch. From X-
ray scattering studies it is estimated that ξch ≈ 70a [24].
This justifies the neglect ofHinter. Conversely, any defect
in the charge-stripe order spoils the symmetry responsi-
ble for the exact cancellation of the Josephson coupling
between neighboring planes. Finite T ordering of an XY
spin-glass is possible in 3D. We tentatively identify the
temperature at which ρc → 0 as a 3D glass transition. A
SC glass would result in the existence of equilibrium cur-
rents (spontaneous time-reversal breaking) and in glassy
long-time relaxations of the magnetization or ρc.
For x 6= 1/8, there is a tendency to develop discom-
mensurations in the stripe order, which, in turn, produce
regions of enhanced (or depressed) SC order with relative
sign depending on the number of intervening stripe peri-
ods. So long as the stripes are dilute, the energy depends
weakly on their precise spacing. Thus, to gain inter-
layer condensation energy, the system can self-organize
so that there are always an even number of intervening
stripes, thus producing an interplane Josephson coupling
J1,1 ∼ |x − 1/8|
2. This, in turn, will lead to a dramatic
increase of the 3D SC Tc. An enhancement of interplane
coherence in any range of T triggered by the magnetic
field induced spin-flop transition would be a dramatic
confirmation of the physics discussed here.
Note added: It was pointed out to us that the state dis-
cussed here was considered by A. Himeda et al.[25] They
found that this is a good variational state for a t− t′ −J
model at x ∼ 1/8 for a narrow range of parameters.
We thank P. Abbamonte, S. Chakravarty, R. Jamei,
A. Kapitulnik, and D. J. Scalapino for discussions.
This work was supported in part by the National Sci-
ence Foundation, under grants DMR 0442537 (EF),
DMR 0531196 (SAK), DMR 0342832 (SCZ), and by
the Office of Science, U.S. Department of Energy un-
der Contracts DE-FG02-91ER45439 (EF), DE-FG02-
06ER46287 (SAK) DE-AC02-98CH10886 (JT) and DE-
AC03-76SF00515 (SCZ), by the Stanford Institute for
Theoretical Physics (EAK), and by a Yale Postdoctoral
Prize Fellowship (VO).
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  In the stripe-ordered state of a strongly-correlated two-dimensional
electronic system, under a set of special circumstances, the superconducting
condensate, like the magnetic order, can occur at a non-zero wave-vector
corresponding to a spatial period double that of the charge order. In this
case, the Josephson coupling between near neighbor planes, especially in a
crystal with the special structure of La_{2-x}Ba_xCuO_4, vanishes identically.
We propose that this is the underlying cause of the dynamical decoupling of the
layers recently observed in transport measurements at x=1/8.

Dynamical layer decoupling in a stripe-ordered, high T
superconductor
E. Berg,1 E. Fradkin,2 E.-A. Kim,1 S. A. Kivelson,1 V. Oganesyan,3 J. M. Tranquada,4 and S. C. Zhang1
Department of Physics, Stanford University, Stanford, California 94305-4060
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080
Department of Physics, Yale University, New Haven, Connecticut 06520-8120
Brookhaven National Laboratory, Upton, New York 11973-5000
(Dated: October 11, 2018)
In the stripe-ordered state of a strongly-correlated two-dimensional electronic system, under a
set of special circumstances, the superconducting condensate, like the magnetic order, can occur at
a non-zero wave-vector corresponding to a spatial period double that of the charge order. In this
case, the Josephson coupling between near neighbor planes, especially in a crystal with the special
structure of La2−xBaxCuO4, vanishes identically. We propose that this is the underlying cause of
the dynamical decoupling of the layers recently observed in transport measurements at x = 1/8.
High-temperature superconductivity (HTSC) was first
discovered [1] in La2−xBaxCuO4. A sharp anomaly [2] in
Tc(x) occurs at x = 1/8 which is now known to be indica-
tive [3, 4] of the existence of stripe order and of its strong
interplay with HTSC. Recently, a remarkable dynamical
layer decoupling has been observed [5] associated with
the superconducting (SC) fluctuations below the spin-
stripe ordering transition temperature, Tspin = 42K.
While Tc(x), as determined by the onset of a bulk
Meissner effect, reaches values up to Tc(x = 0.1) = 33 K
for x somewhat smaller and larger than x = 1/8, Tc(x)
drops to the range 2–4 K for x = 1/8. However, in other
respects, superconductivity appears to be optimized for
x = 1/8. The d-wave gap determined by ARPES has
recently been shown [6] to be largest for x = 1/8. More-
over, strong SC fluctuations produce an order of magni-
tude drop [5] in the in-plane resistivity, ρab, at T ≈ Tspin,
which is considerably higher than the highest bulk SC.
The fluctuation conductivity reveals heretofore un-
precedented characteristics (as described schematically
in Fig. 1): 1) ρab drops rapidly with decreasing tem-
perature from Tspin down to TKT ≈ 16K, at which
point it becomes unmeasurably small. In the range
Tspin > T > TKT , the temperature dependence of ρab
is qualitatively of the Kosterlitz-Thouless form, as if the
SC fluctuations were strictly confined to a single copper-
oxide plane. 2) By contrast, the resistivity perpendicular
to the copper-oxide planes, ρc, increases with decreasing
temperatures from T ⋆ >∼ 300 K, down to T
⋆⋆ ≈ 35 K.
For T < T ⋆⋆, ρc decreases with decreasing temperature,
but it only becomes vanishingly small below T3D ≈ 10 K.
Within experimental error, for TKT > T > T3D, the re-
sistivity ratio, ρc/ρab, is infinite! 3) The full set of usual
characteristics of the SC state, the Meissner effect and
perfect conductivity, ρab = ρc = 0, is only observed be-
low Tc = 4K. Thus, for T3D > T > Tc, a peculiar form
of fragile 3D superconductivity exists.
The above listed results are new, so an extrinsic
explanation of some aspects of the data is possible.
Here we assume that the measured properties do re-
flect the bulk behavior of La2−xBaxCuO4. We show
that there is a straightforward way in which stripe or-
35 K=
Meissner  State
~ 0 ~ 0
ab ~ 0 ρc ~
10 m Ω cm
~ 10 x  10 m Ω cm
}max { ~ 8 x 10 m Ω cm
~ 8 x 10
5 x 10
2 x 10 3
FIG. 1: Summary of the thermal phase transitions and trans-
port regimes in x = 1/8 doped La2−xBaxCuO4.
der can lead to an enormous dynamical suppression
of interplane Josephson coupling, particularly in the
charge ordered low-temperature tetragonal (LTT) phase
of La15/8Ba1/8CuO4, i.e. T ≤ Tco = 54 K.
The LTT structure has two planes per unit cell. In
alternating planes, the charge stripes run along the x or
y axes, as shown in Fig. 3. Moreover, the parallel stripes
in second neighbor planes are thought to be shifted over
by half a period (so as to minimize the Coulomb interac-
tions [7]) resulting in a further doubling of the number of
planes per unit cell, as seen in X-ray scattering studies.
Below Tspin, the spins lying between each charge stripe
have antiferromagnetic (AFM) order along the stripe di-
rection, which suffers a π phase shift across each charge
http://arxiv.org/abs/0704.1240v2
stripe, resulting in a doubling of the unit cell within the
plane, see Fig. 2c. Hence, the Bragg scattering from the
charge order in a given plane occurs at (2π/a)〈±1/4, 0〉
while the spin-ordering occurs at (2π/a)〈1/2± 1/8, 1/2〉.
SC order should occur most strongly within the charge
stripes. Since it is strongly associated with zero center-of-
mass momentum pairing, one usually expects, and typi-
cally finds in models, that the SC order on neighboring
stripes has the same phase. However, as we will discuss,
under special circumstances, the SC order, like the AFM
order, may suffer a π phase shift between neighboring
stripes if the effective Josephson coupling between stripes
is negative. Within a plane, so long as the stripe order
is defect free, the fact that the SC order occurs with
k = (2π/a)〈±1/8, 0〉 has only limited observable conse-
quences. However anti-phase SC order within a plane
results in an exact cancellation of the effective Josephson
coupling between first, second and third neighbor planes.
This observation can explain an enormous reduction of
the interplane SC correlations in a stripe-ordered phase.
Before proceeding, we remark that there is a preexist-
ing observation, concerning the spin order, which sup-
ports the idea that interplane decoupling is a bulk fea-
ture of a stripe-ordered phase. Specifically, although the
in-plane spin correlation length measured in neutron-
scattering studies in particularly well prepared crystals
of La2−xBaxCuO4 is ξspin ≥ 40a [8], there are essentially
no detectable magnetic correlations between neighboring
planes. In typical circumstances, 3D ordering would be
expected to onset when (ξspin/a)
2J1 ∼ T , where J1 is
the strength of the interplane exchange coupling. How-
ever, the same geometric frustration of the interplane
couplings that we have discussed in the context of the
SC order pertains to the magnetic case, as well. Thus,
we propose that the same dynamical decoupling of the
planes is the origin of both the extreme 2D character of
the AFM and SC ordering.
We begin with a caricature of a stripe ordered state,
consisting of alternating Hubbard or t–J ladders which
are weakly coupled to each other (Fig. 2). Such a car-
icature, which has been adopted in previous studies of
superconductivity in stripe ordered systems [9, 10, 11],
certainly overstates the extent to which stripe order pro-
duces quasi-1D electronic structure. However, we can
learn something about the possible electronic phases and
their microscopic origins, in the sense of adiabatic conti-
nuity, by analyzing the problem in this extreme limit. As
shown in the figure, distinct patterns of period 4 stripes
can be classified by their pattern of point group symme-
try breaking as being “bond centered” or “site centered.”
Numerical studies of t–J ladders [12] suggest that the dif-
ference in energy between bond and site centered stripes
is small, so the balance could easily be tipped one way or
another by material specific details, such as the specifics
of the electron-lattice coupling.
The simplest caricature of bond centered stripes is an
array of weakly coupled two-leg ladders with alternately
larger and smaller doping, as illustrated in Fig. 2a. This
a) Bond centered b) Site centered
c) Magnetic striped superconductor
Figure 2
(a) Bond centered
a) Bond centered b) Site centered
c) Magnetic striped superconductor
Figure 2
(b) Site centered
a) Bond centered b) Site centered
c) Magnetic striped superconductor
Figure 2
(c) Magnetic striped
FIG. 2: a) Pattern of a period 4 bond centered and b) site
centered stripe, with nearly undoped (solid lines) and more
heavily doped (hatched lines) regions. c) Sketch of the pair-
field (lines) and spin (arrows) order in a period 4 site centered
stripe in which both the SC and AFM order have period 8 due
to an assumed π phase shift across the intervening regions.
Solid (checked) lines denote a positive (negative) pair-field.
problem was studied in Ref. 10. Because a strongly inter-
acting electron fluid on a two-leg ladder readily develops
a spin-gap,[13] i.e. forms a LE liquid, this structure can
exhibit strong SC tendencies to high temperatures. Weak
electron hopping between neighboring ladders produces
Josephson coupling which can lead to a “d-wave like” SC
state.[14] However, the spin-gap precludes any form of
magnetic ordering, even when the ladders are weakly cou-
pled, and there is nothing about the SC order that would
prevent phase locking between neighboring planes in a 3D
material. For both these reasons, this is not an attractive
model for the stripe ordered state in La15/8Ba1/8CuO4.
(There is, however, evidence from STM studies on the
surface of BSCCO [15] of self-organized structures sug-
gestive of two-leg ladders.)
By contrast, a site-centered stripe is naturally related
to an alternating array of weakly coupled three and one
leg ladders, as shown in Fig. 2(b). Because the zero-point
kinetic energy of the doped holes is generally large com-
pared to the exchange energy, it is the three-leg ladder
that we take to be the more heavily doped. The three leg
ladder is known [9, 16] to develop a spin-gapped LE liquid
above a rather small [16] critical doping, xc (which de-
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FIG. 3: Stacking of stripe planes.
pends on the interactions). An undoped or lightly doped
one-leg ladder, by contrast, is better thought of as an
incipient spin density wave (SDW), and has no spin-gap.
Where the one-leg ladder is lightly doped it forms a Lut-
tinger liquid with a divergent SDW susceptibility at 2kF .
The phases of a system of alternating, weakly coupled
LE and Luttinger liquids were analyzed in [11]. How-
ever, the magnetic order in La15/8Ba1/8CuO4 produces a
Bragg peak at wave-vector (2π
〉 in a coordinate
system in which y is along the stripe direction. There-
fore, it is necessary to consider the case in which, in the
absence of inter-ladder coupling, the one-leg ladder is ini-
tially undoped, and the three leg ladder has x = 1
> xc.
Our model of the electronic structure of a single charge-
stripe-ordered Cu-O plane is thus an alternating array of
LE liquids, with a spin-gap but no charge gap, and spin-
chains, with a charge gap but no spin gap. None of the
obvious couplings between nearest-neighbor subsystems
is relevant in the renormalization group sense, because of
the distinct character of their ordering tendencies. How-
ever, certain induced second neighbor couplings, between
identical systems, are strongly relevant, and, at T = 0,
lead to a broken symmetry ground-state.
The induced exchange coupling between nearest-
neighbor spin-chains leads to a 2D magnetically ordered
state. The issue of the sign of this coupling has been
addressed previously [17, 18, 19] and found to be non-
universal, as it depends on the doping level in the inter-
vening three-leg ladder. For x = 0, the preferred AFM
order is in-phase on neighboring spin-chains, consistent
with a magnetic ordering vector of (2π/a)〈1/2, 1/2〉. For
large enough x (probably, x > xc), the ordering on
neighboring spin-chains is π phase shifted, resulting in
a doubling of the unit-cell size in the direction perpen-
dicular to the stripes, and a magnetic ordering vector
(2π/a)〈1/2± 1/8, 1/2〉. This ordering tendency has also
been found in studies of wide t–J ladders [12].
A question that has not been addressed systematically
until now is the sign of the effective Josephson coupling
between neighboring LE liquids. In the case of 2-leg lad-
ders, it was found [10, 12] that the effective Josephson
coupling is positive, favoring a SC state with a spatially
uniform phase. It is possible, in highly correlated sys-
tems, especially when tunneling through a magnetic im-
purity [20], to encounter situations in which the effective
Josephson coupling is negative, therefore producing a π-
junction. Zhang [21] has observed that, regardless the
microscopic origin of the anti-phase character of the mag-
netic ordering in the striped state, if there is an approx-
imate SO(5) symmetry relating the antiferromagnetism
to the superconductivity, one should expect an anti-phase
ordering of the superconductivity in a striped state. The
example of tunneling through decoupled magnetic impu-
rities [20] is a proof in principle that such behavior can
occur. However, interplane decoupling associated with
the onset of superconductivity is not seen in experiments
in other cuprates, and states with periodic π phase shifts
of the SC order parameter have not yet surfaced in nu-
merical studies of microscopic models [12]; this suggests
anti-phase striped SC order is rare.
The new proposal in the present paper is that, for
the reasons outlined above, the SC striped phase of
La15/8Ba1/8CuO4 has anti-phase SC and anti-phase
AFM order, whose consequences we now outline. We
can express the most important possible interplane
Josephson-like coupling terms compactly as
Hinter =
∆⋆j∆j+m
+ h.c.
where ∆j is the j-th plane SC order parameter. The
term proportional to the usual (lowest order) Josephson
coupling, J1,1, and indeed, J1,2 and J1,3 all vanish by sym-
metry. The most strongly relevant residual interaction is
the Josephson coupling between fourth-neighbor planes,
J1,4. Double-pair tunnelling between nearest-neighbor
planes, J2,1, is more weakly relevant, but it probably has
a larger bare value since it involves half as many pow-
ers of the single-particle interplane matrix elements than
J1,4. J1,4 and J2,1 have scaling dimensions 1/4 and 1
at the (KT) critical point of decoupled plains, so both
are relevant. Thus, they become important when the in-
plane SC correlation length ξ ∼ ξ1,4 ∼ [Jo/J1,4]
1/4 and
ξ2,1 ∼ [Jo/J2,1], where Jo is the in-plane SC stiffness.
We can make a crude estimate of the magnitude of
the residual interplane couplings by noting that the
same interplane matrix elements (although not neces-
sarily the same energy denominators) determine the in-
terplane exchange couplings between spins and the in-
terplane Josephson couplings. Defining Jm to be the
exchange couplings between spins m planes apart, this
estimate suggests that Jn,m/J0 ∼ [Jm/J0]
n. In undoped
La2CuO4, it has been determined [22] that J1/J0 ≈ 10
which is already remarkably small.
Although in-plane translation invariance forbids direct
Josephson coupling between adjacent planes, there is an
allowed biquadratic inter-plane coupling involvingM and
∆, the SDW and the SC order parameters,
δHinter = J1,s
∆∗j∆j+1Mj ·Mj+1 + h.c.
Even though M 6= 0 for T < Tspin, this term vanishes
because, not only the direction of the stripes, but also
the axis of quantization of the spins (due to spin-orbit
coupling) rotates [23] by 90◦ from plane to plane, i.e.
Mj · Mj+1 = 0. However, a magnetic field, H ∼ 6T ,
induces a 1st order spin-flop transition to a fully collinear
spin state [23] in which Mj ·Mj+1 6= 0.
Thus, for perfect stripe order, the anti-phase SC or-
der would depress, by many orders of magnitude, of the
interplane Josephson couplings, which explains the exis-
tence of a broad range of T in which 2D physics is ap-
parent. Accordingly, there still would be a transition
to a 3D superconductor at a temperature strictly greater
than TKT , when ξ(T ) ∼ ξ1,4 or ξ2,1, whichever is smaller.
The only evidence for the growth of ξ comes indirectly
from the measurement of ρab; by the time ρab is “un-
measurably small,” it has dropped by about 2 orders of
magnitude from its value just below Tspin, which implies
(since ρab ∼ ξ
−2) that ξ has grown by about 1 order
of magnitude. Thus, if some other physics cuts off the
growth of in-plane SC correlations at long scales, we may
be justified in neglecting the effects of Hinter.
Defects in the pattern of charge stripe order have con-
sequences for both magnetic and SC orders. A dislo-
cation introduces frustration into the in-plane ordering,
resulting in the formation of a half-SC vortex bound to it.
For the single-plane problem, this means that the long-
distance physics is that of an XY spin-glass. Since there
is no finite T glass transition in 2D, the growth of ξ will
be arrested at a large scale determined by the density
of dislocations. The same is true of the in-plane AFM
correlations. Both ξ and ξspin should be bounded above
by the charge stripe correlation length, ξch. From X-
ray scattering studies it is estimated that ξch ≈ 70a [24].
This justifies the neglect ofHinter. Conversely, any defect
in the charge-stripe order spoils the symmetry responsi-
ble for the exact cancellation of the Josephson coupling
between neighboring planes. Finite T ordering of an XY
spin-glass is possible in 3D. We tentatively identify the
temperature at which ρc → 0 as a 3D glass transition. A
SC glass would result in the existence of equilibrium cur-
rents (spontaneous time-reversal breaking) and in glassy
long-time relaxations of the magnetization or ρc.
For x 6= 1/8, there is a tendency to develop discom-
mensurations in the stripe order, which, in turn, produce
regions of enhanced (or depressed) SC order with relative
sign depending on the number of intervening stripe peri-
ods. So long as the stripes are dilute, the energy depends
weakly on their precise spacing. Thus, to gain inter-
layer condensation energy, the system can self-organize
so that there are always an even number of intervening
stripes, thus producing an interplane Josephson coupling
J1,1 ∼ |x − 1/8|
2. This, in turn, will lead to a dramatic
increase of the 3D SC Tc. An enhancement of interplane
coherence in any range of T triggered by the magnetic
field induced spin-flop transition would be a dramatic
confirmation of the physics discussed here.
Note added: It was pointed out to us that the state dis-
cussed here was considered by A. Himeda et al.[25] They
found that this is a good variational state for a t− t′ −J
model at x ∼ 1/8 for a narrow range of parameters.
We thank P. Abbamonte, S. Chakravarty, R. Jamei,
A. Kapitulnik, and D. J. Scalapino for discussions.
This work was supported in part by the National Sci-
ence Foundation, under grants DMR 0442537 (EF),
DMR 0531196 (SAK), DMR 0342832 (SCZ), and by
the Office of Science, U.S. Department of Energy un-
der Contracts DE-FG02-91ER45439 (EF), DE-FG02-
06ER46287 (SAK) DE-AC02-98CH10886 (JT) and DE-
AC03-76SF00515 (SCZ), by the Stanford Institute for
Theoretical Physics (EAK), and by a Yale Postdoctoral
Prize Fellowship (VO).
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