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09

Theory of the striped superconductor

Erez Berg,1 Eduardo Fradkin,2 and Steven A. Kivelson1

1Department of Physics, Stanford University, Stanford, California 94305-40602Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080

(Dated: February 25, 2009)

We define a distinct phase of matter, a pair density wave (PDW), in which the superconductingorder parameter, φ(~r, ~r ′), varies periodically as a function of position such that when averaged overthe center of mass position, (~r +~r ′)/2, all components of φ vanish identically. Specifically, we studythe simplest, unidirectional PDW, the “striped superconductor,” which we argue may be at theheart of a number of spectacular experimental anomalies that have been observed in the failed hightemperature superconductor, La2−xBaxCuO4. We present a solvable microscopic model with strongelectron-electron interactions which supports a PDW groundstate. We also discuss, at the level ofLandau theory, the nature of the coupling between the PDW and other order parameters, and theorigins and some consequences of the unusual sensitivity of this state to quenched disorder.

I. A NEW PHASE OF MATTER

Superconductivity which arises from the pairing ofelectrons in time reversed states is well understoodas a weak coupling “Fermi surface” instability – aconsequence of arbitrarily weak effective attractiveinteractions.1 In this paper we explore a new type ofsuperconducting state, a “pair density wave” (PDW),which is a distinct state of matter, and which does notoccur under generic circumstances in the weak couplinglimit – it requires interactions in excess of a criticalstrength. The PDW spontaneously breaks the globalgauge symmetry, in precisely the same way as a conven-tional superconductor does. Thus, the order parameteris a charge 2e complex scalar field, φ. However, it alsospontaneously breaks some of the translational and pointgroup symmetries of the host crystal, in the same way asa conventional charge-density wave (CDW). In a PDW,for fixed ~r − ~r ′, the order parameter

∆(~r, ~r ′) ≡ 〈ψ†↑(~r)ψ

†↓(~r

′)〉 (1.1)

is a periodic function of the center of mass coordinate,~R ≡ (~r + ~r ′)/2. Here, ψ†

σ(~r) is the electron creationoperator at position ~r with spin polarization σ. We willfocus on the case in which translational symmetry is bro-ken only in one direction, i.e. a unidirectional PDW orequivalently a “striped superconductor.”

Accompanying the PDW there is, as we will show be-low, induced CDW order with half the period of thePDW. However, the PDW differs from a state of coex-isting superconducting and CDW order2, which has alsobeen previously called a “pair-density wave”.3 Unlike thepair-density-wave state of Ref.[2], the average value of thesuperconducting order parameter vanishes for the PDWstate we discuss here. This is a defining symmetry prop-erty of this state. The PDW is more closely analogousto the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state5

which arises, under appropriate circumstances, when theelectron gas is partially polarized by an applied magneticfield. However, explicit time-reversal symmetry break-ing is an essential ingredient of the FFLO state, and

is responsible for lifting the degeneracy (“nesting”) be-tween time-reversed pairs of quasiparticle states. In theabsence of quenched disorder, the PDW state preservestime-reversal symmetry.6

In general, there is no necessary relation between spin-density wave (SDW) and PDW order – whatever rela-tion there is derives from common microscopic physicsrather than from general symmetry conditions. How-ever, the two orders can be linked if one postulates alarger, emergent SO(5) symmetry12 which unifies thesuperconducting and SDW orders. In this context, anearly speculation13 concerning the existence of a unidi-rectional SO(5) spiral density wave state is particularlyinteresting.14 Such a state would consist of interleavingSDW and superconducting spirals, both with the sameperiod. The fact that this state is a spiral implies thatthe phase of φ varies as a function of position, and hencethat there are equilibrium currents. Thus, the SO(5)spiral, though similar to a state of coexisting PDW andcollinear SDW order, is not the same. We have not foundany microscopic model that supports a spiral-PDW phase- whether coexisting with SDW order, or not. However,it is an interesting state and deserves further study.16

Many features of the superconducting state depend onlyon the existence of a charge 2e order parameter. Thisincludes zero resistance, the Meissner effect, flux quanti-zation and, in 3D, the existence of a finite critical current.Therefore, these properties are expected in a PDW (inthe absence of quenched disorder).

However, other properties are very different. For in-stance, in an s-wave superconductor, there is a full gapin the quasiparticle spectrum. For a d-wave superconduc-tor, there are typically nodal points on the Fermi surfacewhich support gapless quasiparticle excitations, but thedensity of states still vanishes at the Fermi energy, evenin the presence of coexisting CDW or most forms of SDWorder18. In contrast, in a PDW portions of the Fermi sur-face are typically ungapped, which as in the case of manyCDWs, results in the reconstruction, but incomplete gap-ping of the underlying Fermi surface18,19. In a conven-tional superconductor, the anisotropy of the supercon-ducting coherence length is determined by the anisotropy

2

of the single-electron effective mass; in a PDW, depend-ing on the precise character of the translation symmetrybreaking, the anisotropy of the superconducting state canbe parametrically larger than the normal state resistivityanisotropy.

Perhaps the most significant new feature of a PDWsuperconductor is its anomalous sensitivity to (non-magnetic) quenched disorder, in stark comparison to aconventional superconductor. Specifically, although thedisorder potential does not couple directly to the super-conducting order parameter, it produces random varia-tions in the magnitude and sign of the local superfluiddensity. Hence quenched disorder can readily drive aPDW into a superconducting XY glass phase. (A similareffect was recently predicted20 in a d-wave superconduc-tor near the critical point of the superconductor to metaltransition.)

Table I characterizes the PDW, CDW, and uniformsuperconducting (SC) phases by their order parametersand their sensitivity to quenched disorder. Here, ∆0 and∆Q are, respectively, the uniform SC and PDW (mod-ulated SC) order parameters (where Q is the orderingwavevector), and ρQ and ρ2Q are the two relevant CDWorder parameters. By definition, the pure PDW phasehas non-zero ∆Q and a vanishing ∆0. It is thus differentfrom a phase of coexisting SC and CDW order, which hasa non-zero ∆0 and ∆Q (as well as a subsidiary ρQ order).(In various places in Refs. [2], as an aid to intuition, theSC+CDW and/or the CDW states have also been re-

ferred to as “Cooper pair crystals”) As mentioned above,the main new characteristic of the PDW (as opposed to auniform SC) is its sensitivity to disorder. Even weak dis-order destroys the long range PDW order, turning it intoa glassy state. The properties of these phases and theirinter-relations will be derived from an order parameterLandau theory in Section V.

In a recent publication22 we argued that the remark-able properties of La2−xBaxCuO4 near x = 1/8 fillingcan be explained as a consequence of the symmetries of astriped superconducting state. The main purpose of thepresent paper is to develop the theory of this state.

This paper is organized as follows. In Section II wegive an overview of the physics the striped superconduc-tor La2−xBaxCuO4 evidenced in recent experiments21.In Section III we discuss microscopic mechanisms thatgive rise to a PDW. In Section IV we present severalsolvable models that shed light on the physical origin ofa PDW superconducting state. Here we investigate therole of different model junctions separating two super-conducting regions, including an empty barrier (a trivialinsulator) and an antiferromagnetic insulating barrier.In both cases, under certain conditions, we find that aπ phase shift between phases of the two superconduc-tors is favored. We then build on these insights to con-struct a model with a striped superconducting groundstate. We discuss qualitatively the quasiparticle spec-trum of a PDW state. In Section V we present a Landautheory of this state and discuss in detail the symmetry-

TABLE I: Summary of the different phases discussed in thetext and their order parameters, as well as their sensitivityto quenched non-magnetic disorder. A X specifies that in aparticular phase, the corresponding order parameter is non-zero. See text for details.

∆0 ∆Q ρQ ρ2Q Sensitivity to disorderPDW 0 X 0 X FragileCDW 0 0 X X FragileCDW’ 0 0 0 X Fragile

SC X 0 0 0 RobustSC+CDW X X X X Becomes equivalent to SC

dictated couplings between the PDW order parameterand the nematic, charge density wave (CDW) and spindensity wave (SDW) order parameters characteristic ofa stripe state. In a recent publication, Agterberg andTsunetsugu4 discussed a Landau theory for a pair den-sity wave which complements the results presented in thisSection. In Section VI we present a statistical mechanicalmodel that describes a layer decoupled striped supercon-ductor, which is proposed to describe the phenomenologyof La2−xBaxCuO4. In Section VII we discuss the implica-tions of this theory. Further details of the Landau theoryare presented in Appendix A. The case of coexisting uni-form and striped d-wave superconductivity is discussedAppendix B.

II. STRIPED SUPERCONDUCTOR IN

LA2−xBAxCUO4

La2−xBaxCuO4 is currently the most promising can-didate experimental system as a realization of a stripedsuperconductor21,22. Considerable indirect evidence infavor of the existence of PDW order in this material hasrecently been gathered and compiled in Ref. [23]. Theputative PDW order has the same period as the unidi-rectional (spin-stripe) SDW order which is known, fromneutron scattering studies, to exist in this material24,25.Strong evidence that a striped SC can be induced in un-derdoped La2−xSrxCuO4 by the application of a trans-verse magnetic field, which is also known to induce spinstripe order, has also recently been presented in Ref. [26].

The behavior of La2−xBaxCuO4 is very striking, andrather complex. We will not elaborate upon it here (seeRef. [23]). However, there are two qualitative featuresof the data on which we would like to focus: 1) Withthe onset of stripe spin order at 42 K,27 there is a large(in magnitude) and strongly temperature dependent en-hancement of the anisotropy of the resistivity and otherproperties, such that below 42 K the in-plane chargedynamics resembles those of a superconductor, while inthe c-direction the system remains poorly metallic. Themost extreme illustration of this occurs in the temper-ature range 10 K < T < 16 K, in which the in-planedirection resistivity is immeasurably small, while the c-axis resistivity is in the 1-10 mΩ range, so the resistivity

3

FIG. 1: Schematic model of a striped superconductor consist-ing of an array of superconducting regions (S.C.) separatedby correlated insulator regions (I). The Josephson couplingbetween neighboring superconducting regions is J . If J < 0,the system forms a striped superconducting (or PDW) state.

anisotropy ratio is consistent with infinity. 2) Despitethe fact that many clear signatures of superconductivityonset at temperatures in excess of 40 K, and that angleresolved photoemission has inferred a “gap”29,30 of order20 meV, the fully superconducting state (i.e. the Meiss-ner effect and zero resistance in all directions) only occursbelow a critical temperature of 4 K. It is very difficult toimagine a scenario in which a strong conventional su-perconducting order develops locally on such high scales,but fully orders only at such low temperatures in a sys-tem that is three dimensional, non-granular in structure,and not subjected to an external magnetic field.

We will see that both these qualitative features of theproblem are natural consequences of the assumed exis-tence of a PDW. Indeed, analogous features have longbeen known to be a feature of the spin ordering in thesame family of materials25. Specifically, unidirectionalspin-stripe order is observed to occur under a numberof circumstances in the 214 family of cuprate supercon-ductors. However, when it occurs: 1) Although the in-plane correlation length can be very long, in the range100 − 400A, the inter plane correlation length is nevermore than a few A, a degree of anisotropy that can-not be reasonably explained simply31 on the basis of theanisotropy in the magnitude of the exchange couplings.2) Despite the presence of long correlation lengths, truelong-range spin-stripe order has never been reported. Itwill be made clear, below, that interlayer decoupling dueto the geometry of the stripe order, and the rounding ofthe ordering transitions by quenched disorder can be un-derstood as arising from closely analogous considerationsapplied to incommensurate SDW and PDW order22.

III. MICROSCOPIC CONSIDERATIONS

At first, the notion that a PDW phase could be sta-ble sounds absurd. Intuitively, the superconducting statecan be thought of as the condensed state of charge 2ebosons. However, in the absence of magnetic fields, theground-state of a bosonic fluid is always node-less, inde-pendent of the strength of the interactions, and thereforecannot support a state in which the superconducting or-der parameter changes sign. Thus, for a PDW state

to arise, microscopic physics at scales less than or of or-der the pair-size, ξ0, must be essential. This physics re-flects an essential difference between superfluids of pairedfermions and preformed bosons32.

Our goal in this section is to shed some light on themechanism by which strongly interacting electrons canform a superconducting ground-state with alternatingsigns of the order parameter. We will consider the case ofa unidirectional (striped) superconductor, but the sameconsiderations apply to more general forms of PDW or-der. We will not discuss the origin of the pairing whichleads to superconductivity. Likewise, we will not focuson the mechanism of translation symmetry breaking bythe density wave, as that is similar to the physics ofCDW and SDW formation. Our focus is on the signalternation of φ. Thus, in much of this discussion, wewill adopt the model shown schematically in Fig. 1, inwhich we have alternating stripes of superconductor andcorrelated insulator. The system looks like an array ofextended superconductor-insulator-superconductor (SIS)junctions, and we will primarily be concerned with com-puting the Josephson coupling across the insulating bar-riers. If the effective Josephson coupling is positive, thena uniform phase (normal) superconducting state is fa-vored, but if the coupling is negative (favoring a π junc-tion), then a striped superconducting phase is found.

A. Previous results

So long as time reversal symmetry is neither sponta-neously nor explicitly broken, the Josephson coupling,J between two superconductors must be real. If it ispositive, as is the usual case (for reasons that will beexpanded upon in later subsections), the energy is min-imized by the state in which the phase difference acrossthe junction is 0; if it is negative, a phase difference of πis preferred, leading to a “π junction.” π junctions havebeen shown, both theoretically and experimentally, to oc-cur for two distinct reasons: they can be a consequence ofstrong correlation effects32,33,34 or magnetic ordering35,36

in the junction region between two superconductors, ordue to the internal structure ( e.g. d-wave symmetry)of the superconductors, themselves37,38.

In the present paper, we will build on the first set ofideas. Until now, π junctions have been confined to sys-tems in which the Josephson tunneling is dominated bya single impurity site or quantum dot. It is non-trivialto extend this mechanism to the situation in which thereis an extended barrier in which J is proportional to thecross-sectional “area” of the junction39. Indeed, as weshall see below, under most circumstances, even in thecase in which the Josephson tunneling through an iso-lated impurity would produce a negative J , for a barrierconsisting of an extended area of identical impurities, Jwill typically be positive. One thing that we achieve be-low is to obtain a proof that there are circumstances inwhich J of such an extended barrier is negative, and to

4

−

+−

+−

+

−

+

−+

−+

−

+

−

+

FIG. 2: An example of a π junction due to a rotating domainwalls in a system with a d-wave order parameter. From leftto right, the crystallographic axes rotates by π/6 across eachdomain wall. After three domain walls, the crystal axis hasreturned to itself, but the order parameter has changed sign.

elucidate the conditions under which this occurs.

In the context of the cuprates, there have been severalstudies looking for a striped superconducting state in thet−J or Hubbard models. On the one hand, DMRG calcu-lations by White and Scalapino40 have consistently failedto find evidence in support of any sort of spontaneouslyoccurring π junctions. On the other hand, a number ofvariational Monte Carlo calculations have concluded thatthe striped superconductor is either the ground-state ofsuch a model41, under appropriate circumstances, or atleast close in energy to the true ground state42,43. Thesevariational calculations are certainly encouraging, in thesense that they suggest that there is no large energeticreason to rule out the existence of spontaneously occur-ring PDW order in strongly correlated electronic systems.However, since no such state has yet been observed inDMRG or other “unbiased” studies of these models, webelieve the mechanism of formation of these states, andindeed whether they occur at all in physically reason-able microscopic models, remain unsolved problems. Itis these problems that we aim to address.

B. π junctions from d-wave symmetry

In the case in which the preferred superconductingstate has a non-trivial internal pair-structure, as in ad-wave superconductor, it is well known both theoreti-cally and experimentally that π junctions can be achievedby suitable orientation of the crystal fields across grainboundaries. This is not the physics we have in mind.However, since it is simple and well understood, we startwith a highly artificial model in which d-wave symmetrygives rise to a striped superconductor.

We consider the case of a d-wave superconductor in asquare lattice, with a strong crystal field coupling whichlocks the lobes of the pair wave-function along the crys-tallographically defined x and y directions, as shown inFig. 2. We imagine we have made a striped array inwhich the crystallographic axis rotates by π/6 betweenneighboring strips. Since rotation by π/2 is a symme-try, every third strip is identical. However, the d-waveorder parameter changes sign under this rotation. Conse-

quently, the ground-state is a striped superconductor inwhich the period of the superconducting order parameteris twice the period of the lattice structure.

This is a very contrived example, in which the origin ofthe PDW does not require any strong coupling effects -it is derived from the internal structure of the pair-wavefunction. For the point of principle, i.e. to establish theviability of a PDW as a phase of matter, this analysisis useful. However, for practical applications, a less con-trived mechanism is needed.

IV. SOLVED MODELS

A. Model of a single junction

To start with, we consider a system consisting of threestrips - a “left” superconductor described by a mean-field

Hamiltonian with gap function ∆L(~k) (not to be con-fused with the order parameter of Eq. (1.1))and quasi-

particle energy EL(~k) =

√

[ǫL(~k)]2 + |∆L(~k)|2, a “right”

superconductor which is also described by a mean-fieldHamiltonian with the index L replaced by R, and be-tween them a strongly correlated insulating “ barrier”

region. Here ~k is either a 1D vector (if this is a line junc-tion) or a 2D vector if this is a planar junction. Notethat we have assumed the left and right superconductorsare thin in the transverse direction - otherwise we wouldhave to include a transverse band index, as well. We as-sume that both superconductors, by themselves, preservetime-reversal symmetry, so in both cases the phase of thesuperconducting order (modulo π) can be defined so that

∆α(~k)e−iθα is real.The three decoupled subsystems are thus described by

the Hamiltonian

H0 = HL +HB +HR, (4.1)

where, in the discussion below, we will consider severaldifferent examples for the insulating barrier. However, inall cases, we will assume that the superconducting Hamil-tonians have the quadratic forms

Hα =∑

~kσ

ǫα(~k)a†α,~k,σ

aα,~k,σ

+∑

~kσ

[

∆α(~k)a†α,~k,σ

a†α,−~kσ

+ h.c.]

(4.2)

with α = L or R in which

aα,~k,σ = N−1/2∑

~r

ei~k·~rcα,~r,σ (4.3)

where c†α,~r,σ creates an electron of spin polarization σ at

position ~r on, respectively, the left superconductor, thebarrier, the right superconductor, for α = L, B, and R,and N =

∑

~r 1 is the “area” of the junction.

5

The three subsystems are coupled together by a single-particle hopping term,

H ′ = −∑

~r,σ

[tLc†L,~r,σc~r,σ + tRc

†R,~r,σc~r,σ + h.c.] (4.4)

(with the convention that c~r,σ ≡ cB,~r,σ are the electronannihilation operators in the barrier region). Our pur-pose is to determine how the ground-state energy de-pends on the difference in phase between the supercon-ducting order parameter on the left and right side of thebarrier, θR − θL. Specifically, we will consider the limitin which the matrix elements in H ′ tend to zero, so thatthe leading θ dependent term in the ground-state energycan be computed in fourth order perturbation theory,

∆E = −2N(tLtR)2 [J cos(θL − θR) + . . .] (4.5)

where . . .→ 0 as |tα| → 0, and 2|tRtL|2J is the Josephson

coupling density across the barrier.An explicit expression can be obtained for J in terms

of imaginary time-ordered correlation functions, as wasshown in Ref. [44]. This can be derived either by makinga Laplace transform of the ordinary perturbative expres-sion, or directly from an Euclidean path integral:

J =1

Nβ

∫

d1 d2 d3 d4 FL(1, 2)F ⋆R(4, 3)Γ(1, 2; 3, 4)

(4.6)where 1 ≡ (τ1, ~r1),

∫

d1 ≡∑

~r1

∫ β

0

dτ1 (4.7)

(in the limit β → ∞) and

Fα(1, 2) ≡⟨

Tτ

[

c†α,~r1,↑(τ1)c†α,~r2,↓(τ2)

]⟩

(4.8)

Γ(1, 2; 4, 3) ≡⟨

Tτ

[

c†~r1,↑(τ1)c†~r2,↓(τ2)c~r3,↓(τ3)c~r4,↑(τ4)

]⟩

It is convenient to express Γ as the sum of a non-interacting piece, corresponding to processes in whichtwo electrons tunnel through the barrier independently ofeach other, plus a correction term, Γ, which expresses theeffects of interactions within the barrier region betweenthe tunneling electrons:

Γ(1, 2; 3, 4) = G↑(1, 4)G↓(2, 3) + Γ(1, 2; 3, 4) (4.9)

where

Gσ(1, 2) ≡⟨

Tτ

[

c†~r1,σ(τ1)c~r2,σ(τ2)]⟩

. (4.10)

Finally, we define two contributions to J :

J = J1 + J2 (4.11)

where J1 is the non-interacting portion, and J2 is theportion proportional to Γ.

B. General considerations

Under most circumstances, J1 > 0 since it is propor-tional to F 2G2 and hence does not depend on the sign ofG (see [45]). In Ref. 44, the situation in which there areattractive interactions between electrons in the barrierwas investigated. In this case, J2 is also, generally, pos-itive. This earlier study dealt with the conditions underwhich J2 > J1, leading to anomalously large values of theIcR product of the junction. In a sense, we are investigat-ing the converse problem, in which repulsive interactionslead to J2 < 0, while at the same time |J2| > J1.

In most cases with conventional superconductors, inwhich |∆| is the smallest energy scale and ξ0 (the super-conducting coherence length) is the largest length scale inthe problem, it is typically the case that J1 ≫ |J2|. Thedominant processes that contribute to J1 involve pairstunneling within an (imaginary) time interval of eachother ∼ 1/|∆| and within a radius ∼ ξ0 of each other.In contrast, the processes that dominate J2 involve elec-trons tunneling within an “ interaction time” τint and aninteraction range, rint of each other. Thus, one generallyexpects |J2/J1| is small in proportion to a positive powerof |∆|τint and rint/ξ0. In conventional superconductors,where these factors are very small, this is the decidingfactor, so that except under very special circumstances,one expects positive (“ferromagnetic”) coupling acrossany junction. Indeed, as discussed in Ref. 44, in thelimit that ∆ is small, Eq. (4.6) reduces to the famil-iar form for a tunnelling Hamiltonian with an effectivehopping teff = tLgtR across the barrier, where

g ≡

∫

d1 d2

NβGσ(1, 2). (4.12)

Conversely, in “high temperature superconductors,” inwhich ∆ is not all that small, it is reasonable to expect,at least under some circumstances, that the coupling isnegative – the long sought π-junction.

There is one other factor that can suppress J1 relativeto J2: When the two electrons tunnel across the barrierindependently of each other, their individual crystal mo-menta are conserved in the process. Thus, if we remove a

pair of electrons at momentum ~k and −~k from the rightsuperconductor, they must be injected into the left su-perconductor at the same momenta. If the right and leftsuperconductors are the same, this is not a problem –electrons can be removed from near the Fermi energy onthe left and added on the right at low energy. However,if the left and right superconductors have different valuesof kF , then no such low energy process is possible. Con-versely, for correlated tunneling, only the center of massmomentum of the tunnelling electrons is conserved, so a

pair of electrons removed from the left with momenta ~k

and −~k, can be injected into the right superconductorwith different momenta ~q and −~q.

6

C. Explicit model “barriers”

To make the considerations explicit, we will computeJ for a model of the barrier in several limits. Specifically,we will consider the case in which the barrier consists ofa Hubbard chain:

HB =∑

~r

ǫn(~r) − t∑

<~r,~r′>,σ

[

c†~r,σc~r,σ + h.c.]

+U∑

~r

c†~r,↑c†~r,↓c~r,↓c~r,↑ (4.13)

in the strong coupling limit, where the hopping matrixelement t → 0, with site energy ǫ and on-site repulsionbetween two electrons on the same site, U > 0. We willconsider two cases – the empty chain case in which ǫ > 0,and the half-filled case, in which ǫ < 0 < U − |ǫ|. Inthe half-filled case, the groundstate of the barrier is 2N

fold degenerate; we resolve this degeneracy by applying a

staggered magnetic field, h∑

~r,σ ei ~Q·~rσc†~r,σc~r,σ where ~Q

is the Neel ordering vector, and we take the limit h→ 0at the end of the calculation.

For simplicity, we will also assume that ∆L and ∆R are

independent of ~k, and that over the relevant range of en-ergies, the density of states in both superconductors canbe approximated by a constant, ρL and ρR, respectively.

1. The empty barrier

For the case of an empty barrier, ǫ > 0, a straightfor-ward (but tedious) calculation reveals that

J1 =2

N

∑

~k

(

∆L,~k

2EL,~k

)(

∆R,~k

2ER,~k

)[

1

EL,~k + ǫ

][

1

ER,~k + ǫ

]

×

[

1

EL,~k + ER,~k

+1

ǫ

]

. (4.14)

If the left and right superconductors are the same

J1 =ρ

2ǫ∆

[

f(y) − y

1 − y2

]

. (4.15)

where

f(yα) =cos−1(yα)√

1 − y2α

,

yα ≡ ǫ/|∆α|, and in this case y = yL = yR. Here f(y)is a smooth function with f(0) = π/2, f(1) = 1, andf(y) ∼ ln(2y)/y as y → ∞.

If the two superconductors have quite different valuesof kF , then J1 gets substantially suppressed. Let EL andER be, respectively, the energy of the left quasiparticleat kF,R and the energy of the right quasiparticle at kF,L.So long as EL, ER ≪ ∆, the magnitude of J1 is hardly

changed. However, if the two superconductors are suffi-ciently different that EL, ER ≫ ∆, then

J1 = ρLf(yL)

(

∆R

E2Rǫ

)

+ ρRf(yR)

(

∆L

E2Lǫ

)

. (4.16)

The interaction correction derives from terms thatwould have contributed to J1, but are suppressed dueto the repulsive interaction, and so has the opposite sign:

J2 = −2f(yL)f(yR)ρLρR

[

U

ǫ(U + 2ǫ)

]

. (4.17)

In conventional superconductors, typically the band-width, W , is the largest energy in the problem, and henceρ ∼ W−1 is “small.” Since J1 ∼ ρ and J2 ∼ ρ2, thismeans that J1 ≫ |J2|. The exception to this rule occursin cases in which ∆EF ≡ min[ER, EL] is a substantialfraction of the bandwidth.

However, in high temperature superconductors, we ex-pect to find that ∆ is not too much smaller than W . Inthis case, it will depend on details whether or not J1 orJ2 dominate. Clearly, this situation is more likely thelarger U . In the case of tunnelling between two identicalsuperconductors, and in the limit U → ∞, |J2| > J1 solong as

ρ∆ > xc ≡[f(y) − y]

4f2(y)(1 − y2). (4.18)

For y ≫ 1, xc ∼ y/ ln2 y, i.e. condition (4.18) requiresimpossibly large gap scales. However, xc = 19/96 for y =1 and xc → 1/2π as y → 0.46 Manifestly, J1 is reduced inmagnitude and J2 is unaffected by a substantial value of∆EF . For instance, if we consider the case in which theleft and right superconductors differ only in the positionof the Fermi surface, J2 dominates so long as

[ρ∆EF ]2f(y) > ρ∆ (4.19)

2. The antiferromagnetic barrier

We now turn to discuss the case ǫ < 0 < U − |ǫ|,in which the barrier is half filled. As mentioned previ-ously, we introduce a staggered Zeeman field h to lift thegroundstate degeneracy, and we take h → 0 at the endof the calculation. The lowest order perturbative term inthe Josephson coupling consists of all processes in whicha pair of electrons tunnels through the barrier, subject tothe constraint that the barrier has to return to its Neelordered ground state at the end of the process.

For simplicity, we consider the U → ∞ case, in whichno doubly occupied sites are allowed in the barrier. (Wehave obtained similar expressions for finite U , but theycontain no qualitatively different physics.) In this case,processes in which electrons tunnel though different sitescontribute to J1 and those in which they tunnel throughthe same site contribute to J2. As discussed in Ref. 32,

7

0.5 0.75 1 1.25 1.50

0.05

0.1

0.15

0.2

0.25

⟨n⟩

∆ / W

J>0 J>0

J<0

FIG. 3: Phase diagram for the antiferromagnetic barrier prob-lem, as a function of the pairing gap ∆ (in units of the band-width W = 4t) and the average density 〈n〉 in the supercon-ductors on either side of the barrier. When ∆ is large or 〈n〉is close to 1 (half filling), then a negative Josephson coupling(J < 0) is favored.

the single site process necessarily involves an exchangeof two fermions, and so makes a negative contribution toJ . A new feature of the present problem is that therean antiferromagnetic “Umklapp” contribution to J1, inwhich the tunneling electron exchanges momentum π/awith the antiferromagnet; this makes a negative contribu-tion to J1 which, under appropriate circumstances, canlargely cancel the usual positive contribution.

It is a straightforward exercise to obtain the resultingexpressions:

J1 =1

2N

∑

~k,~k′

(

∆L,~k

2EL,~k

)(

∆R,~k′

2ER,~k′

)

(

δ~k−~k′ − δ~k−~k′+πy

)

×

[

1

EL,~k + ER,~k′

+1

|ǫ|

][

1

EL,~k + |ǫ|

] [

1

ER,~k′ + |ǫ|

]

(4.20)

where δ~k is the Kronecker delta function, and

J2 = −1

N2

∑

~k,~k′

(

∆L,~k

2EL,~k

)(

∆R,~k′

2ER,~k′

)

(4.21)

×

[

1

EL,~k + |ǫ|

][

1

ER,~k′ + |ǫ|

][

1

EL,~k + ER,~k′

]

.

Generically, the magnitude of the umklapp term (pro-portional to δ~k−~k′+πy) is considerably smaller than the

momentum conserving term, since it is typically not pos-

sible for both ~k or ~k+πy to be close to the Fermi surfaceunless a special “nesting” condition is satisfied. For ex-ample, if the two superconductors are one dimensionalwires close to half filling, then the separation betweenthe two Fermi points is indeed close to π. In that case,the umklapp term tends to cancels the momentum con-

serving term. (If particle-hole symmetry is present, thiscancellation is exact.)

We have evaluated the expressions in Eqs. (4.20) and(4.21) numerically, for the case in which the two super-conducting strips are identical chains with a k indepen-dent gap function, ∆α,k = ∆ and the band-structure isthe simplest tight-binding result, ǫα,k = −2t cos(k) − µ.The results are summarized in Fig. 3, where the x axis isthe average density of electrons 〈n〉 in the superconduc-tors, and the y-axis is the magnitude of ∆; the solid linesseparate the region of net negative from the regions of netpositive Josephson coupling. The general trends alludedto in the above discussion are clear from this figure.

D. A model with a striped superconducting

ground state

The basic ingredients necessary for finding a modelwith a striped superconducting ground-state are presentin the above discussed examples. However, in the interestof making everything explicit, we consider the followingproblem. Firstly, we generalize the problem to d+ 1 di-mensions. We consider an array of parallel d dimensionalhyper-planes with weak attractive interactions, and aparticle-hole symmetric dispersion. Each hyperplanethus has a uniform superconducting order parameter. Forweak attractions between electrons and large enough d,the superconducting correlations in each hyperplane can,presumably, be accurately treated in the context of BCSmean-field theory. The particle-hole symmetry impliesthat E~k = E~k+~π, where ~π is the d-dimensional antiferro-magnetic Neel ordering vector. Sandwiched between eachsuperconducting hyperplane there is a d-dimensional in-sulting hyperplane, which is the d-dimensional general-ization of the antiferromagnetic barrier discussed above.Finally, we couple neighboring hyperplanes with an arbi-trarily weak hopping matrix, t. The Josephson couplingper unit hyperarea is then computed as above.

The particle-hole symmetry of the model insures thatJ1 = 0, and that therefore the Josephson coupling perunit hyper-area is proportional to J2, and hence is neg-ative. Thus, the ground-state of this system is a statewith coexisting SDW and striped superconducting or-der. Note that the phase of the antiferromagnetic orderis determined by the sign of h. If we take h > 0 inall cases, then the antiferromagnetic order is “in-phase.”In this case, the Hamiltonian is invariant under transla-tions in the direction perpendicular to the hyperplanes bythe distance between insulating hyperplanes, and hencethe striped superconductor spontaneously breaks trans-lational symmetry. However, we can also imagine stag-gering the sign of h from one hyperplane to the next, soas to mimic the antiphase spin-stripe order seen in thecuprates. In this case, the underlying unit cell is twice aslarge, and the striped superconductor does not sponta-neously break translational symmetry. However, in bothcases, the spatial average of any component of the super-

8

conducting order parameter is 0, so this state conformsto our definition of a PDW.

This model is, admittedly, somewhat contrived, espe-cially in the assumed particle-hole symmetry. However,since at this point J2 has a finite magnitude, the state isrobust with respect to small deformations of the model.

The construction of solvable microscopic models witha striped superconducting ground-state, and the elucida-tion of the ingredients necessary to get anti-phase super-conducting order across a correlated insulating barrier,are the principle results of the present paper.

E. Quasiparticle spectrum of a striped

superconductor

The spectrum of an s-wave superconductor is typicallyfully gapped, while a d-wave superconductor has a gapthat vanishes only on a few isolated points (lines in 3d)on the Fermi surface. These features persist even in thepresence of coexisting CDW or SDW order of varioussorts18. In particular, the density of quasiparticle statesis either zero for energies less than a non-zero minimumvalue (in the s-wave case), or vanishes (linearly) as theenergy approaches the Fermi surface. This effect of su-perconducting order is in contrast to most other orders,including SDW and CDW order, in which, so long asthe effective ordering potential is weak compared to theFermi energy, at most a portion of the Fermi surface isgapped leaving the system with a (smaller) reconstructedFermi surface, but a Fermi surface none-the-less.

The gap character in a PDW is more similar to thatof a CDW than to a uniform superconductor. Specif-ically, so long as the order parameter is not too large,an ungapped reconstructed Fermi surface remains18,19.Thus, a PDW typically has a finite density of states inthe superconducting phase.

For band-structure parameters characteristic of a typ-ical underdoped or optimally doped curpate, the quasi-particle spectrum of a striped superconductor has a large(d-wave-like) gap near the “ antinodal points” (i.e. near(π, 0) and (0, π)), but has a closed, Fermi-surface pocketwhich closely follows the contours of the portion of thenormal-state Fermi surface near the “ nodal points” [i.e.where the Fermi surface crosses the (0, 0) to (π, π) cord].In short, it closely resembles certain phenomenologicaldescriptions47 of the “pseudo-gap.” The low energy spec-tral weight is substantial only on a portion of the bareFermi surface near the nodal (π, π) direction31, forminga “Fermi arc”48,49.

V. COUPLED ORDER PARAMETERS

In this section, we explore the aspects of the theory ofa PDW that can be analyzed without reference to mi-croscopic mechanisms. We focus on the properties of or-dered states at T = 0, far from the point of any quantum

phase transition, where for the most part fluctuation ef-fects can be neglected. (The one exception to the generalrule is that, where we discuss effects of disorder, we willencounter various spin-glass related phases where fluctu-ation effects, even at T = 0, can qualitatively alter thephases.) For simplicity, most of our discussion is couchedin terms of a Landau theory, in which the effective freeenergy is expanded in powers of the order parameters;this is formally not justified deep in an ordered phase,but it is a convenient way to exhibit the consequences ofthe order parameter symmetries.

Specifically, the emphasis in this section is on the inter-relation between striped superconducting order and otherorders. There is a necessary relation between this orderand CDW and nematic (or orthorhombic) order, since thestriped superconductor breaks both translational and ro-tational symmetries of the crystal. From the microscopicconsiderations, above, and from the phenomenology ofthe cuprates, we also are interested in the relation of su-perconducting and SDW order. We assume that the hostcrystal is tetragonal, and that there are therefore two po-tential symmetry related ordering wave-vectors, Q andQ, which are mutually orthogonal. Spin-orbit couplingis assumed to be negligible. We introduce the followingorder parameter fields: the nematic, N , the PDW ∆Q,

the SDW ~SQ, and the CDW ρK (where K = 2Q), and,of course, the corresponding orders at the symmetry re-lated wave-vector, Q. The Landau effective free energydensity can then be expanded in powers of these fields:

F = F2 + F3 + F4 + . . . (5.1)

where F2, the quadratic term, is simply a sum of decou-pled terms for each order parameter,

F3 = γs[ρ⋆K~SQ · ~SQ + ρ⋆

K~SQ · ~SQ + c.c.] (5.2)

+γ∆[ρ⋆K∆⋆

−Q∆Q + ρ⋆K∆⋆

−Q∆Q + c.c.]

+g∆N [∆⋆Q∆Q + ∆⋆

−Q∆−Q − ∆⋆Q∆Q − ∆⋆

−Q∆−Q]

+gsN [~S⋆Q · ~SQ − ~S⋆

Q · ~SQ]

+gcN [ρ⋆KρK − ρ⋆

KρK ],

and the fourth order term, which is more or less standard,is shown explicitly in Appendix A. (Detailed discussionsof the microscopic definition of the PDW order parameteris also discussed in this Appendix.)

The effect of the cubic term proportional to γs onthe interplay between the spin and charge componentsof stripe order has been analyzed in depth in Ref. 50.Similar analysis can be applied to the other terms. Inparticular, it follows that the existence of superconduct-ing stripe order (∆Q 6= 0, and ∆Q = 0), implies the exis-tence of nematic order (N 6= 0) and charge stripe orderwith half the period (ρ2Q 6= 0). However, the conversestatement is not true: while CDW order with orderingwave-vector 2Q or nematic order tend to promote PDWorder, depending on the magnitude of the quadratic termin F2, PDW order may or may not occur.

9

One new feature of the coupling between the PDWand CDW order is that it produces a sensitivity to disor-der which is not normally a feature of the superconduct-ing state. In the presence of quenched disorder, thereis always some amount of spatial variation of the chargedensity, ρ(r), of which the important portion for our pur-poses can be thought of as being a pinned CDW, that is,a CDW with a phase which is a pinned, slowly varyingfunction of position, ρ(r) = |ρK | cos[K · r + φ(r)]. Be-low the nominal striped superconducting ordering tem-perature, we can similarly express the PDW order interms of a slowly varying superconducting phase, ∆(r) =|∆Q| exp[iQ · r+ iθQ(r)] + |∆−Q| exp[−iQ · r+ iθ−Q(r)].The resulting contribution to F3 is

F3,γ = 2γ∆|ρK∆Q∆−Q| cos[2θ−(r) − φ(r)]. (5.3)

where

θ±(r) ≡ [θQ(r) ± θ−Q(r)]/2; (5.4)

θ±Q(r) = [θ+(r) ± θ−(r)].

The aspect of this equation that is notable is that the dis-order couples directly to a piece of the superconductingphase, θ−. No such coupling occurs in usual 0 momentumsuperconductors.

It is important to note that the condition that ∆(r) besingle valued imposes a non-trivial topological constrainton possible vortices. Specifically, an isolated half-vortex(about which the phase winds by π) is forbidden in eitherθ+ or θ−; vortices must occur either as full (2π)-vorticesin one or the other phase field, or as a bound pair of a θ+and a θ− half-vortex. An important consequence of thisphase coupling is that the effect of quenched disorder,as in the case of the CDW itself, destroys long-rangesuperconducting stripe order. (This statement is true51,even for weak disorder, in dimensions d < 4.) Naturally,the way in which this plays out depends on the way inwhich the CDW state is disordered.

In the most straightforward case, the CDW order ispunctuated by random, pinned dislocations, i.e. 2π vor-tices of the φ field. The existence of the coupling inEq. (5.3) implies that there must be an accompanyingπ vortex in θ−. The condition of single-valued-ness im-plies that there must also be an associated half-vortexor anti-vortex in θ+

52. If these latter vortices are fluc-tuating, they destroy the superconducting state entirely,leading to a resistive state with short-ranged striped su-perconducting correlations. If they are frozen, the re-sulting state is analogous to the ordered phase of an XYspin-glass: such a state has a non-vanishing Edwards-Anderson order parameter, spontaneously breaks time-reversal symmetry, and, presumably, has vanishing re-sistance but no Meissner effect and a vanishing criticalcurrent. In 2D, according to conventional wisdom, a spin-glass phase can only occur at T = 0, but in 3D there canbe a finite temperature glass transition53.

In 3D there is also the exotic possibility that, for weakenough quenched disorder, the CDW forms a Bragg-glass

phase, in which long-range order is destroyed, but no freedislocations occur54,55,56. In this case, φ can be treatedas a random, but single-valued function - correspond-ingly, so is θ−. The result is a superconducting Bragg-glass phase which preserves time reversal symmetry and,presumably, acts more or less the same as a usual super-conducting phase. It is believed that a Bragg-glass phaseis not possible in 2D55.

A summary of the characterization of the PDW, SCand CDW phases in terms of their order parameters, theirinter-relations and their sensitivity to quenched disorderappears in Table I.

Another perspective on the nature of the supercon-ducting state can be obtained by considering a compos-ite order parameter which is proportional to ∆Q∆−Q.There is a cubic term which couples a uniform, charge4e superconducting order parameter, ∆4, to the PDWorder:

F ′3 = g4∆

⋆4[∆Q∆−Q + ∆Q∆−Q] + c.c. (5.5)

This term implies that whenever there is PDW order,there is also necessarily charge 4e uniform superconduct-ing order. However, since this term is independent of θ−,it would be totally unaffected by Bragg-glass formationof the CDW. The half-vortices in θ+ discussed above cansimply be viewed as the fundamental (hc/4e) vortices ofa charge 4e superconductor.

Turning now to the quartic terms in F4, there are anumber of features of the ordered phases which dependqualitatively on the sign of various couplings. Again,this is very similar to what happens in the case of CDWorder - see, for example, Ref. [57]. For instance, de-pending on the sign of a certain biquadratic term, eitherunidirectional (superconducting stripe) or bidirectional(superconducting checkerboard) order is favored.

Finally, we comment on the case of coexisting uniformand striped superconducting order parameters. Such astate is not thermodynamically distinct from a regular(uniform) superconductor coexisting with a charge den-sity wave, even if the uniform superconducting compo-nent is in fact weaker than the striped component. There-fore, we expect many of the special features of the stripedsuperconductor (such as its sensitivity to potential dis-order) to be lost. In Appendix B, we extend the Landaufree energy to include a uniform superconducting com-ponent, and show that this is indeed the case.

VI. MODEL FOR A LAYERED STRIPED

SUPERCONDUCTOR

We now discuss a low energy effective theory for a lay-ered, disordered striped superconductor. As we saw inthe previous section, disorder inevitably nucleates halfvortices which are pinned to dislocations in the CDW.The system therefore has Ising-like degrees of freedomwhich are the “charges” (or vorticities) of these π vor-tices. We assume that their positions are quenched, ran-

10

dom variables. As we discussed in Ref. [22], a PDW ina layered superconductor can give rise to a frustrationof the inter–layer Josephson coupling, given a structurein which the CDW is rotated by 90 between layers [asoccurs in the low temperature tetragonal (LTT) phase ofLa2−xBaxCuO4]. To make the model relevant to this sys-tem, we consider the case where the inter–layer Josephsoncoupling vanishes identically.

Under these assumptions, the interaction energy be-tween half vortices is composed of two parts: the mag-netic energy and the kinetic energy associated with thescreening currents that surround the vortices. The to-tal interaction energy for a given configuration of halfvortices is58

U =1

2

∑

ijαβ

u(

~Riα − ~Rjβ

)

qiαqjβ+1

2

∑

αβ

v (zα − zβ)QαQβ .

(6.1)

Here, qiα = ±1 and ~Riα are the “charge” (or vorticity)and the position of the ith vortex in layer α, respectively,zα is the z coordinate of layer α, and Qα =

∑

i qiα. The

interaction potentials u(~R) and v (z) are given by

u(

~R)

=B2

0λ2d

µ0

∫

d2k

(2π)2

[

δαβ

k2−

d

2λ

e−α(k)|z|

λ

α (k) k2

]

[

ei~k·~ρ − 1]

,

(6.2)where ~ρ and z are the radial and the z axis separation be-

tween the two vortices, respectively, α (k) ≡

√

1 + (λk)2,

B0 ≡ h4eλ2 , δαβ is a Kronecker delta function of the layer

indices α, β of the two vortices, λ is the in-plane pene-tration depth, d is the inter-layer distance, and

v (z) =B2

0λ2d

2πµ0

δαβ lnL

ξ−

d

2λe−

|z|λ ln

L

λ(6.3)

+d

4λe−

2|z|λ

[

ln|z|

2λ+ γ + e

2|z|λ E1

(

2 |z|

λ

)]

,

where L is a long wavelength cutoff of the order of thelinear dimension of the system, ξ is the coherence length,

E1 (x) =∫∞

xe−t

t dt is the exponential integral function,and γ is Euler’s constant. The second term in Eq. (6.1)diverges as lnL, unless Qα = Q = const. for all α.In the absence of an external magnetic field, this termconstrains Qα = 0. This is the usual “charge neutrality”condition which comes from the infrared divergence ofthe vortex self energy.

For√

ρ2 + z2 ≫ λ, Eq. (6.2) reduces to

u(

~R)

≈B2

0λ2d

2πµ0

[

δαβ −d

2λe−

|z|λ

]

lnξ

ρ. (6.4)

The statistical mechanics problem of a finite density ofhalf vortices with quenched random positions, and whoseinteraction is given by Eq. (6.1), is an interesting un-solved statistical mechanics problem.59,60

As we mentioned, the putative superconducting glassphase (in which the half vortices are frozen) necessar-ily involves time reversal symmetry breaking. It should

therefore be detectable by measuring the magnetic fieldsassociated with the spontaneous half vortices which oc-cur at dislocations in the charge order. For example,consider a half vortex at the surface of the sample (takento be at z = 0). The asymptotic form of the magneticfield above the surface in the limit 0 < z ≪ λ, is

Bz (z ≪ λ) ≈~

8λ2e

(

d

z

)

, (6.5)

where d is the interlayer distance. [For z & λ, screeningby diamagnetic currents in the other planes becomes con-siderable, and Bz crosses over to ∼ ~d

8λe

(

1z2

)

.] Assuming

λ ≈ 2000 A and d ≈ 15 A, Eq. (6.5) gives Bz ≈ 300(

Az

)

G. At a distance of about 1000 A from surface, the result-ing characteristic fields of the order of 0.3 G (assuming ahalf vortex right at the surface) are well within the reso-lution of current local magnetic field measurement tech-niques. The onset of a spontaneous, random magneticfields of this order at the glass transition temperature(which is presumably also signaled by the vanishing ofthe linear resistivity) would be a dramatic confirmationof the PDW scenario in 1

8 doped LBCO.

VII. FINAL THOUGHTS

The strong suppression of the three-dimensional super-conducting Tc in La2−xBaxCuO4 at x = 1/8, and other“1/8 anomalies” in the La2−xSrxCuO4 family of cupratesuperconductors, have long been interpreted in the liter-ature as evidence that charge order competes with hightemperature superconductivity. (More recently61,62,63,clear evidence of a 1/8 anomaly has been adduced inYBa2Cu3O7−δ, as well.) However, the remarkable prop-erties of La2−xBaxCuO4, particularly the fact that theanti-nodal gap is largest at x = 1/8, where the dynam-ical layer decoupling is observed, strongly suggests thatcharge stripe order can be part of the mechanism of su-perconductivity as argued in Ref.[64].

The superconducting state presented here representsa new face of the interplay between superconductivityand spin and charge order. If the SC and SDW orderssimply competed, the PDW would be a rather unnaturalstate; it is natural if self-organized inhomogeneities arean essential feature of the mechanism. In this sense, thePDW state should also be regarded as an electronic liquidcrystal phase.65 Numerous and important implicationsfollow from the properties of this state.

A. Implications

Glassy superconductors: The most dramatic conse-quence of the PDW physics is that, in the presenceof weak, quenched disorder, the superconducting phasegives way to a regime of glassy behavior, where stronglocal superconducting correlations extend up to a finite

11

correlation length set by the strength of the disorder. Inthe regime of temperature below the onset of substantiallocal superconducting coherence but above the transitionto a fully superconducting state (if one occurs), the sys-tem can be characterized as a “failed superconductor”30.Clearly, in this regime, the longitudinal resistivity will besmall compared to normal state values, as will the quasi-particle contributions to the thermopower and linear Hallresistance. Moreover, one expects to see strong local indi-cations of superconductivity, especially the formation ofa superconducting pseudo-gap in the single-particle spec-trum with a form similar to that of the ordered phase.

Taking into account that the materials are ultimately3D, there will generally be a glass transition at T = Tg

in this regime. In the absence of an applied magneticfield, the glass phase can be characterized by its spon-taneously broken time-reversal symmetry. Presumably,the glass phase has zero resistance, zero critical current,and fails to exhibit a full Meissner effect. More gener-ally, for a range of temperatures, including temperaturesabove Tg, the magnetic response of the PDW will becharacterized by slow dynamics, a broad distribution ofrelaxation rates, and probably a certain degree of historydependence; all the dramatic and confusing features ofspin-glasses, but amplified by the coherent orbital orga-nization of a superconducting state.

Interlayer decoupling: When a PDW state occurs in aquasi-2D (layered) material, it will frequently be the case(depending on the interlayer geometry) that the usualJosephson coupling between neighboring layers vanishes.This feature was explored at length in our earlier pa-per, Ref. [22]. In the case of the LTT structure ofLa2−xBaxCuO4, the fact that the stripes in neighbor-ing planes run at right angles to each other insures that(in the absence of disorder), the Josephson coupling be-tween neighboring planes vanishes. Strictly speaking,this does not mean that there is no coupling betweenplains, as would occur in a putative “floating phase”66.There are always couplings between farther neighbor lay-ers, and higher order Josephson couplings between neigh-boring planes (which couple ∆4 in 3D), and these inter-actions are relevant in the renormalization group sense atany temperature below the putative Kosterlitz-Thouless(KT) transition temperature. However, at the very least,it means that the 3D superconducting state is enormouslymore anisotropic than would be expected on the basis ofthe bare electronic anisotropy. Moreover, as long as su-perconducting coherence in a given plane is limited dueto quenched disorder, if these higher order couplings areweak enough, they can be essentially ignored.

Striped superconductors in strongly correlated models:While we have established, as a point of principle, thatwell defined models exist that support a PDW phase, itstill makes sense to make predictions that can be testedin “numerical experiments” on t− J and Hubbard mod-els. The fact that spontaneously occurring π-junctionshave not yet been reported in extensive previous DMRGstudies40 is worrisome, in this regard.

Based upon the insights gained in the above, we pro-pose DMRG studies of microscopic models in which weexpect indications of PDW formation can be observed.Two examples are:

i) A three leg Hubbard ladder in which the outer twolegs have a negative U and the inner-leg a positive U .The chemical potential and on-site energies should bechosen so that the inner leg is near half-filling, and sostrongly antiferromagnetically correlated. The density ofelectrons on the outer (“superconducting”) legs can bevaried, and need not be identical. This model is thus aclose relative of the model that we treated in Sec. VI.C.2,except that we do not treat the superconducting legs inmean-field theory and we are not restricted to consider-ing parametrically small coupling between the legs. Thetendency to a striped SC phase can be tested by studyingthe sign of the pair-field correlations between the upperand lower legs. When the outer legs are near half filling,we expect to see negative correlations, indicative of PDWformation. Indeed, we have already seen such behaviorin preliminary DMRG studies of this model67.

ii) A five leg Hubbard ladder with all repulsive interac-tions, constructed to consist of two outer two-leg laddersand an inner Hubbard chain. Again, the chemical po-tential and on-site energies should be chosen so that theinner chain is near half-filling, making it strongly antifer-romagnetically correlated, while the density of electronson the outer ladders can be varied, and need not be iden-tical. By making the densities on the outer legs differentenough, we are confident that a PDW state can be in-duced. However, we are very curious to learn how robustthis state is under less contrived conditions. We are nowundertaking such an investigation67.

B. Speculations

Striped SC phases in La2−xBaxCuO4 at x = 1/8: Wewere originally motivated in this study by experimentsin La2−xBaxCuO4, and we remain optimistic that theyare, indeed, evidence of the existence of a PDW phase.In order to make more than an impressionistic compar-ison with experiment, the nature of the superconduct-ing glass phase will need to be understood theoretically,much better than we do at present. However, in the ab-sence of such theoretical control, we can still make somespeculative statements concerning the relation betweenexperiment and theory.

Tentatively, we would like to identify the point atwhich resistivity vanishes in the c-direction as the truepoint of the 3D glass transition, Tg ≈ 10K. If this is thecase, the linear-response resistivity at lower temperaturesis truly zero, although various non-linear and hystereticprocesses may complicate the measurements. We expectvery long time-scales and a degree of history dependenceof macroscopic measurements to begin being significantat considerably higher temperatures, T > Tg.

It is natural to associate the point at which the in-

12

plane resistivity apparently vanishes TKT ≈ 16K, withwhat would have been the KT transition of a single planein the absence of quenched disorder. We do not thinkthis is a true transition, and would expect that if experi-ments could be carried out with higher precision than hascurrently been possible, the in-plane resistivity would befound to have a finite value for Tg < T < TKT . Inrelatively clean samples, simple scaling arguments sug-gest that the residual resistivity should be proportionalto ξ−2, where ξ is the coherence length, which in turn isroughly the distance between dislocations.

The sharp drop in the resistivity at the spin-orderingtemperature, TSDW ≈ 42K, is probably not a true phasetransition, either, but rather marks the sudden onset ofsignificant intermediate scale superconducting coherence.However, we suspect that local pairing correlations per-sist to higher temperatures; as long as the spin order isstrongly fluctuating at T > TSDW , we imagine that phasecoherence between neighboring superconducting stripesis prevented, while for T < TSDW , antiphase supercon-ducting correlations extend over multiple stripes. We fa-vor this viewpoint for several reasons, most importantlybecause the gap-features seen in ARPES persist to highertemperatures.30

ARPES spectrum of La2−xBaxCuO4: A recent study30

of the ARPES spectrum of La2−xBaxCuO4 confirmed theconclusion of an earlier29 ARPES/STM study concerningthe existence of a gap consistent with a generally d-waveangle-dependence. However, the higher resolution studyrevealed that this gap has what appears to be a two-component structure, in that the gap near the anti-nodalpoint at (π, 0), is considerably larger (by a factor of 2-3) than a simple extrapolation from the nodal directionwould suggest. Since there are several forms of densitywave order known to be present in La2−xBaxCuO4, it isnot straightforward to unambiguously identify particularspectral features with particular types of order. Thisis particularly problematic, since fluctuating order canalso lead to a pseudo-gap with many similarities to thegap that would be produced in the corresponding orderedphase.

That having been said, it is striking how much the ob-served spectrum in La2−xBaxCuO4 resembles the mean-field spectrum found under the assumption that thereis large amplitude PDW (striped) order coexisting withsmall amplitude uniform superconducting order. ThePDW has a large gap along the anti-nodal [(π,0) or (0,π)]direction, and a Fermi pocket in the nodal [(π,π)] direc-tion, whose spectral weight is considerable only along anopen “Fermi arc” region which nearly coincides with thebare Fermi surface. If, in addition, there is a uniform d-wave component to the order parameter, this completesthe gapping of the Fermi surface. A spectrum that resem-bles the measured ARPES spectrum in La2−xBaxCuO4

can be obtained18 under the assumption that the PDWhas a gap magnitude ∆PDW = 40meV, and uniform gapmagnitude ∆d = 8meV. The ARPES spectrum is alwaysmeasured at temperatures well above the bulk supercon-

ducting Tc = 4K, so to the extent that this identificationis correct, what is being observed is a pseudo-gap. More-over, as mentioned above, the gap structure persists totemperatures above TSDW , although at the higher tem-peratures, the gap in the nodal regions decreases ap-preciably. Much more detailed analysis of the energy,momentum, and temperature dependence of the ARPESspectrum will be necessary to test this interpretation.

Relevance to other cuprates: We have summarizedthe evidence for a state with the symmetries of aPDW state (or the glassy version of it) in 1

8 dopedLa2−xBaxCuO4. There is also evidence for dynamicallayer decoupling effects in La1.6−xNd0.4SrxCuO4

68, andalso in La2−xSrxCuO4 in a magnetic field.26,69 (For thepossible relevance of these ideas to certain heavy fermionand organic superconductors see [70].)

The quasiparticle spectrum of a PDW state has strik-ing qualitative resemblance to the spectra seen in ARPESin other cuprates, especially BSCCO 2212 and 2201. Thispossibly sheds new light on the issue of the “nodal-antinodal dichotomy”: According to this interpretation, boththe nodal and the anti-nodal gaps are superconductinggaps, the first being uniform and the other modulated.There are two distinct types of order (“two gaps”), butthey are both superconducting, and so they can smoothlyevolve into one another. Perhaps, as the temperature isdecreased, the PDW gradually decreases and the uniformorder parameter increases, while their sum (which is de-termined by relatively high energy microscopic physics)remains approximately constant. (Note that an earlystudy74 of modulated structures seen in STM75,76 con-cluded that they could be understood in terms of justsuch a two-superconducting-gap state.)

More generally, one of the most remarkable featuresof the pseudo-gap phenomena is the existence of effectsof superconducting fluctuations, detectable77,78,79 for in-stance in the Nernst signal, over a surprisingly broadrange of temperatures and doping concentrations. At abroad-brush level80, these phenomena are a consequenceof a phase stiffness scale that is small compared to thepairing scale. However, it is generally difficult to under-stand the existence of such a broad fluctuational regimeon the basis of any sensible microscopic considerations.The glassy nature of the ordering phenomena in a PDWmay hold the key to this central paradox of HTC phe-nomenology, as it gives rise to an intrinsically broadregime in which superconducting correlations extend overlarge, but not infinite distances.

Acknowledgments

We thank John Tranquada, Hong Yao, RuihuaHe, Srinivas Raghu, Eun-Ah Kim, Vadim Oganesyan,Kathryn A. Moler and Shoucheng Zhang for great discus-sions. This work was supported in part by the NationalScience Foundation, under grants DMR 0758462 (EF)and DMR 0531196 (SAK), and by the Office of Science,

13

U.S. Department of Energy under Contracts DE-FG02-07ER46453 through the Frederick Seitz Materials Re-search Laboratory at the University of Illinois (EF) andDE-FG02-06ER46287 through the Geballe Laboratory ofAdvanced Materials at Stanford University (SAK).

APPENDIX A: ORDER PARAMETERS

We will now define the various order parameters intro-duced in Sec. V and discuss their symmetry properties.The striped superconducting order parameter ∆Q is acharge 2e complex scalar field, carrying momentum Q.To define it microscopically, we write the superconduct-ing order parameter as

∆ (~r, ~r ′) ≡⟨

ψ†↑ (~r)ψ†

↓ (~r ′)⟩

= F (~r − ~r ′)[

∆0 + ∆Qei ~Q·~R + ∆−Qe

−i ~Q·~R]

,

(A1)

where R = (~r + ~r ′) /2, F (~r − ~r ′) is some short rangefunction (for a “d-wave like” striped superconductor,F (~r) changes sign under 90 rotation), and ∆0 is a uni-form order parameter. In the remaining of this appendix,we set ∆0 = 0. The effect of ∆0 is discussed in AppendixB. In cases where a ∆Q is also non-zero (where Q is re-lated to Q by a 90 rotation) analogous terms with ∆Q

replaced by ∆Q have to be added.

The order parameters that may couple to ∆Q and theirsymmetry properties are as follows. The nematic orderparameter N is a real pseudo-scalar field; the CDW ρK

with K = 2Q is a complex scalar field; ~SQ is a neu-tral spin-vector complex field. All these order parametersare electrically neutral. Under spatial rotation by π/2,

N → −N , ρK → ρK , ~SQ → ~SK , and ∆Q → ±∆Q, where± refers to a d-wave or s-wave version of the striped su-perconductor. Under spatial translation by r, N → N ,

ρK → eiK·rρK , ~SQ → eiQ·r ~SK , and ∆Q → eiq·r∆Q.

With these considerations, we write all the possiblefourth order terms consistent with symmetry:

F4 = +u(

~SQ · ~SQ∆⋆Q∆−Q~+SQ · ~SQ∆⋆

Q∆−Q + c.c.)

+(

v+[~S⋆Q · ~SQ + ~S⋆

Q · ~SQ] + v+[|ρK |2 + |ρK |2])

(

|∆Q|2 + |∆−Q|2 + |∆Q|2 + |∆−Q|

2

+(

v−[~S⋆Q · ~SQ − ~S⋆

Q · ~SQ] + v−[|ρK |2 − |ρK |2])

(

|∆Q|2 + |∆−Q|2 − |∆Q|2 − |∆−Q|

2)

+vN2(

|∆Q|2 + |∆−Q|2

)

+(

|∆Q|2 + |∆−Q|2

)

+λ+

(

|∆Q|2 + |∆−Q|2)2

+(

|∆Q|2 + |∆−Q|2)2

+ λ−

(

|∆Q|2 − |∆−Q|2)2

+(

|∆Q|2 − |∆−Q|2)2

+ . . .

(A2)

where we have explicitly shown all the terms involving∆Q, while the terms . . . represent the remaining quarticterms all of which, with the exception of those involvingN , are exhibited explicitly in Ref. 50.

On physical grounds, we have some information con-cerning the sign of various terms in F4. The term propor-tional to u determines the relative phase of the spin andsuperconducting stripe order - we believe u > 0 whichthus favors a π/2 phase shift between the SDW and thestriped superconducting order, i.e. the peak of the su-perconducting order occurs where the spin order passesthrough zero. The other interesting thing about this termis that it implies an effective cooperativity between spinand striped superconducting order. The net effect, i.e.whether spin and striped superconducting order cooper-ate or fight, is determined by the sign of |u| − v+ − v−,such that they cooperate if |u|− v+ − v− > 0 and opposeeach other if |u| − v+ − v− < 0. It is an interesting pos-sibility that spin order and superconducting stripe order

can actually favor each other even with all “ repulsive”interactions. The term proportional to λ− determineswhether the superconducting stripe order tends to bereal (λ− > 0), with a superconducting order that sim-ply changes sign as a function of position, or a complexspiral, which supports ground-state currents (λ− < 0).

APPENDIX B: COEXISTING UNIFORM AND

STRIPED ORDER PARAMETERS

In this appendix, we analyze the coupling of a stripedsuperconducting order parameter ∆Q to a uniform orderparameter, ∆0. In this case, we have to consider in ad-dition to the order parameters introduced in Sec. V aCDW order parameter with wavevector Q, denoted byρQ. The additional cubic terms in the Ginzburg-Landaufree energy are

14

F3,u = γQ∆⋆0

[

ρQ∆−Q + ρ⋆Q∆Q + ρQ∆−Q + ρ⋆

Q∆Q

]

+ c.c.+ gρ

[

ρ⋆2Qρ

2Q + ρ⋆

2Qρ2Q + c.c.

]

. (B1)

Eq. (B1) shows that if both ∆0 and ∆Q are non-zero, there is necessarily a coexisting non-zero ρQ, through the γQ

term. The additional quartic terms involving ∆0 are

F4,u = u∆

(

∆⋆20 ∆Q∆−Q + ∆⋆2

0 ∆Q∆−Q + c.c.)

+ δ|∆0|2[|∆Q|

2 + |∆Q|2]

+ |∆0|2[

uρ

(

|ρQ|2

+∣

∣ρQ

∣

∣

2)

+ u′ρ

(

|ρ2Q|2

+∣

∣ρ2Q

∣

∣

2)]

+ v′|∆0|2[~S⋆

Q · ~SQ + ~S⋆Q · ~SQ]. (B2)

Let us now consider the effect of quenched disor-der. Following the discussion preceding Eq. (5.4), wewrite the order parameters in real space as ∆ (r) =|∆0| e

iθ0 +|∆Q| ei(θQ+Q·r)+|∆−Q| ei(θ−Q−Q·r) and ρ (r) =|ρQ| cos (Q · r + φQ) + |ρ2Q| cos (2Q · r + φ). Let us as-sume that the disorder nucleates a point defect in theCDW, which in this case corresponds to a 2π vortex inthe phase φQ. By the gρ term in Eq. (B1), this in-duces a 4π vortex in φ. (Note that a in the presence

of ρQ, a 2π vortex in φ is not possible.) The γ∆ termin Eq. (5.3) then dictates a 2π vortex in the phaseθ− = (θQ − θ−Q) /2. However, unlike before, this vor-tex does not couple to the global superconducting phaseθ+ = (θQ + θ−Q) /2. Therefore, an arbitrarily small uni-form superconducting component is sufficient to removethe sensitivity of a striped superconductor to disorder,and the system is expected to behave more or less like aregular (uniform) superconductor.

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