Spin rotationally symmetric domain flux phases in underdoped cuprate superconductors

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Spin-rotationally symmetric domain flux phases in underdoped cuprates

Marcin RaczkowskiMarian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Krakow, Poland

Didier PoilblancLaboratoire de Physique Theorique UMR5152, CNRS & Universite de Toulouse III, F-31062 Toulouse, France

Raymond FresardLaboratoire CRISMAT, UMR 6508 CNRS–ENSICAEN,

6 Boulevard du Marechal Juin, F-14050 Caen Cedex, France

Andrzej M. OlesMarian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Krakow, Poland and

Max-Planck-Institut fur Festkorperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

(Dated: February 6, 2008)

We propose a new form of inhomogeneous phases consisting of out-of-phase staggered flux domainsseparated by diagonal charged domain walls centered either on bonds or on sites. Remarkably, suchdomain flux phases are spin-rotationally symmetric and exhibit cone-like quasiparticle dispersionnear the Fermi energy, as well as incommensurate order of orbital currents. Such features areconsistent with the pseudogap behavior and the diagonal stripes observed experimentally in lightlydoped cuprates. A renormalized mean field theory shows that these solutions with coexisting chargemodulation and charge currents are competitive ground state candidates within the t–J model.

PACS numbers: 74.72.-h, 71.45.Lr, 74.20.Mn, 75.40.Mg

I. INTRODUCTION

Among numerous new ideas and concepts that havebeen put forward to explain the unusual properties thehigh temperature superconductors (HTS), which go be-yond the conventional Fermi liquid theory,1 the staggeredflux (SF) phase2 attracts much attention as a candi-date for the pseudogap normal phase of the underdopedcuprates.3 Such a state is characterized by a checkerboardpattern of plaquette currents circulating clockwise andanticlockwise on two different sublattices so that the cor-responding flux flowing through each plaquette alternatesin sign.

On the one hand, using the SU(2) gauge invarianceof the Heisenberg model one can show that at half-fillingthe SF phase is equivalent to the d-wave superconductingwave function4 which has correctly reproduced severalkey experimental properties of the HTS.5 Moreover, itsGutzwiller-projected energy is in a very good agreementwith the best estimate for the ground-state energy of thetwo-dimensional undoped Heisenberg antiferromagnet.1

On the other hand, even though a finite doping removesthis degeneracy and stabilizes d-wave superconductivityin the ground state,6 the SF phase is the lowest-energyGutzwiller-projected nonsuperconducting state that hasbeen constructed so far,7 and its energy spectrum re-mains similar to the d-wave superconductor. Signa-tures of the SF pattern in the current-current correla-tion have been seen in the Gutzwiller-projected d-wavesuperconducting phase8 and in the exact ground-statewave-function of the t–J model.9 It has also been pro-posed that the hidden d-density wave (DDW) order of

the doped SF phase could be the origin of the mysteriouspseudogap behavior.10 Finally, it has been shown thatunder some circumstances the SF phase can coexist withd-wave superconductivity in the underdoped regime.11

However, the physics of the hole-doped cuprates seemsto be even more involved as the competition betweenthe superexchange interaction which stabilizes the anti-ferromagnetic (AF) long-range order in the parent Mottinsulator, and the kinetic energy of doped holes, mightlead to the formation of stripe phases with hole-rich re-gions and locally suppressed magnetic order, which wassuggested in early Hartree-Fock studies.12 In a stripephase two neighboring AF domains are separated by aone-dimensional domain wall (DW), where a phase shiftof π occurs in the AF order parameter. Later on, ex-perimental confirmation of the stripe phases has trig-gered a large number of studies devoted to their prop-erties within a number of methods which go beyond theHartree-Fock approach.13 Moreover, even though staticcharge and spin orders have only been observed in layeredcuprates, e.g., in La1.6−xNd0.4SrxCuO4 (Nd-LSCO) (seeRef. 14) and La2−xBaxCuO4 (see Ref. 15), while in bi-layered YBa2Cu3O6+δ (YBCO) only a stripe-like chargeorder and incommensurate spin fluctuations have beenreported,16 stripe phases quickly joined the list of can-didates for the pseudogap phase in the cuprates as theyare compatible with many experimental results.17

Although numerical simulations of microscopic modelsof correlated fermions, such as the t–J model (see later),are especially difficult, various signatures consistent with(i) DDW states, and (ii) stripe phases have been de-tected. In particular, the emergence of strong staggered

2

current correlations under doping the Mott insulator hasbeen reported in exact diagonalizations by Leung,9 andattributed to the formation of spin bipolarons.18 Thesefindings are consistent with an early observation of stag-gered spin chirality19 since charge degrees of freedomstrongly couple to spin scalar chirality. Interestingly,spin chirality/charge currents seem to compete with holepairing,20 and this issue requires a further careful consid-eration. Simultaneously with those findings, the obser-vation of stripes and checkerboard patterns (which alsoinclude some form of charge ordering) has also been con-firmed by density matrix renormalization group (DMRG)computations for some boundary conditions.21

We also note that an exotic SF phase with long-rangeorbital current order at half-filling (in contrast to thefully projected SF phase, see Ref. 3) was stabilized in var-ious extended Hubbard-like models (which include someform of charge fluctuations not present in the simplermodel discussed above) within ladder22 or bilayer23 ge-ometries. It was also shown that such a long-range DDWorder could survive with the emergence of stripe-like fea-tures under doping.24

Unfortunately, even though stripe phases seem to playimportant role in the physics of HTS, it is still not clearhow the stripes are connected, as a competing state, tod-wave superconductivity. Therefore, in this paper weintroduce a new class of wave functions with compos-ite order in a form of filled domain flux (FDF) phases,with one doped hole per one DW atom. In additionto capturing essential properties of the SF phases, theFDF structure accounts for the incommensurate diag-

onal spin peaks observed in lightly (x < 0.06) dopedLa2−xSrxCuO4 (LSCO)25 and Nd-LSCO.26 Thus, ourphase should allow one to obtain a smooth transitionfrom the insulating state at half-filling to the d-wave su-perconductor above a critical doping xc, with a concomi-tant change of the DW orientation into vertical stripesjust at xc, as observed experimentally in LSCO.27 Theexistence of such phases is suggested by recent variationalMonte-Carlo calculations which show an instability of theSF states towards phase separation,7 and we argue thatself-organization into flux domains separated by DWs isgeneric in the doped t–J model. Most pronounced fea-tures of these phases shown in Fig. 1(a,b) are: (i) dopedholes self-organize into diagonal DWs, (ii) DWs separateweakly doped SF domains with a smoothly modulatedmagnitude of the flux within them, (iii) DWs introducea phase shift of π in the flux phase and the SF domainsalternate, and finally (iv) in contrast to the so-called com-mensurate flux (CF) phases, the total flux vanishes, andtherefore no asymmetry of the magnetic response is ex-pected when reversing the direction of an applied mag-netic field. In fact, these FDF phases have strong similar-ities with the solution obtained in Ref. 28 using uniform(i.e., site independent) Gutzwiller factors.

The paper is organized as follows. The t-J model andits treatment in the Gutzwiller approximation are intro-duced in Sec. II. The properties of locally stable domain

flux phases with either bond-centered or site-centered do-main walls are presented in Sec. III. The paper is con-cluded in Sec. IV by pointing out certain possibilities ofexperimental verification of the suggested type of orderand by a short summary of main results.

II. MODEL AND FORMALISM

We consider the t-J model,29

H = −∑

〈ij〉,σ

tij(c†iσ cjσ + h.c.) + J

∑

〈ij〉

Si · Sj , (1)

which is believed to describe the physics of the HTS.5

Here the summations include each bond 〈ij〉 only once.Next, the local constraints that restrict the hopping pro-

cesses ∝ c†iσ cjσ to the subspace with no doubly occu-

pied sites are replaced by statistical Gutzwiller weights,30

while decoupling in the particle-hole channel yields thefollowing mean field (MF) Hamiltonian,

HMF = −∑

〈ij〉,σ

tijgtij(c

†iσcjσ + h.c.) − µ

∑

iσ

niσ

− 3

4J

∑

〈ij〉,σ

gJij(χjic

†iσcjσ + h.c. − |χij |2), (2)

with the self-consistency conditions for the bond-orderparameters

χji = 〈c†jσciσ〉. (3)

In principle, simultaneous decoupling in the particle-particle channel is also possible,31 but since we are inter-ested in the diagonal DWs similar to the ones observedin the underdoped LSCO family,25,26 we focus here onnonsuperconducting solutions. In particular we choosex = 1/16, one of the magic doping fractions at whichlow-temperature in-plane resistivity of LSCO is weaklyenhanced suggesting a tendency towards charge order.32

Here, to allow for small non-uniform charge modulations,the Gutzwiller weights have been expressed in terms oflocal doped hole densities

nhi = 1 −∑

σ

〈c†iσciσ〉 (4)

as follows:33

gtij =

√zizj, gJ

ij = (2 − zi)(2 − zj), (5)

with zi = 2nhi/(1 + nhi). For simplicity, results shownbelow correspond to nearest neighbor hopping tij = tonly.34 Thanks to developing an efficient reciprocal spacescheme by making use of the symmetry,35 the calcula-tions were carried out on a large 256× 256 cluster at lowtemperature βJ = 500, which eliminates the finite sizeeffects.

3

++

+

+ +

+

++

+

+ +

++

+

+

+

0 +

+ +

+

+ +0

0

0 +

+

+

+

++ + +

Φ Φ

Φ Φ Φ Φ

Φ Φ Φ Φ

Φ Φ Φ Φ Φ

Φ Φ Φ Φ Φ Φ Φ Φ

ΦΦΦΦ

Φ

Φ Φ

(a)

(b)

(c)

FIG. 1: (color online) Spatial modulation of the hole density nhi (circles), bond-order parameter χij (lines with arrowsindicating the direction of charge currents), and flux Φ� defined by Eq. (6) (positive/negative flux indicated by symbols +/−)distribution found in two FDF phases at hole doping x = 1/8 and t/J = 3. Circle diameters are proportional to the dopedhole densities; widths of bond lines connecting them are proportional to the magnitudes of the bond-order parameters χij ,while the magnitude of flux flowing through each plaquette is represented by the size of +/− symbol. Two distinct phases are:(a) bond-centered FDF phase with a vanishing current (dashed lines) at the DW bonds; (b) site-centered FDF phase with avanishing flux (indicated by 0) at the DW plaquettes. Panel (c) shows the self-consistent CF phase (t = 0) characterized bythe uniform fictitious flux Φ� = 1

2(1 − x), as well as by homogeneous charge distribution.

Our starting point is the CF phase, a wave functionwhich, away from half-filling, displays remarkable com-mensurability effects at special fillings and fulfills the self-consistency condition at t = 0.28 Indeed, in the limitxt/J → 0, the magnetic (superexchange) energy in theCF phase exhibits a minimum when the fictitious flux(in unit of the flux quantum), flowing through each pla-quette and defined by a sum over the four bonds of theplaquette

Φ� =1

2π

∑

〈ij〉∈�

Θij , (6)

where Θij is the phase of χij , follows exactly the fillingfraction, i.e., Φ� = 1

2(1 − x). In this case, Hamiltonian

(2) reduces to the Hofstadter Hamiltonian describing themotion of an electron in a uniform magnetic flux assumedto be rational Φ� = p/q.36 Therefore, the peculiar prop-erty of the superexchange energy follows from the CFphase band structure with q bands and the Fermi levellying in the largest gap above the pth subband. As aresult, the modulus of the bond-order parameter χij (3),the spin correlation and the hole density are all spatiallyuniform [see Fig. 1(c)]. However, infinitesimally smallxt/J selects a special arrangement of the phases {Θij}so as to optimize the kinetic energy term ∝

∑ij cosΘij

and should produce an inhomogeneous structure.28

Within this class of singlet (nonmagnetic) wave func-tions, competing with possible inhomogeneous solutions(see later), the uniform SF phase also offers a very good

compromise between the magnetic (EJ ) and kinetic (Et)energy. For small t and x, the kinetic energy is mini-mized (within the MF approach) when all phases of χij

are set to a constant Θij = ±π/4, corresponding to al-ternating fluxes Φ� = ±0.5 (SF phase). Increasing xt/Jgradually reduces |Φ�| and drives the system towards aFermi liquid state (with real χij) in a continuous way.

III. DOMAIN FLUX PHASES

Starting with initial parameters corresponding to auniform CF phase, the self-consistent procedure leadsto new FDF solutions which could explain a diagonalspin modulation observed experimentally in the insu-lating regime of LSCO25 and Nd-LSCO,26 usually in-terpreted in terms of diagonal stripes, even though nosignatures of any charge modulation were observed yet.This conjecture is also supported by the recent neutronscattering studies of the Ni impurity effect on the diago-nal incommensurability in LSCO.37 Indeed, doping by Niquickly suppresses the incommensurability and restoresthe Neel state. This indicates a strong effect on hole lo-calization and thus favors the presence of charge stripeswith mobile holes rather than the spiral order with local-ized hole spins.

Interestingly, we found two types of topologically dif-ferent but nearly degenerate solutions which both havethe same size of the unit cell (see Fig. 1): (i) a bond-

centered FDF phase, very similar to the original CF one,

4

where each DW is characterized by a zero current stair-case and by a maximum of the hole density spread overthe related bonds [Fig. 1(a)], as well as (ii) a site-centered

FDF phase, where the DWs are characterized by zero flux

plaquettes ordered along a diagonal line and by a maxi-mum of the hole density centered at two of their cornersites [Fig. 1(b)]. Apart from local doped hole densities{nhi}, bond quantities are needed for a full characteriza-tion of both phases (here we use a short-hand notation):— the spin correlation

Si = −3

2gJ

i,i+x|χi,i+x|2, (7)

— the bond charge hopping

Ti = 2gti,i+xRe{χi,i+x}, (8)

— the charge current

Ii = 2gti,i+xIm{χi,i+x}, (9)

— as well as the modulated flux

Φπi = (−1)ix+iyΦi,i+x, (10)

with a phase factor (−1)ix+iy compensating the modula-tion of the flux within a single domain of the SF phase.Typical profiles of the above defined observables at lowdoping are depicted in Fig. 2.

The stability of the FDF phases originates from a sub-tle competition between the magnetic EJ and kinetic en-ergies Et. Let us first focus on the t/J → 0 limit wherethe site-centered SF phase is stable and very competitive(among the nonmagnetic states), in contrast to the bond-

centered one. This extreme case corresponds to the lo-calization of doped holes at DWs and the superexchangeenergy in the SF domains is best optimized. Indeed, byexpelling holes from the SF domains one reinforces locallythe AF correlations with a concomitant reduction of bothbond charge and current correlations. On the contrary,due to a large hole density, both these tendencies are re-versed around the DWs. However, increasing t/J leadsto a much broader charge spatial distribution in the unitcell as a larger fraction of holes enters the SF domains(see Fig. 2). Nevertheless, both FDF phases remain com-petitive even in the regime of large (realistic) values oft/J ∼ 3 due to: (i) enhanced short-range AF correla-tions deep in the SF domains (Si ≃ −0.33 compared toS ≃ −0.28 in the uniform phase), where the fictitiousflux approaches the special value Φ = 1

2(local minimum

of the kinetic energy in the limit xt/J → 0), and (ii)strongly enhanced bond charge accumulated around theDWs, typically three times larger than that in the SFphase, due to both amplification of the gt

ij factors andreduced (vanishing) fictitious flux flowing through thebond-centered (site-centered) plaquettes at the DWs.

Of particular interest is whether one can also stabilizewithin the present formalism the so-called half-filled do-main flux (HDF) phases, analogous to half-filled stripes

0.00

0.05

0.10

0.15

T

0.0

0.1

0.2

0.3

0.4

n h

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

S

0 8 16 24 32i

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6Φ

π

0.0

0.1

0.2

0.3

0.4

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

0.00

0.05

0.10

0.15

0 8 16 24 32i

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

FIG. 2: (color online) (a,e) Hole density nhi (4), (b,f) spincorrelation Si (7), (c,g) bond charge Ti (8), and (d,h) modu-lated flux Φπi (10) in the bond-centered (left) and site-centered

(right) FDF phases at x = 1/16 for: t/J = 1 (triangles), andt/J = 3 (squares). For comparison, circles depict the relatedt/J → 0 solutions: the CF phase with uniform fictitious fluxΦ = 15/32 (left) and a two-domain |Φ| = 1

2SF phase (right).

with one hole per two atoms in a DW as observed in thecuprates around x = 1/8.14,15 On the one hand, bothself-consistent bond- and site-centered HDF phases foundat x = 1/16 and t/J = 3 have a somewhat higher totalenergy per site (F ≃ −1.03J) than those obtained forboth degenerate FDF ones (F ≃ −1.07J), and for theuniform SF phase (−1.09J). However, Table I showsthat all domain flux phases become very competitive atx = 1/8, not only with respect to the SF phase but alsowith respect to a recently proposed nonuniform 4 × 4superstructure.31 Note also that while the FDF phasesoptimize mainly EJ , the HDF ones are characterized byrather low Et. Therefore, we predict that large t/J ratherfavors the domain flux phases with partially filled DWs.We argue that quantum fluctuations are likely to stabi-

5

TABLE I: Kinetic energy per hole Eh (in units of t), andkinetic energy Et, magnetic energy EJ , free energy F (all persite in units of J) for the locally stable phases: bond-centered

HDF(1) site-centered HDF(2), 4× 4 checkerboard, FDF, andSF one, as found at hole doping x = 1/8 and t/J = 3. FDF(1)and FDF(2) phases are fully degenerate. The lowest energyincrements are given in bold characters.

phase Eh Et EJ FHDF(1) −2.7856 −1.0446 −0.4028 −1.4474HDF(2) −2.7843 −1.0441 −0.4026 −1.4467

4 × 4 −2.7128 −1.0173 −0.4348 −1.4521FDF −2.7067 −1.0150 −0.4418 −1.4568SF −2.7587 −1.0345 −0.4246 −1.4591

lize them, in analogy to the half-filled stripe phases,13 orto the fully projected 4 × 4 checkerboard wave functionwhich was recently shown to be more stable than theuniform SF phase.38 This suggests that other inhomoge-neous solutions might be stable as well. Unfortunately, adirect comparison of our singlet wave functions to theoriginal (magnetic) stripe phases12 is not possible yetsince both are described within two entirely different for-malisms. Hence further studies using more sophisticatedmethods (like projected wave functions as in Ref. 38) areneeded.

An experimental support of the FDF phases fol-lows from angle-resolved photoemission (ARPES) exper-iments on lightly doped LSCO that show a strongly sup-pressed spectral weight near the pseudogapped X =(π, 0) and Y = (0, π) points, and a quasiparticle bandcrossing the Fermi energy µ along the nodal Γ−M direc-tion, with M = (π, π).39 Both features are qualitativelyreproduced in the FDF phases – the electronic bands arealmost dispersionless along the X−Y direction, and a gapopens at ω = µ (Fig. 3), indicating that transport across

Γ S M X S Y Γ

-4

-2

0

2

4

(ω−µ

)/J

-0.2

0.0

0.2

FIG. 3: (color online) Electronic structure of the site-centered

FDF phase (solid lines) and SF phase (dashed lines) alongthe main directions of the Brillouin zone for x = 1/16 andt/J = 3. Inset shows a pseudogap between the FDF bandsalong the X − Y direction near the Fermi energy µ (thindashed line).

the DWs is suppressed. However, the most salient featureof the electronic structure in FDF phases is a relativis-tic cone-like dispersion around the S = (π/2, π/2) point.Indeed, massless Dirac excitations are at the heart of thequantum electrodynamics in (2+1) dimensions (QED3)theory of pseudogap in the cuprates.40 This feature isalso found in the SF phase, but for the uniform flux andhole distribution it occurs away from the Fermi energy µ.The shape of the electronic structure in the FDF phasedepends on the actual value of t/J . Firstly, a strong lo-calization of holes at DWs in the limit t/J → 0 pushesthe top of the lower band cone well below µ. Secondly,finite t weakens the stripe order so that the gap betweenthe lower and upper band at the S point is reduced. Afurther increase of t pushes some lower band states aboveµ enabling transport along the DWs.

IV. DISCUSSION AND SUMMARY

For possible experimental verification of the presentproposal it is important to realize that orbital cur-rents of the domain flux phase give rise to weak mag-netic fields (that should be experimentally distinguish-able from the copper spins). Muon spin rotation (µSR)technique is an extremely sensitive local probe especiallysuited to study small modulations of local fields. Earlierestimations41 give 10 to 100 Gauss corresponding roughlyto 0.03 to 0.25 µB in cuprates. In fact, incommensu-rate order in the LSCO family seen in neutron scatteringmeasurements,25,26 (with a large but finite correlationlength) might be attributed, at least partly, to the ex-istence of orbital moments. Finally, note that althoughthe phases considered here do not break SU(2) symme-try and do not exhibit AF long range order, on generalprinciple they can still sustain AF correlations on largedistances (i.e., beyond nearest neighbor sites) betweencopper spins.

In summary, we have introduced and investigated anew class of flux phases that unify the remarkable prop-erties of the SF uniform phase with the incommensu-rate magnetic correlations established in the underdopedcuprates. Bond- and site-centered FDF phases are nearlydegenerate which indicates strong fluctuations which areexpected to be amplified, either for increasing t/J or forincreasing doping x. As these phases are only marginallyunstable at the MF level, they might be stabilized byquantum effects and explain the low temperature physicsof the cuprates in the low doping regime, where a pseu-dogap phase forms at higher temperature. Therefore,the solutions presented here could be viewed as a low-temperature instability of the nearby DDW pseudogapphase (stable at higher temperature but below T ∗) in thesame way as the ”ordinary” stripe phases could be seenas an instability of the nearby doped AF Neel state atinfinitesimal x. Therefore, our proposal calls for a searchof experimental signatures of domain flux phases in theunderdoped cuprates, especially in the LSCO family.

6

Acknowledgments

We thank M. M. Maska and Z. Tesanovic for insight-ful discussions. We acknowledge support by the thePolish Ministry of Science and Education under ProjectNo. 1 P03B 068 26, by the internal project granted by

the Dean of Faculty of Physics, Astronomy and AppliedComputer Science of the Jagellonian University, as wellas by the Ministere Francais des Affaires Etrangeres un-der POLONIUM contract No. 09294VH. D.P. thanksthe “Agence Nationale pour la Recherche” (ANR) forsupport.

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