Magnetism and superconductivity of strongly correlated electrons on the triangular lattice

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Magnetism and superconductivity of strongly correlated electrons

on the triangular lattice

Cedric Weber,1 Andreas Laeuchli,1 Frederic Mila,2 and Thierry Giamarchi3

1Institut Romand de Recherche Numerique en Physique des Materiaux (IRRMA), CH-1015 Lausanne, Switzerland2Institute of Theoretical Physics, EPFL, CH-1015 Lausanne, Switzerland

3DPMC, University of Geneva, Quai Ernest Ansermet 24, CH-1211 Geneva, Switzerland(Dated: February 2, 2008)

We investigate the phase diagram of the t−J Model on a triangular lattice using a VariationalMonte-Carlo approach. We use an extended set of Gutzwiller projected fermionic trial wave-functions allowing for simultaneous magnetic and superconducting order parameters. We obtainenergies at zero doping for the spin-1/2 Heisenberg model in very good agreement with the bestestimates. Upon electron doping (with a hopping integral t < 0) this phase is surprisingly stablevariationally up to n ≈ 1.4, while the dx2

−y2 + idxy order parameter is rather weak and disappearsat n ≈ 1.1. For hole doping however the coplanar magnetic state is almost immediately destroyedand dx2

−y2 + idxy superconductivity survives down to n ≈ 0.8. For lower n, between 0.2 and 0.8,we find saturated ferromagnetism. Moreover, there is evidence for a narrow spin density wave phasearound n ≈ 0.8. Commensurate flux phases were also considered, but these turned out not to becompetitive at finite doping.

PACS numbers: 74.72.-h, 71.10.Fd, 74.25.Dw

I. INTRODUCTION

The discovery of high-Tc superconductivity, and theobservation2 that strong correlations are important inconnection with these compounds has led to a tremen-dous interest in understanding strongly correlated elec-tron physics. In particular the two simplest models forstrongly correlated electrons, namely the Hubbard andt−J models, have been the subject of intensive studies.

For example, one question of crucial interest is theinterplay between superconductivity and antiferromag-netism close to the insulating phase in the t−J model.The ground state of this model on the square latticeis known to be antiferromagnetic at half-filling and oneof the important questions is what happens upon dop-ing. All the approaches to these strong coupling prob-lems involve approximations, and it is sometimes difficultto distinguish the artefact due to approximations fromthe true features of the model. However, for the caseof the square lattice, both the variational Monte-Carlomethod (VMC)14,35 and mean-field theories21 have founda d-wave superconducting phase in the the t−J model.A wavefunction combining antiferromagnetism and su-perconductivity was proposed for the Hubbard and t−Jmodels12,13, allowing to reconcile the variational resultsbetween these two models. This wavefunction allowed foran excellent variational energy and order parameter anda range of coexistence between superconductivity andanti-ferromagnetism was found. Further investigationsof this class of wavefunctions has been very fruitful forthe square lattice. This allowed to successfully compareto some of the experimental features with the high-Tc

cuprates3,23, even if of course many questions remain re-garding the nature of the true ground state of the system.

The resonating valence bond (RVB) scenario proposedby Anderson2 was argued to be even more relevant in

the geometry of the triangular lattice. At half-filling, thelattice is a frustrated magnet: the competition betweenthe exchange integrals leads to unsatisfied bonds. Theoriginal expectation is that quantum fluctuations mightlead to a spin-liquid behavior. However, at half-filling, itappears by now that the spin-1/2 triangular lattice hasa three sublattice coplanar magnetic order7,9,18. Quan-tum fluctuations are nevertheless strong, and the sub-lattice magnetization is strongly reduced due to thesefluctuations. It is therefore expected that the magnetismis fragile and quickly destroyed by doping and that astrong RVB instability is present. Indeed, RVB meanfield theories4,22,32 were used for the t−J model, anddx2−y2 +idxy pairing was found over a significant range ofdoping. The same approach and questions that arise inthe framework of the t−J model on the square lattice arethus very relevant in the present frustrated lattice. Thesuccess of the variational approach for the square lat-tice suggests to investigate the same class of variationalwavefunctions for the triangular one.

Besides its own theoretical interest, another motiva-tion for understanding the physics of electrons on a tri-angular lattice is provided by the recent discovery ofsuperconductivity at low temperature in the CoO2 lay-ered compounds30 (NaδCoO2.yH2O). In these systems,superconductivity is observed in a range of electron dop-ing δ between 25% and 33%26. NaδCoO2.yH2O consistsof two dimensional CoO2 layers separated by thick in-sulating layers of Na+ ions and H2O molecules. It isa triangular net of edge sharing oxygen octahedra; Coions are at the center of the octahedra forming a 2Dtriangular lattice. Takada et al.30 speculated that thissystem might be viewed as a doped spin-1/2 Mott insu-lator. Based on LDA calculations28, a simplified singleband t−J picture with negative t and electron dopingwas put forward4,22,32. Such systems might thus be the

2

long-sought low-temperature resonating valence bond su-perconductor, on a lattice which was at the basis of An-derson’s original ideas on a possible quantum spin liquidstate1,11.

We propose in this paper to study the t−J modelwithin the framework of the Variational Monte-Carlo(VMC) method, which provides a variational upperbounds for the ground state energy. In contrast to mean-field theory, it has the advantage of exactly treating theno double-occupancy constraint. VMC using simple RVBwave-functions has been used for the triangular lattice34

and it was found that dx2−y2 + idxy superconductivity isstable over a large range of doping. However, in the previ-ous study the fact that the t−J is magnetically orderedat half-filling was not taken into account. We expectthat the frustration in the triangular lattice may leadto a richer phase diagram and to many different insta-bilities. We thus propose here to study extended wave-functions containing at the same time magnetism, fluxphase and RVB instabilities, in a similar spirit as for thesquare lattice12,13, in order to study in detail the inter-play between frustrated magnetism and superconductiv-ity. Given the non-collinear nature of the magnetic orderparameter compared to the case of the square lattice,the task is however much more complicated. We thuspresent in this work a general mean-field Hamiltonianwhich takes into account the interplay between magnetic,RVB and flux-phase instabilities. The resulting varia-tional wave-function is sampled with an extended VMC,which uses Pfaffian updates rather than the usual de-terminant updates. We show that the interplay betweenthe different instabilities leads to a faithful representa-tion of the ground-state at half-filling, and we also findgood variational energies upon doping. To benchmarkour wave-function, we carry out exact diagonalizationson a small 12 sites cluster and compare the variationalenergies and the exact ones. Finally, a commensuratespin density wave is considered, and is shown to be rele-vant for the case of hole doping.

The outline of the paper is as follows: in Sec. IIwe present the model and the numerical technique. InSec. III we show the variational results both for the caseof hole and electron doping. Finally Sec. IV is devotedto the summary and conclusions.

II. MODEL AND METHOD

We study the t−J model on the triangular lattice de-fined by the Hamiltonian:

Ht−J = −t∑

〈i,j〉,σ

(

c†i,σcj,σ + h.c.)

+ J∑

〈i,j〉

(

Si · Sj −1

4ninj

)

(1)

The model describes electrons hopping with an amplitudet, and interacting with an antiferromagnetic exchange

term J between nearest neighbor sites (denoted 〈i, j〉)of a triangular lattice. Si denotes the spin at site i,

Si = 12c

†i,α~σα,βci,β and ~σ is the vector of Pauli matri-

ces. Ht−J is restricted to the subspace where there areno doubly occupied sites. In order to simplify the connec-tion to the Cobaltates we set t = −1 in the following andpresent the results as a function of the electron densityn ∈ [0, 2], half-filling corresponding to n = 1. n > 1 cor-responds to a t−J model at n = 2−n for t = 1, by virtueof a particle-hole transformation. In the first part of thissection, we emphasize on the method to construct a vari-ational wave-function containing both superconductivityand non-collinear magnetism. The wave-function allowsto consider 3-sublattice magnetism, however, since thelatter wavefunction is restricted to a 3 site supercell, webriefly comment on a second simpler variational wave-function type, which allows to describe commensuratespin order. In the second part of the section, we definethe relevant instabilities and the corresponding order pa-rameters.

A. Variational wave-function

In order to study this model we use a variational wave-function built out of the ground state of the followingmean-field like Hamiltonian:

HMF =∑

〈i,j〉,σ

(

−teiθσi,jc†iσcjσ + h.c.

)

+∑

〈i,j〉,σ,σ′

(

{∆σ,σ′}i,j c†iσc

†jσ′ + h.c.

)

+∑

i

hi · Si − µ∑

i,σ

ni,σ (2)

HMF contains at the same time BCS pairing (∆i,j ={∆σ,σ′}i,j), an arbitrary external magnetic field (hi),

and arbitrary hopping phases (θσi,j), possibly spin depen-

dent. These variational parameters are unrestricted onthe A,B,C sites and the corresponding bonds of a 3-site supercell, as shown in Fig. 1. We allow both singlet

(∆(S=0)i,j ) and general triplet (∆

(S=1)i,j ) pairing symme-

tries to be present. They correspond to choosing:

∆(S=0)i,j =

(

0 ψi,j

−ψi,j 0

)

∆(S=1)i,j =

(

ψ2i,j ψ1

i,j

ψ1i,j ψ3

i,j

) (3)

Since HMF is quadratic in fermion operators it canbe solved by a Bogoliubov transformation. In the mostgeneral case considered here, this gives rise to a 12 × 12eigenvalue problem, which we solve numerically. We thenfind the ground state of HMF

|ψMF 〉 = exp

∑

i,j,σi,σj

a(i,j,σi,σj)c†iσic†jσj

|0〉 (4)

3

A B

C

a1

CC

A

B A

B

C AB

a2a3

FIG. 1: 3-site supercell of the triangular lattice. The onsitemagnetic variational parameters can vary independently oneach of the site A,B and C of the supercell. The BCS pairingas well as the flux vary independently on each of the differentdashed bonds.

Here a(i,j,σi,σj) are numerical coefficients. Note that|ψMF 〉 has neither a fixed number of particles due tothe presence of pairing, nor a fixed total Sz due to thenon-collinear magnetic order. Thus in order to use itfor the VMC study we apply to it the following projec-tors: PN which projects the wave-function on a statewith fixed number of electrons and PSz which projectsthe wave-function on the sector with total Sz = 0. Fi-nally we discard all configurations with doubly occupiedsites by applying the complete Gutzwiller projector PG .The wavefunction we use as an input to our variationalstudy is thus:

|ψvar〉 = PGPSzPN |ψMF 〉

= PGPSz

∑

i,j,σi,σj

a(i,j,σi,σj)c†iσic†jσj

N/2

|0〉 (5)

Although the wavefunction (5) looks formidable, it can bereduced to a form suitable for VMC calculations. Using

〈α| = 〈0| ck1,σ1...ckN ,σN

, (6)

we find that

〈α | ψvar〉 = Pf (Q)

Qi,j = a(ki,kj ,σi,σj) − a(kj ,ki,σj ,σi)(7)

where Pf (Q) denotes the Pfaffian of the matrix Q. Usingthis last relation, the function (5) can now be evaluatednumerically using a Monte Carlo procedure with Pfaffianupdates, as introduced in Ref. 8. In the particular casewhere ak,l,↑,↑ = ak,l,↓,↓ = 0 and at Sz = 0 (this happensif the BCS pairing is of singlet type and the magneticorder is collinear), the Pfaffian reduces to a simple de-terminant, and the methods becomes equivalent to thestandard Variational Monte-Carlo10 technique.

The above mean field Hamiltonian and wavefunctioncontain the main physical ingredients and broken sym-metries we want to implement in the wavefunction. Inorder to further improve the energy and allow for out ofplane fluctuations of the magnetic order we also add anearest-neighbor spin-dependent Jastrow19 term to thewave-function:

PJ = exp

α∑

〈i,j〉

Szi S

zj

, (8)

where α is an additional variational parameter. Our finalwavefunction is thus:

|ψvar〉 = PJPSzPNPG |ψMF 〉 (9)

When α < 0 the Jastrow factor favors all configurationswhich belong to the ground state manifold of a classicalIsing antiferromagnet on the triangular lattice. Such amanifold is exponentially large33, and this Jastrow fac-tor thus provides a complementary source of spin fluctu-ations.

In what follows we use the wavefunction (9) directly forthe VMC, but we also examine improved wavefunctionswith respect to (9) that can be obtained by applying oneor more Lanczos steps5,15,16:

|1Ls〉 = (1 + λHt−J) |ψvar〉 (10)

with optimized29 λ. Since the calculation of Lanczos stepwave functions beyond the first step is very time consum-ing, most of the results we will present here were obtainedusing a single Lanczos step.

In the following, to clearly indicate which wavefunc-tion we use, we will denote them in the following way:MF / J / nLs, where MF denotes the fields present inthe mean-field like Hamiltonian HMF , J is present if weuse the Jastrow factor, n Ls denotes the presence andthe number of Lanczos steps applied on top of the barewave function.

As usual with the VMC procedure, these generalwave functions are now used to minimize the expecta-tion value of the total energy 〈Ht−J〉 by changing thevariational parameters. We used a correlated measure-ment technique12,13,31 combined with parallel processingto smoothen the energy landscape and use a steepest-descent type routine to locate the minimum of energy.We then define the condensation energy ec of the opti-mal wave function as

ec = evar. − eGutzwiller, (11)

where eGutzwiller is the energy of the Gutzwiller wavefunction, i.e. the fully projected Fermi sea at zero mag-netization. In some cases we had to keep a small BCSpairing field to avoid numerical instabilities. Let us notethat the linear size of the Q matrix is two times largerthan in the simpler case of determinantal update VMC.Therefore our largest 108 sites cluster with Pfaffian up-dates corresponds roughly to a 200 sites cluster usingstandard updates.

4

B. Commensurate order

Since the mean-field Hamiltonian (2) is restricted toa 3 site supercell, we investigated also a second class ofmean-field Hamiltonians based on collinear commensu-rate structures, which are not contained in the previousHamiltonian. For this type of phase, we used a simplermean-field ansatz along the lines of Ref. 13. The mean-field Hamiltonian written in k-space is

HSDW =∑

k,σ

(

(ǫk − µ) c†kσckσ + f(Q, σ)c†k+Qσckσ

)

+∑

k

(

∆kc†k↑c

†−k↓ + h.c.

)

, (12)

where k does run over the Brillouin zone of the originaltriangular lattice, ǫk is the dispersion of the free electronHamiltonian, and ∆k is the Fourier transform of ∆i,j .Depending on f(Q, σ), the ground state of the Hamilto-nian is a commensurate charge density wave (f(Q, σ) =f(Q)) or a spin density wave (f(Q, σ) = σf(Q)).

At half-filling, we considered also several commensu-rate flux phases with 2π× q

p flux per plaquette, using the

Landau gauge36 with p ∈ {2, . . . , 10} and q < p. The onewith the lowest energy was found to be the q = 1, p = 4,as predicted theoretically24, giving an energy close to thesimple dx2−y2 + idxy wave-function. Upon doping how-ever the energy of such commensurate flux phases arerapidly much worse than the energies of our best wavefunctions. The main reason for this poor performanceupon doping is the rather bad kinetic energy of thesewave functions.

C. Characterization of the encountered instabilities

By minimizing all the variational parameters of themean-field Hamiltonians (2) and (12) on a 12 and a 48site lattice, we find that the relevant instabilities presentat the mean field level consist of:

• a 120◦ coplanar antiferromagnetic order (AF ), rep-resented in Fig. 2.

• a concomitant staggered spin flux phase instability(SFL) with:

θi,i+a1,σ = θσ

θi,i+a2,σ = −θσθi,i+a3,σ = θσ.

These bond phase factors correspond to a spin cur-rent in the z direction which is staggered on ele-mentary triangles of the triangular lattice. Thisinstability follows rather naturally, since the 120◦

antiferromagnetic state itself already displays thesame staggered spin currents (Si×Sj)

zon the near-

est neighbor bonds (see Fig. 2). The effect of this

A B

C

C

A

C A

+

+

+-

- -

FIG. 2: The variational parameters hi for the coplanar 120◦

antiferromagnetic order. The spins lie in the x−y plane. Thez-component of the vector chirality (±1) on each triangularplaquette forms a staggered pattern.

instability was rather small and visible only at half-filling.

• a translationally invariant superconducting phasewith dx2−y2 + idxy singlet pairing symmetry (d+),as well as the dx2−y2 − idxy (d−). We have alsolooked extensively for triplet pairing for both elec-tron and hole dopings and low J/|t| ≤ 0.4 on a 48site cluster, but with no success. The minimum en-ergy was always found for singlet pairing symmetry.

• a ferromagnetic state with partial or full polariza-tion (F ),

• a commensurate collinear spin density wave17

(SDW ) instability with wavevector QN =

(π,−π/√

3).

D. Order parameters

In order to characterize the phases described by the op-timal wave functions after projection, we have calculatedthe following observables:

• the sublattice magnetization of the 120◦ coplanarantiferromagnetic order:

MAF =1

N

∑

i

∥

∥

∥

∥

〈ψvar|Si|ψvar〉〈ψvar | ψvar〉

∥

∥

∥

∥

(13)

We have checked that the projected magnetizationhas the correct 120◦ symmetry. To simplify the cal-culations, this expectation value has been sampledusing wavefunctions without the projector PSz . Wehave checked that this gives the same result as thecorrelation function limr→∞

√

Si · Si+r calculatedwith the projector PSz .

5

• the z component of the vector chirality (spin twist)on nearest neighbor bonds:

χ =1

3N

∑

i,α

∣

∣

∣

∣

〈ψvar| (Si×Si+aα)z |ψvar〉

〈ψvar | ψvar〉

∣

∣

∣

∣

(14)

We have checked that the measured vector chiralityhas the symmetry of the staggered currents derivedfrom the 120◦ coplanar structure (c.f. Fig. 2).

• the amplitude of the absolute value of the collinearmagnetization in the spin density wave wave-function :

MSDW =1

N

∑

i

∣

∣

∣

∣

〈ψvar |Szi |ψvar〉

〈ψvar|ψvar〉

∣

∣

∣

∣

(15)

• the amplitude of the singlet superconducting orderparameter:

∆ =

√

√

√

√

√

1

4N

∣

∣

∣

∣

∣

∣

limr→∞

∑

i

⟨

ψvar|∆†i,α∆i+r,β |ψvar

⟩

〈ψvar | ψvar〉

∣

∣

∣

∣

∣

∣

, (16)

where

∆†i,α = c†i,↑c

†i+aα,↓ − c†i,↓c

†i+aα,↑. (17)

The angular dependence of the real space correla-tions corresponds to those of the unprojected pair-ing symmetry. We have also checked that the valueof ∆ is independent of the choice of α and β.

III. RESULTS AND DISCUSSION

A. Half-filling

We consider in this section the Heisenberg model(which is the limit of the t−J model at half-filling, upto a constant) The comparison with the large body ofexisting results for the Heisenberg model allows us tobenchmark the quality of our wave-function.

Let us first briefly discuss the symmetry of the vari-ational parameters at half-filling. The variational mag-netic field minimizes the energy for the two degenerate120◦ configurations. We found that the BCS pairingsymmetry in the presence of AF order is of d+ type,whereas the d−, the dx2−y2 and the dxy pairings haveclose but higher energies. Finally, a staggered spin fluxvariational order improves a little bit the energy. In-terestingly, this latter variational order is present in theground state of the classical Heisenberg model. However,this instability was only relevant at half-filling, and theenergy gain when δ > 0 is not significant. The variousenergies for these wavefunctions are shown in Fig. 3. TheAF +d+ +SFL/J/Ls wave-function is thus the best ap-proximation, within our variational space, of the ground

0 0.002 0.004

N-3/2

-0.56

-0.54

-0.52

-0.50

-0.48

3<S

i.Sj>

d+

AF+d+ / J

AF+d++SFL / J

AF+d+ / J / 1Ls

AF+d++SFL / J / 1Ls

Sindzingre et al. & Capriotti et al.

FIG. 3: Energy per site e = 3〈Si · Sj〉 of the differentvariational wave functions for the Heisenberg model versusthe system size N−3/2, with N = 36, 48, 108 sites. Or-dered by increasing condensation energy we find: d+ (opensquares), AF + d+/J (open triangles), AF + d+ + SFL/J(open diamonds), AF + d+/J/Ls (full triangles) and theAF + d+ + SFL/J/Ls (full diamonds). The stars are thebest estimates of the ground state energy available in theliterature9,27.

TABLE I: Comparison of the average energy 3〈Si ·Sj〉 and theaverage magnetization MAF for the Heisenberg model (t−Jmodel at half-filling) in different recent works for the 36 sitecluster and the extrapolation to the thermodynamic limit.The energy and the sublattice magnetization are measuredfor our best wave-function (AF + d+ + SFL/J/1Ls) at half-filling.

〈3Si · Sj〉 MAF

36 sites latticeour best wf -0.543(1) 0.38

Capriotti et al.9 -0.5581 0.406exact diag6,7 -0.5604 0.400

∞×∞our best wf -0.532(1) 0.36

spin-wave results9 -0.540 0.25Capriotti et al.9 -0.545 0.21

state of the Heisenberg model. We compare its energywith other estimates of the ground state energy in theliterature (see Fig. 3 and Table I). The mixture of AFand d+ instabilities is improving a lot the energy, andour wave-function has significantly lower energies thanthe simple d+ wave-function, and has energies very closeto the best ones available. More precisely, we find in thethermodynamic limit an energy per site of e = −0.52Jfor our variational wave-function. Applying one Lanczosstep leads to a small further improvement of the energyto e = −0.53J . A summary of the energies and of the120◦ magnetization for the Heisenberg model are givenin Table I and in Fig. 3.

6

Inspection of these results shows that our wave-function has in the thermodynamic limit an energy only0.013J higher than the estimates of more sophisticated,but restricted to the undoped case, methods9. Indeed,these latter methods use pure spin variational wave-function as a starting point, restoring quantum fluctua-tions with a non-variational method. Let us point outhowever that these methods are not giving an upperbound on the true ground state energy, so the groundstate energy could in principle lie between this result andour variational one. Since our variational wavefunctionalready gives an excellent energy it would be interestingto check how the (non variational) methods used to im-prove the energy starting with a much cruder variationalstarting point would work with our variational wavefunc-tion and which energy it would give. We leave this pointfor future investigation however.

Our wavefunction shows a reduced but finite magneticorder that survives in the triangular Heisenberg antifer-romagnet (THA). The 120◦ magnetization of our wave-function is reduced by the BCS pairing down to 72% ofthe classical value (see Fig. 7) which is somewhat largerthan the spin-wave result. Thus in addition to havingan excellent energy, our wavefunction seems to capturethe physics of the ground state of the Heisenberg systemcorrectly. Let us note that the BCS order of the wave-function is destroyed by the full Gutzwiller projector athalf-filling. So, despite the presence of a variational su-perconducting order parameter, the system is of coursenot superconducting at half filling. Somehow the BCSvariational parameter helps to form singlets, which re-duces the amplitude of the AF order. This is very similarto what happens for the t−J model on the square lattice:the inclusion of a superconducting gap decreases the en-ergy and decreases also the magnetization from M ≈ 0.9down to M ≈ 0.7, which is somewhat larger than thebest QMC estimates (M ≈ 0.6, see Refs.13,25). Thus thewave-function mixing magnetism and a RVB gap seemto be interesting variationally, both in the square andtriangular lattice, to restore spin fluctuations that werefrozen in the pure classical magnetic wave-function. Forthe triangular lattice, the present work is the first at-tempt to reproduce the magnetic order in the THA interms of a fermionic representation, which gives resultsin good agreement with other methods. The great ad-vantage of this approach is of course that the fermioniclanguage allows to directly consider the case of hole andelectron doping in the AF background, which is the casewe consider in the following sections.

B. Electron doping: n ∈ [1, 2]

Very few results exist away from half-filling, so in or-der to have a point of comparison for our variational ap-proach we will compare it with exact diagonalizations onvery small clusters. Having ascertained that our wave-function is indeed in good agreement with the exact re-

sults on a small cluster, we can then use it with confidenceto describe much larger systems and extract the physicsof the thermodynamic limit.

Therefore, we start by comparing on a 12 site clusterdifferent wave-functions with the exact-diagonalizationresults for the case of electron doping (see Fig. 4), sincelarger lattices are not readily available. Interestingly, itwas found that even with only 2 Lanczos steps on ourbest wave-function (AF + d+/J) the energy has almostconverged to the exact ground state energy at half-filling.

Note that small system size is the worst possible casefor a VMC method since the simple variational wavefunc-tion is not expected to reproduce well the short distancecorrelations, as we fix the long-range magnetic correla-tions in our variational ansatz by imposing an on-sitemagnetic field, but we do not introduce short range cor-rections. Variational Monte-Carlo instead focus on thelong distance properties, which will become dominant inthe energy as the lattice size increases. Nevertheless, theshort range correlations contributes significantly to theenergy on small lattices. One can thus expect on gen-eral grounds the energies to become increasingly good asthe system size increases, provided that the correct longrange order has been implemented in the wavefunction.The Lanczos iterations allow to correct this local struc-ture of the wave-function. Here we see that by changingthis local structure our wave-function is converging veryfast to the ground state. This is a good indication thateven away from half filling our wavefunction is quite ef-ficient in capturing the physics of the system. Actually,the variance of the energy per site σ2 reaches its maxi-mum value for doping x = 1

3 (σ2 = 0.006), but apply-ing one Lanczos step reduces drastically the variance :σ2 = 0.0004. At half-filling, a variance-energy plot forthe three functions AF + d+/J/pLs (p = 0, 1, 2) allowsto extrapolate the energy at 0 variance, and we get anenergy per site of e = −0.61(1), which is very close tothe exact result e = −0.6103.

Let us now use our wavefunction to describe large sys-tems away from half filling. We now focus on a 108 sitecluster, which is the largest cluster we can treat with areasonable effort. We have first measured the conden-sation energy per site (see Fig. 5) for different types ofinstabilities. Very interestingly, the AF/J is even betterthan a simple dx2−y2 + idxy RVB state. Moreover, theRVB order is only weakly increasing the condensationenergy in presence of the antiferromagnetic background(AF + d+/J). This is suggesting that superconductivityis only weakly present in the t−J model when n > 1which is also confirmed by the measurement of the su-perconducting gap (see Fig. 6). The superconductingorder of our best wave-function is approximately 4 timessmaller in amplitude and in range of stability than thed-wave pairing in a 10 × 10 square lattice with the sameboundary conditions. For electron doping δ > 0.04 wefind that the d+ BCS pairing symmetry has the same en-ergies as the d− one, and also as the wavefunctions withdx2−y2 and dxy pairings.

7

1 1.1 1.2 1.3Electron density n

-0.10

-0.08

-0.06

-0.04

-0.02

0.00e c

d+

AF / JAF+d

+ / J

AF+d+ / J / 1Ls

AF+d+ / J / 2Ls

ED

N=12

FIG. 4: Condensation energy per site versus the electron dop-ing for a 12 site cluster for the different variational wave-functions. We have done exact diagonalization (ED) for a 12sites cluster with same periodic boundary conditions.

1 1.1 1.2 1.3 1.4 1.5Electron density n

-0.08

-0.06

-0.04

-0.02

0

e c d+

AF / JAF+d

+ / J

AF+d+ / J / Ls

N=108

FIG. 5: Condensation energy per site ec versus electron dop-ing for the 108 sites lattice. We show different wave-functionsand also the best estimate in the literature9 at half-filling(open diamond).

Very strikingly, the 120◦ magnetic order parameteris surviving up to high doping δ = 0.4, see Fig. 7.Long-range magnetic order at finite doping is potentiallycaused by a limitation of the VMC method, in that itis not possible in our calculation to model wavefunctionwith a finite correlation length, i.e. short range 120◦ mag-netic order. In our calculation, we can either totally sup-press the long-range 120◦ and get back to the Gutzwillerwave-function, or use the long-range 120◦ magnetic orderthat is highly stabilized by the potential energy. No inter-mediate scenario, such as incommensurate structures, isyet available in our calculations, but one can only expectthe optimization of the magnetic structure to increasethe region of stability for magnetism. We interpret thisfinding as an indication that the hole motion is not dras-

1 1.2 1.40

0.01

0.02

0.03

0.04our best wfd

+

square lattice dx2-y

2

0.6 0.8 1Electron density n

0

0.01

0.02

0.03

0.04

∆

N=108

FIG. 6: Superconducting order parameter ∆ for a 108 sitetriangular cluster in our best wave-function (full triangles),in the d+ wave-function (open squares). For comparison weshow the amplitude of the d-wave gap in a 10 × 10 squarelattice (dashed line).

1 1.1 1.2 1.3 1.4 1.5

Electron density n

0

0.1

0.2

0.3

0.4

0.5M

AF

AF+d+ / J

1 1.2 1.40

0.3

χN=108

FIG. 7: Amplitude of the 120◦ magnetic order MAF measuredin the AF + d+/J for a 108 site cluster. Inset: the amplitudeof the staggered spin-current χ in the same wave-function.

tically modified by the presence of the non-collinear mag-netic structure, so that short range magnetic correlationswill survive up to high electron doping. For the t−Jz

model on the square lattice, it is commonly understoodthat the Ising Neel order is not surviving high dopingbecause of its costs in kinetic energy: whenever a holewants to move in a antiferromagnetic spin background,it generates a ferromagnetic cloud. Therefore, good ki-netic energies and Neel Ising order are not compatible.In our case, the 3-sublattice order imposes no such con-straint on the kinetic energy of the holes, because of the120◦ structure. We see this fact in our wave-functionenergies: the potential energy is improved when start-

8

ing from the Gutzwiller wavefunction and adding 120◦

correlations, but the kinetic energy is unchanged. Ourbest wavefunction has a better potential energy than theGutzwiller wavefunction (and also than the different CFPwavefunction), but it also keeps the same kinetic energy.Therefore, this qualitatively explains why one can stabi-lize the 3-sublattice magnetic phase for a large set of Jvalues. Finally, we note that the staggered spin-currentpattern is also present for doping δ = [0, 0.3] (see Fig. 7).

Interestingly, the Jastrow variational parameter α (8)is changing sign at δ = 0.4 when MAF ≈ 0 : for δ < 0.4(δ > 0.4) the Jastrow factor favors classical Ising (ferro-)anti-ferromagnetic states. The VMC results show thatthe competition between the classical Ising configurationon the triangular lattice and the 120◦ order is improvingthe energy. We argue that the classical Jastrow simu-lates with a good approximation quantum fluctuationsaround the 120◦ order. Note that the Jastrow does notplay the same role as the BCS pairing: the BCS pairingforms configurations of resonating singlets, and the Jas-trow factor forms classical Ising configurations. It is alsoworth noting that at higher doping the Jastrow parame-ter is leading to a small condensation energy of 0.01t for alarge range of doping (δ = [0.4, 0.8]). It was checked thatthis gain in energy does not decrease with the size of thelattice and is also present for a square lattice geometry.We found also that for the small clusters (12 and 48 sites)the system was gaining a significant amount of energywhen having a weak ferromagnetic polarization. There-fore, this is suggesting that the Gutzwiller wavefunctionis not the best approximation of the ground state of thet−J model in the high doping limit. Nevertheless, theJastrow factor does not introduce long-range correlationand the variational wavefunction we introduce here is stilla Fermi-Liquid.

Note that the variance of the energy per site σ2 reachesits maximum value for doping δ = 0.4 for the AF +d+/Jwith σ2 = 0.0008, and applying one Lanczos step leadsto σ2 = 0.0004.

C. Hole doping: n ∈ [0, 1]

For hole doping the scenario is strikingly different. The120◦ order is weakened in a strong d+ RVB backgroundand disappears at doping δ = 0.08 (see Fig. 8 and Fig. 9).

When superconductivity disappears, there is a firstorder transition to a commensurate spin density wave.No coexistence between superconductivity and the spindensity wave was found. Then, a ferromagnetic phasesemerges with a strong gain of condensation energy. In-deed the polarized states are leading to a strong gain inkinetic energy. This can be understood in the simplepicture of the Stoner model, which gives a critical onsiterepulsion related to the density of states: UF

cr = 1/ρ(ǫF ).Ferromagnetism becomes favorable if ǫF is sitting at asharp peak of ρ(ǫ). In the triangular lattice the tightbinding (TB) density of states is strongly asymmetric

0 0.2 0.4 0.6 0.8 1

Electron density n

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

e c d+

AF+d+ / J

FerroSDW

N=108

FIG. 8: Condensation energy per site ec for different wave-functions in a 108 site cluster.

0

1

MF

/MS

AT

0 0.5 1Electron density n

0

0.1

0.2

0.3

0.4

MA

F,M

SD

W

MAFMSDWMF/MSAT

N=108

FIG. 9: Amplitude of the 120◦ magnetic order measured inour best wavefunction for the 108 site cluster (left scale, fulltriangles) and the ratio of the polarization MF on the satu-rated polarization Msat in our best wavefunction (right scale,full circles). We show also the absolute magnetization MSDW

for the spin density wave wavefunction (left scale, see alsoFig. 10).

and has a sharp peak at the n = 0.5 electronic densitylying at the Van Hove singularity. Note also that the sim-ple t−J model of a 3 site cluster with 2 electrons showsthat in the t > 0 the ground state is a singlet, whereasthe ground state is a triplet in the t < 0 case. This showsthat the negative sign of t with hole doping is inducingferromagnetic correlations on a very small cluster. Wefind again trace of these correlations and ferromagnetictendencies in the range of electronic density n ∈ [0.2, 0.8]in our 108 site lattice. Such a ferromagnetic instabilitywas also predicted in Ref. 34 by comparing the energyof the RVB wavefunction with an analytical calculationof the energy of the fully polarized state. We see that

9

FIG. 10: On-site magnetization for each site of a 108 sitelattice for the spin density wave wavefunction. Open (filled)circles denotes down (up) spins. The size of each circle isproportional to the respective amplitude of the on-site mag-netization. We find that the spins forms a stripe-like pattern,alternating ferromagnetic bonds in the a2 direction, and anti-ferromagnetic bond in the two other directions. The averageon the lattice sites of the absolute value of the local magneti-zation is shown in Fig. 9.

minimizing the energy by changing the variational onsitemagnetic field leads to similar results.

Moreover, at δ = 0.5 doping, there is a nesting of theFermi surface, with three possible Q vectors. Thus, itis reasonable to expect that a particle-hole instability ofcorresponding pitch vector Q is stabilized close to thisdoping. We have investigated the following instabilities :a commensurate charge density wave, and a spin densitywave. Interestingly, the commensurate spin density wavewas stabilized. Indeed, we have found that the scatter-ing between the k and k + Q vectors introduced in theHamiltonian HSDW (particle-hole channel) allow to gainkinetic energy in the range of doping δ = [0.15, 0.6]. Forsake of simplicity, we have only considered the mean-fieldHamiltonian containing one of the three possible nestingvectors : QN = (π,−π/

√3). Finally, the phase is sta-

bilized, when compared to the RVB and ferromagneticphases, in the window δ = [0.16, 0.24]. Nonetheless, nocoexistence between superconductivity and the spin den-sity wave was found : the energy is minimized eitherfor (∆k = 0, f(Q) 6= 0), or (∆k 6= 0, f(Q) = 0) depend-ing on the doping, with ∆k of dx2−y2 + idxy symmetrytype in the latter case. Measuring the on-site magnetiza-tion value, we found that the spin density wave is form-ing a collinear stripe-like pattern in the spins degrees offreedom, whereas the charge is found to be uniformly dis-

0 0.2 0.4 0.6 0.8 1 1.2 1.4

electron density n

0.5

1

Ord

er

pa

ram

ete

r (a

.u.)

120˚ F Fermi Liquid

t<0

SD

W

1.6

SC

SC

FIG. 11: (color online) Cartoon picture of the phase diagramof the t−J model we get with t < 0. Here we sketch onan arbitrary scale the order parameter amplitude of the 120◦

magnetic phase, the ferromagnetic phase (F ), the supercon-ducting dx2

−y2 + idxy phase (SC), the commensurate spindensity wave (SDW ). Note that for electron density n > 1.04and n < 0.96, the energy is degenerated, within the error barsdue to the Monte-Carlo sampling, with the pairings dx2

−y2 ,dxy and dx2

−y2 − idxy . The pitch vector of the commensu-

rate spin density wave is QN = (π,−π/√

3), and this phaseis depicted more in details in Fig. 10.

tributed among the lattice sites, as expected (see Fig. 10).The amplitude of the on-site magnetization is shown inFig. 9 as a function of doping.

D. Phase diagram of the model

Based on our wavefunction we can now give the phasediagram for the doped system on the triangular lattice.The phase diagram, summarizing the various instabili-ties discussed in the previous sections is show in Fig. 11.This phase diagram prompts for several comments. Firstone notices immediately that the competition betweenmagnetism and superconductivity in this model dependscrucially on the sign of the hopping integral (see Fig. 11).

For both hole and electron doping, the triangular lat-tice has a very different phase diagram from the squarelattice one37. In the square lattice, the AF order disap-pears at δ = 0.1 and the d-wave RVB dies at δ = 0.4 forthe same value of J . In the triangular lattice, a similarstability of superconductivity exists on the hole side, butthe electronically doped side is resolutely dominated byantiferromagnetic instabilities. Our results, based on animproved class of wavefunctions, present marked differ-ences with previous approximate results for the dopedsystem. On the electron side, mean-field theories wouldhave suggested that the long-range magnetic order stateundergoes a first order phase transition32 into a uniformd+ superconducting state at δ ≈ 3% for values of J sim-ilar to those considered here. A rationalization of theseresults would be that the frustration of the lattice, which

10

was from the start the motivation of RVB as a compet-ing state, disfavors magnetic order. Our results, wherethe Gutzwiller projection is treated exactly within theresidual error bars due to the statistics, are in strong dis-agreement with this mean-field theory. Contrarily to themean-field result, magnetism is dominant and the super-conducting order is not favored on the electronic side. Inaddition, the t−J model on the triangular lattice was ex-pected to have a strong and large RVB instability, sincethe coordination number is higher than on most of theother lattices, and naively we would expect this to pro-vide an easy way to form singlets. In this work we showthat it is not the case: for electron doping the systemis magnetic, and for hole doping the system is supercon-ducting, but the range of superconductivity is not ex-traordinary large (δ < 0.16), and smaller than on thesquare lattice.

Previous variational approaches34 were restricted topure superconducting wavefunctions dx2−y2 + idxy on at− t′ square lattice with t = t′. In that work it was foundthat superconductivity is stabilized up to electron dopingδ ≈ 0.24 and hole doping δ ≈ 0.2 for similar albeit slightlydifferent values of J/t (J/t = 0.3). In our work, for thecase of electron doping, which corresponds to the dopingin the cobaltite experiments, our phase diagram, usingthe larger class of wavefunctions, is clearly completelydifferent from this previous result, and the stabilizationof the superconductivity in that case was clearly an arte-fact of the too restricted variational subspace. As showin Fig. 11 superconductivity is strongly weakened by thepresence of 3-sublattice magnetization and is present onlyin the range of electron doping δ = [0, 0.12]. On the con-trary, for the case of hole doping, superconductivity hadhigher energy than ferromagnetic and spin density wavephases for δ > 0.16. We thus confirm that the previousresults are not an artefact of their restricted variationalsubspace, and find an acceptable agreement for the phasediagram. However we emphasize the presence of the spindensity wave wavefunction that was not considered inthe mentioned work and implies a small reduction of thesuperconductivity range.

Our calculation thus clearly prompts for a reexamina-tions of the arguments on the nature of superconductivityin a frustrated lattice. Clearly the non-collinear natureof the order parameter helps making the AF order muchmore stable to electron doping than initially anticipated.Understanding such issues is of course a very crucial andchallenging question. Moreover, on the triangular lat-tice, no significant enhanced cooperative effect betweenmagnetism and superconductivity seems to be observed:the electron doped side has a magnetic signature, andthe hole doped side a superconducting one, but the twoorders seem to exclude each other as much as they can,contrarily to what happens for the square lattice. Evenin the parts of the phase diagram where coexistence isobserved, coexistence between magnetism and supercon-ductivity in the electron doped case shows again thatsuperconductivity is decreased in the presence of strong

long range magnetic correlations.

IV. CONCLUSION

In this paper we have presented a variational Monte-Carlo study of the t−J (J/t = 0.4 and t < 0) model onthe triangular lattice, using extended wavefunction con-taining both superconductivity and non-collinear mag-netism, as well as flux phase instabilities. The methodwe used to construct and sample the wavefunction isquite general and applicable to other lattices (honey-comb, kagome, ladders...) as well as other symmetries(e.g. triplet superconductivity). It thus provides a gen-eral framework to tackle the competition between anti-ferromagnetism and superconductivity in frustrated sys-tems. We obtained very good variational energies at half-filling when comparing with other more sophisticatedmethods, specialized to the half-filled case. The fermionicrepresentation of our wavefunction allows to consider holeand electron doping. The most stable pairing corre-sponds to singlet pairing. We find that dx2−y2 + idxy

superconductivity is only weakly stabilized for electrondoping in a very small window (δ = [0, 0.12]) and is muchstronger and also appears in a wider range (δ = [0, 0.16])in the case of hole doping. A commensurate spin densitywave phase is leading to a gain in kinetic energy and isstabilized in the small window δ = [0.16, 0.24] hole dop-ing. Finally, ferromagnetism emerges in a wide rangefor hole doping δ = [0.24, 0.8]. Very surprisingly, the 3-sublattice magnetism which is present at half-filling ex-tends to a very wide range of electron doping δ = [0, 0.4]and is suppressed very fast in the case of hole dopingδ = [0, 0.08]. The large extent of 120◦ order for elec-tron doping is responsible for the suppression of super-conductivity. This feature was neither observed in pre-vious VMC calculations, nor predicted by the mean fieldtheories, and prompts for a reexamination of the questionof the stability of magnetic order on a triangular system.

Our results show that, for electron doping, the squareand triangular lattices behave in a very different way. Forthe square lattice, the t−J Hamiltonian finds a domainof stability of superconductivity and a pairing symmetrythat is very consistent with other methods. It is thus anatural candidate to investigate superconducting phasesin systems like the cuprates. For the case of the trian-gular lattice, the predicted phase diagram is dominatedby antiferromagnetic instabilities, and superconductiv-ity, albeit slightly present, is strongly suppressed. Thisclearly indicates that, contrarily to what was suggestedby mean-field and previous variational calculations, thet−J model itself is not a good starting point to tackle thesuperconductivity of the cobaltite compounds, where su-perconductivity is observed in the range of electron den-sity n =

[

1 + 14 , 1 + 1

3

]

. This model must be completedby additional ingredients to obtain a faithful descriptionof the experimental system. Two missing ingredients inthe simple t−J model could solve this discrepancy and

11

perhaps allow to obtain a superconducting instability.On one hand, a strong Coulomb repulsion is expectedin this type of compound. Such a long-range interac-tion is not taken into account in the t−J model. Thus acoulomb V term should be added to get a t−J−V model.On the other hand, in this paper, we have used a single-band model as a first step to study the Co-based oxides.However, it is quite possible that the multi-band effectplays an essential role for superconductivity20. The in-teraction between the three bands of the compound couldplay a non trivial role in the physics of the t−J model.Therefore, a study of the 3-band model could also be of

interest. Such an analysis can be done by extending themethods exposed in this paper to these more complicatedmodels.

Acknowledgments

We are grateful to Federico Becca and Antoine Georgesfor useful discussions. This work was supported by theSwiss National Fund and by NCCR MaNEP.

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trivial role in the projected wave-function. Indeed, the ki-netic energy of the projected wave-function depends on thechosen gauge. One cannot exclude that another choice ofthe gauge could lead to a better wave-function.

37 In the square lattice the sign of t plays no role because ofthe particle-hole symmetry of a bipartite lattice.