Magnetism and superconductivity in the t− t ′ −J model
Leonardo Spanu, Massimo Lugas, Federico Becca, Sandro Sorella
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Magnetism and superconductivity in the t − t ′ − J model Leonardo Spanu, Massimo Lugas, Federico Becca, and Sandro Sorella
INFM-Democritos, National Simulation Center and International School for Advanced Studies (SISSA), I-34014 Trieste, Italy (Dated: November 19, 2018)We present a systematic study of the phase diagram of the t − t ′ − J model by using the Green’sfunction Monte Carlo (GFMC) technique, implemented within the fixed-node (FN) approximationand a wave function that contains both antiferromagnetic and d-wave pairing. This enables usto study the interplay between these two kinds of order and compare the GFMC results withthe ones obtained by the simple variational approach. By using a generalization of the forward-walking technique, we are able to calculate true FN ground-state expectation values of the pair-pair correlation functions. In the case of t ′ = 0, there is a large region with a coexistence ofsuperconductivity and antiferromagnetism, that survives up to δ c ∼ .
10 for
J/t = 0 . δ c ∼ . J/t = 0 .
4. The presence of a finite t ′ /t < δ c . .
03, for
J/t = 0 . t ′ /t = − .
2) and pairing correlations. In particular, the latter ones aredepressed both in the low-doping regime and around δ ∼ .
25, where strong size effects are present.
PACS numbers:
I. INTRODUCTION
After more than twenty years from the discoveryof high-temperature superconductivity, a comprehensivedescription of the cuprate materials is still lacking. Oneof the main concern is about the origin of the electronpairing, namely if it is due to electron-phonon coupling,like in the standard theory by Bardeen, Cooper andSchrieffer (BCS), or it can be explained by alternativemechanisms, based on the electronic interaction alone.From one side, though the isotope effect in cuprates (ifany) is much smaller than the one observed in BCS su-perconductors, there are experiments suggesting a strongcoupling between electrons and localized lattice vibra-tions. On the other side, besides a clear experimentaloutcome showing unusual behaviors in both metallic andsuperconducting phases, there is an increasing theoreticalevidence that purely electronic models can indeed sus-tain a robust pairing, possibly leading to a high criticaltemperature.
Within the latter scenario, the minimalmicroscopic model to describe the low-energy physics hasbeen proposed to be the Hubbard model or its strong-coupling limit, namely the t − J model, which includes anantiferromagnetic coupling between localized spins anda kinetic term for the hole motion. In this respect,Anderson proposed that electron pairing could naturallyemerge from doping a Mott insulator, described by a res-onating valence bond (RVB) state, where the spins arecoupled together forming a liquid of singlets. Indeed,subsequent numerical calculations for the t − J model, showed that, though the corresponding Mott insulator(described by the Heisenberg model) has magnetic or-der, the RVB wave function with d-wave symmetry inthe electron pairing can be stabilized in a huge regionof doping close to the half-filled insulator. These calcu-lations have been improved by studying the accuracy ofsuch a variational state, giving solid and convincing argu-ments for the existence of a superconducting phase in the t − J model. However, other numerical techniques, like Density Matrix Renormalization Group (DMRG), pro-vided some evidence that charge inhomogeneities can oc-cur at particular filling concentrations.
These stripesare probably enhanced by the strong anisotropic bound-ary conditions used in this approach and can be alsofound by allowing anisotropies in the hopping and inthe super-exchange coupling. Coming back to the pro-jected RVB wave functions, it is worth mentioning thatan approximate and simplified description of these statescan be obtained by the renormalized mean-field theory(RMFT), the so-called “plain vanilla” approach. Whenthis approach is applied to the t − J model, it is pos-sible to describe many unusual properties of the high-temperature superconductors and capture the most im-portant aspects of the cuprate phase diagram. However,at present, most of the calculations have been done byneglecting antiferromagnetic correlations, that are defini-tively important at low doping. Within RMFT and mostof the variational calculations, the magnetic correlationsare omitted, implying a spin liquid (disordered) state inthe insulating regime. Although antiferromagnetism canbe easily introduced in both approaches, it is often notsatisfactorily described, since the presence of an antifer-romagnetic order parameter in the fermionic determinantimplies a wrong behavior of the spin properties at smallmomenta, unless a spin Jastrow factor is used to de-scribe the corresponding spin-wave fluctuations. Indeed,it is now well known that the accurate correlated de-scription of an ordered state is obtained by applying along-range spin Jastrow factor to a state with magneticorder.
The important point is that the Gaussianfluctuations induced by the Jastrow term must be orthog-onal to the direction of the order parameter, in order toreproduce correctly the low-energy spin-wave excitations.Moreover, by generalizing the variational wave functionto consider Pfaffians instead of simple determinants, it is possible to consider both electron pairing and mag-netic order, that are definitively important to determinethe phase diagram of the t − J model in the low-dopingregime.The interplay between superconductivity and mag-netism is the subject of an intense investigation in therecent years. In most of the thermodynamic measure-ments these two kinds of order do not coexist, thoughelastic neutron scattering experiments for underdopedYBa Cu O x could suggest a possible coexistence, witha small staggered magnetization. On the contrary,in the t − J model, there is an evidence in favor of a co-existence, the antiferromagnetic order surviving up to arelatively large hole doping, i.e., δ ∼ . J/t = 0 . Therefore, the regime of magnetic order predicted bythese calculations extend to much larger doping thanthe experimental results and also the robustness of thecoexistence of superconductivity and antiferromagnetismseems to be inconsistent with the experimental outcome.Of course, disorder effects, which are expected to be im-portant especially in the underdoped region, would affectthe general phase diagram. However, without invokingdisorder, one is also interested to understand if alterna-tive ingredients can modify the phase diagram of the sim-ple t − J model. For instance, band structure calculationssupport the presence of a sizable second-neighbor hop-ping t ′ in cuprate materials, showing a possible connec-tion between the value of the highest critical temperatureand the ratio t ′ /t . Moreover, an experimental analysissuggests an influence of the value of t ′ /t on the pseudo-gap energy scale. From a theoretical point of view, theeffect of t ′ is still not completely elucidated, though there are different calculations providing evidencethat a finite t ′ could suppress superconductivity in thelow-doping regime. On the other hand, recent MonteCarlo calculations suggest that the presence of t ′ (as wellas a third-neighbor hopping t ′′ ) could induce an enhance-ment of pairing in optimal and overdoped regions. In this paper, we want to examine the problem of theinterplay between magnetism and superconductivity inthe t − J model and its extension including a next-nearest-neighbor hopping t ′ by using improved variational andGreen’s function Monte Carlo (GFMC) techniques. In-deed, especially the latter approach has been demon-strated to be very efficient in projecting out a very accu-rate approximation of the exact ground state and, there-fore, can give useful insight into this important issue re-lated to high-temperature superconductivity.The paper is organized as follows: In Sec. II we de-scribe the methods we used, in Sec. III we show the nu-merical results for antiferromagnetism and superconduc-tivity, and in Sec. IV we draw our conclusions. II. MODEL AND METHODA. Model and variational wave function
We consider the t − t ′ − J model on a two-dimensionalsquare lattice with L sites and periodic boundary condi- tions on both directions: H = J X h i,j i (cid:18) S i · S j − n i n j (cid:19) − t X h i,j i σ c † i,σ c j,σ − t ′ X hh k,l ii σ c † k,σ c l,σ + h.c. (1)where h . . . i indicates the nearest-neighbor sites, hh . . . ii the next-nearest-neighbor sites, c † i,σ ( c i,σ ) creates (de-stroys) an electron with spin σ on the site i , S i = ( S xi , S yi , S zi ) is the spin operator, S αi = P σ,σ ′ c † i,σ τ ασ,σ ′ c i,σ ′ , being τ α the Pauli matrices, and n i = P σ c † i,σ c i,σ is the local density operator. In the fol-lowing, we set t = 1 and consider t ′ = 0 and t ′ /t = − . L = l × l sites and 45 ◦ tilted latticeswith L = 2 × l sites. Besides translational symmetries,both of them have all reflection and rotational symme-tries.The variational wave function is defined by: | Ψ V MC i = J s J d P N P G | Φ MF i , (2)where P G is the Gutzwiller projector that forbids doubleoccupied sites, P N is the projector onto the subspacewith fixed number of N particles. Moreover, J s is a spinJastrow factor J s = exp X i,j v ij S zi S zj , (3)being v ij variational parameters, and finally J d is a den-sity Jastrow factor J d = exp X i,j u ij n i n j , (4)being u ij other variational parameters. The above wavefunction can be efficiently sampled by standard varia-tional Monte Carlo, by employing a random walk of aconfiguration | x i , defined by the electron positions andtheir spin components along the z quantization axis. In-deed, in this case, both Jastrow terms are very simpleto compute, since they only represent classical weightsacting on the configuration.As previously reported, the main difference from pre-vious approaches is the presence of the spin Jastrow fac-tor and the choice of the mean-field state | Φ MF i , definedas the ground state of the mean-field Hamiltonian H MF = X i,j,σ t i,j c † i,σ c j,σ + h.c. − µ X i,σ n i,σ + X h i,j i ∆ i,j ( c † i, ↑ c † j, ↓ + c † j, ↑ c † i, ↓ + h.c. ) + H AF , (5)where we include both BCS pairing ∆ i,j [with d -wave symmetry, i.e., for nearest-neighbor sites ∆ k =∆(cos k x − cos k y )] and staggered magnetic field in the x − y plane H AF = ∆ AF X i ( − R i ( c † i, ↑ c i, ↓ + c † i, ↓ c i, ↑ ) , (6)where ∆ AF is a variational parameter that, together withthe chemical potential µ and the next-nearest-neighborhopping of Eq. (5), can be determined by minimizing thevariational energy of H . Whenever both ∆ and ∆ AF arefinite, the projection h x | Φ MF i of the mean-field state ona given configuration | x i can be described in terms ofPfaffians, instead if ∆ = 0 or ∆ AF = 0 it can be de-scribed by using determinants. Moreover, only in thecase where the magnetic order parameter is in the x − y plane, the presence of the spin Jastrow factor (3) canintroduce relevant fluctuations over the mean-field orderparameter ∆ AF , leading to an accurate description ofthe spin properties. A detailed description of the wavefunction of Eq. (2) and its physical properties can befound in Ref. 21. The variational parameters containedin the mean-field Hamiltonian (5) and in the Jastrow fac-tors (3) and (4) are calculated by using the optimizationtechnique described in Ref. 34, that makes it possible tohandle with a rather large number of variational param-eters. B. GFMC: beyond the Variational Monte Carlo
The optimized variational wave function | Ψ V MC i canbe also used within the GFMC method to filter out anapproximation of the ground state | Ψ F N i . Indeed, dueto the presence of the fermionic sign problem, in order tohave a stable numerical calculation, the GFMC must beimplemented within the fixed-node (FN) approach, thatimposes to | Ψ F N i to have the same nodal structure of thevariational ansatz. Here, we recall the basic definitionsof the standard FN method. A detailed description ofthis technique can be found in Ref. 21.Starting from the original Hamiltonian H , we definean effective model by H eff = H + O. (7)The operator O is defined through its matrix elementsand depends upon a given guiding function | Ψ i , that isfor instance the variational state itself, i.e., | Ψ V MC i : O x ′ ,x = (cid:26) −H x ′ ,x if s x ′ ,x = Ψ x ′ H x ′ ,x Ψ x > P y,s y,x > H y,x Ψ y Ψ x for x ′ = x, where Ψ x = h x | Ψ i , | x i being an electron configurationwith definite z -component of the spin. Notice that theabove operator annihilates the guiding function, namely O | Ψ i = 0. Therefore, whenever the guiding function isclose to the exact ground state of H , the perturbation O is expected to be small and the effective Hamiltonianbecomes very close to the original one. The most im-portant property of this effective Hamiltonian is that its ground state | Ψ F N i can be efficiently computed by us-ing GFMC. The distribution Π x ∝ h x | Ψ ih x | Ψ F N i issampled by means of a statistical implementation of thepower method: Π ∝ lim n →∞ G n Π , where Π is a start-ing distribution and G x ′ ,x = Ψ x ′ (Λ δ x ′ ,x − H eff,x ′ ,x ) / Ψ x ,is the so-called Green’s function, δ x ′ ,x being the Kro-necker symbol. The statistical method is very efficientsince all the matrix elements of G are non-negativeand, therefore, G can represent a transition probabilityin configuration space, apart for a normalization factor b x = P x ′ G x ′ ,x . Since | Ψ F N i is an exact eigenstate of theeffective Hamiltonian H eff , the corresponding ground-state energy can be evaluated efficiently by computing E MA = h Ψ V MC |H eff | Ψ F N ih Ψ V MC | Ψ F N i , (8)namely the statistical average of the local energy e L ( x ) = h Ψ V MC |H| x i / h Ψ V MC | x i over the distribution Π x . Thequantity E MA ≤ E V MC because, by the variational prin-ciple E MA ≤ h Ψ |H eff | Ψ i / h Ψ | Ψ i = E V MC . Moreover, E MA represents an upper bound of the expectation value E F N of H over | Ψ F N i , as it is shown in Ref. 35 or itcan be simply derived by considering that the opera-tor O is semi-positive definite, namely all its eigenval-ues are non-negative. In the following, we will denoteby FN the (variational) results obtained by using theGFMC method with fixed-node approximation, whereasthe standard variational Monte Carlo results obtained byconsidering the wave function of Eq. (2) will be denotedby VMC.Summarizing the FN approach is a more general andpowerful variational method than the straightforwardvariational Monte Carlo. Within the FN method, thewave function | Ψ F N i , the ground state of H eff is knownonly statistically, and, just as in the variational approach, E F N depends explicitly on the variational parametersdefining the guiding function | Ψ i . This is due to thefact that H eff depends upon | Ψ i through the operator O . The main advantage of the FN approach is that itprovides the exact ground-state wave function for theundoped insulator (where the signs of the exact groundstate are known), and therefore it is expected to be par-ticularly accurate in the important low-doping region.Moreover, the FN method is known to be very efficient invarious cases: For instance, it has allowed to obtain a ba-sically exact description of the three-dimensional systemof electrons interacting through the realistic Coulomb po-tential (in presence of a uniform positive background). Therefore, it represents a very powerful tool to describecorrelated electronic systems.
III. RESULTSA. Phase separation
Before showing the results on magnetic and supercon-ducting properties, we briefly discuss the stability againstphase separation. In order to detect a possible phase sep-aration, it is very useful to follow the criterion given inRef. 39 and consider the energy per hole: e h ( δ ) = e ( δ ) − e (0) δ , (9)where e ( δ ) is the energy per site at hole doping δ and e (0) is its value at half filling. In practice, e h ( δ ) repre-sents the chord joining the energy per site at half fill-ing and the one at doping δ . For a stable system, e h ( δ )must be a monotonically increasing function of δ , becausethe ground-state energy of a short-range Hamiltonian isa convex function of the doping. By performing exactenergy calculations on finite clusters, the phase separa-tion instability is marked by a minimum of e h ( δ ) at agiven δ c and by a flat behavior up to δ c in the thermody-namic limit. For an approximate variational techniquebased on a spatially homogeneous ansatz, this flat be-havior is never reached in the phase separated region, sothat e h ( δ ) remains with a well defined minimum even forvery large sizes; in this case, δ c can be estimated with theMaxwell construction, provided the variational asnatz isaccurate enough. In Ref. 21, we demonstrated the exis-tence of an homogeneous state for t ′ = 0 and J/t . . provides a substantial lowering ofthe VMC energy, especially away from half filling andfor a finite t ′ . This is a first indication that, for t ′ /t < t ′ /t enhances the stability of the homogeneous phase,whereas positive values of t ′ favor phase separation. Inthis work, we do not want to address in much detail thisissue and we will focus our attention on the more inter-esting magnetic and superconducting properties.
B. Antiferromagnetic properties
Here we present the results for the magnetic propertiesof the t − t ′ − J model and compare the FN approach withthe VMC one, based upon the wave function (2). Asalready discussed in Ref. 21, the optimized wave func-tion (2) breaks the SU(2) spin symmetry, because of themagnetic order parameter ∆ AF of Eq. (6) and the spinJastrow factor (3). It turns out that at half-filling and inthe low-doping regime, the variational state (2) has anantiferromagnetic order in the x − y plane, whereas thespin-spin correlations in the z axis decay very rapidly.Therefore, in order to assess the magnetic order at the δ -2.15-2.125-2.1-2.075-2.05-2.025-2 e h ( δ ) L=98 J/t=0.2L=162 J/t=0.2L=98 J/t=0.2 t’/t=-0.2L=162 J/t=0.2 t’/t=-0.2
FIG. 1: (Color online) Energy per hole e h ( δ ), calculated byusing the FN method, as a function of the doping δ for L = 98and 162 and two values of the next-nearest-neighbor hopping t ′ /t = 0 and − . J/t = 0 . t ′ = 0 (third and fourth columns),and t ′ /t = − . L = 98 and 162 and different hole concentrations N h = L − NL N h E V MC /L E
F N /L E
V MC /L E
F N /L
98 0 -0.233879(1) -0.23432(1) -0.233879(1) -0.23432(1)98 2 -0.274144(5) -0.27752(1) -0.27290(1) -0.27808(1)98 4 -0.31429(1) -0.32053(1) -0.31189(1) -0.32123(1)98 6 -0.35482(1) -0.36328(1) -0.35132(1) -0.36405(1)98 8 -0.39550(1) -0.40563(2) -0.39028(1) -0.40575(1)98 10 -0.43581(1) -0.44728(2) -0.42814(1) -0.44561(1)162 0 -0.233707(1) -0.23409(1) -0.233707(1) -0.23409(1)162 2 -0.258002(5) -0.26020(1) -0.257260(5) -0.26012(1)162 4 -0.282117(5) -0.28621(1) -0.28067(1) -0.28698(1)162 6 -0.306324(5) -0.31212(1) -0.30429(1) -0.31307(1)162 8 -0.33060(1) -0.33793(1) -0.32807(1) -0.33925(2)162 10 -0.35498(1) -0.36360(2) -0.35207(1) -0.36514(2)162 12 -0.37954(1) -0.38912(2) -0.37567(1) -0.39079(2)162 14 -0.40406(1) -0.41446(2) -0.39939(1) -0.41520(2)162 16 -0.42838(1) -0.43946(2) -0.42232(1) -0.43936(2) variational level, we have to consider the isotropic spin-spin correlations: h S · S r i = h Ψ V MC | S · S r | Ψ V MC ih Ψ V MC | Ψ V MC i . (10)The FN approach alleviates the anisotropy between the x − y plane and the z axis; in this case, we find that arather accurate (and much less computational expensive)way to estimate the magnetic moment can be obtainedfrom the z component of the spin-spin correlations: h S z S zr i = h Ψ F N | S z S zr | Ψ F N ih Ψ F N | Ψ F N i . (11)This quantity can be easily computed within the forward-walking technique, because the operator S z S zr is diago-nal in the basis of configurations used in the Monte Carlo M L=98 VMCL=162 VMCL=98 FNL=162 FN δ J/t=0.2J/t=0.4
FIG. 2: (Color online) Magnetization obtained from the spin-spin correlations at the maximum distance calculated for the t − J model with J/t = 0 . J/t = 0 . z axis (see text). sampling. From the spin-spin correlations at the maxi-mum distance, it is possible to extract the value of themagnetization. In particular, the variational wave func-tion is not a singlet when the antiferromagnetic order setsin, and the magnetization has to be computed with thespin isotropic expression M = lim r →∞ p h S · S r i . Onthe other hand, the FN ground state is almost a perfectsinglet for all the cases studied and the magnetization canbe estimated more efficiently by M = lim r →∞ p h S z S zr i .The spin isotropy of the FN wave function can be explic-itly checked by computing the mixed-average of the totalspin square h S i MA = h Ψ V MC | S | Ψ F N ih Ψ V MC | Ψ F N i , (12)that vanishes if | Ψ F N i is a perfect singlet, even when | Ψ V MC i has not a well-defined spin value.In Fig. 2 we report the results of the magnetizationin the t − J model with J/t = 0 . .
4. At finitedoping, it is not possible to perform a precise size scal-ing extrapolation since it is very rare to obtain the samedoping concentration for different cluster sizes. More-over, though the FN approach is able to recover an exactsinglet state at half filling, h S i MA increases by doping,reaching its maximum around δ ∼ .
06, e.g., h S i MA ∼ δ ∼ . J/t = 0 .
2. Definitively, both the VMC andFN wave functions are almost spin singlets close to thetransition point, because the mean-field order parameter∆ AF goes to zero together with the parameters definingthe spin Jastrow factor. Therefore, we are rather confi-dent in the estimation of the critical doping δ c , where the S ( q )
12 holes16 holes18 holes20 holes24 holes32 holes42 holes VMCFN
L=256J/t=0.2 Γ X M X M
16 holes
FIG. 3: (Color online) Static spin structure factor S ( q ) for L = 16 ×
16 cluster and different hole concentrations for the t − J model with J/t = 0 .
2. Γ = (0 , X = ( π, π ), and M = ( π, S ( q ) for the variational state (emptysymbols) and for the FN approximation (full symbols). q S ( q , π ) L=64L=144L=256L=400L=256 Hubbard S ( π / L - π , π ) J/t=0.2 π /2 π FIG. 4: (Color online) Spin structure factor S ( q ) for the t − J with J/t = 0 . δ = 1 / L = 8 ×
8, 12 ×
12, 16 ×
16, and 20 × U/t = 4 and L = 16 ×
16 is also reportedfor comparison. Inset: Size scaling of the peak as a functionof 1 /L . long-range antiferromagnetic order disappears. In partic-ular, we find δ c = 0 . ± .
01 and δ c = 0 . ± .
02 for
J/t = 0 . J/t = 0 .
4, respectively.At low doping, we have evidence that long-range orderis always commensurate, with a (diverging) peak at X =( π, π ) in the static spin structure factor, defined as S ( q ) = 1 L X l,m e iq ( R l − R m ) S zl S zm . (13)This outcome is clear for all kinds of cluster considered,namely both for standard l × l and 45 ◦ tilted lattices.By contrast, close to the critical doping δ c , we have theindication that some incommensurate peaks develop. Re-markably, we do not find any strong doping dependenceof the peak positions. We show the results of S ( q ) forthe 16 ×
16 cluster and
J/t = 0 . δ isreported. By increasing the hole doping, the commen-surate peak at X reduces its intensity and eventuallythe maximum of S ( q ) shifts to a different k-point, i.e.,( π, π − π/L ). It should be stressed that this outcome isobtained only when the FN projection is applied to thelowest-energy ansatz contaning a sizable BCS parame-ter, and the FN calculation with a fully projected free-electron determinant cannot reproduce an incommensu-rate peak in S ( q ). Moreover, the variational wave func-tions always show commensurate correlations, see insetof Fig. 3. The strong dependence of this feature on thevariational ansatz may also indicate that more accuratecalculations are necessary to clarify this important as-pect of the phase diagram of the t − J model. In order tosupport the validity and the accuracy of our results, wehave applied the same method to the Hubbard model at U/t = 4, where essentially exact calculations are avail-able for S ( q ). In this case, we have reproduced both theposition and the intensity of the incommensurate peak onthe 10 ×
10 lattice. It is interesting to notice that, withinthe FN approximation, the intensity of the incommen-surate peak at
U/t = 4 is much smaller than the corre-sponding one for the t − J model, see Fig. 4. This clearlyindicates that in the t − J model the magnetic correlationsare much more pronounced than the corresponding onesof the Hubbard model, possibly explaining the origin ofthe large extention of the antiferromagnetic region foundin the t − J model.We now discuss whether these incommensurate spincorrelations remain in the thermodynamic limit. For allthe cluster sizes we considered, i.e., up to L = 20 × S ( q ) always appears at( π, π − π/L ), namely the closest k-point to X along theborder of the Brillouin zone. This indicates that, in thethermodynamic limit, the peak should be located veryclose to X and it is not compatible with ( π, π − πδ ),found in cuprate materials. Although size scaling ex-trapolations are not possible for a generic hole doping, wedo not have evidence that the incommensurate peak di-verges, implying no incommensurate long-range order atfinite doping concentrations. Nevertheless, once the com-mensurate magnetic order is melted, the ground state ischaracterized by short-range incommensurate spin corre-lations. In Fig. 4, we show the results for
J/t = 0 . δ = 1 /
8, where different clusters with the same dopingare available. Similar calculations with t ′ /t = − . S ( q ).Coming back to the commensurate magnetic orderclose to half-filling, we stress that the pure t − J modelshows robust antiferromagnetic correlations, with a crit-ical doping much larger than the one observed in thehole-doped cuprates materials, where the long-range or-der disappears at δ c ∼ . This smaller value of thecritical doping cannot be explained by reducing the an- δ M L=98 VMCL=162 VMCL=242 VMCL=98 FNL=162 FNL=242 FN
J/t=0.2t’/t=-0.2AF
FIG. 5: (Color online) The same as in Fig. 2 for the t − t ′ − J model with J/t = 0 . t ′ /t = − .
2. For the VMC calcu-lations the error-bars are smaller than the symbol sizes. Thedashed line indicates a tentative estimation for the thermo-dynamic limit. tiferromagnetic super-exchange J , given the fact thateven for J/t = 0 . δ c ∼ .
1. Besides dis-order effects that can be important in the underdopedregime, one important ingredient to be considered in amicroscopic model is the next-nearest-neighbor hopping,that was shown to have remarkable effects on both mag-netic and superconducting properties. In partic-ular, in spite exact diagonalization calculations suggest asuppression of antiferromagnetic correlations for negative t ′ /t , more recent Monte Carlo simulations (also includ-ing a further third-neighbor hopping t ′′ ) do not confirmthese results, pointing instead toward a suppression ofsuperconducting correlations. In Fig. 5, we report the magnetization for
J/t = 0 . t ′ /t = − .
2. The first outcome is that the VMCresults, though renormalized with respect to the case t ′ = 0, present a critical doping δ c which is very simi-lar to the one found for the pure t − J model. By con-trast, the FN approach strongly suppresses the spin-spincorrelations, even very close to half filling. In this case,the FN results have rather large size effects, that preventus to extract a reliable estimate for the thermodynamiclimit. However, it is clear that the antiferromagnetic re-gion is tiny and we can estimate that δ c . .
03. It shouldbe emphasized that for t ′ /t = − . t − J model with t ′ = 0, but nevertheless the projection technique, even ifapproximate, is able to reduce the bias (e.g., the presenceof a large magnetic order up to δ ∼ . C. Superconducting properties
In the following, we want to address the problem ofthe superconducting properties of the Hamiltonian (1).In particular, we would like to obtain an accurate de-termination of the pair-pair correlations as a function ofthe hole doping and clarify the role of the next-nearest-neighbor hopping t ′ . The effect of such term has been re-cently considered by using different numerical techniques.DMRG calculations for n -leg ladders (with n = 4 and 6)showed that the effect of a negative t ′ is to stabilize ametallic phase, without superconducting correlations. Moreover, improved variational Monte Carlo techniquessuggested that t ′ could suppress pairing at low doping,whereas some increasing of superconducting correlationscan be found in the optimal doping regime. A furthervariational study, suggested the possibility that a suffi-ciently large ratio t ′ /t can disfavor superconductivity andstabilize charge instabilities (stripes) near 1 / µ,ν ( r ) = S r,µ S † ,ν , (14)where S † r,ν creates a singlet pair of electrons in the neigh-boring sites r and r + µ , namely S † r,µ = c † r, ↑ c † r + µ, ↓ − c † r, ↓ c † r + µ, ↑ . (15)For the first time, we implemented the forward-walkingtechnique in order to compute true expectation values ofthe pairing correlations over the FN state: h ∆ µ,ν ( r ) i = h Ψ F N | ∆ µ,ν ( r ) | Ψ F N ih Ψ F N | Ψ F N i . (16)Indeed, given the fact that ∆ µ,ν ( r ) is a non-diagonaloperator (in the basis of configurations defined before),within the FN approach the previous calculations werebased upon the so-called mixed average, where, similarlyto Eqs. (8) and (12), the state on the left is replaced bythe variational one. Now, by using Eq. (16), it is possibleto verify the fairness of the variational results against amuch more accurate estimation of the exact correlationfunctions given by the FN approach.The superconducting off-diagonal long-range order im-plies a non-zero value of h ∆ µ,ν ( r ) i at large distance r .In the following, we consider the pair-pair correlation atthe maximum distance and µ = ν (parallel singlets) bothfor the VMC and the FN approximations and denote P d = 4 lim r →∞ h ∆ y,y ( r ) i . It is worth noting that, as faras the superconducting correlations are concerned, thereis no appreciable difference between the results obtainedwith and without the antiferromagnetic order parame-ter and the long-range spin Jastrow factor. The resultsfor the pure t − J model are reported in Fig 6, where wereport two different values of the antiferromagnetic cou-pling, i.e., J/t = 0 . J/t = 0 .
4. In this case, VMCand FN calculations are in fairly good agreement, giv-ing a similar superconducting phase diagram. Interest-ingly, the optimal doping, i.e., the doping at which the P d2 VMC L=98VMC L=162FN L=98FN L=162 δ J/t=0.4
J/t=0.2J/t=0.2
No Jastrow
FIG. 6: (Color online) Pair-pair correlations at the maximumdistance as a function on the doping for
J/t = 0 . J/t = 0 . δ P d2 L=98 VMCL=162 VMCL=98 FNL=162 FN
J/t=0.2 t’/t=-0.2 SC FIG. 7: (Color online) The same as in Fig. 6 for the t − t ′ − J model with J/t = 0 . t ′ /t = − .
2. The dashed lineindicates a tentative estimation for the thermodynamic limit. maximum in the pair-pair correlations takes place, oc-curs in both cases at δ ∼ .
2, whereas the actual value ofthe correlations is proportional to
J/t . At high doping,where antiferromagnetic fluctuations play a minor role,the behavior of the pairing is unchanged when J is var-ied. Although in this region there are some size effects,we can safely estimate that superconductivity disappears P d2 L=98 VMCL=162 VMCL=98 FNL=162 FN δ L=98 VMCL=162 VMCL=98 FN
J/t=0.2
J/t=0.2t’/t=-0.2
FIG. 8: (Color online) Detail of the pair-pair correlationsreported in Figs. 6 and 7 at low doping. around δ ∼ .
35 and δ ∼ . J/t = 0 . J/t = 0 . t − J model we clearly obtain a linear be-havior of the pair-pair correlations with δ , indicatinga superconducting phase as soon as the Mott insulatoris doped, in the case of a finite t ′ , the FN results couldbe compatible with a finite critical doping, below whichthe system is not superconducting. This outcome is inagreement with earlier Monte Carlo calculations done byone of us, where it was suggested that the extended t − J model with hoppings and super-exchange interac-tions derived from structural data of the La CuO com-pound could explain the main experimental features ofhigh-temperature superconducting materials, with a fi-nite critical doping for the onset of electron pairing.Remarkably, from δ ∼ . δ ∼ . δ ∼ .
3, small clusters, e.g., L = 98, indicate stronger pairing correlations than the δ P d2 δ ~0.3 L=242
L=162 L=98N=72 N=50 N=30
FIG. 9: Size scaling of the Pair-pair correlations at themaximum distance for t − t ′ − J model with J/t = 0 . t ′ /t = − . δ ∼ . pure t − J model without t ′ , larger clusters point out alarge reduction of P d . Nonetheless, we have a ratherclear evidence that for δ ∼ . t ′ , that could eventually stabilizecompeting phases with modulation in the charge distri-bution and/or a magnetic flux through the plaquettes. However, for t ′ /t = − .
2, the homogenous variationalansatz (2) provides a lower energy when compared tothe one used in Ref. 45. At present, the most accurateFN calculations based on the lowest-energy variationalansatz, do not show any tendency towards charge in-homogeneities for δ . .
4. This outcome is importantbecause in principle the FN approach can spontaneouslyinduce charge-density wave modulations in the groundstate, even when the variational wave function before theFN projection is translationally invariant. IV. CONCLUSION
In this paper, we considered the magnetic and su-perconducting properties of the t − t ′ − J model withinVMC and FN approaches. We showed that for t ′ = 0the ground-state properties can be accurately reproducedby a state containing both electron pairing and suit-able magnetic correlations, namely a magnetic order pa-rameter in the mean-field Hamiltonian that defines thefermionic determinant and a spin Jastrow factor for de-scribing the spin fluctuations. In this case, we obtain arather large magnetic phase, with a critical doping thatslightly depends upon the super-exchange coupling J ,i.e., δ c = 0 . ± .
01 and δ c = 0 . ± .
02 for
J/t = 0 . J/t = 0 .
4, respectively. The superconducting corre-lations show a dome-like behavior and vanish when theMott insulator at half filling is approached. Interestingly,compared to the RMFT that predicts a quadratic behav-ior of the pair-pair correlations as a function of the doping δ , here we found that a linear behavior is more plausible.Then, we also reported important modifications due tothe presence of a finite ratio t ′ /t . The first effect of thisfurther hopping term is to strongly suppress antiferro-magnetic correlations at low doping, shifting the criticaldoping to 0 .
03 for t ′ /t = − .
2. This is a genuine effectof the FN method, since, within the pure variational ap-proach, though the spin-spin correlations are suppressedwith respect to the case of t ′ = 0, the values of the crit-ical doping for these two cases are very similar. Mostimportantly, the presence of a finite value of the next-nearest-neighbor hopping has dramatic effects on the su-perconducting properties. At small doping, i.e., δ . . Moreover, for 0 . . δ . . t − J model, largerclusters show huge size effects that strongly renormalizethe pairing correlations at large distance. However, forthe value of t ′ considered in this work, we are rather con- fident that superconducting off-diagonal long-range ordertakes place in a considerable hole region. In any case, thehuge renormalization of the electronic pairing for δ ∼ . t ′ /t .In conclusion, the main qualitative features of thecuprate phase diagram appear rather well reproduced bythe t − t ′ − J model with a sizable next-nearest-neighborhopping. 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