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7

Competition among various charge-inhomogeneous states and d-wave superconducting

state in Hubbard models on square lattices

Kota Ido, Takahiro Ohgoe and Masatoshi ImadaDepartment of Applied Physics, University of Tokyo,

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

We study competitions between charge uniform and inhomogeneous states in two-dimensionalHubbard models by using a variational Monte Carlo method. At realistic parameters for cupratesuperconductors, emergent effective attraction of carriers generated from repulsive Coulomb interac-tion leads to charge/spin stripe ground states, which severely compete with uniform superconductingexcited states in the energy scale of 10 K for the cuprates. Stripe period increases with decreasinghole doping δ, which agrees with the experiments for La-based cuprates at δ = 1/8. For lower δ, wefind a phase separation. Implications of the emergent attraction for the cuprates are discussed.

I. INTRODUCTION

After the discovery of the high temperature supercon-ductivity in the cuprates1, its mechanism remains one ofthe most challenging issues in condensed matter physics.A necessary condition of high-temperature superconduc-tivity for strongly correlated electron systems is a largeeffective attractive interactions between electronic car-riers emerging from strong Coulomb repulsions. How-ever, this strong attraction can also enhance the tendencyof electron aggregations in real space. This means thatthe strong attractive interaction induces diverging chargecompressibility2,3 as well as charge inhomogeneous statessuch as phase separations (PS) and stripe states4–16. Infact, the competition between the superconductivity andthe charge inhomogeneity as a stripe state has been ob-served and well discussed in La-based cuprates17–21. Re-cently, such phenomena were also reported in Y-22–27,Hg-23,28 and Bi-based cuprates29–31, indicating a ubiqui-tous feature in the cuprate superconductors32,33.

To understand the origin of superconductivity in thecuprates, the Hubbard model on a square lattice hasbeen studied for long time. Although many theoret-ical studies have been devoted to understanding theground states of the Hubbard model, they are still un-der debate2,12–15,34–46. To gain insight into the chargeinhomogeneous phases including the stripes, detailedanalyses of their existence and competitions with thed-wave superconductivity are desired, particularly ontheir dependences on the hole doping concentration δ,band structure and the interaction. Most numericalstudies based on variational calculations or dynamicalmean-field theory showed that charge uniform statesare the ground states or macroscopic phase separationappears12,13,35–37,40,41,47. However, in these calculations,the possibility of long-period stripe states are ignored.Recent studies using infinite projected entangled pairstates, the density matrix embedding theory (DMET),constrained path auxiliary field quantum Monte Carlomethod and density matrix renormalization group all re-ported the stripe ground state, but studied systemati-cally only for a special choice of band structure (onlywith nearest neighbor transfer t = 1) at δ = 0.125, with

8/16 period for charge/spin stripes46. Recent variationalMonte Carlo (VMC) calculations combined with tensornetwork states also found stripe states with 8/16 (forδ < 0.15) and 4/8 (for δ > 0.15) periods for charge/spinas ground states below δ ∼ 0.2515. However, the stripeperiod extensively studied at δ = 0.125 in these calcula-tions is different from that observed in La-based cuprates,which is 4 charge and 8 spin periods17,18. These resultsimply that more systematic and realistic study is neededto understand the real cuprate systems.One of the missing ingredients in the simple Hub-

bard model is hopping parameters beyond the nearest-neighbor pairs. The previous DMET study showed thatthe stripe state in the experiments has a lower energythan the charge uniform state in the system with thenext-nearest hopping43. However, since the sizes of em-bedded clusters are restricted, the competitions withother stripe states are still unclear at a finite hole con-centration.In this paper, by using the VMC method, we study the

competitions among stripe states with different periodic-ities in addition to charge uniform states. We show thatthe ground states has stripe orders, the period of whichdecreases with increasing δ in a wide range. In the lowerdoping region, the PS occurs between the antiferromag-netic insulator and the stripe state. More importantly,we find that the stripe state experimentally observed atδ = 0.125 is indeed the ground state for a realistic valueof next-nearest-neighbor hopping. We clearly see thatthe superconducting (SC) long-range order is stronglysuppressed due to the emergence of stripe orders, whilecharge uniform and strong superconducting states existas excited states with tiny excitation energies.

II. MODEL AND METHOD

We study t− t′ Hubbard model on square lattices un-der the antiperiodic-periodic boundary condition. TheHamiltonian is defined by

H =−∑

i,j,σ

tijc†iσcjσ + U

Ns∑

i

ni↑ni↓, (1)

2

where the hopping amplitude tij is taken as tij = t forthe nearest-neighbor pairs, tij = t′ for the next-nearest-neighbor pairs and otherwise tij = 0. U is the onsite

repulsive interaction, Ns = L × L is the system size, c†iσ(ciσ) is a creation (annihilation) operator of an electron

with spin σ on the site i, and niσ = c†iσciσ. The latticeconstant is taken as the length unit. We mainly per-formed the calculations for U/t = 10 because it is closeto proposed ab initio estimate for the cuprates48.

To study the ground states of the Hubbardmodel, we have used the VMC method. Asa trial wave function, we adopted the generalizedpair product wave function with correlation factors:|ψ〉 = PGPJPex

d−h |φ〉49. Here Gutzwiller factorPG = exp (−g∑i ni↑ni↓), Jastrow factor PJ =

exp(

−∑i,j vijninj

)

, and the doublon-holon correlation

factor Pexd−h = exp

(

−∑5

m=0

∑

l=1,2 α(l)(m)

∑

i ξ(l)i(m)

)

are

considered and |φ〉 =(

∑Ns

i,j fijc†i↑c

†j↓

)N/2

|0〉, where ni =

ni↑ + ni↓ and N is the number of electrons. ξ(l)i(m) is 1

when a doublon (holon) exists at the i-th site and mholons (doublons) surround at the l-th nearest neighbor.

Otherwise, ξ(l)i(m) is 0. In this study, we treat g, vij , α

(l)(m)

and fij as variational parameters. To describe inhomoge-neous stripe states, we assume that fij has the ls×2 sub-lattice structure, which enables the ls period spin stripe.In our calculations, we treat several tens of thousandsof variational parameters for the largest systems. Allthe variational parameters are optimized by using thestochastic reconfiguration method50.

To clarify physical properties of the ground states,we measured the spin structure factor Ss(q) =1

3Ns

∑

i,j 〈Si · Sj〉 e−iq·(ri−rj), the charge structure fac-

tor Sc(q) = 1Ns

∑

i,j 〈ninj − ρ2〉 e−iq·(ri−rj) and thelong-range part of dx2−y2-wave SC correlation func-

tions P∞d = 1

M

∑

|r|≥rmax/2Pd(r), where M is

the number of vectors satisfying |r| ≥ rmax/2.

Here, ρ =∑

i,σ 〈niσ〉 /Ns, rmax = L/√2 and

Pd(r) = 12Ns

∑

i 〈∆†d(r)∆d(r + ri) + ∆d(r)∆

†d(r + ri)〉

with ∆d(ri) =1√2

∑

rg(r)(cri↑cri+r↓ − cri↓cri+r↑). The

form factor g(r) is defined as g(r) = δrx,0(δry,1 +δry,−1) − δry,0(δrx,1 + δrx,−1), where r = (rx, ry). Wedefine the spin/charge order parameter as ∆S/C =√

Ss/c(qpeak)/Ns, where Ss/c(qpeak) represents the peakvalue of the spin/charge structure factor. We also definethe SC order parameter as ∆SC =

√

P∞d .

III. RESULTS

A. Ground-state phase diagram of the t− t′

Hubbard model

The main results are summarized in Fig. 1, whichshows the ground-state phase diagram in the δ− t′ planefor U/t = 10. Throughout this paper, the stripe statewith charge (spin) period lc(ls) is denoted as “ClcSls”for simplicity. Charge uniform states are obtained underthe 2×2 sublattice structures and energies are comparedwith inhomogeneous states obtained under longer sublat-tices. As shown in Fig. 1, charge inhomogeneous statesexist as the ground states in a wide range of δ for any t′/t.The wavelength of the charge lc becomes longer with thedecrease of δ, and eventually the PS, whose wavelength isinfinite, occurs between the antiferromagnetic insulatorand a stripe state. For −0.3 ≤ t′/t . −0.15, which is arealistic range of t′/t for the cuprates, the ground stateat δ = 1/8 is the C4S8 state which has been observedin La-based cuprates17,18. However, charge inhomoge-neous states are stabilized even in the highly overdopedregime and thus a uniform d-wave superconducting statedoes not appear as the ground state of the single-bandHubbard model at strong coupling. We will discuss ournumerical results in comparison with the experiments inSec. IV.

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5

Uniform

(PM)

C2S4C3S3

C4S8

C5S5

C6S12

C7S7

PSAF

FIG. 1. (Color online) Ground-state phase diagram of theHubbard model on a square lattice for U/t = 10. Note thatt′/t is a negative value. At δ = 0, the ground state is the an-tiferromagnetic (AF) Mott insulator (green bold line). Crosssymbols indicate the boundary of the phase separation (PS).Solid black circles represent the boundaries of “ClcSls” stripestates with lc/ls period for charge/spin. Dashed line showsδ = 0.125. Solid lines and painted regions are guides for theeyes. In the unpainted (white) region, the ground state is acharge uniform paramagnetic (PM) state.

3

B. Ground states and excited states

First, we show results for t′/t = 0 as a simplest model.Figure 2 (a) shows the energies of uniform and stripe

0.0 0.1 0.2 0.3 0.4 0.5

L=20, Uniform

L=24, Uniform

L=16, C2S4

L=24, C2S4

L=18, C3S3

L=24, C3S3

L=16, C4S8

L=24, C4S8

L=20, C5S5

GS

0.00

0.05

0.10

0.15

-0.05

0.0 0.1 0.2 0.3

0.00

0.01

0.02

0.03

L=18, Uniform

L=20, Uniform

L=16, C4S8

L=20, C5S5

L=24, C6S12

L=14, C7S7

L=16, C8S16

L=18, C9S9

L=22, C11S11

GS

FIG. 2. (Color online) Doping concentration dependenceof energies for several different states in two-dimensionalHubbard model with U/t = 10 at (a) t′/t = 0 and (b)t′/t = −0.3. A linear function f(δ) = −1.835δ − 0.4211 org(δ) = −1.5δ − 0.4222 is subtracted for better visibility. Forclarity, we draw yellow thick line to represent the energies ofthe ground states. Types of states and system sizes are de-scribed in the legend. Error bars indicate the statistical errorsarising from the Monte Carlo sampling, but most of them aresmaller than the symbol sizes here and in the following fig-ures. Dashed black line and gray region show the tangent lineof the energy curve drawn from δ = 0 and PS, respectively.In panel (a), commensurate fillings δ = 1/lc are indicated bycolored arrows.

states with different periodicities as functions of hole-doping concentration δ = 1 − N/Ns. We will show evi-dences for the stripe long-range order described in Fig. 2(a) later in Fig. 4. From Fig. 2 (a), we see that stripestates are the ground states below δ ≈ 0.25. The max-imum value of energy difference between uniform and

stripe states is the order of ∼ 0.01t at δ ≈ 0.125, whichis consistent with the recent results by other numericalcalculations such as the tensor network states15,46. Byincreasing the hole concentration, the wavelength of thecharge lc becomes shorter. This is naturally related to themean distance between holes, which decreases with in-creasing doping concentrations. Stripe states with lc ≤ 3were not found as the ground states.To clarify the possibility of PS, we performed a

Maxwell construction for the energy curve of the groundstates (dashed line in Fig. 2 (a)). We find that a PSappears for 0 < δ ≤ 0.125. This region is narrower thanthat obtained in the previous VMC study, where onlyuniform states were assumed13. Then we conclude thatthe stripe states are stable ground states in the region0.125 < δ < 0.25. At δ ≈ 0.125, several stripe statesfor lc = 6 − 8 are nearly degenerate, which is also con-sistent with recent studies by state-of-the-art numericalmethods46. The charge and spin configurations of theC8S16 state at δ = 0.125 are plotted in Figs. 3 (a) and(b), respectively.

0.8

0.9

1

-0.5

0

0.5

-0.5

0

0.5

0.75

0.85

0.95

FIG. 3. (Color online) Charge density n(r) = 〈nr↑ + nr↓〉and spin density along z-direction Sz(r) = 0.5 〈nr↑ − nr↓〉 forthe ground state for L = 16 and U/t = 10 at δ = 0.125. Thenext-nearest-neighbor hopping in (a-b) and (c-d) are t′/t = 0and t′/t = −0.3, respectively.

Next, we show the results for t′/t = −0.3, which isa realistic value for the cuprate superconductors48. Fig-ure 2 (b) shows the hole-doping dependence of the en-ergies for U/t = 10. We find essential similarity to thecase t′/t = 0, indicating the robust stability of the stripeground state irrespective of the band structure. A quan-titative difference is, however, that the stripe states asground states extends in a wider region 0.1 < δ < 0.5.Moreover, the ground state at δ = 0.125 shows C4S8 or-der, which is consistent with the experiments of La-based

4

cuprates17,18. The charge and spin configurations of theC4S8 ground state at δ = 0.125 are shown in Figs. 3(c) and (d), respectively. This C4S8 state stably existsas the ground states for 0.11 ≤ δ ≤ 0.15 although itseverely competes with other stripe order such as C3S3and C5S5. The locking of stripe period has been re-cently observed in the scanning-tunneling-microscope ex-periment combined with phase resolved electronic struc-ture visualization technique29. Below δ ∼ 0.1, a PS be-tween antiferromagnetic and stripe states occurs as withthe case of t′/t = 0.

C. Spin, charge and superconducting orders

The δ-dependence of ∆2S and ∆2

C for t′/t = 0 are shownin Figs. 4 (a) and (b), respectively. We see that ∆2

S

decreases as δ increases. On the other hand, ∆2C has a

dome structure around the maximum at δ ∼ 0.1. Thedome-like stripe order exists even after the extrapolationto the thermodynamic limit as shown in Appendix B.

0.0 0.1 0.2 0.3

0.00

0.02

0.04

0.06

0.00

0.02

0.04

0.00

0.02

0.06

0.04

L=20, Uniform

L=16, C4S8

L=20, C5S5

L=24, C6S12

L=14, C7S7

L=16, C8S16

L=18, C9S9

L=22, C11S11

GS

FIG. 4. (Color online) δ-dependence of (a) ∆2S, (b) ∆

2C and

(c) ∆2SC for U/t = 10 and t′/t = 0. Notations are the same

as in Fig. 2 (a). Enlarged view for ∆2SC will be shown in

Appendix A.

Figure 4 (c) shows δ-dependence of ∆2SC. We see that

∆2SC in the stripe states is substantially smaller than

those of charge uniform states. The previous VMC studyshowed that the strong superconductivity obtained by as-suming the charge uniformity emerges in accord with theregion of the PS, and therefore is mostly preempted bythe PS13. In the present study, we have shown that ifmicroscopic inhomogeneity is allowed, large portion ofthe PS is compromised by the formation of stripes. Thesuperconductivity is anyhow weakened by the stripe for-mation, because of its character, where carrier rich stripsare weakly coupled by the Josephson tunneling. How-ever, it should be remarked that the uniform stronglySC state also survives as an excited state with the exci-tation energy in the order of 0.01t (in the cuprate scale∼ 10− 100K) as one sees in Figs. 2 (a) and (b). ∆2

SC inthe uniform state has a dome structure13 similar to ∆2

C

in the ground state as one sees in Figs. 4(b) and (c).

Figure 5 plots physical quantities for the case of t′/t =−0.3, which are again similar to the case of t′/t = 0.Note that the stripe order parameters remain finite in thethermodynamic limit below δ ∼ 0.4 (see also AppendixB). However, in the experiments, the stripe state hasbeen observed only below δ ∼ 0.233. This discrepancywill be discussed later.

0.0 0.1 0.2 0.3 0.4 0.50.00

0.03

0.06

0.00

0.01

0.02

0.00

0.03

0.06L=24, Uniform

L=24, C2S4

L=24, C3S3

L=24, C4S8

GS

FIG. 5. (Color online) δ-dependence of (a) ∆2S, (b) ∆

2C and

(c) ∆2SC for U/t = 10 and t′/t = −0.3. Notations are the

same as in Fig. 2 (b). Enlarged view for ∆2SC will be shown

in Appendix A.

5

D. Interaction-dependence for t′/t = −0.3

Finally, we show the interaction dependence of the en-ergy difference between the uniform and inhomogeneousstates for t′/t = −0.3 in Fig. 6. The stripe states arethe ground states above U/t ∼ 4 and the stripe phaseextends with the increase in U . For U/t = 6, the stripeand the uniform strongly SC states are nearly degeneratearound δ ∼ 0.3. The stripe and uniform SC order param-eters become smaller compared with those for U/t = 10but the δ-dependence is similar, and we do not find aclear indication of PS. (See Appendix C.) At U/t = 4,the charge uniform state is nearly degenerate with thestripe state but the order parameters for the stripe andSC are all nearly zero in the both states, implying thatthe ground state is a paramagnetic metal. Although thestability changes, the stripe and SC orders have similartrend in the dependences on U and δ.

0.1 0.2 0.3 0.4 0.5

-0.02

-0.01

0.00

FIG. 6. (Color online) Interaction dependence of the stabilityof uniform and inhomogeneous states (the energy difference∆E = Estripe − Euniform) for t′/t = −0.3. Here, Estripe andEuniform are the energies of stripe and uniform states, respec-tively. Circle, square and triangle symbols show the energiesof C2S4, C3S3, and C4S8 stripes, respectively. Red, greenand blue symbols represent ∆E for U/t = 10, 6, and 4, re-spectively. Curves are guides for the eyes.

IV. DISCUSSION

The same trend between the stripe and SC orders isnaturally understood because the emergent and strongeffective attractive interaction of carriers, which arisesfrom the originally repulsive interaction, generates bothof the order. The stripe as a consequence of aggrega-tion of carriers in the real space, and the strong cou-pling superconductivity both requires strong effective at-traction of carriers. The effective attraction may haveboth static and retarded pieces. It is possible that thelatter may be contributed from bosonic glues includingspin fluctuations51–55 and reinforced by hidden fermion

excitations56,57. The static effective attraction is a directconsequence of the negative quadratic coefficient b < 0in the energy expansion E = E0 + aδ + bδ2 + · · · as seenin Figs.2 (a) and (b). b < 0 is caused by the Mottness,where the kinetic energy decreases nonlinearly upon dop-ing13.

In the presence of realistic values of t′/t and U/t forthe cuprates, our calculations show the severe competi-tion among stripe states with lc = 3 − 7 below δ ∼ 0.2.The charge wavelengths lc = 3− 7 have been observed ina number of cuprates for 0.05 . δ . 0.217–29. The wave-length of charge lc = 4 is consistent with the observationsnot only in La-based cuprates17,18 but also in a Bi-basedcuprate29. The charge inhomogeneity with lc = 5−7 hasbeen observed in La-based cuprates below δ ∼ 0.119–21.The wavelength lc = 3 is close to the experimental obser-vations for a Y-based cuprate22–27. The charge wave-lengths observed in a single-layered Hg-based cuprateare lc ≈ 3.5823 and 4.3528, which is located withinlc = 3−5. Recent first-principles studies have shown thatthe single-layered Hg-based cuprate has weaker effectiveCoulomb interactions than the single-layered La-basedcuprate48,58. Our results support these studies becausethe inhomogeneities become weaker with weakening ofthe interaction, which is consistent with the experimentswhere the charge order in the Hg-based cuprate is muchweaker than that in the La-based cuprate20,21,23.

The parameter values t′/t = −0.3 and U/t = 10 wereproposed as realistic values for the cuprates48,59,60. How-ever, our results show that the stripe phase is extended ina much wider range of δ compared with the experiments.On the other hand, by weakening U/t, the stripe orderparameters and the energy difference between the stripestates and the uniform SC state becomes small. These re-sults imply that an appropriate description of single-bandeffective hamiltonians for the cuprates is found in the re-gion of intermediate on-site interactions rather than thestrong coupling region at least in terms of the stabilityof the stripe and SC phases.

The reason why the d-wave SC ground state does notclearly appear in contradiction to the experimental re-sults is speculated to be the oversimplification of theHubbard models we studied. As recent numerical resultsare consistent with each other46, the discrepancy doesnot seem to originate from the limitation of the accuracyof our calculations (see also the last paragraph of this sec-tion). In order to make a more quantitative and reliablecomparison with experiments beyond our present analy-sis, we should analyze the ab initio effective Hamiltoni-ans, which include long-range Coulomb interactions andhopping integrals and, if necessary, the electron-phononcoupling missing in the simplified Hubbard model. Forexample, in the ab initio single-band effective Hamilto-nian for the Hg-based cuprate48, the nearest-neighborCoulomb interaction is about 20% of the on-site inter-action. The third-nearest-neighbor hopping t′′ in theeffective Hamiltonian has also a non-negligible value oft′′/t ∼ 0.1548. A tiny energy difference between the su-

6

perconducting and stripe states is subject to be easily re-versed by such realistic factors. We are now at the stagethat allows quantitative comparisons between model cal-culations and the experimental results, because of theachieved accuracy of the solver. The origin of the quan-titative discrepancy will be discussed elsewhere based onfirst-principles studies.One may be concerned about the accuracy of the

present calculation. However, our trial wave function canbe systematically improved by using methods such as thepower Lanczos and/or tensor network15,61–64. These ad-ditional refinements indeed lower the energies. However,the energies are nearly equally lowered among competingstates, and other physical quantities such as stripe andsuperconducting orders only slightly change13,15. (Seealso Appendix D.)

V. SUMMARY

Our VMC calculations show stripe ground states of theHubbard models irrespective of the amplitude of the nextnearest neighbor hopping. Its stability and stripe orderparameter substantially increases with increasing U inthe strong coupling region beyond U/t = 5 and becomesextended in a wider range of hole doping concentrationwith a dome-like δ dependence. With increasing holedoping, the stripe period decreases. The stripe period isroughly proportional to the mean hole distance for t′/t =0.0, whereas it is not for t′/t = −0.3. This detaileddifference may be ascribed to the difference in the Fermisurface nesting vectors especially in the antinodal region.This issue will be studied in future studies. The period att′/t = −0.3 agrees with that observed in the experimentsat δ = 0.125.In the static stripe ground states, the superconduc-

tivity is substantially suppressed. On the other hand,metastable excited states with the uniform and stronglySC order, whose excitation energy is tiny (∼ 0.01t), ap-pear with dome-like δ dependence similarly to the domeof charge stripe order. The superconducting order, inboth excited and ground states decreases for smaller U/tand numerically invisible for U/t . 4 which again hastrend essentially similar to the charge order.

The same trend between the SC and stripe states andtheir severe competition are a consequence of the strongeffective attraction originating from the strong repulsiveinteraction. Understanding their common route and dis-tinctions revealed here will help designing ways of sup-pressing the stripe and stabilizing the SC state simul-taneously. Some attempts were already made16,65, andextensive studies along this line are intriguing challengingissues in the future.An interesting future issue is to more quantitatively

analyze effective low-energy hamiltonians of the cupratesobtained from ab initio calculations48 to understand themechanisms and materials dependence in the light of thepresent severe competitions. In particular, the validity ofthe single-band description has to be seriously examinedbecause the present elucidation suggests a weaker corre-lation than the parameters proposed in the literature48

if one sticks to the single-band description.

ACKNOWLEDGMENTS

The authors thank the Supercomputer Center, the In-stitute for Solid State Physics, the University of Tokyofor the facilities. The calculations were performed by us-ing the open-source software mVMC66. We thank thecomputational resources of the K computer provided bythe RIKEN Advanced Institute for Computational Sci-ence through the HPCI System Research project, aswell as HPCI Strategic Programs for Innovative Research(SPIRE), the Computational Materials Science Initiative(CMSI), and Social and scientific priority issue (Creationof new functional devices and high-performance mate-rials to support next-generation industries; CDMSI) tobe tackled by using post-K computer, under the projectnumber hp130007, hp140215, hp150211, hp160201, andhp170263 supported by Ministry of Education, Culture,Sports, Science and Technology, Japan (MEXT) . Thiswork was also supported by Grant-in-Aids for Scien-tific Research (No. 22104010, No. 22340090 and No.16H06345 ) from MEXT. KI was financially supportedby Grant-in-Aid for JSPS Fellows (No. 17J07021) andJapan Society for the Promotion of Science through Pro-gram for Leading Graduate Schools (MERIT).

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8

Appendix A: Enlarged view of δ dependence of ∆SC

Figures 7 (a) and (b) show the enlarged views of Figs.2 (c) and 2 (c) in the main text which plot the holeconcentration dependence of the superconducting orderparameters for t′/t = 0 and t′/t = −0.3, respectively.The maximum value of the SC order paramters for theground states is the order of 10−4 − 10−3 at most, whichis much smaller than that of the uniform excited state.

0

4

8

0

5

0.0 0.1 0.2

0.0 0.40.1 0.2 0.3 0.5

FIG. 7. (Color online) Doping concentration dependence ofsuperconducting order parameter for U/t = 10 at (a) t′/t = 0and (b) t′/t = −0.3. Notations in the panels (a) and (b) arethe same as Figs. 4 and 5, respectively.

Appendix B: Size-dependence of stripe and

superconducting order parameters for stripe states

To clarify the thermodynamic properties of the groundstates, we show the size-dependence of physical quanti-ties for t′/t = 0 and U/t = 10 within the stripe groundstate at several doping concentrations in Fig. 8. Here,following the convention in the literature67, we estimatedthe extrapolated order parameter ∆ by fitting the severalpoints with a + bL−1. Even when we employ the scal-ing a′ + b′L−1/2, the results do not essentially change.Figure 8 shows that both the spin and charge order pa-rameters remain finite even after the extrapolations be-

low δ ∼ 0.2. At commensurate fillings, one hole fills ina one charge wavelength, i.e. δ = 1/lc. The bottompanel of Fig.8 shows, in the thermodynamic limit, clearstronger suppression of long-range superconducting orderat commensurate fillings δ = 1/lc than the case δ 6= 1/lcincommensurate to the stripe period. In the latter in-commensurate fillings, the superconducting order likelyremains nonzero in the thermodynamic limit.

C6S12, = 0.153

C6S12, = 0.167

C5S5, = 0.18

C5S5, = 0.2

C4S8, = 0.25

0.00 0.120.080.040.00

0.08

0.04

0.00

0.06

0.04

0.02

0.00

0.24

0.16

0.08

FIG. 8. (Color online) System-size dependece of order pa-rameters for t′/t = 0 and U/t = 10. In the legend, types ofquantum states and hole concentrations are described. Solidsymbols correspond to the commensurate fillings in which onehole fills per one charge-stripe unit cell. Solid and dashed linesrepresent the linear-extrapolation fittings by a+ bL−1.

We also show size-dependences of physical quantitiesfor t′/t = −0.3 in Fig. 9. As we mentioned in themain text, the extraporated values of stripe orders havenonzero values below δ ∼ 0.4.On the other hand, we donot find any non-positive extrapolated values of the SCorder parameter at this stage, which is different from thecase of t′/t = 0. To understand this difference, we needfurther analysis of the size dependence of the SC orderparameter and its doping dependence in the thermody-namic limit for both t′/t = 0 and t′/t = −0.3, but it isleft for a future study.

9

0.00 0.120.080.040.00

0.08

0.04

0.00

0.08

0.04

0.00

0.20

0.10

FIG. 9. (Color online) System-size dependece of order param-eters for t′/t = −0.3 and U/t = 10. In the legend, types ofquantum states and hole concentrations are described. Solidand dashed lines represent the linear-extrapolation fittings.

Appendix C: Physical quantities for t′/t = −0.3 and

U/t = 6

Figure 10 compares the hole-doping dependence of theenergies between U/t = 6 and U/t = 10 below δ ∼ 0.15.We do not find an evidence for the PS between the an-tiferromagnetic state and the stripe state at U/t = 6,where a tangent line from δ = 0 to the energy curvecannot be drawn, distinctly from the case U/t = 10.

Figure 11 plots the δ-dependence of the spin, chargeand superconducting order parameters for U/t = 6 andt′/t = −0.3. We see that the results are qualitativelysimilar to the case of U/t = 10, but the stripe orderparameters become smaller. This means that the inho-mogeneity is weakened by the decrease of the on-site in-teraction. This tendency is also seen in Fig. 12, wherethe electron distribution in real space is depicted. Thesuperconductivity in the uniform excited states has the

0.00 0.05 0.10 0.15

0.00

-0.01

0.01

0.02

0.00 0.05 0.10 0.15-0.08

-0.04

0.00

FIG. 10. (Color online) Doping dependence of the energy ofseveral different states for t′/t = −0.3 below δ = 0.15. Weset f(δ) = −0.8δ − 0.640 and g(δ) = −1.7δ − 0.4211. Typesof states and system sizes are described in the legend. Forclarity we draw yellow thick line for the energies of the groundstates for L = 24. Dashed black line and gray region showthe tangent line of the energy curve and PS, respectively.

same trend as the case of the stripe orders. At U/t = 4,the stripe and superconducting orders are scaled to zerowithin the numerical accuracy.

Appendix D: Power lanczos method

The power lanczos method is one of the systematicways to improve a trial wave function in the VMCmethod61. In the N -th power Lanczos method, we multi-ply the Hamiltonian to the optimized trial wave function|ψopt〉 as

|ψ(N)〉 =(

1 +

N∑

n=1

αnHn

)

|ψopt〉 , (D1)

10

0.0 0.1 0.2 0.3 0.4 0.5

0.00

0.01

0.02

0.00

0.04

0.06

0.02

0.00

0.04

0.06

0.02

FIG. 11. (Color online) Doping dependence of order param-eters for (a) spin and (b) charge stripes for U/t = 6 andt′/t = −0.3. Dashed and dotted curves represent the resultsof the ground states and the charge uniform state for U/t = 10for comparison, respectively. Notations are the same as in Fig.5.

0.022(5) 0.641(4) 0.767(5) 0.658(6) 0.011(9) 0.645(2) 0.753(4) 0.632(6)

0.205(5) 0.117(4) 0.057(5) 0.115(6) 0.205(9) 0.112(3) 0.060(4) 0.119(6)

0.177(8) 0.095(7) 0.086(7) 0.133(6) 0.165(6) 0.105(7) 0.089(8) 0.143(7)

0.187(8) 0.573(7) 0.610(7) 0.438(6) 0.147(6) 0.574(7) 0.614(8) 0.390(7)

FIG. 12. (Color online) Spin density along z-direction Sz

i =〈ni↑ − ni↓〉 and hole density 1 − 〈ni〉 = 1 − 〈ni↑ + ni↓〉 fort′/t = −0.3 at δ = 0.125 for the ground state with C4S8 forL = 24. The radius of every red circle is propotional to thehole density 1−ni. The length of every black arrow is propor-tional to the amplitude of the spin density |Sz

i |. The valuesof |Sz

i | and 1− ni averaged over y-direction are shown aboveand below the plots, respectively. Note that the simulationswere performed for finite size systems. Nevertheless, the vari-ational wavefunctions show translational symmetry breakingwhen the momentum projection is not operated. Althougha better ground-state wavefunction is obtained after the mo-mentum projection, the overlap of the two functions spatiallytranslated each other is negligible in the size L = 24 and theorderparameter is expected to be close to the thermodynamiclimit.

where αn are the variational parameters. We use the 1ststep Lanczos method (N = 1) since the numerical costsgrow exponetially with increasing N .

0 2 4 6 8 10 12

VMC1st Lanczos

10-3

10-2

10-1

100

0

0.4

0.8

0 0.2 0.4 0.6 0.8 1

VMC1st Lanczos

FIG. 13. (Color online) Superconducting correlation functionPd(r) (a) and charge structure factor Sc(qpeak) at qpeak =(qx, 0) (b) of the C8S16 state for L = 16, U/t = 10 andt′/t = 0 at δ ≈ 0.11 . Blue line and red dashed line are theresults obtained by using the VMC method and the 1st stepLanczos method, respectively.

Table I shows the energies of competing states forvarious doping concentrations δ. The Lanczos methodimproves the energies of competing states but does notchange character of the ground states and only slightlyalters physical properties as below. We have checked theeffects by the Lanczos operation to other physical quan-tities such as the superconducting correlation functionand the charge structure factor. However, these are onlyslightly changed as shown in Fig. 13.

11

TABLE I. Energies per site of competing states obtainedfrom the VMC and 1st Lanczos calculations for several systemsizes L and the hole-doping concentrations δ at U/t = 10 andt′/t = 0. The number in brackets represents the error on thelast digit.

L δ state VMC 1st Lanczos

20 0.180 Uniform -0.7384(2) -0.7591(4)

20 0.180 C5S5 -0.74820(4) -0.7639(8)

14 0.143 Uniform -0.6665(5) -0.6900(7)

14 0.143 C7S7 -0.68315(5) -0.6992(3)

16 0.109 Uniform -0.60744(9) -0.6272(4)

16 0.109 C8S16 -0.62232(4) -0.6377(1)