Using the variational Monte-Carlo method we find that a relatively weak
long-range electron-phonon interaction induces a d-wave superconducting state
of doped Mott-Hubbard insulators and/or strongly-correlated metals with a
condensation energy significantly larger than can be obtained with Coulomb
repulsion only. Moreover, the superconductivity is shown to exist for infinite
on-site Coulomb repulsion, removing the requirement for additional mechanisms
such as spin fluctuations to mediate d-wave superconductivity. We argue that
the superconducting state is robust with respect to a more intricate choice of
the trial function and that the true origin of high-temperature
superconductivity lies in a proper combination of strong electron-electron
correlations with poorly screened Froehlich electron-phonon interaction.Comment: 4 pages, 4 figure

A. S. Alexandrov

I. G. Lang

J. H. Samson

J. P. Hague

T. M. Hardy

English

arXiv

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Superconductivity in a Hubbard-Fröhlich model and in cuprates

Using the variational Monte Carlo method, we find that a relatively weak long-range electron-phonon
interaction induces a d-wave superconducting state in doped Mott-Hubbard insulators and/or strongly correlated
metals with a condensation energy significantly larger than can be obtained with Coulomb repulsion only.
Moreover, the superconductivity is shown to exist for infinite on-site Coulomb repulsion without the need for
additional mechanisms such as spin fluctuations to mediate d-wave superconductivity. We argue that our
superconducting state is robust with respect to a more intricate choice of the trial-wave function and that a
possible origin of high-temperature superconductivity lies in a proper combination of strong electron-electron
correlations with poorly screened Fröhlich electron-phonon interaction

Thomas M. Hardy (7171691)

J.P. Hague (7170866)

John Samson (1251294)

A.S. Alexandrov (7170284)

Loughborough University Institutional Repository

   This item was submitted to Loughborough’s Institutional Repository (https://dspace.lboro.ac.uk/) by the author and is made available under the following Creative Commons Licence conditions.      For the full text of this licence, please go to: http://creativecommons.org/licenses/by-nc-nd/2.5/  Superconductivity in a Hubbard-Fröhlich model and in cupratesT. M. Hardy,1 J. P. Hague,1,2 J. H. Samson,1 and A. S. Alexandrov11Department of Physics, Loughborough University, Loughborough LE11 3TU, United Kingdom2Department of Physics and Astronomy, The Open University, Milton Keynes MK7 6AA, United KingdomReceived 3 February 2009; revised manuscript received 6 May 2009; published 2 June 2009Using the variational Monte Carlo method, we find that a relatively weak long-range electron-phononinteraction induces a d-wave superconducting state in doped Mott-Hubbard insulators and/or strongly corre-lated metals with a condensation energy significantly larger than can be obtained with Coulomb repulsion only.Moreover, the superconductivity is shown to exist for infinite on-site Coulomb repulsion without the need foradditional mechanisms such as spin fluctuations to mediate d-wave superconductivity. We argue that oursuperconducting state is robust with respect to a more intricate choice of the trial-wave function and that apossible origin of high-temperature superconductivity lies in a proper combination of strong electron-electroncorrelations with poorly screened Fröhlich electron-phonon interaction.DOI: 10.1103/PhysRevB.79.212501 PACS numbers: 74.40.k, 71.38.k, 72.15.JfIt is now over 20 years since the discovery of the firsthigh-temperature superconductor1 and yet, despite intensiveeffort, the origin of the superconductivity remains fundamen-tally unknown with no widely accepted theory. The absenceof consensus on the physics of the cuprates and the recentdiscovery of iron-based compounds with high transition tem-peratures has re-emphasized the importance of understandingthe origins of superconductivity in quasi-two-dimensional2D materials.2The canonical BCS-Migdal-Eliashberg theory cannot ac-count for the well-documented non-Fermi-liquid propertiesof cuprate superconductors. Moreover, calculations based onthe local-density approximation LDA often predict negli-gible electron-phonon interaction EPI insufficient to ex-plain a kink in the quasiparticle energy dispersion observedby angle-resolved photoemission spectroscopies ARPES.3Hence, it is not surprising that a large number of researchersheld the view that the repulsive Hubbard model would havethe essential physics to account for the superconducting andnon-Fermi-liquid normal states of cuprates. The idea behindthis originally proposed by Anderson4 is that mobile holepairs are created via a strong on-site repulsion U. Results byParamekanti et al.5 and Yamaji et al.,6 using a variationalMonte Carlo VMC simulation with a projected BCS-typetrial-wave function, appeared to back this up. More recentlyYamaji et al.7 found a condensation energy in a Hubbardmodel with next- and third-neighbor hopping but not in thesimplest nearest-neighbor Hubbard model. Using the VMCmethod, with a trial function that includes virtual hoppingprocesses, Yokoyama et al.8 found that superconductivitywas present in a wide range of U. In addition, work by Spanuet al. using the Green’s function MC technique for thet− tJ model9 and earlier work using a combination of theVMC and Lanczos method10 are also in agreement withthese findings. Further support was provided by dynamicalmean-field theory DMFT and dynamical cluster approxi-mation DCA results for a review, see Ref. 11.However, recent studies by Aimi and Imada,12 using asign-problem-free Gaussian-basis Monte Carlo GBMCalgorithm13 showed that the Hubbard model does not accountfor high-temperature superconductivity either. This remark-able result is in line with earlier numerical studies using theauxiliary-field quantum Monte Carlo Ref. 14 andconstrained-path Monte Carlo15,16 methods, none of whichfound superconductivity in the Hubbard model. Furthermore,the validity of the Lanczos extrapolation used in Ref. 10 wasquestioned by Lee et al.,17 who suggested that the robustnessof the superconductivity was overestimated.On the other hand, compelling experimental evidence foran EPI has arrived from isotope effects,18 high-resolutionARPES Ref. 19 and a number of optical,20 neutronscattering,21,22 and some other spectroscopies ofcuprates.23,24 Previous numerics have shown that d-wave or-der could exist in basic models of electron-phonon interac-tions as a consequence of second-order effects which looksimilar to those in repulsive models, with d selected overs-wave order by the Hubbard repulsion, which acts as a kindof filter.25 Here we show that even a weak long-rangeFröhlich EPI Ref. 26 combined with the Hubbard U pro-vides sizable superconducting order in doped Mott-Hubbardinsulators and/or strongly correlated metals.Our Hubbard-Fröhlich model HFM contains the usualnearest-neighbor electron hopping tn and electron-electronon-site repulsive correlations U. In addition, there is a termto describe lattice vibrations, with frequency  and mass M,and a term to describe the long-range interaction of the elec-trons with ion displacements,H = − n,n,=↑,↓tn − ncn†cn + Unnˆn↑nˆn↓+ m Pˆ m22M+M2m22 − m,n,fmncn† cnm. 1Here cn†and cn create and annihilate the electron with spin at sites n and n, respectively, Pˆ m=−i /m is the ionmomentum operator at site m, and m is the ion displace-ment. The long-range Fröhlich EPI is characterized by aforce function of the form27fmn =m − n2/a2 + 13/2exp− 	m − n	Rsc  , 2PHYSICAL REVIEW B 79, 212501 20091098-0121/2009/7921/2125014 ©2009 The American Physical Society212501-1where m , n are the lattice vectors, a is the lattice constant,and Rsc is the screening radius. The Fröhlich EPI Eq. 2routinely neglected in the Hubbard U and t-J models of cu-prate superconductors28 is on the order of 1 eV as estimatedfrom optical data.26 The force function Eq. 2 describes theinteraction between c-axis-polarized vibrations of out-of-plane ions and in-plane carriers. This interaction is poorlyscreened since the maximum out-of-plane plasmon fre-quency in cuprates29 is well below the characteristic fre-quency of optical phonons. In-plane ions can also contributeto this interaction if the symmetry is broken, for example, bybuckling of the plane.30,31The carrier mass and the range of the applicability ofanalytical weak- and strong-coupling expansion effectivelydepend on the EPI radius. In particular, the exact carriermass calculated with the Fröhlich EPI using the continuous-time QMC algorithm27 was found to be several orders ofmagnitude smaller than in the Holstein model in the relevantregion of  / t ratio. The mass is well fitted by a singleexponent remarkably close to that obtained using the Lang-Firsov transformation and subsequent averaging overphonons, for any strength of the Fröhlich EPI, justifying ouruse of the Lang-Firsov transformation in this work. Here weuse the transformation32 to integrate out phonons for tech-nical details see, for example, Ref. 26 mapping the electronpart of the transformed Hamiltonian H˜ to an extended U˜ −Hubbard model with renormalized hopping integrals t˜n, adiminished on-site repulsion U˜ and a long-range effectiveattraction W,H˜ = − nn,t˜n − ncn†cn + U˜nnˆn↑nˆn↓− W nn,	n − nnˆnnˆn. 3Here t˜n= tnexp−g2n, g2n= Ep−W	n /,	n=−2mfm0fmn, U˜ =U−2Ep, W=zt˜a is the renor-malized half bandwidth, z is the coordination number, and=2 /2M2W is proportional to the conventional BCSelectron-phonon coupling constant. The latter is proportionalto the single-particle density of states DOS and could belarger than  due to the van-Hove singularity of DOS andcorrelation-enhanced effective mass of carriers. The po-laronic shift Ep= 2 /2M2	0 of atomic levels is in-cluded in the chemical potential. In the following, onlynearest-neighbor hops are allowed. As shown by Bonča andTrugman,33 the physical properties of bipolaronic carriersdepend predominantly on the EPI coming from the first twosites in Eq. 2 so that we take Rsc= in our simulations.When EPI is strong compared with the renormalized ki-netic energy 1, one can apply the 1 / perturbation ex-pansion reducing the multipolaron problem to a chargedBose gas of small mobile bipolarons.26 Intermediate andweak-coupling regimes, which might be separated from thestrong-coupling bipolaronic regime by a phase boundary,with 1 and large U˜ require a variational approach. Herewe use a standard VMC method as, for example, in Ref. 5 tominimize the energy 	H˜ 	 /  	. The difference is thatwe make an additional measurement of the long-range attrac-tion in the extended Hubbard Hamiltonian 3 described inTable I.A BCS-type trial-wave function is used, which has theform	T = PNPGkuk + vkck↑† c−k↓† 	0 , 4where PN=inˆi,N projects onto a fixed particle number stateand PG=n1− 1−gnˆn↑nˆn↓, 0g1, suppresses doubleoccupancy.6 Our variational parameters are g and the chemi-cal potential  that enters through the kinetic energy k−2t˜cos kx+cos ky− via uk2= 1+k /k2 +k2 /2 and vk2= 1−k /k2 +k2 /2. We then vary the parameters to mini-mize the energy for values of the electron density , U˜ , su-perconducting order parameter k and  keeping t˜=1 anda=1. We place 42 spin-up and 42 spin-down electrons on a1010 2D square lattice to simulate optimal doping, keep-ing the electron density fixed at =0.84. Two cases are in-vestigated: a an on-site repulsion U˜ =8 is used for compari-son with results of Ref. 6 and b an infinite U˜ is used to seeif the attraction induced by EPI, alone, is enough for thehigh-temperature superconductivity.We examine the d-wave k=cos kx−cos ky, extendeds-wave k=cos kx+cos ky, and s-wave k= order pa-rameters. Following previous work,6 to avoid zeros in thewave function, periodic boundary conditions are used in onedirection and antiperiodic boundary conditions are used inthe other. The antiperiodic boundary conditions lead to aphase factor that is accounted for in the algorithm. The valueof  is varied to study the effect of increasing EPI and wefind the value of  at which the energy minimum occurs. Wealso measure the energy of clustered states to ensure ourresults are stable against the formation of immobile clusters.In our results, all energies quoted are per electron; note thatRef. 6 used energy per site.In the following, all energies are quoted in units of t˜. Forthe =0 and U˜ =8 case, with a d-wave order parameter,shown in Fig. 1 top, we recover Yamaji’s result6 witha minimum at 0.08 and energy per electron of−0.878 00.000 4 compared with the normal-state energyof −0.876 30.000 4. It can clearly be seen that the effectof increasing EPI is to increase the depth of the minimumand therefore the stability of the superconducting state. For=0.075, we find the minimum energy per electron to be−4.400 80.000 4 at =0.16. The maximum condensa-tion energy gain E0−E is 0.001 70.000 6 when =0 and 0.004 90.000 6 when =0.075.A surprising result is found for infinite Coulomb repul-sion, where mapping to the t-J model would result inTABLE I. 2D x axis long-range electron attraction.n 0,0 1,0 2,0 3,0 4,0 5,0	n 1.000 0.666 0.324 0.155 0.080 0.043BRIEF REPORTS PHYSICAL REVIEW B 79, 212501 2009212501-2	J	=4t2 /U=0; here no virtual hopping is present so morecomplicated trial wave functions such as that used byYokoyama et al.8 will not change our result. We find a mini-mum energy of −7.551 70.000 3 for =0.15. Figure 1bshows an increasing condensation energy gain with  forU˜ = and d-wave order parameter: the maximum condensa-tion energy gain is 0.000 560.000 04 per electron at =0.16 with =0.15. In this limit, the effect of increasing  isweaker and in order to produce sizable condensation energygains the strength of the EPI has been doubled. These resultsdemonstrate that a d-wave superconducting state does existfor U˜ = with strong enough ; that is, d-wave order can beinduced without spin fluctuations.We examined the energies of various static configurationsthat are diagonal in real space. We placed 84 electrons on a1010 square lattice in various configurations, with nodouble occupancy and calculated the energy. The hoppingcorrection vanishes in the thermodynamic limit and is there-fore neglected. The lowest static energy per electron is−3.682 4 for =0.075 and −7.364 8 for =0.15, with theholes in a circular configuration, so that the system is stableagainst the formation of clusters. No superconducting statewas found for the s-wave or extended s-wave order param-eter for either U˜ =8 or U˜ =, with =0.075 see Fig. 2.Accurate QMC studies have indicated that the Hubbard Uand t-J models are unlikely to account for the high criticaltemperatures of cuprate superconductors.12,14–16 On the otherhand, ab initio LDA calculations of EPI have led to a con-clusion that phonons fail to explain high-temperature super-conductivity either.3 It is well known that LDA badly under-estimates the role of the Coulomb correlations, incorrectlypredicting that the parent compounds of cuprate supercon-ductors such as La2CuO4 are metallic rather than insulat-ing. Also, the anisotropy of the electron response functionsdetermined with LDA is much smaller than the experimen-tally observed value in these layered materials. It is not sur-prising at all that the EPI turns out to be very weak in this“metallic” picture due to electron screening of the long-rangeelectron-ion interaction. Moreover, it has been noted that theinclusion of Hubbard U via the LDA+U scheme signifi-cantly enhances the EPI strength34 since the system becomesa doped Mott insulator with poor screening in at least thec-axis direction as anticipated in Refs. 35 and 26 see alsoRefs. 30, 31, 36, and 37.Here, we showed, using the VMC method, that even arelatively weak Fröhlich EPI is sufficient to induce a d-wavesuperconducting state with substantial condensation energyin a doped Mott-Hubbard insulator and/or strongly correlatedmetals. While the exact energy gain may be overestimated,-0.01-0.00500.0050.010.0150.020.08 0.16 0.24 0.32 0.4E(∆)-E(0)∆λ = 0.075λ = 0.05λ = 0.025λ = 0.00-0.001-0.000500.00050.0010.00150.0020.04 0.08 0.12 0.16 0.2E(∆)-E(0)∆λ = 0.15λ = 0.10λ = 0.05λ = 0.00(b)(a)FIG. 1. Condensation energy per electron in units of t˜ versus the amplitude of the superconducting d-wave order parameter for U˜ =8 aand U˜ = b with different EPI coupling .00.00020.00040.00060.00080.0010 0.01 0.02 0.03 0.04E(∆)-E(0)∆U-2Ep = ∞U-2Ep = 8(b)00.0010.0020.0030.0040 0.01 0.02 0.03E(∆)-E(0)∆U-2Ep = ∞U-2Ep = 8(a)FIG. 2. Condensation energy per electron in units of t˜ versus the amplitude of the superconducting s-wave order parameter a for U˜ =and U˜ =8 with =0.075, showing no s-wave ground state. The condensation energy versus the amplitude of the extended s-wave orderparameter b for U˜ = and U˜ =8 with =0.075, showing no extended s-wave ground state.BRIEF REPORTS PHYSICAL REVIEW B 79, 212501 2009212501-3due to the accuracy of the wave function for the normal state,our results clearly demonstrate that increasing the EPIstrength increases the condensation energy. The supercon-ducting energies are lower than the static energies in a wideregion of  so that our superconducting state is robustagainst the formation of clusters. As a result, we concludethat a combination of strong electron-electron correlationswith long-range electron-phonon interactions is a mechanismfor high-temperature superconductivity in the cuprates.We would like to thank Pavel Kornilovitch for long-standing collaboration and helpful discussions. We acknowl-edge support of this work by EPSRC U.K. under Grant No.EP/C518365/1.1 J. G. Bednorz and K. A. Müller, Z. Phys. B 64, 189 1986.2 Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am.Chem. Soc. 130, 3296 2008.3 R. Heid, K. P. Bohnen, R. Zeyher, and D. Manske, Phys. Rev.Lett. 100, 137001 2008.4 P. W. Anderson, Science 235, 1196 1987.5 A. Paramekanti, M. Randeria, and N. Trivedi, Phys. Rev. B 70,054504 2004.6 K. Yamaji, T. Yanagisawa, T. Nakanishi, and S. Koike, Physica C304, 225 1998.7 K. Yamaji, T. Yanagisawa, M. 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Superconductivity in a Hubbard-Frohlich model and in cuprates

Using the variational Monte Carlo method, we find that a relatively weak long-range electron-phonon interaction induces a d-wave superconducting state in doped Mott-Hubbard insulators and/or strongly correlated metals with a condensation energy significantly larger than can be obtained with Coulomb repulsion only. Moreover, the superconductivity is shown to exist for infinite on-site Coulomb repulsion without the need for additional mechanisms such as spin fluctuations to mediate d-wave superconductivity. We argue that our superconducting state is robust with respect to a more intricate choice of the trial-wave function and that a possible origin of high-temperature superconductivity lies in a proper combination of strong electron-electron correlations with poorly screened Fröhlich electron-phonon interaction

Superconductivity in a Hubbard-Froehlich Model and in cuprates

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