We present a detailed numerical study of the Hubbard-Holstein model in one
dimension at half filling, including full finite-frequency quantum phonons. At
half filling, the effects of the electron-phonon and electron-electron
interactions compete, with the Holstein phonon coupling acting as an effective
negative Hubbard onsite interaction U that promotes on-site electron pairs and
a Peierls charge-density wave state. Most previous work on this model has
assumed that only Peierls or U>0 Mott insulator phases are possible at half
filling. However, there has been speculation that a third metallic phase exists
between the Peierls and Mott phases. We present results confirming the
intermediate metallic phase, and show that the Luttinger liquid correlation
exponent K_rho>1 in this region, indicating dominant superconducting pair
correlations. We explore the full phase diagram as a function of onsite Hubbard
U, phonon coupling constant, and phonon frequency.Comment: 4 pages, 4 EPS figures. v2: typos corrected. To appear in Phys. Rev.
  Let

R. P. Hardikar

R. T. Clay

T. Ishiguro

English

arXiv

We present a numerical study of the Hubbard-Holstein model in one dimension at half filling, including finite-frequency quantum phonons. At half filling, the effects of the electron-phonon and electron-electron interactions compete with the Holstein phonon coupling acting as an effective negative Hubbard on-site interaction U that promotes on-site electron pairs and a Peierls charge-density wave state. Most previous work on this model has assumed that only Peierls or Mott phases are possible at half filling. However, there has been speculation that a third metallic phase exists between the Peierls and Mott phases. We confirm the intermediate phase, and show that the Luttinger liquid correlation exponent Kρ\u3e1 in this region, indicating dominant superconducting pair correlations. We explore the full phase diagram as a function of Hubbard U, phonon coupling constant, and phonon frequency

Clay, R. T.

Hardikar, Rahul

Digital Commons @ Butler University

Butler UniversityDigital Commons @ Butler UniversityScholarship and Professional Work - LAS College of Liberal Arts & Sciences2005Intermediate Phase of the One Dimensional Half-Filled Hubbard-Holstein ModelR. T. ClayRahul HardikarButler University, rhardika@butler.eduFollow this and additional works at: http://digitalcommons.butler.edu/facsch_papersPart of the Atomic, Molecular and Optical Physics Commons, and the Quantum PhysicsCommonsThis Article is brought to you for free and open access by the College of Liberal Arts & Sciences at Digital Commons @ Butler University. It has beenaccepted for inclusion in Scholarship and Professional Work - LAS by an authorized administrator of Digital Commons @ Butler University. For moreinformation, please contact fgaede@butler.edu.Recommended CitationClay, R. T. and Hardikar, Rahul, "Intermediate Phase of the One Dimensional Half-Filled Hubbard-Holstein Model" Physical ReviewLetters / (2005): -.Available at http://digitalcommons.butler.edu/facsch_papers/818Intermediate Phase of the One Dimensional Half-Filled Hubbard-Holstein ModelR. T. Clay and R. P. HardikarDepartment of Physics and Astronomy and ERC Center for Computational Sciences, Mississippi State University,Mississippi State, Mississippi 39762, USA(Received 7 May 2005; revised manuscript received 24 June 2005; published 26 August 2005)We present a numerical study of the Hubbard-Holstein model in one dimension at half filling, includingfinite-frequency quantum phonons. At half filling, the effects of the electron-phonon and electron-electroninteractions compete with the Holstein phonon coupling acting as an effective negative Hubbard on-siteinteraction U that promotes on-site electron pairs and a Peierls charge-density wave state. Most previouswork on this model has assumed that only Peierls or Mott phases are possible at half filling. However,there has been speculation that a third metallic phase exists between the Peierls and Mott phases. Weconfirm the intermediate phase, and show that the Luttinger liquid correlation exponent K > 1 in thisregion, indicating dominant superconducting pair correlations. We explore the full phase diagram as afunction of Hubbard U, phonon coupling constant, and phonon frequency.DOI: 10.1103/PhysRevLett.95.096401 PACS numbers: 71.10.Fd, 71.30.+h, 71.45.LrElectron-phonon (e-ph) interactions can give rise to anumber of interesting effects in low-dimensional materials,including superconductivity as well as charge-densitywave and insulating phenomena. Frequently these materi-als feature strong electron-electron (e-e) interactions aswell, leading to very rich phase diagrams that combinelattice, charge, and spin (magnetic) orderings. We focusspecifically on materials where the electrons are coupled tolocalized vibrational modes, which may be of relativelyhigh frequency. This type of e-ph interaction is moststudied in molecular crystal materials, including thequasi-one- and quasi-two-dimensional organic supercon-ductors [1] and fullerene superconductors [2]. In all ofthese materials, a fundamental question is whether theeffects of e-e and e-ph interactions compete or cooperatewith each other. In this Letter, we examine this issue withinone of the most basic models. We find that, despite the twointeractions each separately favoring insulating states, to-gether they can mediate an unexpected metallic phase withsuperconducting (SC) pair correlations.The model we consider is the one-dimensional (1D)Hubbard-Holstein model (HHM), with HamiltonianH  tXj;cyj1;cj;  H:c: UXjnj;"nj;# gXj;byj  bjnj; !Xjbyj bj; (1)where cyj; (cj;) are fermionic creation (annihilation) op-erators for electrons on-site j with spin , byj (bj) arebosonic creation (annihilation) operators for phonons atsite j, and nj;  cyj;cj;. The dispersionless phononshave frequency ! and are coupled to the local electrondensity with coupling strength g [3]. U is the Hubbard on-site e-e interaction energy. All energies are given below inunits of the hopping integral t.The properties of the 1D 1=2-filled HHM are well under-stood in two limits: the static or !! 0 and the !! 1limit [4]. First, in the static limit, the ground state is Peierlsdistorted for any nonzero e-ph coupling g  0. As thephonons couple to the electron density, the Peierls state is a2kF charge-density wave (CDW) consisting of alternatinglarge and small site charges. In the !! 1 limit, theretarded interaction between electrons mediated by thephonons becomes instantaneous in imaginary time, andthe phonons may be integrated out. This leads to an effec-tive renormalized Hubbard interaction Ueff  U 2g2=!.While strictly at !! 1 the Peierls state cannot occur, forfinite ! the Peierls state may again occur, although themapping to an effective negative U is no longer exact.However, based on the !! 1 mapping, it has beenbelieved that the ground state phase of the HHM may bedetermined through Ueff . Ueff should correspond to theU > 0 Hubbard model, which in 1D at 1=2 filling has afinite charge gap for any U > 0 and no spin gap. We shallrefer to this state as the Mott state. On the other hand, ifUeff < 0, e-ph interactions dominate over e-e interactions,and the ground state is Peierls CDW distorted.Hence at !  0, the Peierls distortion occurs uncondi-tionally, while at !  1 there is no Peierls state. Muchless is known in the intermediate ! region. In the U  0model, the Peierls state may be viewed as a traditional bandinsulator for small !, and as a bipolaronic insulator com-posed of tightly bound pairs in the large ! limit, with acrossover between these pictures for ! t [5]. If !> 0,the Peierls distortion occurs only for e-ph coupling g abovesome critical value gc [6,7]. For g < gc, the ground state isthen metallic at 1=2 filling (at U  0). For U  0, gc hasbeen calculated using a functional integral method [6] andalso via the density matrix renormalization group (DMRG)[7]. For intermediate U as well as !, far less is known.However, it has been recently proposed that a metallicground state exists intermediate between the Mott andPRL 95, 096401 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending26 AUGUST 20050031-9007=05=95(9)=096401(4)$23.00 096401-1  2005 The American Physical SocietyPeierls states at 1=2 filling, i.e., when Ueff is close to zero[8]. This metallic phase occurs for intermediate ! andhence cannot be predicted from the small and large !limits. Our goal in this Letter is to confirm this metallicstate and investigate its properties. Its existence is perhapsnot surprising given the known existence of a metallicphase for g < gc at U  0; this region of the phase dia-gram continues to exist for finite U. We further show thatthis metallic region persists for a substantial range ofparameters provided ! is not too small.Hamiltonian Eq. (1) is difficult to analyze in the inter-mediate coupling region due to the presence of both elec-trons and phonons. The numerical method we use is thestochastic series expansion (SSE) quantum Monte Carlomethod with directed loops [9]. SSE is a powerful methodfor nonfrustrated quantum spin systems or 1D electron lat-tice models where no sign problem occurs. Importantly,there are no approximations in the method besides finitesystem size and temperature. Electron-phonon interactionshave been incorporated in SSE for both spin models [10]and electron models [11]. As in these references, we treatthe phonons in the occupation-number basis. The numberof phonons per lattice site is unbounded in the thermody-namic limit, but for a finite system at a finite temperature,the number of phonons may be truncated. We choose thistruncation in a similar manner as the truncation of se-quence length in the SSE method [9]: in the equilibrationphase of the calculation, the phonon truncation is increasedto exceed the current number of phonons on any givenlattice site whenever necessary. All results shown belowused periodic lattices of N sites, with inverse temperaturesof at least =t  2N and phonon cutoffs of up to 30 pho-nons per lattice site. In this Letter we focus on results forthe HHM model, and details on the SSE implementationwill be published separately. Our code was checked exten-sively against Lanczo¨s exact diagonalization results forseveral different observables. We also implemented thequantum parallel tempering algorithm [12], where differ-ent processors of a parallel computer are assigned differentmodel parameters (U, g, and !). A Metropolis probabilityis then computed to switch configurations between pro-cessors with adjacent parameters. As in Ref. [12], we foundthis technique essential in obtaining smooth data acrossquantum phase transition boundaries.The low energy properties of any 1D gapless interactingelectron model may be mapped to an effective continuummodel, or Luttinger liquid (LL). The properties of the LLand, in particular, the decay with distance of differentcorrelation functions are then described by two correlationexponents, K for charge properties and K for spin [13].K values greater than 1 indicate dominant attractive SCcorrelations (no SC long-range order is possible in strictly1D systems), while K < 1 corresponds to repulsivecharge correlations. For models with spin-rotation symme-try, K is always equal to 1. K  0 then indicates thepresence of a spin gap. These exponents are most easilycomputed via the SSE data from the static structure factors:S;q  1NXj;keiqjkhnj"  nj#nk"  nk#i (2)K and K are then proportional to the slope of thecorresponding structure factor in the long-wavelength limitq! 0 [14]:K;  1qS;q! 0: (3)These values once finite-size scaled may then be used todetermine the quantum phase boundaries. A second ob-servable we use are the charge and spin stiffnesses, c ands, measured in the SSE method via the winding number[12]. A zero stiffness indicates a gap in the correspondingsector, while nonzero  indicates no gap. We also verifieddirectly that charge-charge (spin-spin) correlation func-tions showed staggered order in the Peierls (Mott) phases.We first present results for U  0 and !  1. Figure 1shows the slope of charge and spin structure factorsS;q1=q1 evaluated at the smallest wave vector q1 2=N, plotted versus e-ph coupling g. For g! 0 both Kand K tend to exactly 1, as required for free electrons.Finite-size effects become very small in this limit, consis-tent with the expected vanishing of logarithmic correctionswhen K  K  1 [11,15]. At a critical coupling gc, Kcrosses 1 and tends to zero, marking the transition to thePeierls state. From finite-size scaling of 16, 32, and 64 sitesystems vs 1=N (see Fig. 1 inset), we find the criticalcoupling gc  0:66 0:01, with the uncertainty estimatedfrom the linear fit. A previous DMRG study, which did notattempt finite-size scaling, found gc  0:8 [7]. For g < gc,0 0.2 0.4 0.6 0.8g0.70.80.911.1πS(q 1)/q10 0.02 0.04 0.06 1/N00.20.40.60.81πSρ(q1)/q1FIG. 1. U  0, !  1 results for long-wavelength charge(open symbols) and spin (filled symbols) structure factors versusg for periodic systems of N  16 (diamonds), N  32 (circles),and N  64 (squares) sites. Statistical errors are smaller than thesymbols. The inset shows finite-size scaling of the criticalcoupling gc (indicated by arrow) where K  1.PRL 95, 096401 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending26 AUGUST 2005096401-2the LL exponent K > 1, and K scales to zero withincreasing N. Therefore, for U  0 and g < gc, the groundstate is metallic, with dominant SC pair correlations and aspin gap.For any finite U at g  0 and 1=2 filling, the dominantground state correlations of Eq. (1) are spin-density wave(SDW). In this ground state K  1, and K  0 indicat-ing an insulating state with no spin gap. As this Mott stateis not equivalent to the metallic state found for g < gc inFig. 1, we may then expect the possibility of three differentphases for the HHM with U > 0 and g > 0: Mott, Peierls,and metallic SC. Figure 2 shows the charge and spinstructure factors again versus g with U  2. We first dis-cuss results for small phonon frequency !  0:5 shown inFig. 2(a). We find that for small g, K tends toward 1 as thesystem size increases. At a critical coupling gc, K be-comes less than 1, indicating the opening of a spin gap andthe transition to the Peierls state. The charge exponent Kscales towards zero on both sides of the transition, devel-oping a peak at g  gc. Similar results are found for the1=2-filled 1D extended Hubbard model (EHM), where Kalso peaks at the transition between CDW and bond-orderwave (BOW) phases [16]. The finite value of K on theboundary indicates that the transition is of continuousnature [16]. Like the CDW-(BOW)-SDW transition in the1=2-filled EHM, we find similar behavior for small ! inthe HHM as U increases: the value of K at the transitiondecreases with U, indicating that for large U the transitionis first order rather than continuous. The correspondence isnot surprising, as the effect of a nearest-neighbor interac-tion V  U can be viewed as an effective negative U [14].Charge and spin response typical for large ! are shownin Fig. 2(b), here shown for U  2 and !  5. Again, forsmall g, we find K tends toward 1 as system size in-creases, and K tends toward 0, consistent with the Mottstate. However, at a critical coupling gc1 K exceeds 1 andK becomes less than one, indicating the dominant-SCstate found for U  0, g < gc. The value of gc1 is veryclose to the expected value where Ueff  0. Increasing gfurther, for g  gc2 K becomes less than 1, indicating theopening of a charge gap and the Peierls state. In theintermediate region gc1 < g< gc2, the properties of themodel are identical to the U  0 model for g < gc as seenin Fig. 1, i.e., metallic with dominant SC correlations.While the second transition point gc2 is finite-size depen-dent, finite-size effects are very weak at the first transitionsince K  K  1. It is then clear that due to the crossingof K and K curves at exactly 1 at g  gc1, a region ofK > 1 must exist for g > gc1.To confirm the structure factor results indicating twotransitions at gc1 and gc2, we present the charge (c) andspin (s) stiffnesses in Fig. 3. As with the structure factors,the presence of two transitions may be detected only fromc and s after finite-size scaling has been performed. Forclarity, we plot only two system sizes in Fig. 3. In Fig. 3(a)we see that, for small g, c decreases with system size,indicating a charge gap. s is nearly constant with systemsize, indicating no spin gap. In the intermediate phase, thisis reversed, with c remaining constant or increasing withsystem size, and s decreasing with system size. This againindicates a spin gap but no charge gap in the SC region.0.5 0.6 0.7 0.8g0.60.70.80.911.1πS(q 1)/q12 2.5 3g0.70.80.911.1πS(q 1)q 1charge N=16spin N=16charge N=32spin N=32(a)(b)FIG. 2. Long-wavelength charge (open symbols) and spin(filled symbols) structure factors versus g. (a) U  2, ! 0:5; arrow indicates finite-size scaled gc. (b) U  2, !  5;arrows indicate finite-size scaled transitions gc1 and gc2.00.51ρN=32 ρcN=16 ρc0 1 2 3g00.51ρN=32 ρsN=16 ρs(a)(b)FIG. 3. Charge (a) and spin (b) stiffnesses versus g for U  2,!  5. Open (closed) symbols are for N  32 (16) site lattices.Arrows mark the transition points gc1 and gc2 determined fromFig. 2(b).PRL 95, 096401 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending26 AUGUST 2005096401-3Finally, for g > gc2, both stiffnesses go to zero in thePeierls state.Figure 4 shows the evolution of the intermediate phaseas a function of !, with phase boundaries determined fromfinite-size scaling of 16, 24, and 32 site structure factordata. In all cases, the SC phase exists for g < gc exactly atU  0. As U is increased at fixed !, the SC region thenshrinks. For small enough !, we see two different sequen-ces of phases, either Mott-SC-Peierls for small U [as inFig. 2(b)] or Mott-Peierls [as in Fig. 2(a)] for large U.Numerically it is difficult to precisely determine the pointwhere all three phases meet, but we find that the SC phasedisappears at U  1 for !  0:5, and at U  2 for ! 1:0. In Figs. 4(a) and 4(b), we have not plotted points forthe Mott-Peierls boundary for large U, as in this region thetransition becomes strongly first order, making exact de-termination of the boundary difficult. However, the tran-sition appears to remain close to g2=!  U=2. For !  5,we were not able to access large enough U to determine theupper cutoff U necessary to suppress the SC state, but theintermediate phase appears to persist up to at least U  7for !  5. In all cases, we find the line Ueff  0 to veryaccurately predict either the Mott-SC or Mott-Peierlsboundary, but not the SC-Peierls boundary. Our phasediagram is slightly different from Ref. [8], where themetallic region was found to extend to either side of theUeff  0 line.In conclusion, we have shown that a metallic SC regionexists intermediate between Peierls and Mott phases in the1D 1=2-filled HHM. While in the 1=2-filled model withsmall phonon frequency !< t the intermediate phaseoccupies a relatively small region of the phase diagram,we expect that in the presence of doping (in fact, most ofthe organic SC’s are 1=4-filled [1]), the size of this regionwill be greatly enhanced. Furthermore, while the SC pair-ing found here exactly at 1=2-filling consists of on-sitepairs or bipolarons, with doping both on-site and nearest-neighbor pairing may be mediated by the Holstein phonons[17]. We are currently exploring these possibilities in thedoped model.This work was supported by Oak Ridge AssociatedUniversities. We thank A. Sandvik and P. Sengupta fordiscussions regarding the SSE method. Numerical calcu-lations were performed at the Mississippi State UniversityERC Center for Computational Sciences.[1] T. Ishiguro, K. Yamaji, and G. Saito, Organic Super-conductors (Springer-Verlag, New York, 1998).[2] O. Gunnarsson, Rev. Mod. Phys. 69, 575 (1997).[3] T. Holstein, Ann. Phys. (N.Y.) 8, 325 (1959).[4] E. Fradkin and J. E. Hirsch, Phys. Rev. B 27, 1680 (1983).[5] H. Fehske, G. Wellein, G. Hager, A. Weie, and A. R.Bishop, Phys. Rev. B 69, 165115 (2004).[6] C. Wu, Q. Huang, and X. Sun, Phys. Rev. B 52, R15 683(1995).[7] E. Jeckelmann, C. Zhang, and S. White, Phys. Rev. B 60,7950 (1999).[8] Y. Takada and A. Chatterjee, Phys. Rev. B 67, 081102R(2003).[9] O. F. Syljuasen and A. W. Sandvik, Phys. Rev. E 66,046701 (2002).[10] A. W. Sandvik and D. K. Campbell, Phys. Rev. Lett. 83,195 (1999).[11] P. Sengupta, A. W. Sandvik, and D. K. Campbell, Phys.Rev. B 67, 245103 (2003).[12] P. Sengupta, A. W. Sandvik, and D. K. Campbell, Phys.Rev. B 65, 155113 (2002).[13] J. Voit, Rep. Prog. Phys. 58, 977 (1995).[14] R. T. Clay, A. W. Sandvik, and D. K. Campbell, Phys.Rev. B 59, 4665 (1999).[15] S. Eggert, Phys. Rev. B 54, R9612 (1996).[16] A. W. Sandvik, L. Balents, and D. K. Campbell, Phys. Rev.Lett. 92, 236401 (2004).[17] J. Bonc˘a, T. Katras˘nik, and S. A. Trugman, Phys. Rev.Lett. 84, 3153 (2000).00.511.522.53g2/ω00.511.522.53g2/ω0 1 2 3 4 5 6U00.511.522.53g2/ωPeierlsMott(a) ω=0.5(b) ω=1.0(c) ω=5.0PeierlsMottPeierlsMottSCSCSCFIG. 4. Evolution of the intermediate phase versus !.(a) !  0:5; (b) !  1; (c) !  5. Open circles indicate theboundary gc1 between Mott and SC phases, filled diamonds theboundary gc2 between SC and Peierls phases, and open squaresthe boundary between Mott and Peierls phases. Dashed lineindicates U  2g2=!.PRL 95, 096401 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending26 AUGUST 2005096401-4

Intermediate phase of the one dimensional half-filled Hubbard-Holstein model

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