Orbital-selective Mott-Hubbard transition in the two-band Hubbard model R. Arita and K. Held Max-Planck-Institut fu¨r Festk¨orperforschung, 70569 Stuttgart, Germany (Dated: October18, 2005) 6 Recent advances in the field of quantum Monte Carlo simulations for impurity problems allow 0 –within dynamical mean field theory– for a more thorough investigation of the two-band Hubbard 0 modelwithnarrow/widebandandSU(2)-symmetricHund’sexchange. Thenatureofthistransition 2 has been controversial, and we establish that an orbital-selective Mott-Hubbard transition exists. n Thereby,thewide band still shows metallic behaviorafter thenarrow band became insulating -not a a pseudogap as for an Ising Hund’s exchange. The coexistence of two solutions with metallic wide J band and insulating or metallic narrow band indicates, in general, first-ordertransitions. 7 2 PACSnumbers: 71.27.+a,71.30.+h,71.10.Fd ] h By virtue of dynamical mean field theory (DMFT) [1, a single first-order Mott-Hubbard transition with simi- c 2],ourunderstandingoftheMott-Hubbardtransition[3] lar changes in both bands. On the insulating side, the e m in the one-bandHubbard model has greatly improvedin wide band has a pseudogapwhich gradually amplifies to the last years. The bandwidth-controlled Mott-Hubbard a real gap with increasing U. In principle, the QMC - t transition is, at least within DMFT [2, 4], of first-order is more suitable for addressing the Mott-Hubbard tran- a t atlowtemperatures(T)andbecomesasmoothcrossover sition since ED only gives discrete peaks in the spectra, s fortemperaturesaboveacriticalpoint,whichterminates makingitdifficulttounambiguouslyidentifyagap. How- . t a thefirst-orderline. Afurthercomplicationarisesexactly ever, the QMC simulations are restricted to relatively m at zero temperature where two solutions coexist like for hightemperatures and there is a sign-problem[13] if the - low Ts. But at T = 0, the insulating solution is always Hund’s exchange coupling is taken into account in full, d higherinenergythanthemetallicone,i.e.,theinsulating i.e., not only the Ising but also the spin-flip component. n solution is metastable throughout the whole coexistence Since the same limitations as in [12] also apply to all o region. TheDMFTMott-Hubbardtransitionisofsecond previous LDA+DMFT(QMC) calculations [5], there is c [ order at T=0 despite the coexistence of two solutions. an urgent need to clarify whether and how the details of the Mott-Hubbard transition are affected. Another im- 2 For making contact with experiments, orbital realism portant aspect is whether two solutions coexist. Liebsch v has to be taken into account, e.g., within the merger of 0 local density approximation and DMFT (LDA+DMFT findstwocoexistingsolutionsatasingletransition,while 4 Kogaetal. [10]donotaddressthisquestion. Iftherewas approach [5]). In the case of transition metal oxides, 0 a first-ordertransitiontwo consecutivetransitions might typically either the three t or the two e bands cross 4 2g g even be bridged into a single one. 0 theFermienergy. Attheveryleast,theseorbitalsshould In this paper, we study this transition by employing 5 be included. Fordegenerateorbitals,the situationseems 0 to be clear, at least within DMFT: there is a first-order the most recent advances in the field of QMC simu- / lations for DMFT. The recently introduced projective t Mott-Hubbard transition [6]. Most transition metal ox- a QMC (PQMC) method [14] enables us to address T=0. ides are, however, non-cubic. Hence, the orbital degen- m Furthermore, the new Hubbard-Stratonovich decoupling eracy is lifted. Take, for example, the unconventional - of [15] allows for the calculation with the full SU(2)- d superconductor Sr2RuO4 [7] which has a wide dxy band symmetric Hund’s exchange at a still-manageable sign- n and narrowd bands [8] and which becomes a Mott- xz,zy o Hubbard insulator upon substituting Sr by Ca [9]. problem, in particular in combination with PQMC. c Model. Startingpoint is the two-bandHubbardmodel : For such a situation with wide and narrow bands the v details of the Mott-Hubbard transitions are so far in- 2 Xi conclusive, even within DMFT and even for a simple H = − X tm X cˆ†imσcˆjmσ (1) r two-band Hubbard model with Coulomb interaction U m=1 hi,jiσ a aetndal.Hu[1n0d]’semexpclhoyaendgethJebsoe-tcwaelelendtehxeactwtodiabgaonndas:lizKatoiogna +U Xnˆim↑nˆim↓+ X(U′−δσσ′J)nˆi1σnˆi2σ′ imσ i;σ<σ′ (ED) method to solve the DMFT equations and obtain J J twoMott-Hubbardtransitions: firstthenarrowbandbe- cˆ† cˆ cˆ† cˆ cˆ† cˆ† cˆ cˆ . −2 X ilσ ilσ¯ imσ¯ imσ − 2 X ilσ ilσ¯ imσ imσ¯ comes insulating, then the wide band. This scenario iσ;l6=m iσ;l6=m has been coined orbital-selective Mott-Hubbard transi- tion [11]. In contrast, Liebsch [12] uses quantum Monte | ≡{Hz2 } Carlo (QMC) simulations and the iterated perturbation Here,cˆ† andcˆ arecreationandannihilationopera- imσ imσ theory (IPT) to solve the DMFT equations and finds torsforelectronsonsiteiwithinorbitalmandwithspin 2 σ. Thefirstlinedescribesthekineticenergyforwhichwe operator are calculated as: O employthe semi-elliptic non-interactingdensity ofstates N0(ε)= πW1m2/8p(Wm/2)2−ε2 (orbital-dependent hop- Trhe−β˜HTe−θH/2Oe−θH/2i pingamplitudestm onaBethelattice). Forthefollowing hOi0 = θl→im∞β˜l→im∞ Tr e−β˜HTe−θH , (3) calculations, we use different widths for the two bands: h i W =4 for the wide and W =2 for the narrow band as 1 2 in [10, 12]. Note that there is no hopping/hybridization where HT is an auxiliary Hamiltonian (its ground state between the two bands. The second line describes the ΨT is the trial wave function which is assumed to be | i intra- (U) and inter-orbital (U′) Coulomb interaction as non-orthogonalto the ground state Ψ0 of H [14]). | i well as the Ising-component of the Hund’s exchange J For HT, it is convenient to take the one-body part (U′=U 2J by symmetry; we setJ=U/4asin [10, 12]). of the Hamiltonian, because the limit β˜ can be Thethir−dlineconsistsofthespin-flipcontributiontothe taken analytically in this case. Then, the s→tar∞ting point Hund’s exchange (yielding together with the second line is the zero-temperature non-interacting Green function anSU(2)-symmetriccontributionwhichcanalsobewrit- G0(τ,τ′)truncatedto 0 τ,τ′ θ anddiscretizedasan ≤ ≤ tenasJSi1Si2,whereSim denotes the spinfororbitalm L×L matrix (L=θ/∆τ). Fromthis G0(τ,τ′), the zero- andsitei). Thelasttermrepresentsapair-hoppingterm temperature interacting Green function G(τ,τ′) is ob- of same strength J. tained via the same updating equations for the auxiliary Hubbard-Stratonovichfieldsasforthefinite-temperature Method. QMC calculations which take the spin- Hirsch-Fye algorithm. flip component of Hund’s exchange term into account While PQMC gives G(τ) only for a limited number have been a challenge. Although a straight-forward Hubbard-Stratonovich decoupling, exp(J∆τc†c c†c ) = of (not too large) τ-points, we need G(iω) to solve the 1 2 3 4 (1/2) exp[s√J∆τ(c†c c†c )], is possible, it has DMFTself-consistencycycle. Tothisend,themaximum Ps=±1 1 2− 3 4 entropy method (MEM) is employed to yield the spec- been recognized that it leads to a serious sign prob- tralfunctionA(ω)whichallowsforcalculatingG(iω )= lem [13]. Therefore, it was neglected in almost every n dω A(ω) at any frequency iω . This makes a crucial DMFT(QMC) calculation so far, including [12]. R iωn−ω n difference to finite-temperature calculations. The large To overcomethis problem,severalattempts havebeen statistical errors occurring at τ β/2 for finite temper- made [15, 16, 17]. Among these, Sakai, RA, and Aoki ∼ atures now occur for rather large τ’s. But even if there proposed a new discrete transformation for the spin-flip is a large statistical error for larger τ’s, the maximum contribution of the exchange and pair-hopping term[15]: entropy method can extract sufficient information from thefirstτ points,discardingthelargerτ’swithexcessive 1 statistical error. e−∆τH2 = eλr(f↑−f↓)ea(N↑+N↓)+bN↑N↓, (2) 2 X One of the main advantages of the PQMC method is r=±1 that the convergence w.r.t. θ is much faster than that w.r.t. β in the Hirsch-Fye algorithm[14]. (Note that the Here, λ 1log(e2J∆τ+√e4J∆τ 1), a log(cosh(λ)), ≡ 2 − ≡− calculation time increases cubically for θ and β.) Hence, b≡log(cosh(J∆τ)), fσ≡c†1σc2σ+c†2σc1σ,Nσ≡n1σ+n2σ− we take in the following PQMC calculations a finite θ = 2n1σn2σ. The advantage of this decoupling is that the 20(L=64),whichshouldbesufficientlyclosetotheT = auxiliaryfield r is realin contrastto thatof [16]. Hence, 0result: quantitativelysmalldeviationsareexpectedfor it is expected to yield better statistics in general [15]. larger θ’s; qualitatively the behavior should not change However, even with this decoupling, we note that the anymoreasin[14]. Similarlyasin[14],thecentral =20 L usualHirsch-FyeQMCalgorithm[18]doesnotworkvery arefor measurementand =(L )/2=22time slices P −L wellforDMFT calculationin the strongcouplingregime on the right and left side of the measuring interval for or at low temperatures. For instance, for Hamiltonian projection. Typically, we performed 7 106 to 3 107 × × (1) and J = U/4, we found it to be infeasible to obtain QMC sweeps. a self-consistent DMFT solution for U > 2.2 when β(= Results. An indicator for the Mott-Hubbard transi- 1/T) > 50 because the Green function G(τ) has a large tionisthequasiparticleweightZ whichisplottedinFig. statistical error at τ β/2. Therefore, it is difficult 1(a). We clearly see that for the narrow band Z = 0 ∼ to clarifywithout ambiguitywhether anorbitalselective for U 2.6, while Z is still finite for the wide band. ≥ Motttransitionindeedoccursinmulti-orbitalsystemsat This means that there is a first Mott-Hubbard transi- low T by means of finite-temperature Hirsch-Fye QMC tion in which only the narrow band becomes insulating calculations; also see [19]. at U 2.5. This is consistent with the result of the ≈ Anotherrecentadvancementwasthedevelopmentofa DMFT(ED) calculation of [10], in which the critical U c new projective QMC (PQMC) algorithm by Feldbacher, is estimated to be about 2.6. In contrast, there is a sin- KH, and Assaad [14]. In this algorithm, ground state glefirst-orderMott-Hubbardtransitionatasmallervalue expectation values Ψ Ψ / Ψ Ψ of an arbitrary U 2.1 if only the Ising-component of Hund’s exchange 0 0 0 0 c h |O| i h | i ≈ 3 (a) 1 1 U=2.0 2 (a) U=2.2 (b) 0.8 U=2.4 Z0.6 w) U=2.6 w) ( ( 0.2 A A 0.4 0.5 1 0.1 0.2 0 2.5 2.7 0 0 0.5 1 1.5 2 2.5 3 0 0 U -4 0 4 -4 0 4 0.25 w w (b) FIG. 2: (Color online) Spectral functions A(ω) for (a) the 0.2 widebandand(b)thenarrowband. ForU =2.6,thenarrow 0D.15 band is insulating while thewide band is metallic. 0.1 readyinsulating with a pronouncedgap. While the wide 0.05 bandshowsa pseudogapfor anIsing-type ofHund’s ex- 0 0 0.5 1 1.5 2 2.5 3 change[12],ourSU(2)symmetricresultrevealsametallic peak in Fig. 2. U FIG. 1: (Color online) (a) Quasiparticle weight Z and (b) Let us now study the possibility of first-order Mott- doubleoccupancyDasafunctionofU (J=U/4). Red(blue) Hubbard transitions. The first question is whether at squares(circles)arethedataforthenarrow(wide)band. For U=2.6(wherewefindametallicwideandinsulatingnar- U =2.4, two solutions are found: the wide band is metallic row band) a secondsolution in which both bands are in- forbothsolutionswhereasthenarrowbandismetallic(closed sulating (co)exists. Starting the DMFT self-consistency symbols) or insulating (open symbols). The solid triangle in (a)and(b)istheUcestimatefromDMFT(ED)[10];theinset cyclewithaninsulatingself-energyforthewide-band,we enlarges thebehavior around thetransition. obtainhowevertheverysame(single)solutionasinFigs. 1 and 2. This demonstrates that the orbital-selective Mott-Hubbardtransitionisnotmergedintoasinglefirst- is taken into account (at T=0.03; between U 1.8 and order transition. There are two distinct Mott-Hubbard c ≈ 2.1 there are two coexisting solutions/hysteresis)[12]. In transitions. our DMFT(PQMC) results,the wide bandis still metal- Thesecondquestionis,Aretheorbital-selectiveMott- lic at U=2.7. But eventually, also the wide band has to Hubbard transitions (generally) of first-order? In this become insulating at larger Coulomb interactions, since case,twosolutionsshouldcoexistforU .2.6: onewitha in the atomic limit both bands are insulating. (The cal- metallicandonewithaninsulatingnarrowband. Special culation for larger Coulomb interactions unfortunately care is necessary for insulating solutions in the PQMC became computationally too expensive as even in the with a very narrow charge gap. For such small charge PQMC the statistical error brought about by the spin gaps, θ might not be sufficiently large to project –via flip term of Hund’s exchange increases dramatically.) e−θH/2–fromthemetallictrialwavefunctionontothein- Nonetheless, we can conclude from the data available sulating ground state solution. We then note systematic that there are two different Mott-Hubbardtransitions in errorsevenforintermediateτ’s,andsubstantialnoiseap- which first the narrow and then the wide band become pears in the charge gap of the spectrum calculated with insulating. We have an orbital-selective Mott-Hubbard the maximum entropy method. This makes the stabi- transition. lization of a small-gap insulating solution delicate. This InFig.1(b), the doubleoccupancyD = n n forthe problem can be mitigated however by doing the max- ↑ ↓ h i two different bands is plotted as a function ofU. We see imum entropy calculation with a reduced number of τ that for the narrow band D 0.01 for U 2.6. A sim- points. Therefore, we used τ points up to τ 3.2 and c ≈ ≥ ∼ ilar value of D 0.01 was reported [4] for the one-band 1.6 for the following results. ≈ ∼ HubbardmodelabovetheMott-Hubbardtransition,i.e., ForalmostallvaluesofU,onlyametallicoronlyanin- forU/W 5.9/4. This suggestsa Mott-Hubbardtransi- sulating solution is obtained for both τ 3.2 and 1.6. ≥ c∼ ∼ tion very similar to the one-bandHubbard model, albeit However, for U = 2.4, we find both a solution with a only for the narrow band. metallic and with an insulating narrow band (the wide Final evidence for the orbital-selective Mott-Hubbard band is metallic in both solutions with only minor dif- transition is obtained from the spectral functions shown ferences). In Fig. 3, the spectral function of these two inFig.2: We canunambiguouslysaythatthewideband solutions are shown; the value of Z and D for the in- isstillmetallicatU =2.6,whereasthenarrowbandisal- sulating solution is plotted in Fig. 1 as open circles and 4 wide band narrow band oretical side, we hope to stimulate further experiments onthe orbital-selectiveMott-Hubbardtransition,e.g.,in 1 2 (a) (b) Sr2RuO4 where results were so-far negative in this re- spect [21]. w) w) A( A( WeacknowledgeveryfruitfuldiscussionswithM.Feld- 0.5 1 bacher, S. Sakai, and A. Toschi as well as support by the Alexander von Humboldt foundation (RA) and the Emmy Noether program of the Deutsche Forschungsge- meinschaft (KH). 0 0 -4 0 4 -4 0 4 During the completion of our work, we learned about w w several related studies [19, 22]. FIG. 3: (Color online) Spectral functions A(ω) for (a) the wideand(b)thenarrowbandatU =2.4. Twosolutionswith insulating/metallic narrow band coexist. [1] W. Metzner and D. 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