Table Of ContentOrbital-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. Vollhardt, Phys. Rev. Lett. 62, 324
squares. The DMFT(PQMC) data suggest that two so- (1989); G. Kotliar and D. Vollhardt Physics Today 57,
lutions with metallic and insulating narrow band coex- 53 (2004).
ists for U 2.4, so that the Mott-Hubbard transition in [2] A. Georges et al., Rev.Mod. Phys. 68, 13 (1996).
whichthe∼narrowbandbecomesinsulating(andinwhich [3] F. Gebhard, The Mott Metal-Insulator Transition
(Springer, Berlin, 1997); M. Imada, A. Fujimori, and Y.
the wide band stays metallic) is in general of first-order.
Tokura, Rev.Mod. Phys. 70, 1039 (1998).
Possibly, the insulating solution is metastable at T=0. [4] G. Moeller et al. Phys. Rev. Lett. 74, 2082 (1995); R.
Discussion. For understanding the DMFT results it is Bulla,Phys.Rev.Lett.83,136(1999);M.J.Rozenberg,
instructive to remind ourselves of what is known for the R. Chitra, and G. Kotliar, Phys. Rev. Lett. 83, 3498
two-orbital Anderson impurity model (AIM). If J > T (1999); N. Blu¨mer, Ph. D. thesis, Universit¨at Augsburg
K
2002 (ShakerVerlag, Aachen,2003).
(the AIM Kondo temperature) the impurity spins of the
[5] V. I. Anisimov et al., J. Phys.: Cond. Matt. 9,
twoorbitalsformasteadfastspin-1(triplet). Forsuchan
7359 (1997); A. I. Lichtenstein and M. I. Katsnel-
AIMandinequivalentorbitalsitis knownthatthis spin- son, Phys. Rev. B 57, 6884 (1998); for a review see
1 is screened in two stages: first only by one orbital to a K. Held et al., Psi-k Newsletter #56, 65 (2003) [psi-
spin-1 at T1, and then by the second orbital to a spin-0 k.dl.ac.uk/newsletters/News 56/Highlight 56.pdf].
2 K
at T2 [20]. Within DMFT we now have to solve AIMs [6] M. J. Rozenberg, Phys. Rev. B 55, R4855 (1997); J.
self-cKonsistently: TheAIM’sT ’sofoneDMFTiteration E. Han, M. Jarrell, and D. L. Cox, Phys. Rev. B 58,
K
R4199 (1998); Y. Ono, M. Potthoff, and R.Bulla, Phys.
(crudely T ZW) sets the hybridization strength for
K ≈ Rev. B 67, 35119 (2003); Th. Pruschke and R. Bulla,
the next DMFT iteration. Hence, we can interpret our
cond-mat/0411186 (unpublished).
DMFT results as following: Given the two inequivalent [7] Y. Maeno et al. Nature 372, 532 (1994); Y. Maeno, T.
KondoscalesoftheAIM,thereisaU-intervalwhereonly M. Rice, and M. Sigrist, Physics Today 54, 42 (2001).
thehybridizationstrength(andTK)ofthenarroworbital [8] T. Oguchi, Phys. Rev. B 51, R1385 (1995); I. I. Mazin
is renormalized to zero by the DMFT iterations. Only and D.Singh, Phys. Rev.Lett. 79, 733 (1997).
the narrow band is insulating. [9] S. Nakatsuji and Y. Maeno, Phys. Rev. Lett. 84, 2666
(2000).
If only the Ising-component of Hund’s exchange is
[10] A. Koga et al.,Phys.Rev. Lett.92, 216402 (2004).
takenintoaccount,thebehavioroftheAIMiscompletely [11] A. I. Anisimov et al., Eur. Phys.J. B 25, 191 (2002).
different. Insteadofatriplet,theimpurityspinsallignto [12] A. Liebsch, Phys. Rev. Lett. 91, 226401 (2003), Phys.
SZ= 1(noSZ=0component). ForJ>TK (J 0.5and Rev. B 70, 165103 (2004).
T ±ZW 0.4 at the Mott-Hubbard transition≈of [12]), [13] K.HeldandD.Vollhardt,Eur.Phys.J.B5,473(1998).
K
ther≈eisno≈spinKondoeffectanymoresinceitrequiresa Generally, the statistical error in QMC simulations be-
comeshugeifpositiveandnegativecontributionsalmost
spin-flipoftheconductionelectronsand,hence,achange
compensate. This is theso-called sign-problem.
of S by 1. As soonas one orbital becomes insulating,
Z ± [14] M. Feldbacher, K. Held, and F. F. Assaad, Phys. Rev.
there is also no orbital Kondo effect anymore: the whole Lett. 93, 136405 (2004).
system is unscreened, i.e., insulating. It is certainly in- [15] S.Sakai,R.Arita,andH.Aoki,Phys.Rev.B70,172504
terestingtostudywhetherthiskindofphysicsisrelevant (2004).
for magnetically anisotropic materials. [16] Y. Motome and M. Imada, J. Phys. Soc. Jpn. 66, 1872
(1997); ibid 67, 3199 (1998).
Conclusion. Taking the full SU(2)-symmetry of
[17] J. E. Han, Phys.Rev.B 70, 054513 (2004).
Hund’s exchange into account in the PQMC calcula-
[18] J. E. Hirsch and R. M. Fye, Phys. Rev. Lett. 56, 2521
tion, we conclude that there are two consecutive Mott-
(1986).
Hubbard transitions, whereby -at least around the first [19] A. Koga et al.,cond-mat/0503651 (unpublished).
transition- two solutions coexist. By clarifying the the- [20] C. Jayaprakash, H. R. Krishna-murthy, and J. W.
5
Wilkins, Phys. Rev. Lett. 47, 737 (1981); W. Izumida, [22] M. Ferrero et al., cond-mat/0503759 (unpublished);
O. Sakai, and Y. Shimizu, J. Phys. Soc. Jap. 67, 2444 L. de’ Medici, A. Georges, and S. Biermann,
(1998). cond-mat/0503764 (unpublished).
[21] See, e.g., S.-C. Wanget al.,Phys. Rev.Lett. 93, 177007
(2004) and references therein.
