Strong Correlations fromHund’s CouplingAntoine Georges,1,4,5 Luca de’ Medici,2,3 andJernej Mravlje1,4,6
1Centre de Physique Théorique, École Polytechnique, CNRS, 91128 Palaiseau Cedex,France; email: [email protected], [email protected] de Physique des Solides, UMR8502, CNRS-Université Paris-Sud, 91405Orsay Cedex, France3Laboratoire de Physique et Etude des Matériaux, UMR8213, CNRS/ESPCI/UPMC,75231 Paris Cedex, France; email: [email protected]ège de France, 75005 Paris, France5DPMC-MaNEP, Université de Genève, CH-1211 Genève, Suisse6Jo�zef Stefan Institute, SI-1000 Ljubljana, Slovenia
Annu. Rev. Condens. Matter Phys. 2013. 4:137–78
First published online as a Review in Advance onJanuary 25, 2013
TheAnnual Review of Condensed Matter Physics isonline at conmatphys.annualreviews.org
This article’s doi:10.1146/annurev-conmatphys-020911-125045
Copyright © 2013 by Annual Reviews.All rights reserved
Keywords
strongly correlated materials, Hund’s coupling, Mott transition,transition-metal oxides, ruthenates, iron superconductors
Abstract
Strong electronic correlations are often associated with the proximityof a Mott-insulating state. In recent years however, it has become in-creasingly clear that theHund’s rule coupling (intra-atomic exchange)is responsible for strong correlations inmultiorbital metallic materialsthat are not close to aMott insulator.Hund’s coupling has two effects:It influences the energetics of theMott gap and strongly suppresses thecoherence scale for the formation of a Fermi liquid. A global picturehas emerged recently,which emphasizes the importance of the averageoccupancy of the shell as a control parameter. The most dramaticeffects occur away from half-filling or single occupancy. We reviewthe theoretical understanding and physical properties of these Hund’smetals, together with the relevance of this concept to transition-metaloxides (TMOs) of the3d, and especially 4d, series (such as ruthenates),as well as to the iron-based superconductors (iron pnictides andchalcogenides).
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1. INTRODUCTION
The electronic state ofmanymaterials with partially filled d- or f-shells, aswell asmolecular solids,is characterized by strong correlations. Their physical properties are not properly described bySlater determinants of single-particle wave functions.Materialswith strong electronic correlationsdisplay fascinating phenomena, often with a large amplitude, such as metal-insulator transitions,high-temperature superconductivity, colossal magnetoresistance, a large thermoelectric power, orcarriers with a large effective mass and reduced spectral weight.
The Mott phenomenon—the localization of electrons due to strong Coulomb repulsion andreduced bandwidth—has emerged as a central paradigm in this field (1). The parent compounds ofhigh-temperature cuprate superconductors are widely considered to beMott insulators (of the so-called charge-transfer type). The metallic and superconducting states emerge by doping this in-sulator with charge carriers. In this view, strong electronic correlations in themetallic state are dueto the proximity of the Mott insulator. Hence “Mottness” is widely regarded as being key to thestrong correlations observed in oxides and organic compounds.
Cuprates have a single active electronic band at the Fermi level, a rather unique property whichincidentally the organic superconductors of the BEDT family also share.With very few exceptions,known oxides of other transition metals are in contrast multiband materials, and so are the re-cently discovered iron-based superconductors. The Fermi level crosses several bands, whichoriginate from the different orbitals of the transition-metal d-shell hybridizing with ligands (2).Many of these multiorbital materials, such as ruthenates and iron pnictides/chalcogenides, aremetals that display clear signatures of strong correlations while not in close proximity to a Mott-insulating state. This raises a puzzling question: What is the physical origin of the electroniccorrelations in these materials?
In the past few years, there has been increasing awareness that Hund’s coupling may be re-sponsible for these effects. Hund’s coupling is the energy scale associated with intra-atomic ex-change,which lowers the cost in repulsiveCoulomb energywhen placing two electrons in differentorbitals with parallel spin, as opposed to two electrons in the same orbital (3). This shakes theparadigm establishing Mottness as the unique origin of strong correlations and highlights thatanother class of strongly correlated but itinerant systemshasphysical properties distinctly differentfrom doped Mott insulators. Yin et al. (4) coined the term “Hund’s metals” to designate suchmaterials.
There are two distinct effects of the Hund’s rule coupling. The first is a high-energy effect. Asemphasized early on (5, 6), the effective Coulomb repulsion for an isolated atom is increased byHund’s coupling for a half-filled shell, whereas it is decreased for all other fillings. The second isa low-energy effect, revealed in early studies of a single-impurity atom coupled to a conduction-electron gas (the Kondo problem). For a multiorbital shell, Hund’s coupling lowers considerablythe characteristic temperature belowwhich screening of the atomic degrees of freedom takes place(7–13). This is due to the quenching of orbital momentum and the associated loss of exchangeenergy, and explains the sensitive dependence of theKondo temperature on the size of the impurityspin (14–16).
What is remarkable is that these effects, documented for an isolated atom or for a single atomicimpurity in a metallic host, continue to play a crucial role in the context of itinerant systems withabandwidth significantly larger thanHund’s coupling. That this is the case has been demonstratedin several recent studies, which led to the realization that “Hundness” is the key explanation ofelectronic correlations in several families of metallic systems. Two remarkable theoretical studies,in the context of a five-band description of the metallic state of iron pnictides by Haule & Kotliar(17) and in that of a three-band multiorbital Hubbard-Kanamori model by Werner et al. (18),
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revealed that the low-energy quasiparticle coherence scale is considerably reduced by Hund’scoupling. Mravlje et al. (19) also emphasized such a reduction in the context of ruthenates. Thisleaves an incoherent metallic state with frozen local moments in an extended temperature rangeabove the coherence scale, for which the authors of Reference 18 coined the term “spin-freezing”regime. These authors discovered also that this regime displays non-Fermi-liquid properties of theself-energy, characterized by a power-law behavior.
As shown inReferences 20 and 21, the influence of Hund’s coupling on the energetics of chargetransfer in an isolated atom has important consequences for the Mott critical coupling in a solid.The generic effect (when orbital degeneracy is preserved) is that non-half-filledmaterials are drivenfurther away from theMott-insulating state. de’Medici et al. (22) proposed a global picture,whichalso shows how to place many different materials on a map parametrized by the interactionstrength and the filling of the shell. They emphasized that the two key effects compete with oneanother in the generic case of a non-half-filled shell: Hund’s coupling drives the system away fromthe Mott transition but at the same time makes the metallic state more correlated by lowering thequasiparticle coherence scale. As with the Roman god Janus, the Hund’s rule coupling has two
0 1.0 2.0 3.0 4.0 5.0 6.0Number of electrons per site (n)
0
1
2
3
4
5
6
7
8
U/D
Quasiparticle
weight (Z)
Z = 0.2Z = 0.4
Z = 0.6
DO
S
ε–D D
SSrVrVS OOOS OSrVO3SrCrO3SSrCSSrCSSrSS CSrCS rrrOOOO333SrCrO3
SSrSrr2RRuuOOOOOOO44444
SrSrSrS 22RhRhOOOOOO444
SSSrrrrrRuOOOOORR OuOOO
Sr2RuO4
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SrSr2TcOTcO4Sr2TcO4
rrrrrrrSSSSSSSSSSSSSrrrrrrrrrSSSSSSSSSSrTcTTTcOTcOTcOTcOTTTTTTTTTTTTTTTTTcccccccOOOOOOOTTTTTTTTTT OOOOOOOcccOOOOOOO333333SrTcO3
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0
0.2
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Figure 1
Color intensity map of the degree of correlation (as measured by the quasiparticle weight Z—right scale) foraHubbard-Kanamori model with three orbitals appropriate to the description of early transition-metal oxideswith a partially occupied t2g shell. The vertical axis is the interaction strength U normalized to the half-bandwidthD, and a finite Hund’s coupling J¼ 0.15U is taken into account. The horizontal axis is the numberof electrons per site—from zero (empty shell) to six (full shell). Darker regions correspond to good metals andlighter regions to correlated metals. The black bars signal the Mott-insulating phases for U > Uc. The whitearrows indicate the evolution of Uc upon further increasing J and emphasize the opposite trend between half-filling and a generic filling.GrayXmarks denote the values ofUc for J¼0.Among integer fillings, two electrons(two holes) display correlated behavior in an extended range of coupling, with spin-freezing above some lowcoherence scale. Specific materials are schematically placed on the diagram. The materials denoted in blackhave been placed according to the experimental value of g/gLDA. For detailed explanations, see Section 6. Thedynamical mean-field theory calculations leading to a related color intensity map in Reference 22 have beenrepeated here using a more realistic density of states (DOS) for t2g states (lower left inset graph).
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faces! Figure 1 illustrates and summarizes this global picture (which is discussed in much greaterdetail in Section 6).
At a fundamental level, a key lesson is that intra-atomic correlations play a crucial role even initinerant systems with relatively broad bands andmoderate Hubbard repulsion, such as transitionmetals of the 4d and 5d series or iron pnictides and chalcogenides. Dynamical mean-field theory(DMFT) (23, 24) is currently themost appropriate theoretical framework to deal with these issues,because it handles band-like and atomic-like aspects on equal footing. In contrast to moreconventional approaches picturing a solid as an inhomogeneous electron gas towhich interactionsare addedperturbatively,DMFTemphasizes localmany-body correlations byviewing a solid as anensemble of self-consistently hybridized atoms.
This review is organized as follows. Section 2 provides an introduction to Hund’s rules andintra-atomic Coulomb interactions in the multiorbital context. Section 3 explains the influence ofHund’s coupling on the intra-atomic charge gap and theMott critical coupling. Section 4 reviewsthe influence of Hund’s coupling on the Kondo temperature of a multiorbital impurity atom ina metallic host. Section 5 briefly introduces DMFT, which provides a bridge between single-atomphysics and the full solid. Section 6 is the core of this review and puts together the key effects of theHund’s rule coupling in the solid-state context. Sections 7 and 8 consider ruthenates and ironpnictides/chalcogenides, respectively, in the perspective of Hund’s metals.
2. INTRA-ATOMIC EXCHANGE AND THE HUND’S RULE COUPLING
In 1925, in an article dealing with the spectra of transition-metal atoms, Friedrich Hund (3)formulated a set of rules specifying the ground-state configuration of multielectron atomic shells.For N electrons in a shell with orbital degeneracy M (¼ 2l þ 1), the rules state the following:
n Total spin S should first be maximized (rule of maximum multiplicity).n Given S, total angular momentum L should be maximized.n Finally, the lowest J ¼ jL � Sj should be selected forN <M (less than a half-filled shell)
and the highest J ¼ L þ S for N > M.
For example, a d-shell with three electrons will have S ¼ 3/2, L ¼ 3, J ¼ 3/2 (e.g., ↑, ↑, ↑, 0, 0),with six electrons S¼L¼ 2, J¼4 (e.g., ↑↓, ↑, ↑, ↑, ↑), whereas the half-filled shellwith five electrons(e.g., ↑, ↑, ↑, ↑, ↑) has S¼ J¼ 5/2 and a fully quenched angular momentum L¼ 0. These rules aresometimes referred to as the bus-seat rule: Singly occupied spots are filled first, then doubleoccupancies are created when singly occupied spots are no longer available.
The origin of these rules is traditionally attributed to the minimization of the Coulomb in-teraction between electrons. For two electrons, for example, the first rule (S¼ 1 rather than S¼ 0)forces an antisymmetric wave function of the radial part, so that electrons are further apart. Inquantum-mechanical terms, the energy gain associated with Hund’s rule is the intra-atomicexchange energy.1 The third rule is associated with spin-orbit coupling, which we do not considerin this review, although its physical effects have attracted considerable attention recently.
To illustrate these rules in amore quantitative formappropriate to the solid-state context of thisreview, let us consider the Hamiltonian describing the t2g triplet of orbitals, as relevant toa transition-metal ion in a cubic crystal field with an octahedral environment. The appendix
1Besides such a gain, which determines the ordering of multiplets in calculations where the single-electron basis is fixed (25),another term appears in self-consistent calculations in which the single-orbital basis is allowed to vary (26). This other term,which comes from the smaller screening of the electron-nucleus interaction for high-spin and high-orbital momentum states(27), becomes dominant for light-neutral atoms. For a recent discussion and references to further reading, see Reference 28.
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considers in detail the case of two orbitals and an eg doublet. For both eg and t2g, there are onlythree independent Coulomb integrals, which are matrix elements of the screened Coulomb in-teraction in appropriately chosen wave functions of the t2g orbitals in the solid:
U ¼Z
drdr0jfmðrÞ j2Vcðr, r0Þjfmðr0Þ j2,
U0 ¼Z
drdr0jfmðrÞ j2Vcðr, r0Þjfm0 ðr0Þ j2, and
J ¼Z
drdr0fmðrÞfm0 ðrÞVcðr, r0Þfmðr0Þfm0 ðr0Þ.
1:
Indeed, the wave functions can be chosen real-valued [so that the spin-exchange and pair-hoppingintegrals are equal (J¼ J0)], and all other terms in the interaction tensor, e.g., of the type Ummmm0,vanish by symmetry in this case. Because there are no other exchange integrals involved, the fullmany-body atomic Hamiltonian for t2g states takes the Kanamori form (29):
HK ¼ UXm
bnm↑bnm↓ þU0 Xm�m0
bnm↑bnm0↓ þ ðU0 � JÞX
m<m0,s
bnmsbnm0s þ
�JXm�m0
d†m↑dm↓d†m0↓dm0↑ þ J
Xm�m0
d†m↑d†m↓dm0↓dm0↑.
2:
The first three terms involve density-density interactions only between electrons with oppositespins in the same orbital (U), opposite spins in different orbitals (U0 < U), and parallel spins indifferent orbitals. The latter case has the smallest coupling U0 � J, reflecting Hund’s first rule.
It is useful to consider a generalization of this Kanamori multiorbital Hamiltonian to a form inwhich all coupling constants are independent:
HGK ¼ UXm
bnm↑bnm↓ þU0 Xm�m0
bnm↑bnm0↓ þ ðU0 � JÞX
m<m0,s
bnmsbnm0s þ
�JXXm�m0
d†m↑dm↓d†m0↓dm0↑ þ JP
Xm�m0
d†m↑d†m↓dm0↓dm0↑.
3:
Defining the total charge, spin and orbital isospin generators ( t! are the Pauli matrices)
bN ¼Xms
bnms, S!¼ 1
2
Xm
Xss0
d†ms t!ss0dms0 , Lm ¼ iXm0m00
Xs
emm0m00d†m0sdm00s, and 4:
the generalized Kanamori Hamiltonian (Equation 3) can be rewritten as
HGK ¼ 14ð3U0 �UÞbN�bN � 1
�þ ðU0 �UÞ S!2 þ 1
2ðU0 �U þ JÞL!2 þ
�74U � 7
4U0 � J
�bNþ
þðU0 �U þ J þ JPÞX
m�m0d†m↑d
†m↓dm0↓dm0↑ þ ðJ � JXÞ
Xm�m0d
†m↑dm↓d
†m0↓dm0↑. 5:
It thus has full U(1)C Ä SU(2)S Ä SO(3)O symmetry provided JX ¼ J and JP ¼ U � U0 � J, inwhich case the Hamiltonian reduces to the first line in Equation 5. We refer loosely to suchsymmetry as rotational invariance. Rotational invariance ofHGK does not imply thatU0 andU arerelated. In particular, for JX ¼ J and U0 ¼ U � J (JP ¼ 0), one obtains a minimal, rotationally
invariantHamiltonian ðU � 3J=2ÞbNðbN � 1Þ=2� J S!2
, involvingonly bN2and S
!2,whichwediscuss
inmoredetail below (Equations12and27).This actuallyholds for anarbitrarynumberMoforbitals.The physical t2g Hamiltonian (Equation 2) has JX ¼ JP ¼ J and, using Equation 5, is seen to be
rotationally invariant, provided
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U0 ¼ U � 2J, 6:
in which case the Hamiltonian takes the form
Ht2g ¼ ðU � 3JÞ bN� bN � 1�
2� 2JS
!2� J2L!2þ 5
2J bN. 7:
In this form,Hund’s first two rules (maximal S, then maximal L) are evident. The spectrum of thisHamiltonian is detailed in Table 1.
Equation 6 holds in particular when U, U0, and J are calculated assuming a sphericallysymmetric interaction and t2g wave functions from simple crystal-field theory. In this approxi-mation, these integrals can be expressed in terms of Slater parameters F0, F2, F4 (or alternativelyRacah parameters A, B, C) (30):
U ¼ F0 þ 449
F2 þ 449
F4 ¼ Aþ 4Bþ 3C,
U0 ¼ F0 � 249
F2 � 4441
F4 ¼ A� 2Bþ C ¼ U � 2J, and
J ¼ 349
F2 þ 20441
F4 ¼ 3Bþ C.
8:
A rotationally invariant form of the t2g Hamiltonian is obtained when assuming sphericalsymmetry because the orbital angular momentum in the t2g states is only partially quenched, froml ¼ 2 down to l ¼ 1. The orbital isospin generators are thus closely related to those of angularmomentumwith l¼ 1 (up to a sign; cf. Reference 30). In the solid state,Vc is the screenedCoulombinteraction. The spherical symmetry of Vc is of course no longer exact but often considered to bea reasonable approximation so that U0 ¼ U � 2J is often used in the solid as well.
For an entire d-shell, theKanamoriHamiltonian (Equation 2) is not exact, and a full interactiontensor Um1m2m3m4 must be considered. For an isolated atom with spherical symmetry, this tensorcan be parametrized in terms of three independent Slater (Racah) parameters F0, F2, F4, whereasnine parameters are needed in principle in cubic symmetry (30). A word of caution is in order
Table 1 Eigenstates and eigenvalues of the t2g Hamiltonian U bNðbNL1Þ=2L2JS!2
L J L!2
=2 in theatomic limit (U ” U L 3J)a
N S L Degeneracy 5 (2S 1 1) (2L 1 1) Energy
0, (6) 0 0 1 0, [15U]1, (5) 1/2 1 6 �5J/2, [10U � 5J/2]
2, (4) 1 1 9 U � 5J, [6U � 5J]
2, (4) 0 2 5 U � 3J, [6U � 3J]
2, (4) 0 0 1 U, [6U]3 3/2 0 4 3U � 15J/2
3 1/2 2 10 3U � 9J/2
3 1/2 1 6 3U � 5J/2
aThe boxed numbers identify the ground-state multiplet and its degeneracy for J > 0.
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regarding notations. For an entire d-shell, it is customary (5, 6, 31) to define Ud ¼ F0, the Hund’srule coupling JH¼ (F2þ F4)/14, and a third parameter 14Cd¼ 9F2/7� 5F4/7. Those should not beconfused with the U and J couplings defined above for t2g and eg shells. For example, usingEquation 8 (and later Equation 25 of the appendix)U¼Udþ 8JH/7, J¼ 5JH/7þCd/9 for t2g, andJ ¼ 30JH/49 þ 4Cd/21 for eg.
We lack space here to discuss in any detail the important issue of the determination of screenedinteraction parameters in the solid state, which is still a lively topic of current research. On thetheoretical side, progress has been achieved using the first-principles constrained random phaseapproximation (c-RPA)method (32) and its recent developments. This approach has emphasizedthat interaction parameters (especiallyU or F0) are actually functions of the energy scale at whichthey are considered and also depend on the set of states that are retained in the effective descriptionof the solid (e.g., on the energy window used to construct appropriate Wannier functions). Theenergy-scale dependence comes from screening. At high energy, the unscreened values associatedwith an isolated atom are found: The monopole Slater integral F0 (and henceU andU0) is of order15–25 eV. Screening reduces this value considerably, down to a few eV’s at low energy in the solid.The exchange integral, in contrast, does not involve the monopole contribution F0 but only thetwo higher-order multipoles F2 and F4. Therefore, it was pointed out that the Hund’s rulecoupling is reduced by only 20%–30% when going from the atom to the solid [see, e.g., thepioneering work of van derMarel and Sawatzky (5, 6).] Although theHartree-Fock (unscreened)value for a 3d transition element with atomic number Z reads JatH ¼ 0:81þ 0:080ðZ� 21ÞeV,these authors estimated the screened JH ¼ 0.59þ 0.075(Z�21) eV (Cdx 0.52 JH in both cases).This varies from JHx 0.59 eV up to JHx 1.15 eV as one moves along the 3d series from Sc to Ni(for a t2g shell, Jx 0.77 JH). Also, given the Hartree-Fock value F0
at ¼ 15:31þ 1:5ðZ� 21Þ eV,JatH=F
0atx0:053 is fairly constant along the series. It is thus reasonable to expect that the ratio J/U for
a t2g shell is also approximately constant among early transition-metal oxides (TMOs). (Using, forexample, a reduction of F0 by screening down to 20% of its atomic value, one obtains J/Ux 0.13for a t2g shell.)
3. ENERGETICS OF THE MOTT GAP
TheHund’s rule coupling affects the energetics of charge transfer in a major way and in a mannerthat depends crucially on the filling of the shell. This effect is already visible for an isolated atom, asnoted by van der Marel and Sawatzky (5, 6). It has direct consequences for the magnitude of theMott gap in the solid-state context, as discussed below.
Consider first an isolated shell with N electrons. We are interested in the energetic cost forchanging the valence of two isolated atoms from the state withN electrons each to that withN� 1andN þ 1 electrons, i.e., transferring one electron from one of the atoms to the other. This energycost reads
Dat ¼ E0ðN þ 1Þ þ E0ðN � 1Þ � 2E0ðNÞ ¼ ½E0ðN þ 1Þ � E0ðNÞ� � ½E0ðNÞ � E0ðN � 1Þ�, 9:
where E0 is the ground-state energy of the shell with N electrons. The last expression is thedifference between the affinity and ionization energies.
For simplicity, we base the discussion on the KanamoriHamiltonian (Equation 2) appropriate,for example, to a t2g shell. The ground-state energy of this Hamiltonian can be obtained byconsidering simply the density-density terms. Consider the state in the (degenerate) ground-statemultiplet with maximal Sz (¼ þN/2 for N � M, ¼ M � N/2 for N � M), consistent with Hund’srules. For example, for three orbitals, j ↑, ↑, 0æ forN¼ 2<M, j ↑, ↑, ↑æ forN¼M¼ 3, and j ↑↓, ↑, ↑æ
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forN¼ 4. The exchange and pair-hopping terms have no action on those states. So, forN�M, theground-state energy involves only the pairwise interaction between parallel spins:E0(N)¼ (U0 � J)N (N� 1)/2¼ (U� 3J)N (N� 1)/2. As long asN<M (less than half-filled shell), the expression ofthe atomic gap (Equation 9) involves only states with energies of this form. Hence, Ueff ¼ U � 3Jplays the role of the effectiveHubbard interaction (which is seen to be reduced by J), and the atomicgap reads
Dat[Ueff ¼ U0 � J ¼ U � 3J, ðN < M or N > MÞ, 10:
where the expression forN > M stems from particle-hole symmetry. In contrast, for a half-filledshell, the excited state with N þ 1 ¼ M þ 1 electrons involves one doubly occupied orbital, andhence its energy is pushedup.Counting thenumber of each typeof pairs, it readsE0(Mþ1)¼ (U0 �J) 3 M(M � 1)/2 þ U 3 1 þ U0 3 (M � 1) ¼ (U0 � J) M(M þ 1)/2 þ (U � U0 þ M J). The lastexpression emphasizes that the energy of this state is increased byU�U0 þMJ, as compared to thevalue it would have if all interactions were between parallel spins. Hence, theMott gap becomes:2
Dat[Ueff ¼ ðU0 � JÞ þ ðU �U0 þMJÞ ¼ U þ ðM� 1ÞJ, ðN ¼ MÞ. 11:
In contrast toageneric filling (N�M), Hund’s coupling for a half-filled shell (N¼M) increases theintra-atomic gap (or effective U). Here we have considered the Kanamori Hamiltonian. Corre-sponding expressions for a fivefold degenerate d-shell with a full Racah-SlaterHamiltonian can befound in References 5, 6, and 31, with similar qualitative conclusions.
These considerations for an isolated atom suggest that, in the solid-state context, the Hund’s rulecoupling has a strong influence on theMott gap and on the critical couplingUc separating a metallicphase from a Mott-insulating phase. Figure 2 displays the dependence of Uc on J for a Hubbard-Kanamorimodel of three degenerate bands, as obtained fromDMFTcalculations (see alsoSection6).
7
6
5
4
3
2
1
00 0.2 0.4 0.6 0.8
M = 3; N = 1M = 3; N = 2M = 3; N = 3
1.0
U c/D
J/D
Figure 2
Critical coupling separating the metallic and Mott-insulating (paramagnetic) phase, as a function of Hund’scoupling, for a Hubbard-Kanamori model of three degenerate bands with one (red), two (purple), andthree (blue) electronsper site.Themodel is solvedwithdynamicalmean-field theory,witha semicirculardensityofstates of bandwidth 2D for each band. The dashed lines (large-J asymptotes) have a slope corresponding to theatomic limit calculation. The shaded region corresponds to U0 � J < 0 (J > U/3) (see References 21 and 22).
2For a generalization including spin orbit, see Reference 33.
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The figure shows thatUc is strongly reducedas J is increased, in the caseof a half-filled shell (N¼ 3). Incontrast, Uc is increased, with a quasilinear dependence on J, for N ¼ 1. The case of two electrons(and, more generally, the generic case N � 1, M, 2M � 1) is especially interesting. Uc has a non-monotonous dependence on J in that case, first decreasing at small J, then increasing at larger J. Thestrong reductionofUcby J at half-filling has been discussed bymany authors (see, e.g.,References 12,13, 34–37). That J enhances Uc and hence makes the Mott-insulating state harder to reach in thegeneric case of a non-half-filled shell has been, in contrast, clearly appreciated only recently. (Al-though implicit in the results of, e.g., References 38 and 39, it has been recently emphasized inReference 20 and, especially, in Reference 21.)
To rationalize the J dependence ofUc displayed in Figure 2, it is natural to use the atomic limitconsidered above and apply a criterion à laMott-Hubbard for the closing of the gap (21), namelyUat
eff ¼ ~WM,NðJÞ. In this expression, ~W is an estimate of the available kinetic energy forN electronshopping amongM-degenerate orbitals. This leads toUc ¼ ~WM,NðJÞ þ 3J for a non-half-filled shellandUc ¼ ~WM,MðJÞ � ðM� 1ÞJ at half-filling. Assuming that ~W reaches a finite value ~W
1at large
J, this yields a linear increase Uc ∼ ~W1N,M þ 3J for N � M and a linear decrease Uc ∼ ~W
1M,M �
ðM� 1ÞJ at half-filling. Figure 2 shows that these expressions (gray dashed lines) describe thelarge-J behavior ofUc quitewell. Figure 2 shows also that the J dependence of the kinetic energy ~Wis crucial to account for Uc(J): The extrapolations of these dashed lines down to J ¼ 0 fall waybelow the actual value ofUc at J¼0, except in the case of a single electronN¼1.This is because theHund’s rule coupling quenches the orbital fluctuations, which in turn blocks many of the hoppingprocesses contributing to ~W. A deeper perspective on this effect is given in Section 4 in the contextof the Kondo problem of amagnetic impurity in ametallic host. This effect is particularly strong athalf-filling: For J¼0, it iswell established (34, 40–42) that the orbital fluctuations lead to a value ofUJ¼0
c (and ~W), which increases rapidly with orbital degeneracy M. [Within DMFT, the Mott-Hubbard gap-closing transition occurs at Uc1ðJ ¼ 0Þ, proportional to ffiffiffiffiffi
Mp
, and the Brinkman-Rice transition where the quasiparticle weight vanishes at Uc2, proportional to M (see Reference42).] In contrast, ~W is renormalized downward as J is turned on, and a value ~WM,M ∼ ~W1,1 ∼W(whereW is the bare bandwidth) is reached already at moderate values of J, leading toUc ∼Wþ(M � 1) J (L. de’ Medici & M. Capone, in preparation). [Accordingly, J strongly reduces thecoexistence region [Ucl, Uc2] (43).]
For generic filling levels, the reduction of the kinetic energy by orbital blocking is responsiblefor the decrease of Uc at small J, whereas the reduction of the atomic Ueff is responsible for theincrease of Uc at large J, hence the nonmonotonous behavior. In contrast, for a single electron orhole, the orbital blocking does not apply because the Hund’s rule coupling does not lift thedegeneracy of the atomic ground state.
Finally, at J¼ 0, the largest value ofUc is obtained at half-fillingN¼M and the smallest one fora single electron (or hole)N ¼ 1, 2M � 1. This is reversed at moderate and large J, with Uc beingsmallest for ahalf-filled shell (Figure 1). Because of this effect, an insulating state is strongly favoredat half-filling. Indeed,most TMOswith a half-filled shell are insulators (e.g., SrMnO3 andLaCrO3
with three electrons in the t2g states; see Section 6.2). Fujimori et al. (44) proposed the reduction ofthe Mott gap by Hund’s coupling for a non-half-filled shell to explain the paramagnetic metalliccharacter of V5S8 and its photoemission spectrum, which shows an exchange splitting.
4. IMPEDED KONDO SCREENING AND BLOCKING OF ORBITALFLUCTUATIONS
We now consider a single atom hybridized with a Fermi sea of conduction electrons. This is thefamous Kondo problem of a magnetic impurity embedded in a metallic host. As we shall see, the
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generic effect of the Hund’s rule coupling is to drastically suppress the Kondo temperature TK
belowwhich the local moment of the atom is screened. This suppression is due to the combinationof two effects: the blocking of orbital fluctuations as well as the reduction of the effective Kondocoupling within the low-lying multiplet selected by Hund’s rule. For a spin-S impurity, this re-duction follows JK,eff } 1/S, as first derived by Schrieffer (14) [see also Blandin (15)]. The Kondotemperature being exponential in JK thus drops exponentially with S, as is indeed observed ex-perimentally for metals hosting transition-metal impurities with different spin values (16). Okada&Yosida (7) were first to perform a systematic study of the suppression of the Kondo scale by theHund’s rule coupling.
Dworin & Narath (45) introduced a generalization of the Anderson impurity model for Morbitals (e.g., M ¼ 2l þ 1 angular momentum channels), which takes into account Hund’s rulephysics in a minimal way (see also References 46 and 47). This reads
HDN ¼Xk
XMm¼1
Xs¼↑↓
�ɛkc
†kms
cks þ Vkmc†kms
dmsþ V�
kmd†ms
ckms
�þHat, 12:
with the atomic term
Hat ¼ U � J2
Xm1m2s1s2
d†m1s1d†m2s2dm2s2dm1s1 þ
J2
Xm1m2s1s2
d†m1s1d†m2s2dm1s2dm2s1
¼�U � 3
2J� bNd
�bNd � 1�
2� JS
!2
d þ14J bNd,
13:
where bNd and S!
d are, respectively, the total charge and spin operators of the d-shell, as above. Theatomic part of the Dworin-Narath Hamiltonian is rotationally invariant and coincides with thegeneralized KanamoriHamiltonianwith appropriately chosen parameters, as discussed in Section2 and the appendix (Equation 27).
For J ¼ 0, the model has full SU(2M) symmetry. Coqblin & Schrieffer (CS) (48) pioneered thestudy of impurity models with enhanced orbital symmetry by considering the Hamiltonian
HCS ¼Xka
ɛkc†kacka þ JK
Xkk0
Xab
c†kack0bbSba, 14:
where a ¼ {m, s} and b are SU(2M) indices and bSab is the impurity operator corresponding toa specific irreducible representation of SU(2M). At large U and J ¼ 0, a Schrieffer-Wolfftransformation maps the Dworin-Narath Hamiltonian (Equations 12 and 13) onto the CSHamiltonian when the number of electrons Nd ¼ P
msd†msdms is constrained to be an integer
(then, bSab }d†adb).
A well-established result for the CS model (49) is that the orbital degeneracy enhances theKondo temperature. For a half-filled shell Nd ¼ M, one has
TJ¼ 0K,M
.D ¼ expð�1=2MrJKÞ ¼
�TK,1=D
�1=M, 15:
and a similar enhancement applies for all values ofNd (see Reference 50 for a detailed comparisonofNd¼ 1 andNd¼ 2 in the case ofM¼ 2 orbitals). In this expression, r is the conduction-electrondensity of states (per orbital and spin), D is a high-energy cutoff (e.g., the bandwidth for the CSmodel, or ∼
ffiffiffiffiffiffiffiUG
pfor the Anderson model), and TK,1 is the usual Kondo temperature for a single
orbital TK,1/D ∼ exp(�1/2rJK). Intuitively, the enhancement of TK for J ¼ 0 occurs because the
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conduction electrons can exchange both spin and orbital momentum with the impurity spins,which enhances the corresponding gain in exchange energy. The enhancement of the Kondotemperature due to orbital degrees of freedom has been investigated intensively in mesoscopicsystems. SU(4) symmetry with entangled spin and orbital degrees of freedom in carbon nanotubesand quantum dots has been discussed theoretically (51–55), and its effects have been observedexperimentally (56).
Anonzero Jbreaks the SU(2M) symmetry down to SU(2)sÄ SU(M)O. It drasticallymodifies thephysics and reduces TK, as first discussed systematically by Okada & Yosida (7) for the modeldefined by Equations 12 and 13 in the large-U limit. These authors performed a Schrieffer-Wolfftransformation for a fixed integer Nd and obtained a Coqblin-Schrieffer-Hund model, whichbasically consists of adding the term �J S
!2to Equation 14. This model was then analyzed by
further taking J→ 0 or J→1 and projecting onto an appropriate subspace that depends onNd. Avariational wave function approach was used, and the resulting binding energies were related tothe Kondo temperature. For a half-filled shell Nd ¼ M and in the limit of large J, the Kondotemperature is strongly reduced:
TK,M=D ¼ expð�M=2rJKÞ ¼�TJ¼0K,M=D
�M2
¼ �TK,1=D
�M. 16:
A similar reductionwas found also forNd¼M6 1 (M> 2) (7), and in this case it was furthermoreobserved that theorbital fluctuations arequenched at a larger energy scale than the spin fluctuations.The narrowing of the Kondo resonance and suppression of TK due to J was also studied usinga numerical renormalization group (NRG) method for the two-orbital model by Pruschke & Bulla(12). J thus takes the system away from the point of high symmetry and high TK (see Figure 3).
log J/D
–10
–8
–6
–4
–2
0
–6 –4 4 6–2 20
log
TK
/D
SU (2M)
[SU(2)]M
SU(2)large spin
(1)
(2)
Figure 3
Kondo temperatureTK as a function ofHund’s coupling J for the two-orbital Coqblin-Schrieffer-Hundmodel,plotted on a log-log scale. The data (red line) are from the poor-man’s scaling analysis in Reference 58. Jsuppresses the Kondo temperature and lowers the symmetry of the model.
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Again, the reduction of theKondo scale can be understood intuitively. For a half-filled shell, the(degenerate) atomic ground-state for J � 0 has a large spin S ¼ M/2 and a vanishing angularmomentum L¼ 0 (seeTable 1 for three orbitals). Hence, all the orbital exchange energy applyingto the J¼0 case is lost here because orbital exchangeprocesses are blocked. Furthermore, at large J,spin-exchange processes are restricted to the ground-state subspace with S¼M/2 (with, therefore,a smaller degeneracy than the J ¼ 0 ground-state subspace). The impurity spins in each orbitalchannel act in this subspace as S
!m ∼ S
!=M, the proportionality factor 1/M being a Clebsch-
Gordan coefficient. (For a related analysis in the case of an antiferromagneticHund’s coupling, seeReference 57.) Hence, the effective Kondo coupling JK,eff ¼ JK/M is reduced, as first observed bySchrieffer (14) (see also Reference 15).
We can see this clearly, following References 8–11 and 13, by considering as a starting pointa composite spin Kondo (CSK) Hamiltonian,
HCSK ¼Xkms
ɛkc†kms
ckmsþ JK
Xm
S!
m × s!cm � J
�Xm
S!
m
�2
, 17:
which shows M spin-1/2 impurities S!
m, each Kondo-coupled to the spin density s!cm ¼P
kabc†kma
t!abckmb(where t represents the Pauli matrices) of an independent bath. The spins are
coupled by the Hund term favoring S ¼ M/2. A crucial difference with the Dworin-Narath(Equations 12 and 13) and Coqblin-Schrieffer-Hund Hamiltonians is that there are no orbitaldegrees of freedom here. The Kondo coupling is diagonal ∼ JK dmm0 , and the J ¼ 0 Hamiltonianthus has a smaller [SU(2)]M symmetry.When Jbecomes larger than the other scales in the problem,
the large spin is formed. Within this subspace, S!
m ∼ S!=M, and the Hamiltonian (Equation 17) is
equivalent to theM-channel Kondo problemwith spin S¼M/2 and JK,eff¼ JK/M. The low-energyfixed point is a Fermi liquid with an exactly screened impurity spin, given that M ¼ 2. An NRGstudy (22) (Figure 4) of the CSKHamiltonian (Equation 17) shows that the Kondo temperature ofthis model is indeed reduced according to Equation 16 in the large-J limit.
We finally discuss the behavior at intermediate values of Hund’s coupling, which is of directinterest in view of applications to the TMOs discussed later, in which typically J ∼ U/6 < D. InReferences 8 and 13, this was analyzed for the CSK Hamiltonian using perturbative RG, whichresulted in an explicit expression for TK(J). The RG flow was separated into two regions (see theschematic plot in Figure 5). At high energies (region I), L > J, and the impurity spins are not yetlocked into the large-spin state. There, JK(L) grows with diminishing L as in the single-channel,single-impurity case. In region II (L< J), the large spin is assumed tobe established. The keypoint isthat, in this region, theKondo coupling is reduced by a factor 1/M and the speed atwhich it flows isreduced by the same factor. This can be summarized in a single scaling equation (to two-looporder): dgeff=d lnL ¼� 2g2eff=Mþ 2g3eff=M for the effective coupling constant geff ¼ rJK(L)Meff,where the effective number of channelsMeff¼1 in region I andMeff¼M in region II. Because of theslower scaling in region II, the screened Kondo regime (region III), signaled in a perturbative RGtreatment by a diverging coupling constant, occurs at a scalemuch smaller than the single-impurityscale TK,1:
TK,M ¼ TK,1
�TK, 1J S
�M�1
ðfor HCSKÞ, 18:
where S ¼ M/2. This is the same as Equation 16, but with JS playing the role of the high-energycutoffD. This RG analysis emphasizes that starting with a small enough J, the screening process intheCSKmodel proceeds first by the formation of a large spin S,which is then eventually screened at
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a lower scale (Figure 5).NRG results forM¼ 2 displayed inFigure 4 confirm this expectation.Oneobserves there an initial exponential reduction of TK at small J, followed by a power-law de-pendence, which at quite large J does match ∼1/J M�1 ¼ 1/J.
When considering the original model (Equations 12 and 13) or the Coqblin-Schrieffer-Hundmodel, it is qualitatively appealing to think of the reduction ofTK as following a two-stage process(Figure 3): first a projection onto a subspace described by the CSK Hamiltonian in which thedifferent orbital channels are decoupled, followed by a second stage inwhich a large spin is formedand eventually screened at a low-energy scale. However, it is not guaranteed that this two-stageprocess does apply in general, and a direct route (dashed arrow in Figure 3) may apply instead.Indeed, at large scalesL> J, the RG flow of the originalmodel follows the same route as that in theSU(2M) symmetric model. At smaller scales L( J, the quenching of the orbital fluctuations andthe emergence of the high-spin (HS) state occur simultaneously. There is no energy scale at whichthe system is represented by M-independent spins undergoing single-channel Kondo scaling. Asa result, expression 18 for the reduction of TK at intermediate J for the CSK model cannot betrusted in general for the original model (Equations 12 and 13) or the Coqblin-Schrieffer-Hundmodel. Indeed, the poor man’s scaling study of Reference 58 for M ¼ 2 (reproduced in Figure 3)suggests a 1/J2 dependence, instead of 1/J as in Equation 18, whereas recent NRG studies(J.Mravlje, unpublished data) yield an even stronger power law. The authors of References 50 and
a
b
TKJHS
Cutoff ΛT*K
Temperature T
IIINozières FL
IILocked large spins
IPM
8
6
4
2
0
1/g
eff
(Λ)
Tχ(T
)
10-4 10-3 10-2 10-1 100 101 102
μ2II
0
μ2I
Figure 4
Composite-spin Kondo model (Equation 17). (a) Schematic behavior of the running coupling constant geff ¼JK(L)rMeff with Meff ¼ 1 in region I and geff ¼ M in regions II and III. The boundary between I and II isat the scale of Hund’s coupling. The Kondo temperature is reduced due to the slower scaling in region II. (b)Schematic dependence of the effective moment. The large moment formed in region II is screened at a reducedtemperature scale. Figure reproduced from Reference 13.
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59 also performed NRG studies of the Dworin-Narath model (Equation 12 and 13) and reportedan exponential dependence ofTK on J.We suspect that this is because rather small values of Jwereexplored there and that an initial exponential suppression followed by a power law at larger J is thegeneric behavior, as shown in Figure 3. The exponential regime and the characterization of precisepower laws (particularly in the case where the impurity has nonvanishing orbital momentum)deserve further studies.
5. DYNAMICAL MEAN-FIELD THEORY: SOLIDS VIEWED AS EMBEDDEDATOMS
Having considered isolated atoms (Section 3) and a single-impurity atom in a host metal (Section4), we now move to a full solid—a periodic array of atoms exchanging electrons. The mainmessage of this review is that the intra-atomic correlations associated with Hund’s coupling playa crucial role for solids also. DMFT is currently the most appropriate framework in which theseeffects can be revealed and studied (23, 24). Indeed, although more traditional approaches viewa solid as an inhomogeneous electron gas to which interactions are later added, DMFT focuses onthe fact that, after all, solids are made of atoms and that an atom is a small many-body problem initself, with a multiplet structure that must be properly taken into account.
DMFT describes the transfer of electrons between atoms in the solid by focusing on a singleatomic site and by viewing the atom on this site as hybridized with an effective medium throughwhich these electronic transfers take place (Figure 6). This effective medium must be self-consistently related to the rest of the solid. In more technical terms, the main physical observ-able onwhich DMFT focuses is the single-electron Green’s functionGmm0 ðvÞ (or spectral functionA ¼ �ImG/p) for a given atomic shell, e.g., the d-shell of an oxide. This observable can berepresented and calculated by considering an atomic shell coupled to the effective medium viaa hybridization function Dmm0 ðvÞ. The latter can be viewed as an energy-dependent (dynamical)
10-12 10-6 10-4 10-2 100 102 10410-10 10-8 10-6 10-4 10-2
10–4
10–5
10–6
10–7
10–8
10–9
10–10
T/D
a bkT
χ/(g
μ b)2
T K/D
0
0.1
0.2
0.3
0.4
0.5
0.6 Two-channel: S = 12* one-channel: S = 1/2J/D = 42, 41, ..., 4–8
JK /D = 0.1
J/D
J
Figure 5
Numerical renormalization group results for the composite-spin KondoHamiltonian (17) withM¼ 2 (data taken fromReference 22). (a)The impurity contribution to the magnetic susceptibility for several values of the Hund’s rule coupling J. The behavior evolves from twoindependent Kondo problems to the S¼ 1Kondo problem as J is increased. Note the increase in x at intermediate values of J (cf. Figure 4).(b) The correspondingKondo temperature as a function of J. At small J an exponential dependenceTK¼TK,1 exp(�J/8.4TK,1) withTK,1¼TK (J ¼ 0) (purple dashed line) and at larger J a power-law TK } 1/J (red dashed line) are found.
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generalization of the Weiss effective-field concept to quantum many-body systems. The key as-sumption is that the self-energy Smm0 ðvÞ of this effective quantum impurity model can be used asa (local) approximation of the full self-energy of the solid. The Dyson equation, projected onto thelocal orbitals xm(r) defining the correlated shell, then yields a self-consistency condition forGmm0 ,which also determines Smm0 and Dmm0 . DMFT has been successfully combined with electronicstructure calculations for real materials, performed in the density-functional theory framework(DFT), using the local-density approximation (LDA þ DMFT approach). In this approach, thereare twokeyobservables: the all-electron charge density r(r) and the correlated orbital localGreen’sfunction Gmm0 ðvÞ (for reviews, see, e.g., References 60–63).
The energy-dependence of the dynamicalmean-fieldDmm0 ðvÞ is of central physical importance.Indeed, in strongly correlated materials, electrons are hesitant entities with a dual character. Athigh energy, they behave as localized. At low energy in metallic compounds they eventually formitinerant quasiparticles, albeit with a strongly suppressed spectral weight. DMFT describes thehigh-energy behavior by taking full account of the multiplet structure of the atomic shell and of itsbroadening by the solid-state environment. The latter is encoded in the high-frequency behavior ofDmm0 ðvÞ, which controls, for example, the widths of Hubbard satellites. At low energy, the keyissue is whether the solid-state environment fully lifts the degeneracy of the ground-statemultiplet.In metallic systems, the effective hybridization ImDmm0 ðvÞ does not vanish at low energy (incontrast to aMott insulator, where it displays a gap). As a result, Kondo screening of the ground-state multiplet can take place. This self-consistent Kondo screening is the local description ofelectron transfer processes that screen out the multiplet structure in the metallic ground state. Forexample, in the simplest context of a single-orbital Hubbardmodel, a twofold degenerate spin-1/2local moment is found in the paramagnetic Mott-insulating phase, whereas it is Kondo screenedinto a singlet in the metallic phase.
In most cases, the low-energy excitations in the metallic phase can thus be described as quasi-particles of a local Fermi liquid. These are characterized by three key quantities: their quasiparticleweight Z, effective mass m� (or renormalized Fermi velocity y�F=y
bandF ), and lifetime Z=Gqp. For
a single-orbital model, the quantities are given by Z�1 ¼ m�=mband ¼ ybandF =y�F ¼ 1� ∂S=∂vjv¼0
and Gqp ¼ ZjImS(v ¼ 0)j, with proper matrix generalization to the multiorbital context. In
Material (crystalline solid)
Atom Effectivemedium
Figure 6
The dynamicalmean-field theory (DMFT) concept.A solid is viewed as an array of atoms exchanging electrons,rather than as a gas of interacting electrons moving in an inhomogeneous potential. DMFT replaces the solidby a single atom exchanging electrons with a self-consistent medium and takes into account many-bodycorrelations on each atomic site.
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a local Fermi liquid with a momentum-independent self-energy, the effective mass enhancementcoincides with Z (in a multiorbital context, however, the renormalization of the Fermi velocitycan depend on the point along the Fermi-surface through the momentum dependence of theorbital character of the band).
Fermi-liquid behavior applies below a scale TFL related to the self-consistent Kondo scale(although significantly smaller quantitatively). According to DMFT, the study of multiorbitalKondo impurity models in Section 4 is thus directly relevant to a full periodic solid. The resultsestablished in Section 4 for the Kondo temperature of an impurity coupled to a structurelessbath cannot be directly applied to DMFT studies of a correlated solid, however. Indeed, theenergy-dependent structure of the self-consistent hybridization must be properly taken intoaccount (in more technical terms, one has to deal with an intermediate-coupling Kondoproblem). Nonetheless, the strong suppression of the Kondo scale by Hund’s coupling impliesthat low values of the Fermi-liquid scale will be observed in the solid-state context, as detailed inSection 6 below. The Landau description of quasiparticles is thus fragile in a strongly correlatedmetal. Quasiparticle excitations may survive in a range of temperatures above TFL, but theirlifetime no longer obeys the T�2 law of Fermi-liquid theory. Because TFL is low, the un-derstanding of the metallic state for T > TFL is often of direct experimental relevance. At veryhigh temperature, the effective DMFT hybridization D(v x 0, T) is small and the physics ofindependent atoms is recovered, whereas at low temperature D(vx 0, T � TFL) saturates andlocal Fermi-liquid coherence is established. In between, quasiparticles become gradually lesscoherent and strong transfers of spectral weight are observed. By bridging the gap betweenisolated atoms and the low-energy coherent regime, DMFT is currently the tool of choice tohandle the entire crossover from a Fermi liquid at low temperature all the way to a bad metalincoherent regime at high temperature. This is, in our view, essential to the physical un-derstanding of many strongly correlated materials.
6. HUND’S CORRELATED MATERIALS AND THE JANUS-FACEDINFLUENCE OF THE HUND’S RULE COUPLING
In this section, we expose the main physical point of this review: Generally, the Hund’s rulecoupling has a conflicting effect on the physics of the solid-state. On the one hand, it increases thecritical U above which a Mott insulator is formed (Section 3); on the other hand, it reduces thetemperature and the energy scale belowwhich a Fermi liquid is formed, leading to a (bad) metallicregime in which quasiparticle coherence is suppressed (Section 4).
This occurs for any occupancy, the two exceptions being a half-filled shell or a shell witha single electron or a single hole. In the former case, Hund’s coupling strongly decreases the Mottcritical coupling and suppresses the coherence scale, which leads to amore correlated behavior. Inthe latter case, Hund’s coupling tends to decrease correlation effects by enhancing Uc withouta strong effect on the coherence scale, because the ground-state degeneracy of the isolated atom isunchanged by J in this case (Table 1). In all other cases, the Hund’s rule coupling has two faces, asdoes the Roman god Janus. This implies that a large class ofmaterials displays hallmarks of strongelectronic correlations while not being in close proximity to a Mott-insulating state.
6.1. Simplest Model: Three Degenerate Orbitals
The simplest model in which the Janus behavior occurs is the Hubbard-Kanamori model of threedegenerate bands described by the Hamiltonian
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H ¼ �Xij,ms
tijd†imsdjms þ
Xi
HKðiÞ, 19:
where HK ¼ ðU � 3JÞbniðbni � 1Þ=2� 2JS!2
i � JL!2
i =2, the rotationally invariant three-orbital in-teraction of Equation 7. This describes, for example, TMOs with cubic symmetry and a partiallyfilled t2g shell well separated from the empty eg shell. Several authors (see, e.g., 18, 20, 22) studiedthis Hamiltonian using DMFT.
In Figure 1, we display as a color map the value of the quasiparticle spectral weight Z asa function of the filling of the shell ðn ¼ ÆbniæÞ andof the strength of the couplingU/D,whereD is thehalf-bandwidth. We used a fixed ratio J/U ¼ 0.15 (cf. the discussion at the end of Section 3) anda semirealistic t2g density of states (Figure 1, inset). The Mott insulators are indicated by thickvertical bars. Long-range ordering was suppressed in these calculations: Figure 1 displaysproperties of the paramagnetic state only and is not a full phase diagram.
Figure 1 reveals the following interesting features. (a) TheMott-insulating state ismost stable athalf-filling n¼ 3, where theMott critical couplingUc is at least twice as small than at other fillinglevels. In contrast, J enhancesUc for the other filling levels, as indicated by the arrows in Figure 1.For vanishing J,Ucwould instead be largest at half-filling (Figure 1, Xmarks). (b) At n¼ 1 and n¼5, correlation effects are weak except in direct proximity to the Mott state, i.e., close to Uc. (c) Incontrast, at the Janus filling levelsn¼2andn¼4, thewhite regionof smallZ extends to quite smallU, as pointed out in Reference 22. Strongly correlated metallic phases are thus found in a widerange of couplings, without direct proximity to the Mott-insulating state. (d) A pronouncedparticle-hole asymmetry is observed, with stronger correlations on the right-hand side of Figure 1(larger n’s). This is due to the higher value of the t2g density of states close to the Fermi level,associated with the van Hove singularity. This implies smaller kinetic energy, and hence slowerquasiparticles that are easier to localize (19).
These features are in very good agreementwith themap of TMOs put forward in the pioneeringworkofFujimori (64), on the basis of experimental and empirical considerations (see alsoReference1). The calculations of Figure 1, which take into account the key physical role of Hund’s coupling,provide strong theoretical support to such classifications of TMOs, as discussed in detail below.
6.2. A Global View on Early 3d and 4d Transition-Metal Oxides
We now turn to real materials and show that the physical effects revealed at the model level aboveallow one to build a global picture of how the strength of electronic correlations evolves in TMOsas one moves along the 3d and 4d series. Figure 1 shows the correlation strength of several earlyTMOs. For most of the metallic compounds, experiments show reliably the specific heat, and itsenhancement over the band-structure (LDA) value g/gLDA is reliably known from experiments.These are positioned in Figure 1 by demanding that the value of Z�1 obtained in the modelcalculation at the DMFT level (wherem�/m ¼ Z�1) coincides with g/gLDA. Materials in the sameseries are positioned with a slight increase of U/D along the series, because the bandwidthdiminishes, and the screened value of U increases slightly as the atomic number and hence nincreases.3 As expected, significantly larger values of interactions pertain to 3d oxides. Apart fromthis, only a moderate variation of U/D values is needed to account for systematics of the early
3Bandwidths for cubic 3d TMOs are 2.6, 2.5, 2.4 eV for SrVO3, SrCrO3, and SrMnO3, respectively. For the 113 4d series (incubic structure), the values are 3.8, 3.7, 3.6 eV for Mo-, Tc-, and Ru-compounds, respectively. For the 214 4d series (intetragonal structure), the xy (xz) bandwidths are 3.8(2.2), 3.6(1.8), 3.4(1.5), 3.1(1.3) eV for Mo-, Tc-, Ru-, and Rh-compounds, respectively.
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TMOs. [These materials were also explicitly simulated within LDAþDMFT. The main experi-mental properties are properly reproducedwith only amild variation of the interaction parameters(19, 22, 65).]
Consider first t2g oxides of the 3d series SrVO3, SrCrO3, and SrMnO3. These three materialsshare a similar typical couplingU/Dx3� 4 (Ux3� 4 eV,Dx1� 1.5 eV in the t2gdescription).Nevertheless, they have very different physical properties. The origin of these differences can befound in the different nominal fillings of the t2g shell, by one, two, and three electrons, respectively.For materials with a half-filled t32g shell, such as SrMnO3 or LaCrO3, the ratio U/D x 4 exceedssubstantially the Mott-insulating critical value (which is strongly reduced by the effect of J). Thisexplains why no metallic 3t32g oxides are known (1, 66). In contrast, the 3t12g cubic SrVO3 isa moderately correlated metal with g/gLDA ∼m�/mx 2 (1). In this case, Uc is increased by J andindeedLDAþDMFTcalculations explicitly demonstrate (22; see also Section 7) that SrVO3wouldbe significantly more correlated (20) if Jwas 0. For 3t22g materials with comparable values ofU/D,strongly correlated behavior caused by the Janus-faced action of J is expected. Cr-perovskites aresituated there, but unfortunately their synthesis necessitates high pressures, which limits the purityof the samples. The experimental data so far are controversial: Whereas initially SrCrO3 wasreported to be a paramagnetic metal (67), a more recent study (68) finds a semiconducting re-sistivity and strong dependence of magnetic susceptibility on temperature. Overall, the seriesSrVO3-SrCrO3-SrMnO3 beautifully illustrates the importance of the Hund’s coupling and of theband filling as a key control parameter.
Oxides of 4d transition metals are characterized by smaller values ofU/D x 1 – 2, due to thelarger bandwidths and smaller screened interaction associatedwith themore extended 4d orbitals.We consider the series SrMO3andSr2MO4,whereM¼Mo,Tc,Ru,Rh (Figure 1). The technetiumcompounds are special among this series. Because they have a half-filled t2g shell and given therelevant value of U/D, these materials are located very close to the insulator-to-metal transition.We are not aware of transport measurements on these compounds, but a recent study (69) reportsantiferromagnetism with a very large Néel temperature TNx 1,000 K for SrTcO3. Indeed, modelconsiderations suggest that the proximity to theMott critical coupling leads to largest values of theNéel temperature. This qualitative observation, together with quantitative LDAþDMFT calcu-lations supporting it,was recently used to explain theobservedmagnetic properties of SrTcO3 (65).
The Mo-, Ru-, and Rh-based compounds are metallic. Indeed, given the reduced U/D, it isexpected and observed in practice that oxides of the 4d series with a non-half-filled t2g shell aremetallic, as long as the orbital degeneracy is not too strongly lifted. Ca2RuO4, a rare example ofa 4d t42g insulator, has indeed strong structural distortions leading to a complete orbital polari-zation (70, 71). Sr2MoO4 and Sr2RuO4 are symmetrically placed with respect to a half-filled t2gshell, with one less and one more electron, respectively, but their properties differ. Sr2RuO4 isconsiderably more correlated. An orbital average of the measured effective mass enhancementsyieldsm�/m∼ 2 for Sr2MoO4 ð4t22gÞ (72) andm�/m∼4 for Sr2RuO4 ð4t42gÞ (1, 73). This distinctionoccurs because the t2gdensity of states is not particle-hole symmetric: The Fermi level of Sr2RuO4 isclose to a vanHove singularity, and therefore this material has a smaller effective bandwidth (19).The model calculations of Figure 1 show this clearly.
In Sr2RhO4 ð4t52gÞ, the mass enhancement is close to 2 (74, 75). Although this can be accountedfor within the simple model description of Figure 1, recent work (76) suggests that the screenedinteraction in this compound is smaller than in the other 4d oxides but that the substantialrenormalization comes from lifting of the degeneracy as a combined result of distortions and spin-orbit coupling.
Obviously, the simple classification displayed in Figure 1 applies to materials in which the t2gstates are degenerate. In general, it should be complemented with a third axis, indicating the
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strength of crystal fields and other terms that lift the t2g orbital degeneracy. These terms, whichappear due to the rotations of the octahedra (e.g., of theGdFeO3 type) and Jahn-Teller distortions,are not negligible for all the materials considered in Figure 1, but the success of the classificationsuggests their effects are small. Inmany other cases, the lifting of degeneracy is crucial, as discussedin Section6.5. Spin-orbit coupling in4doxides reaches 0.2 eV, and its effects on the correlations formost of them remain to be investigated in detail.
Putting all such refinements aside, the big picture is that the materials at Janus-filling candisplayHund’s coupling–induced correlationswhile not being close to aMott-insulating state, andthat a rich diversity of behavior is observed depending on key control parameters such as filling,coupling strength, crystal field, location of vanHove singularity, etc. In this respect, it is interestingto contrast the properties of ruthenates (discussed in Section 7) to their rhodate equivalents, whichare structurally and chemically close but have a single hole in the 4d shell. Unlike their Ru relatives,Rh-compounds are paramagnets: Sr2RhO4 (74) is not an unconventional superconductor,SrRhO3 (77) is not a ferromagnet, and Sr3Rh2O7 (78) is not ametamagnet with nematic behavior.
6.3. The Non-Fermi-Liquid Spin-Freezing Regime
Here, we discuss the physics of the strongly correlated metallic phase induced by the Hund’s rulecoupling, corresponding to the pale-colored region of Figure 1.
Key features of this phase were pointed out in the pioneering work of References 17 and 18.Deep within this phase, the local moments freeze (hence the name “spin-freezing” coined inReference 18): The local spin susceptibility at low temperatures increases strongly (17) and thelocal spin-spin correlation function ÆSzi ð0ÞSzi ðtÞæ does not decay at long times (18). Furthermore,the authors of Reference 18 discovered that the electronic self-energy at low frequency obtainedfrom DMFT calculations is in strong contrast to that of a Fermi liquid and obeys a power-lawbehavior S00ðvÞ∼Gþ ðv=DÞa þ . . . . Near the boundary of the spin-freezing regime, G is small atlow-T and a x 1/2. This is illustrated in Figure 7, where we display the results of DMFT cal-culations on the boundary of the spin-freezing regime at a Janus filling factor n ¼ 2 and down tovery low temperatures T/D ¼ 1/100, . . ., 1/800. These data also reveal (inset) that the power-lawbehavior actually does not persist down to T¼ 0 and that a crossover to Fermi-liquid behavior isfound for T < TFL. The Fermi-liquid scale TFL is extremely low, however, which is anotherdistinctive feature of this regime (17, 19), and corresponds to the strong suppression of the Kondoscreening scale byHund’s coupling discussed in Section 4. Spin-flip terms are essential in restoringFermi-liquid behavior at low temperature (79, 80).
Besides frozen local moments, the regime T > TFL has anomalous transport and opticalproperties that differ from that of a Fermi liquid. Reference 17 reports a large resistivity exceedingthe Mott-Ioffe-Regel criterion with weak temperature dependence for temperatures much largerthan TFL and a sharp drop upon entering the coherent regime. In the low-T Fermi liquid, a smallvalue of the quasiparticle weight and a large effective mass (Figure 1) are found. Reference 18emphasized the non-Drude low-frequency optical responses(v)∼v�0.5.Manyother properties ofthe spin-frozen regime remain to be worked out in detail, such as a possible enhancement of thethermoelectric power. In Sections 7 and 8, we review the implications of the unconventionalproperties of the bad-metal, spin-frozen phase for the physics of ruthenates (18) and iron-basedsuperconductors (17), in connection with experimental observations.
Finally, let us emphasize that a precise theoretical understanding of the non-Fermi-liquid be-havior S00ðvÞ∼Gþ ðv=DÞa þ . . . and of the other unconventional properties of the spin-freezingmetallic regime is to a large extent an open and fascinating problem. Here, we suggest some hints,which may prove useful for future work. The theoretical study (81) of the relevant three-orbital
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impurity problem with a local atomic Hamiltonian (Equation 7) has established that the low-energy T ¼ 0 fixed point is a Fermi liquid. This is clear at half-filling nd ¼ 3, where the orbitalfluctuations are quenched (L ¼ 0) and the low-energy effective Hamiltonian is an S ¼ 3/2, K ¼3-channel Kondomodel, which is a Fermi liquid given thatK¼ 2S. In contrast, for nd¼ 2 (or nd¼4), the angularmomentum is not completely quenched in the S¼L¼ 1ninefold degenerate groundstate (Table 1). The low-energy fixed point does remain a Fermi liquid, however, due to theappearance of a potential scattering term in the effective Hamiltonian obtained after eliminatinghigh-energy states with a different valence. This is consistent with the observation of Fermi-liquidbehavior at a very low temperature scale (above; see also Figure 7). It is tempting to speculate thatthe anomalous power-law behavior at intermediate temperature is associated with a crossovercontrolled by anon-Fermi-liquid fixed point obtainedwhen this potential scattering term is absent.A strong-coupling analysis of the effective Hamiltonian (à la Nozières and Blandin) indeed reveals(81) a residual pseudospin-1/2 degree of freedom, suggesting that this fixed point could be relatedto an overscreened three-channel, spin-1/2 Kondo problem. Another possibility is the role playedby the continuous line of non-Fermi-liquid fixed points (80) separating the behavior of this model(for 2 � nd � 3) between ferromagnetic and inverted antiferromagnetic Hund’s coupling. Veryrecently, the potential role of a ferromagnetic Kondo coupling emerging at low energy has alsobeen emphasized (82). (For another recent illustration of the potential relevance of non-Fermi-liquid impurity fixed points to the solid state, in the context of iron pnictides, see Reference 83.)
6.4. Spin-Freezing and Magnetic Ordering
Another important issue is the possible development of intersite spin correlations and magneticlong-range order in the spin-freezing regime. With such a low coherence scale for quasiparticleformation, a Doniach-type criterionwould indeed suggest that this phase is prone to various kinds
ImΣ/
D
ImΣ/
D
(ωn/D)0.5
ωn/D
–0.3
0 0.01 0.02 0.03 0.04 0.05
–0.2
–0.1
0
1.00 0.5–1.5
–1.0
–0.5
0
100200400800
βD
2 t2g electrons, semicircular, U/D = 4, J/U = 1/6
–1.48 (ωn/D)0.5
–0.01 – 10 ωn /D
Figure 7
Self-energy in the spin-freezing regime of the three-orbital Hubbard-Kanamori model for two electrons in theband, as calculated by dynamical mean-field theory for U/D ¼ 4, J/U ¼ 1/6. bD [ D/kT is the inversetemperature normalized to the half-bandwidth. The plot displays ImS(ivn) on the Matsubara frequency axisand emphasizes the (non-Fermi-liquid) power-law behavior ∼(v/D)1/2 (18) as well as the very low-energycrossover into a Fermi liquid (inset).
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of ordering. A very small amount of disorder could, for example, freeze the local moments intoaphasewith spin-glass order.Recent observations (84) onCa-substituted Sr2RuO4 indeed provideexperimental support for this possibility.
Another obvious possibility for local moments in the presence of a strong Hund’s coupling isferromagnetic ordering, as observed, e.g., in SrRuO3. Ordering at a critical temperature higherthan the low coherence scale of the paramagnetic state is an efficient way to restore good metallictransport. The direct transition from an incoherent bad metal into an ordered phase is a hallmarkof strong correlations. It is also a major challenge to theory, given that in those circumstances,ordering cannot be described as an instability of interacting Landau quasiparticles. The authors ofReference 85 studied the magnetic and orbitally ordered phases of the three-orbital modelconsidered in this section. Although ferromagnetism is found for large enough U, an extendedparamagnetic spin-freezing region is nonetheless preservedat intermediateU and filling 2(n(4.These issues deserve further study, e.g., in the framework of cluster extensions of DMFT.
6.5. Competition Between Hund’s Coupling and Crystal-Field Splitting
Up to now, we have considered situations with perfect orbital degeneracy. For this reason, thegeneral perspective on early TMOs provided in Section 6.2 applies mostly to materials with onlysmall deviations from perfect cubic symmetry and t2g orbital degeneracy. For many materials,however, it is crucial to take into account the lifting of orbital degeneracy induced by structuraldistortions. The interplay between crystal-field effects and interactions leads to a rich diversity ofpossible behaviors (2). Here, we focus on the interplay with Hund’s coupling.
The key point is that the Hund’s rule coupling J and the crystal-field energy scale D competewith each other (see, e.g., References 86 and 87). The former favors orbital compensation,i.e., tends to equalize the different orbital populations so that the electrons distributed in allavailable orbitals can take full advantage of the reduction of the Coulomb repulsion by the intra-atomic exchange. The latter, in contrast, tends to populate more the lowest-lying orbitals, henceleading to orbital polarization.
At a qualitative or model level, this competition can be discussed in general terms, whether thecrystal-field splitting refers to the splitting (10Dq) between t2g and eg states or to the splittingbetween states within the t2g (or eg) manifold itself, due to, e.g., a rotation of the oxygen octahedraor to a Jahn-Teller distortion of these octahedra. In practice, one should keep inmind that the orderofmagnitude of these two types of crystal-field splitting is quite different inTMOs: 1–2 eV’s for thet2g-eg splitting and (300 meV for the intra-t2g splitting.
The lifting of degeneracy due to crystal-field splitting directly affects theMott critical couplingand hence has important consequences for deciding whether a material is insulating or metallic.The importance of this effect is best illustrated (88, 89) by the series of materials SrVO3, CaVO3,LaTiO3,YTiO3,which all have a nominald1 occupancy of the t2g shell and comparable values ofUand J. Nevertheless, SrVO3 and CaVO3 are metals (the latter more correlated than the former),whereas LaTiO3 and YTiO3 are Mott insulators [the latter with a larger gap (∼1 eV) than theformer (∼0.2 eV)]. This is because the orthorhombic distortion, and the ensuing splitting of t2gstates, increases as one moves along the series (starting with cubic SrVO3) due to the rotations ofthe oxygen octahedra.
A first effect of the crystal-field is to counteract the effect of J on theMott gap (21). In the d1 case,Hund’s coupling enhancesUc, as detailed in Section 3, causing cubic d1 oxides such as SrVO3 andSrNbO3 to be metallic with moderate correlations. The crystal field compensates this effect, thusenlarging again the Mott gap and contributing to the stronger correlations found in CaVO3.
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The distortion has then two further effects: It reduces the t2g bandwidth and also lifts the t2gdegeneracy (by as much as ∼300 meV in YTiO3). The latter effect reduces Uc (Section 3). Botheffects increase correlations and are responsible (88) for LaTiO3 and YTiO3 being insulators.These insulators have a substantial degree of orbital polarization: For thosematerials, the intra-t2gsplittingwins overHund’s coupling. (For adiscussionof these effects in themodel context, see, e.g.,References 90 and 91.)
In contrast, in BaVS3, a d1 material that is metallic at high temperature, the Hund’s rulecoupling wins over the small (x0.1 eV) intra-t2g splitting, leading to a compensation of orbitalpopulations. This has been proposed (92) to play a major role in explaining the development ofa charge-density wave-insulating state at low temperature in this material.
The competition between Hund’s coupling and the crystal field is particularly dramatic in thestrong-coupling large-U regime, where it can induce a transition between two different insulatingground states, fromhigh-spin (HS)whenHund’s rule dominates to low spin (LS)when crystal fielddominates (93). This has been the subject of several recent studies (37, 94–97). It can be simplyillustrated by considering aHubbard-Kanamorimodel of twobands (bandwidth 2D) separated byan on-site crystal-field energy 2D (37, 96, 97). (For model studies of the crystal field versus Hundcompetition involving three orbitals, see, e.g., References 20, 21, and 98.) The generic phasediagram of this model in the half-filled case (two electrons per site) is depicted in Figure 8 asa function of D/D and U/D for a fixed ratio J/U.
The transition from an HS insulator to an LS insulator can be understood from simple ener-getics in the limit of isolated atoms (96, 97). We neglect first, for simplicity, the spin-flip and pair-hopping terms in the Kanamori Hamiltonian. At small enough crystal field, the ground state has
Δ/D
U/D
Metal
J/U = 0 J/U = 0.25
HS insulator
LS insulator
5
4
3
2
1
00 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Figure 8
CompetitionbetweenHund’s coupling and crystal-field splitting: phase diagram (paramagnetic phases only) ofthe two-orbital Hubbard-Kanamori model at half-filling, as a function of crystal-field D/D and interactionstrengthU/D, for a fixed value of J/U¼ 0.25. Two insulating phases are found, one with high-spin (HS; S¼ 1)and onewith low-spin (LS; S¼ 0), togetherwith ametallic phase (blue). The three continuous gray lines denotesimple estimates (see References 96 and 97) of the transitions between these three states, based on the atomiclimit. Also depicted (red dashed line) is the phase boundary separating the metallic phase (left) from theinsulating phase (right) for J ¼ 0 (in this case, an LS insulator is always found except at D ¼ 0). The arrowsindicate how the phase boundaries move as J is increased. Figure adapted from Reference 37.
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S ¼ 1 and orbital isospin T z ¼ 0, corresponding to each orbital occupied by one electron. Theenergy of this state is EHS¼U� 3J� Dþ D¼U� 3J. At higher crystal field, the LS ground statewith two electrons in the lowest orbital (Sz¼ 0,Tz¼�1) hasELS¼U� 2D. Hence, for 2D< 3J, theHS ground state with compensated orbital populations is favored, while the LS orbitally polarizedground state takes over for 2D > 3J. The energies of the lowest excited states with one and threeelectrons, respectively, read E1¼�D andE3¼Uþ (U� 2J)þ (U� 3J)� 2Dþ D¼ 3U�5J� D.Hence, the Mott gap in the zero-bandwidth limit Dat ¼ E3 þ E1 � 2Egs reads, for density-densityinteractions,
Dat ¼ U þ J � 2D ðHS, 2D < 3JÞ, Dat ¼ U � 5J þ 2D ðLS, 2D > 3JÞ. 20:
In the presence of spin-flip and pair-hopping terms, only the expression of the LS energy is
modified:ELS ¼ U �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2DÞ2 þ J2
q. The HS/LS transition occurs at D >
ffiffiffi2
pJ (37), and the atomic
gap in the LS case becomes DLSat ¼ U � 5J � 2Dþ 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2DÞ2 þ J2
q.
Hence, for a half-filled shell, when Hund’s rule dominates (HS regime) the effective U is in-creased (critical Uc decreased) by J (and decreased by D), as explained in Section 3, whereas theopposite applies in the LS regime. By continuitywith themetal-to-band-insulator transition atU¼0, which occurs at 2D ¼ 2D, the phase boundary between the metal and the LS insulator can beapproximately located by DLS
at ¼ 2D. The metal/HS insulator boundary can be approximated byDHSat x2D. On the right-hand side of this expression, we use the bandwidth as a measure of kinetic
energy, as appropriate for J � 0, because of the quenching of orbital fluctuations (Section 3),whereas a larger valuewouldbe appropriate for J¼0.With these choices, the three lines separatingthe HS (LS) insulator and the metallic phase cross at a single point and yield a reasonable ap-proximation to the phase boundaries calculated with DMFT (Figure 8) (37, 96, 97).
In the weak-coupling small-U regime, a calculation in the Hartree approximation (86, 87)yields an effective crystal field Deff ¼ D þ (U � 5J)dn, where dn is the orbital polarization.Correspondingly, the orbital polarizability reads xO ¼ x0
O=½1� cðU � 5JÞx0O�. Hence, the orbital
polarizability is enhanced by interactions if J < U/5 and suppressed if J > U/5 (86, 87). Theseconsiderations explain the slope of the phase boundary in Figure 8 near the metal-to-LS-insulatortransition at small U.
Recently, Kune�s and coworkers have studied in detail HS/LS transitions for three materials,MnO (94), a-Fe2O3 (formal valence d5) (95), and LaCoO3 (formal valence d6) (97), andperformed corresponding LDAþDMFT calculations (see Reference 96 for a review). In botha-Fe2O3 andMnO, a transition is observed between a low-pressure HS-insulating phase ðt32ge2gÞand a high-pressure LS metallic phase (at ∼50 GPa for a-Fe2O3 and ∼100 GPa for MnO). Theauthors of References 94–96 suggest that the transition in a-Fe2O3 is analogous to the HS/metaltransition in Figure 8, whereas that in MnO is more in the ionic limit, analogous to the crossingbetween the HS and LS atomic ground states in Figure 8 (for a d5 shell, the LS ground state is notnecessarily an insulator given that it has one hole in the t2g shell). These authors also suggest (97)that LaCoO3 is an example of a material poised very close to the triple point where phaseboundaries meet (Figure 8). In this circumstance, raising the temperature can lead to an entropy-driven spontaneous disproportionation with translational symmetry breaking, where the HSstates occupy dominantly one sublattice and the LS state the other. An effective Blume-Emery-Griffiths model (retaining only the HS↑, HS↓, and LS states) was introduced to describe thisphysics. This electronic mechanism for disproportionation should be contrasted to the elastic(lattice) mechanism proposed in the early work of Bari & Sivardière (93). Both effects are likelyto conspire in the actual materials.
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In the discussion above, we have limited ourselves to the paramagnetic phases and have notconsidered phases with long-range magnetic or orbital ordering. This is a vast field beyond thelimited goal of this review,with a rich interplaybetween theHund’s rule coupling, crystal-field andstructural effects, and superexchange and double-exchangemagnetic interactions (2). (For studiesof these issues at the level of the two- and three-orbital Kanamori-Hubbard models, see, e.g.,References 85, 99–103.)
6.6. Hund’s Coupling as a Band Decoupler and Orbital-Selective Physics
When orbital degeneracy is lifted by effects such as crystal fields or the bands’ different electronicstructures (e.g., different bandwidths), correlations can affect each band differently. Here, weemphasize that the Hund’s rule coupling enhances such an orbital differentiation and acts es-sentially, in some aspects, as a band decoupler.
Anearlyworkstressingthat J induces orbital differentiation is theNRGstudyof the two-impurityCSKmodel (Equation17)with unequal coupling strengths (11). For thismodel, theKondo screeningproceeds in two stages (8, 9). As temperature is lowered, the system first approaches the unstableunderscreened fixed point, at which only half of the total spin is screened and eventually reaches thefully screened Fermi-liquid stable fixed point. J can greatly reduce the temperature belowwhich fullscreening applies (8, 11). The Hund’s rule coupling not only suppresses both respective Kondotemperatures, but also enhances their ratio (which can be seen also bygeneralizingEquation 18 to anorbitally dependent case) and thereby the tendency toward orbital differentiation.
The extreme form of orbital differentiation is when the carriers on a subset of orbitals get lo-calized, while others remain itinerant, a concept dubbed orbital-selective Mott phase (OSMP)(104). In its simplest, almost trivial, form, one can say that anOSMP is realized indouble-exchangesystems such as the manganites La1�xSrxMnO3, where the t2g electrons form a localized core spin,while the eg electrons are itinerant.
Many model studies have documented the occurrence of an OSMP and an associated orbital-selective Mott transition (OSMT), and that the Hund’s rule coupling promotes these effects. Thesimplest model is the two-band Hubbard-Kanamori model with unequal bandwidthsD1 andD2,which has been thoroughly investigated (see, e.g., Reference 105 for an extensive list of references).As the correlation strength is increased (Figure 9c from top to bottom), the narrower band localizesbefore the wider one, if the bandwidth ratioD1/D2 is larger than a critical value. This critical ratiois quite large (∼5 for J¼ 0), but small values of J/U are needed for OSMT to be possible whenD1
and D2 are of similar magnitude (106, 107).An OSMT can also happen in a system of bands of the same width in which the degeneracy is
broken by the crystal field. Following Reference 108, consider a model of three bands of the samewidth filled with four electrons, with a crystal field tuned such that there are three electrons in thelower two degenerate bands and the higher band is half-filled. The half-filled band gets localizedfirst and as shown in Figure 9a, a robust OSMP is found for J > Jc, whose extent furthermorewidens as J is increased (21, 108). This presumably happens because J diminishes theMott gap ofthe lower twobands (Section 3) occupied by a single hole.Whereas at a small J the increase ofUcbyorbital degeneracy plays a role (108), the main rationale for this robust OSMP is the differentindividual band fillings (21).
The relevance of individual band-filling and the importance of J in promoting orbital-selectivephysics can be understood by recognizing that J blocks orbital fluctuations (108–110). Koga et al.(109, 110) first noted that, for J > 0, an electron added in a specific orbital cannot gain de-localization energy from hopping processes involving an electron in another orbital (Figure 9b).This keeps the respective Hubbard bands and thus the Mott gaps independent. The OSMT then
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follows simply from theMott transitions in each individual band,which happen for distinct valuesofU (see Figure 9c). The band decoupling accounts also for the behavior under doping: TheOSMPis stable (37, 108), until the chemical potential exits from the widest gap (see also References 105,109, and 111).
The spin degrees of freedom become strongly interdependent when approaching the orbital-selective phase (112), however. Indeed, in the OSMP, the system is appropriately described bya double-exchange model and behaves as a non-Fermi liquid, due to the scattering of the itinerantelectrons on the localized ones (80, 113).
The model studies mentioned above aimed at unveiling the basic mechanism of OSMT anddisregarded the possibility of long-range ordering. However, at low temperature, the localmoments present in the OSMP carry extensive entropy and will tend to order (100). Interbandhybridization, which can favor a singlet ground state and replace an OSMP with a heavy Fermiliquid at low temperature (106, 110, 114), offers another possibility for the system to reduce theentropy. However, the coherence temperature of this metallic phase will be very low if hybrid-ization is small, and a selectively localized phase will be restored at finite temperature. Likewise,even on the Fermi-liquid side of the OSMT, the state at finite temperature might resemblethe OSMP. Hence, although the occurrence of an OSMP as a stable zero-temperature phase isquestionable, the general concept has relevance to situations in which an extended finite-temperature regime with strong orbital differentiation is observed.
To conclude our brief survey of OSMT, we turn to materials in which orbital-selective physicsmay be relevant. The concept of OSMT was initially proposed (104) to explain the properties of
c
baJ/
U
U/D
0.3
0.2
0.1
00 2 4 6 8 10 12 14 16 18 20
Metal
Orbital-selectiveMott phase
Mottinsulator
×3 ×2
×1
×2
×1Δ
~UU
Figure 9
Orbital-selective Mott physics promoted by Hund’s coupling. (a) Phase diagram of a three-band Hubbard model populated by fourelectrons, as a functionof interaction strengthU/D andHund’s coupling strength J/U. The crystal field lifts the threefold degeneracy so thatthe upper band is half-filled and the lower two bands that remain degenerate contain three electrons. An orbital-selective Mott phase, inwhich the half-filled band has a gap, is stabilized by J. Panel a is reproduced fromReference 108. (b) Propagation of a charge excitation intwo half-filled bands. The lower process leads to a state with energy larger by 2J and is therefore suppressed (109, 110). (c) As theinteraction strength in the two-bandmodel with unequal bandwidths is increased (top), the narrower band localizes first, and the orbital-selective Mott phase results (middle). A Mott insulator (bottom) is found at a still larger interaction strength only.
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Ca2�xSrxRuO4, in which spin-1/2 local moments coexist withmetallic transport for x< 0.5 (115–117). This is discussed inmore detail in Section 7.3.3.We also consider in Section 8 the relevance oforbital-selective physics to iron-based superconductors.
Other materials for which orbital-selective physics has been discussed are LiV2O4 (118, 119),BaVS3 (92), V2O3 (120, 121), Hg2Ru2O7 (122), and CoO under pressure (123). Following anearly suggestion of Goodenough (124), recent LDAþDMFT studies of elemental metallic a-Fe(125) suggest that the d-electrons in t2g bands are itinerant, in contrast to the ones in eg bands,which form local moments due to Hund’s exchange. A similar situation was proposed for FeOunder pressure (126), but a different result (a high-to-low-spin crossover) is reported from fullycharge self-consistent LDAþDMFT calculations (127). Finally, M. Vojta (128) recently discussedthe relevance of the OSMT concept to heavy-fermion physics.
7. RUTHENATES
In this section, we give a brief overview on perovskite ruthenates Anþ1RunO3nþ1, where A is Ca orSr. In these materials, four electrons occupy the three t2g orbitals. Compared to the 3d TMOs, theextended nature of 4d orbitals gives rise to moderate values of the screened interaction U ∼ 2 eVand broad bandsW¼ 2D∼ 3 eV. The larger overlap of 4d orbitals with oxygen enhances the t2g-egcrystal field splitting. Hence, an HS state is not realized here in contrast to the isoelectronic 3dLaMnO3. In spite of U ( W and threefold t2g orbital degeneracy, these materials are quitecorrelated with specific heat enhancements g/g LDA > 4.
7.1. Ruthenates in a Nutshell
The properties of a few widely investigated ruthenates are listed in Table 2.We start by the single-layern¼ 1 compound. Sr2RuO4 has a body-centered tetragonal unit cell.
Below Tc ¼ 1.5 K, it becomes superconducting. The unconventional superconductivity ina material isostructural with La2�xSrxCuO4 cuprates generated wide interest. The supercon-ductivity andnormal state properties are reviewed inReferences 73 and129.AboveTc, Sr2RuO4 isa paramagnetic metal with Fermi-liquid behavior at low temperatures. The carrier masses areenhanced with g/gLDA 4. Despite the (small) tetragonal splitting, 4/3 of an electron is found ineach of the orbitals.4
The three-dimensional material SrRuO3 is an itinerant ferromagnet with Curie temperatureTc ¼ 160 K (see Reference 131 for a recent review). It crystallizes in a rhombohedral GdFeO3
structure, in which the octahedra are tilted by ∼10 degrees from an ideal cubic structure (see, e.g.,Reference 132). Optical spectroscopy revealedRes(v)}v�0.5 (133, 134). Despite this anomalousdependence, at low temperatures, quantum oscillations (130) and strict T2 resistivity below 15 Khave been found (135). Specific heat enhancements g/gLDA ¼ 3.7 (136) and 4.4 (137) have beenreported.
The bilayer compound Sr3Ru2O7 is a paramagnetic metal. It is situated very close to themetamagnetic quantum critical point, which is reached upon applying amagnetic field of 7.9 Teslaalong the c-axis (138). At very low temperatures, an electronic nematic state forms (see Reference139 for a review). The carrier masses are strongly enhanced, with g/gLDA ∼ 10 at zero-field. A T2
resistivity is observed below 7 K (135).
4In LDA a slight polarization in favor of xz and yz orbitals is found. The discrepancy between theory and quantum-oscillationexperiments (130) is diminished if the atomic physics (Hund’s coupling) is treated appropriately, such as in LDAþDMFT (19).
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Table 2 contains also two Ca-substituted ruthenates. The smaller Ca ion causes a strongerdistortion of the lattice. The infinite-layer compound CaRuO3 has a stronger rhombohedraldistortion than SrRuO3 with octahedra tilted by 17 degrees (132). It is paramagnetic and hasa large g[ ¼ 74 mJ/molK2 (140)], corresponding to an enhancement of ∼7 over LDA value.Compared to SrRuO3, the mass enhancement is larger most likely because CaRuO3 is moredistorted and not ferromagnetic. Similar anomalous dependence of optical conductivity as thatin the Sr-compound is found (141, 142), and the resistivity behaves as r } T1.5 down to a fewKelvin (135).
Ca2RuO4 is the only insulating ruthenate. Following a structural distortion, it becomes in-sulating below 365 K (143) and orders antiferromagnetically below 110 K (144). The insulatingstate has been explained (70, 71) in terms of the complete filling of the xy orbital, which occurs dueto the compression of oxygen octahedra along the c-axis in the low-temperature S-Pbca structure,followed by a transition to a Mott insulator, which occurs in the narrower bands spanned by thexz, yz orbitals, for which W < U. The phase-boundary can be shifted by application of pressure(145). Interestingly, upon substituting a few percent of Ru for Cr, a negative thermal expansion isfound (146).
7.2. Origin of Correlations
Overall, the ruthenates exhibit several remarkable properties signaling a correlated metallic state,with the carrier masses significantly enhanced over the LDA predictions. Where do the strongcorrelations come from?
In several 3d oxides, the strong correlations appear due to the proximity to a Mott-insulatingstate, as revealed, e.g., by the pronounced Hubbard bands observed in photoemission spectros-copy. In Figure 10, we plot the LDAþDMFT t2g density of states for the 3d oxide SrVO3 andcompare it to the data from (inverse-) photoemission spectroscopy. The data show the quasi-particle band (visible to a lesser extent in the inverse photoemission and low-energy photo-emission) and also the signatures of the Hubbard bands. Whereas the upper Hubbard band alsooverlaps with the eg states and can thus not be identified unambiguously, the oxygen contributionto the spectra is easily identifiable (large peak below 3 eV) and has been subtracted out from thedata in Reference 148.
InFigure 10b, the data are plotted for Sr2RuO4. Encouraging agreementwith experiment is alsofound there. Comparing the two materials, one sees that the Hubbard bands have a largerseparation in the case of SrVO3, corresponding to the larger value of the interaction for thiscompound. The peak-to-peak distance between the Hubbard bands in the two compounds differsby an amount corresponding to the respective Ueff ¼ U � 3J values.
Table 2 Ruthenates in a nutshella
Compound Magnetic order g/gLDA r ~ T2 Remarks
Sr2RuO4 PM 4 <25 K Unconventional SC < 1.5 K
SrRuO3 FM < 160 K 4 <15 K s } v�0.5
Sr3Ru2O7 PM 10 <10 K Metamagnetic quantum-critical point and nematicity
CaRuO3 PM 7 T1.5 > 2 K s } v�0.5, g ¼ gFL þ log(T)
Ca2RuO4 AF < 110 K ✗ ✗ Insulator < 310 K
aAbbreviations: AF, antiferromagnet; FM, ferromagnet; PM, paramagnet; SC, superconductor
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Figure 10c displays the imaginary part of the LDAþDMFT self-energies on theMatsubara axisvn¼ (2nþ1)pkT. For SrVO3, the largerU/W induces large values of ImS(ivn) at large frequencies.At smaller frequencies, well-defined quasiparticles are rapidly recovered: The data points arelinearly aligned and intercept the y-axiswith a small slope (corresponding toZ∼0.5) andat a smallvalue corresponding to a scattering rateG¼ ImS(i0þ)� kT. In contrast, Sr2RuO4 displaysweakercorrelations (smaller jImSj) at high frequency, but those correlations turn stronger at low fre-quency, giving rise to a large slope corresponding to Z ¼ 0.2 for the xy and Z ¼ 0.3 for the xzorbital. The correlations are weaker for the xz, yz orbitals in spite of their smaller bandwidth(which is therefore not a crucial physical ingredient here). Indeed, quantumoscillation experiments
J
0.6
0.4
Sr2RuO4
0.2
SrVO3PES 60eVPES 900eVBIS
Sr2RuO4
PES: 30eVPES: 700eVXAS
a
0–4 –3 –2 –1 0 1 2 3 4
0.0, 0.1
0.2
0.3
0.4
>1,000
300
100
60
1.7
2.3
3.2
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1.7
2.0
2.4
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J (eV) m*/m (xy) m*/m (xz/yz) T* (K)
t 2g D
OS
ω (eV)
b
0
0.2
0.4
0.6
0.8
t 2g D
OS
–4 –3 –2 –1 0 1 2 3 4ω (eV)
SrVO3
SrVO3
Sr2RuO4
Sr2RuO4 xySr2RuO4 xz
432100
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0.2
0.3
0.4
0.5
0.6
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0.9
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–lm
Σ(iω)
ωn
0.6
0.40.4 J (eV) (eV)0.4
0
0
J (eV)
J (eV)
c
Figure 10
Strong correlations fromHund’s coupling in ruthenates. (a) The SrVO3 t2gLDAþDMFTdensity of states (DOS) compared to the results ofthe X-ray photoemission (PES) (147, 148) and inverse-photoemission (BIS) (147). High-energy PES (148) is more sensitive to the d-statesand resolves better the quasiparticle DOS. (b) The Sr2RuO4 LDAþDMFT DOS compared to the valence-band PES from Reference 149,high-energy PES (150), and X-ray absorption spectroscopy (XAS) (151). (c) Imaginary part of the Matsubara self-energies. Resultsobtainedwith physical values of the interactionsU¼2.3 eV, J¼0.4 eV for Sr2RuO4 andU¼4.5 eV, J¼ 0.6 eV for SrVO3 are compared tothe results with the same U, but J ¼ 0. (d) Table (from Reference 19) displaying the mass enhancements m�=mLDA ¼ Z�1 ¼ 1�∂SðzÞ=∂zjz ¼ 0þ for each orbital. The coherence temperatureT� is defined as the highest temperature whereZImS(0þ)� kT holds for bothorbitals.
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reveal that the largest mass renormalization (∼5) corresponds, surprisingly (152), to the widestxy band.
The persistence of correlations to low energies in the Ru- but not in the V-compound is sug-gestive of the Hund’s rule coupling. This binds a pair of holes on a Ru-ion into a high-spin state(Table 1) but does not affect the single-electron ground-state multiplet of the n ¼ 1 SrVO3
compound. InFigure 10c, we also show theLDAþDMFT results for J¼0. For SrVO3, suppressingJ increases correlations at all frequencies and brings thematerial in proximity to aMott-insulatingstate. Indeed, at Ueff ¼ 5 eV, a Mott insulator is found within a t2g description. In contrast, fora Sr2RuO4 setting, J¼ 0 does not influence much the correlations at higher energies in spite of theincreasedUeff. However, the low-frequency correlations disappear. Such behavior is found also inLDAþDMFT calculations for other ruthenates, thus indicating that the strong correlations inthese compounds are due to the Hund’s rule coupling.
7.3. Physical Consequences of Correlations Induced by the Hund’s Rule Coupling
The previous section emphasizes the importance of the Hund’s rule coupling for ruthenates. Wenow review some important consequences for physical properties and experimental observables.
7.3.1. Coherence-incoherence crossover in Sr2RuO4. Together with the large mass enhance-ments, the scale below which Fermi-liquid behavior applies is found to be quite low in Sr2RuO4.The crossover out of the Fermi liquid is seen by several experimental probes. Despite a largeanisotropy (with rc/rab > 1,000 at low T), the in-plane rab and out-of-plane rc resistivity bothinitially increase as T2 up to TFL ¼ 25 K (153). At a temperature of 130 K, rc reaches a maximumand diminishes if the temperature is raised further. Conversely, rab retains metallic dependenceand increases up to the highest temperature (1,300 K) measured (154) without any sign ofsaturation. In addition to transport, ARPES (155, 156) and NMR (157) also reveal a low co-herence scale. InARPES, quasiparticles persist up to 150K; inNMRKorringa law, 1/T1}T is seenonly below 50 K.
A theoretical calculation within LDAþDMFT (19) has accounted for many aspects of theexperiments. A coherence scale T� was defined by comparing the inverse quasiparticle lifetime tokT, and the Hund’s coupling was shown to be essential in explaining the low value of T� (Figure10). Quantitative agreement with ARPES and NMR was found. The theoretical curves shown inFigure 10 are based onLDAþDMFT calculations (J.Mravlje, unpublished; see alsoReference 19).Figure 10ddisplays themass renormalizations and coherence scale as a function of J. A largermassrenormalization is found for the xy orbital (g-band), in agreement with experiment. This has beenrelated to the proximity to the van Hove singularity in the xy band. This differentiation betweenthe xy, xz, and yz bands occurs only once theHund’s rule coupling is turned on, due to the orbital-decoupling action of J discussed earlier (Section 6.6). The Hunds’s coupling and the proximity toa van Hove singularity thus cooperate to make ruthenates strongly correlated materials, despitetheir small U/W ratio.
7.3.2. Non-Fermi-liquid behavior in SrRuO3 and CaRuO3. In ruthenates, the resistivity at veryhigh temperatures exceeds (158) theMott-Ioffe-Regel limit. Nevertheless, at low temperaturesT<
TFL, electrons in ruthenates form a Fermi-liquid. The signatures of the Fermi-liquid behavior suchas the observation of quantumoscillations and theT2 law in resistivity have by nowbeen seen in allmetallic ruthenates and also very recently on thin-film samples of CaRuO3 below 2K, which weremeasured in P. Gegenwart’s group at the time this article was written (M. Schneider et al., un-published data). The Fermi-liquid temperature TFL, however, is quite small, and the ruthenates
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provide a tantalizing ground for trying to identify their behavior forT>TFL in terms of a universalbut non-Fermi-liquid regime. So far, the most successful identification has been in the meas-urements of optical conductivity in SrRuO3 and CaRuO3 (131, 133, 134, 142, 159). Figure 11(142) summarizes these data. It shows that v/T scaling applies and that the optical conductivity atlarge enough frequencies obeys s1(v)∼v�1/2. Another signature of the non-Fermi liquid, which isseen in CaRuO3 (160) and in Ca-substituted Sr2RuO4 (117), is a log(T) correction to C/T. Theorigin of this has not been clarified yet. In particular, it remains to be shown whether it is anintrinsic property of the correlated state with a low-coherence scale.We notice that the few lowest-temperature data points of Reference 160 display saturation of C/T and may be indicative of theeventual formation of a Fermi liquid below 3 K.
Overall, it is quite tempting to associate (18) the non-Fermi liquid regimes observed in SrRuO3
and CaRuO3 to the power laws found in the spin-freezing regime for T> TFL discussed in Section6.3. Obviously, this fascinating possibility deserves further investigation.
7.3.3. Ca2Lx Srx RuO4: heavy carriers and orbital selectivity. Partial substitution of Ca intoSr2RuO4 leads to a rich physics and phase diagram (117). The smaller size of Ca induces rotationsof octahedra, which appear first at x ¼ 1.5 and progressively become more pronounced withdiminishing x until reaching almost 13 degrees at x¼ 0.5. A strong ferromagnetic enhancement ofthe magnetic susceptibility with Curie-Weiss behavior corresponding to an S ¼ 1/2 moment isfound in awide range 0.2( x( 1.5 (115). Onewould expect an S¼ 1moment for an isolated Ruatomwith four electrons. At 0.2< x< 0.5, a weak rhombohedral distortion appears (161). Belowx¼ 0.2, stronger rhombohedral distortions with compressed octahedra lead to an insulating state(see discussion on Ca2RuO4 above).
b
σ 1 (ω)
/aω
–1
/2
ω/T (cm–1/K)
00
0.4
0.8
1.2
0.5 1.0 1.5 2.0
100 K200 K300 K400 K500 KSRO 185 KSRO 225 KSRO 250 K
a
T 1/2 (K1/2)
T (K)
ρ (μΩ
cm)
4000 100
CaRuO3
SrRuO3 (Allen et al.)
300 900
300
200
100
00 10 20 30
Figure 11
Non-Fermi-liquid behavior in SrRuO3 and CaRuO3, possibly related to Hund’s coupling physics and spin freezing. (a) Resistivity versusT0.5. (b) Optical conductivity showing 1/v1/2 behavior and v/T scaling. Figure taken from Reference 142.
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Especially interesting is the regime close to the structural transition at x ¼ 0.5. The coexistenceof metallic transport with an S ¼ 1/2 Curie-Weiss magnetic susceptibility has inspired Anisimovet al. (104) to propose that an OSMT occurs. In this scenario, one-third of an electron would betransferred from the metallic xy band, and the three electrons in the narrower xz, yz bands wouldlocalize. However, there is by now much experimental evidence against this proposal, the mostdirect being theARPESobservation of all three Fermi surface sheets (162). The unchangedpositionof the nesting-induced peaks at incommensurate wave vectors in the susceptibility (161) alsosuggests that the charge transfer does not occur.
In fact, it is the xy band that displays the strongest correlations and the heaviest carriers. This isalready the case for Sr2RuO4, as discussed above. With diminishing x, the correlations graduallybecome stronger, as evidenced by the decrease of TFL (identified as the scale below which r } T2)and by the increase of the specific heat coefficient g. Close to x ¼ 0.5, the carriers become veryheavy, with g ∼ 250mJ/mol K2, approximately 20 times the LDA value. The optical spectroscopydata (163) point at a mass enhancement associated mainly with the xy band. Similar indicationsfollow from the polarized neutron diffraction study at x¼ 0.5,which found that, in the presence ofamagnetic field, themoment is on the xy orbitals and the adjacent oxygen sites (164). ARPES dataat x¼ 0.2 are controversial: One study reported all the Fermi sheets (165), whereas another studydid not see the xy sheet (166).
In our view, a possible comprehensive explanation of this rich behavior is reached by recog-nizing that the effects of the Hund’s rule coupling and of the proximity to a van Hove singularityresponsible for heavy carriers and orbital differentiation in the undistorted Sr2RuO4 becomeamplified by structural distortions in Ca2�x SrxRuO4. Certainly, the value of J does not changeupon rotations of the octahedra; however, the effective bandwidths do. Indeed, the dominant effectis the narrowing of the band originating from the in-plane xy orbitals (167). The effects of J on theelectrons with lower Fermi velocity, and its band-decoupling action, lead to poorly screenedmoments on xy orbitals and incoherent carriers. This accounts for the S ¼ 1/2 Curie-Weisssusceptibility even though strict OSMTmay not occur. In fact, at higher temperatures, an S¼ 1/2moment is observed in an extended range 0.2< x< 1.5. Below x¼ 0.5, when the octahedra alsotilt, the xz- and yz-derived bands narrow down, and the corresponding correlations increase, asperhaps indicated by the buildup of incommensurate magnetic fluctuations. These qualitativeideas call for a detailed study using LDAþDMFT techniques.
The poorly screened moments are susceptible to ordering at low temperature. Nakatsuji et al.(117) found a history-dependent magnetization compatible with the buildup of short-rangeferromagnetic ordering close to the x ¼ 0.5 critical point. The phase diagram has very recentlybeen refined in amSR study, which revealed subtle signatures of spin-glass orderingwithmomentsbelow 0.2 mB at most Ca concentrations (84).
Finally, in the Ca-rich region 0 < x < 0.2, an antiferromagnetic insulator is found, withproperties similar to that of the x ¼ 0 end-compound Ca2RuO4 discussed above. The metal-insulator transition coincides with the structural transition from L-Pbca to S-Pbca. The transitiontemperature diminishes with increasing x and vanishes a bit below x¼ 0.2. The rotations and tiltsare not sufficient to induce the insulating state in ruthenates; a compression of the octahedrarealized in the S-Pbca phase, which completely polarizes the orbitals, is needed.
8. IRON-BASED SUPERCONDUCTORS AS HUND’S CORRELATED METALS
The recent discovery of high-temperature superconductivity (168, 169) in iron pnictides andchalcogenides has generated considerable interest (for reviews, see, e.g., References 170 and 171).Obviously, in the limited space of this review we do not attempt to cover the intensive research
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performed on the subject. Rather, we focus on the importance of the Hund’s coupling for thephysics of these compounds.
Right from their discovery, the degree of electronic correlation in these materials has beendebated, with views ranging from the itinerant limit withmagnetic correlations induced by nesting(172, 173), all the way to localized magnetism (174). Early on, the authors of References 17 and175 emphasized the importance of electronic correlations while keeping a metallic description.
In our view, it has now become clear that these materials do display important effects ofelectronic correlations. From a phenomenological standpoint (looking, for example, at the Drudeweight, specific heat enhancement, renormalization of bandwidth, Fermi velocities, etc.), thedegree of correlation clearly increases when going over the different materials, in the order (fromweaker to stronger correlation effects) 1111 pnictides (such as LaFeAsO); 122 (such as BaFe2As2);111 (such as LiFeAs) and; at the more strongly correlated end (176), the 11 chalcogenides (FeSe,FeTe). An issue that is still controversial is whether these differences are mainly due to variationsin the structural properties with similar interaction strengths (Ud, JH) (4, 177) or whether it is
3.0
2.5
2.0
1.5
1.0
0.5
00 100 200
T (K)
χ lo
c (10
–3 e
mu
/mo
l)
300 400 500
a0
JHund (eV)
0.350.400.70
b30
BaFe Ni0.1As2
BaFe As2
20
10
0 0 100E (meV)
χ" (ω
) (μ
2 B e
V–
1 (
pe
r fo
rmu
la u
nit
)
200 300
RPA
DMFT
600
300
450
150
00.4 0.2
Hole doped
Tem
pe
ratu
re (
K)
d6
SC SDW SC
Electrondoped
0 0.2
Non-Fermiliquid Non-Fermi
liquid
Fermiliquid
Fermiliquid
d5
4
3
2
15
Mott FeSe
FeAsLaO
6nDoping x
7
U (
eV
)
c
Figure 12
Iron-based superconductors as Hund’s metals. (a) Temperature-dependence of the local susceptibility for a five-band LDAþDMFTdescription of LaO1�xFxFeAs, revealing the sensitivity toHund’s coupling (fromReference 17). (b) RPA andLDAþDMFTcalculations ofx00locðvÞ in absolute units for BaFe2As2 and BaFe1.9Ni0.1As2 (from Reference 183). (c,d). Spin-freezing region with power-law NFL self-
energy: (c) For doped BaFe2As2, as obtained in the LDAþDMFT study of Reference 184. (d) Schematic boundary in the U versus fillingdiagram (from Reference 182), illustrating the stronger correlations in the chalcogenides. Abbreviations: DMFT, dynamical mean-fieldtheory; RPA, random phase approximation; SC, superconductivity; SDW, spin-density wave.
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essential to take into account an increase of the interactions, especially for the 11 chalcogenides(79, 178–182). By and large, the big picture is nonetheless that the correlations are important.
Thekey role of theHund’s couplingwas recognized early on for thesematerials. In a pioneeringarticle, Haule & Kotliar (17) proposed that Hund’s coupling may be indeed responsible for thecorrelation effects and thus for the unconventional aspects of the metallic state. Within five-bandLDAþDMFT calculations, they found that the Hund’s coupling dramatically reduces the co-herence scaleT� belowwhich ametalwith Pauli susceptibility is found, leaving an incoherentmetalwith local moments for T > T� (see Figure 12a). It was also recognized early on (185) that theHund’s coupling is responsible for the formation of the iron-local moment in these compounds.This is consistent with X-ray spectroscopy (186), which reported a large value of J (∼0.8 eV).
Unexpectedly, in the magnetic state, the LSDA was found to overestimate the size of theorderedmagnetic moments (∼2 mB, whereas experiments yield moments < 1 mB). Because LDAis a static theory, a possibleway of interpreting this is thatmagneticmoments undergo importantdynamical fluctuations. Indeed, in References 187 and 188, Hansmann and coworkers per-formed LDAþDMFT calculations of the local spin-spin correlation function xloc(t) ¼ ÆSz(0) ×Sz(t)æ in the paramagnetic phase and looked at the short-time (high-energy) fluctuating localmoment. They found that its instantaneous value Æ S
!2æ is rather large but rapidly decays
(typically after a few femtoseconds) due to the screening in a metallic environment. The valuemloc¼ gmB[3 xloc(t ¼ 0)]1/2x 3.68 mBwas found for LaFeAsO, with a similar value reported inReference 189 and somewhat larger inReference 4. (The actual values reported in these two articlescorrespond only to g
ffiffiffiffiffiffiffiffiÆS2z æ
pand should be multiplied by
ffiffiffi3
p.) This corresponds [from
m2loc ¼ ðgmBÞ2SeffðSeff þ 1Þ] to an effective spin per iron atom SeffT 1.4. From neutron scattering,
Liu et al. (183) report a smaller valuemlocx1.8mB (Seffx1/2) for BaFe2As2.5 Thex00
locðvÞ they findis compared to LDAþDMFT calculations, and the agreement supports the notion of a localmoment formed at a high energy, with little influence of doping on the high-energy spectrum.Furthermore, the maximum of x00
locðvÞ was found to be at ∼200 meV, corresponding to a fluc-tuation timescale of ∼20 fs. This energy scale (resp. time-scale) is an order of magnitude smaller(resp. longer) than the bare electronic bandwidth (x 4 eV). Indeed, a weak-coupling itinerantpicture based on an RPA calculation (183, 188) would yield a timescale approximately 10 timesshorter, and vertex correctionswere found to be crucial (see Figure 12b). Experimental support forthe formation of a sizeable fluctuating local moment at high energy also stems from fast spec-troscopic probes such as X-ray absorption (190, 191) and core-level photoemission (192). Theimportance of theHund’s coupling in properly accounting also for themagnetic long-range order ofthese compounds6 has been emphasized by theoretical studies both from the strong coupling (4, 189,193, 194) and weak coupling (185, 195) viewpoints.
As mentioned above, it was first proposed in Reference 17 that the Hund’s coupling, besidescausingmoment formation at high energy, is also responsible for the low-energy correlation effectsin the metallic phase of these compounds, hence making them Hund’s metals (a term coined inReference 4). This point of view has been further confirmed and elaborated upon in severaltheoretical studies,mostly based on the LDAþDMFTmethodology. Aichhorn&coworkers (181)and Liebsch& Ishida (182) found that the chalcogenide FeSe displays local moments down to low
5Imxloc(v) can be probed by neutrons, and its integral can be related to the value of the moment. However, because neutronsonly reach frequencies of order a few 100meV’s, which is an order ofmagnitude too low, reduced values of themoment can beexpected from such experiments.6The nesting picture has been shown tobe unable to describe somekey aspects of the SDWordered phase, such as the differencein the magnetic ordering of BaFe2As2 and FeTe (185).
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temperature, together with bad metallic behavior characterized by a large scattering rate for someof the orbitals. This is a manifestation of the spin-freezing behavior discussed in Section 6.3. Thesecalculations also reveal a strong tendency to orbital differentiation [present in all materials butmore pronounced for the chalcogenides (4, 82, 181, 182, 196)], with the t2g-like orbitals morecorrelated than the eg ones (see below). The authors of Reference 196 also emphasized the im-portance of Hund’s coupling for LiFeAs.
The non-Fermi-liquid power-law regime associated with the onset of the spin-freezing behavior(Section 6.3)was revealed very clearly in a recent study of doped BaFe2As2 (184). The crossover linebetween the Fermi-liquid and non-Fermi-liquid power-law behaviors found by these authors asa function of doping and temperature is reproduced in Figure 12c. Very recently, such power lawswere reported and discussed for chalcogenides as well (82). On the basis of these studies and on thegeneral considerations presented earlier in this review, one may want to position the different Fe-basedmaterials in a diagram similar to Figure 1; as a function of the filling of the d-shell and strengthof interaction, seeFigure 12d. It is seen that hole doping takes thesematerials deeper into the stronglycorrelated spin-freezing regime, and electron doping restores amore itinerant Fermi-liquid behavior.With this perspective inmind, some authors have recently viewed the hole-dopedmaterials as beingin the proximity of thed5Mott-insulating state, i.e., as derived from this insulator by electron doping(182, 197, 198). In this respect, the Mn-based materials, with nominal d5 composition, are indeedinsulators (199), as expected from the much lower value of Uc for a half-filled shell.
Insights into the qualitative difference between Mott-correlated and Hund-correlated metalshave been obtained within LDAþDMFT by focusing on atomic histograms (4, 177). Thesehistograms register the probability of occurrence of each atomic state, resolved with respect to theatomic charge and the energy of the state. They reveal that charge fluctuations are substantial inthese materials, in contrast to a metal close to a Mott transition in which valence fluctuations aresuppressed. Here, in contrast, the probability is highest forN¼ 6 andN¼ 7 states, is still sizeableforN¼ 5, and is nonnegligible forN¼ 4 andN¼ 8 states. Furthermore, within a givenN, the HSatomic ground state has the largest probability (177), but other states are also often visited, unlikein heavy fermions. It is also argued (4, 177) that, because themost probableN¼ 6 andN¼ 7 statesspan an energy range of more than 6 eV, the atomic excitations (Hubbard bands) are broadenedand difficult to resolve in photoemission spectroscopy, explaining why they are not observed.These considerations highlight the itinerant nature of these systems, yet dominated by the cor-relation effects due to Hund’s rule coupling. Valence fluctuations on individual sites implya corresponding change of the local effective interaction (Section 3; Reference 5). Local aspects ofthis physics are taken into account fully by DMFT, but intersite correlations (such as the effectsdiscussed in Reference 200) may also play a role and require treatment beyond single-site DMFT.
On the experimental side, optical measurements have been interpreted as revealing the im-portance of the Hund’s rule coupling. Besides a reduction of the Drude weight and thus of theelectron kinetic energy (201), which signals strong correlations, optical measurements on
BaFe2As2 show that the spectral weightZ V
0sðvÞdv is suppressed upon cooling down around
3,000 cm�1, the lost spectral weight being recovered above 8,000 cm�1. This energy scale, firstreported by Hu et al. (202), is interpreted as a signature of Hund’s coupling (203, 204).
Soon after the discovery of high-Tc superconductivity in iron pnictides, it was also pointed out(108, 205) that the electronic structure of these materials constitutes an ideal ground for orbital-selective physics caused by electronic correlations and for the formation of localized magneticmoments coexistingwithmetallic properties. An importantmechanism is theHund’s coupling roleof band decoupler discussed in Section 6.6.
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Indeed, several theoretical studies (4, 181, 196, 206–209) have reported strong orbital dif-ferentiation (e.g., in the degree of coherence), particularly in the more correlated iron chalco-genides, or in pnictides for correlation strengths somewhat larger than the physical estimate (210).In general, t2gorbitals in these calculations showstrongermass enhancements and lower coherencethan the eg’s. In parallel, phenomenological models based on the coexistence of localized anditinerant electrons were developed to explain themagnetic and superconducting properties of ironpnictides (211) and their evolution under pressure (212). However, there is not necessarily a directconnection between these two components (localized and itinerant) and the two types of orbitals(t2g, eg). Superexchange betweenwell-formed localmoments has been suggested as an explanationfor both the collinear AF order coexisting with metallic properties and the linear dependence ontemperature of the magnetic susceptibility in the paramagnetic phase. (Although, Reference 213reproduces this behavior, already in a purely local picture, due to orbital differentiation.) Fluc-tuating local moments also hint at a possible pairing mechanism for superconductivity throughspin fluctuations.
On the experimental side, some evidence for the coexistence of local moments and itinerantelectrons has been reported. Inelastic neutron scattering on FeTe0.35Se0.65 (214) shows a signifi-cant temperature-independent magnetic moment (obtained by integrating the magnetic spectralweight up to 12meV), indicating that a large energy scale (i.e., states at an energymuch larger thanthe temperature) is involved in the formation of this moment. A picture based on itinerant (albeitrenormalized) electrons alone cannot explain such a magnetic response. Analogously, nuclearmagnetic resonance data on FeSe0.42Te0.58 (215) show a Knight shift scaling with the spin sus-ceptibility measured by electron paramagnetic resonance and not with the bulk magnetic sus-ceptibility. This observation is interpreted as arising from intrinsically localized states coupled toquasiparticles. ARPES measurements of the Fermi velocity in each Fermi surface sheet ofpotassium-doped BaFe2As2 reported orbital-dependent mass renormalizations (216). Accord-ingly, a model of two electronic fluids with different coherence properties was needed to interpretthe magnetoresistance data in the cobalt-doped compound (217).
Overall, a substantial orbital differentiation (induced by Hund’s coupling) in the degree ofcorrelation and localization of the conduction electrons associated with the different Fe orbitalsappears to play a role in the physics of iron-based superconductors. To what extent and howstrongly in each family of materials is still an issue for future investigation. Finally, Hund’scoupling–induced correlations are relevant to other iron compounds (218), such as, e.g., FeSi (219,220).
9. CONCLUSION: FUTURE DIRECTIONS
This review highlights the essential role of the Hund’s coupling in multiorbital materials. Thiscoupling induces strong electronic correlations in itinerant materials that are not in closeproximity to a Mott-insulating state. This is especially relevant to TMOs of the 4d series and toiron pnictides and chalcogenides. A global picture has recently emerged, which has beenreviewed in this article.
Some key questions to be addressed in future investigations remain unanswered, however. Asreviewed above, the Fermi-liquid scaleTFL is strongly reduced by the Hund’s coupling, and a non-Fermi-liquid state with frozen local moments and power-law self-energy applies for T > TFL. Aprecise theoretical understanding of this regime is still missing, however. Is this regime associatedwith a specific unstable fixed point of the underlying effective impurity model, within a single-siteDMFT approach? This would yield the fascinating possibility that there is something universal to
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be learned about the crossover between the very high-temperature regime of quasi-isolated atomsand the very low-temperature Fermi liquid.
Much work also remains to be done on the interplay of the effects described in this reviewwithmagnetic long-range order, a topic to which we have devoted only little discussion. How thedevelopment of intersite magnetic correlations modifies the local picture reviewed here should beaddressed using other approaches, such as cluster extensions of DMFT.
Although several indications of the key role played by theHund’s coupling have been reviewedin this article, direct “smoking-gun” evidence would be invaluable.
Finally, there are some important topics that we have not covered in this review. These includethe role of Hund’s coupling in stabilizing the ferromagnetic state (221) in transition metals andother materials (see References 221, 222, 223, 224 and references therein); the physics of negative(antiferromagnetic) Hund’s coupling, which can occur due to the Jahn-Teller coupling and whichis important for the physics of fullerides (225); the possibility ofHund’s coupling–mediatedpairingand superconductivity (see, e.g., References 226, 227, and 228); and the role of the Hund’scoupling in heavy-fermion compounds and in low-dimensional systems. Last but not least, theinterplay of the Hund’s coupling with the spin-orbit coupling is a topic of considerable currentinterest and of special relevance to the physics of TMOs of the 5d series.
Appendix: Two-Orbital Hamiltonian
In this appendix, we provide details on the different Hamiltonians relevant to the case of twoorbitals. The orbital isospin generators in this case (where t! represents the Pauli matrices) read
T!¼ 1
2
Xmm0s
d†ms t!mm0dm0s. 21:
The expression of the five terms in the generalized KanamoriHamiltonianHGK (Equation 3), readin terms of charge, spin, and orbital-isospin generators, isX
mbnm↑bnm↓ ¼ bN2
=4þ T2z � bN=2,
Xm�m0bnm↑bnm0↓ ¼ �S2z � T2
z þ bN=2Xm<m0 ,s
bnmsbnm0s ¼ bN2=4þ S2z � bN=2X
m�m0d†m↑dm↓d
†m0↓dm0↑ ¼
�S!2 � T
!2�
2þ T2z � S2z ,
Xm�m0d
†m↑d
†m↓dm0↓dm0↑ ¼ T2
x � T2y .
22:
Note also the relation
� bN � 2�2þ 2S
!2 þ 2T!2 ¼ 4. 23:
As for t2g, the KanamoriHamiltonian (Equation 2) is exact for an eg doublet, but in this case, cubicsymmetry itself implies that U0 ¼ U � 2J (30). The eg Kanamori Hamiltonian can be written as
Heg ¼ ðU � JÞbN� bN � 1
�2
þ 2J�T2x þ T2
z
�� J bN. 24:
It is seen that no continuous orbital symmetry remains, due to the total quenching of orbital an-gularmomentum for an eg doublet. For a spherically symmetric atom,U and J can again be relatedto Racah-Slater parameters, as (30)
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U ¼ U0 þ 2J ¼ F0 þ 449
F2 þ 449
F4 ¼ Aþ 4Bþ 3C, J ¼ 349
F2 þ 5147
F4 ¼ 4Bþ C. 25:
The generalized Kanamori Hamiltonian (Equation 3) can also be written in terms of the differentgenerators:
14ðU þU0 � J þ JXÞ
�bN � 2�2þ JX T
!2 þ ðU �U0 � JXÞT2z þ ðJX � JÞS2z þ JP
�T2x � T2
y
�,
26:
in which we have focused on the particle-hole symmetric case, hence omitting a termbNðUþ 2U0 � JÞ=2. Two special cases are worth mentioning for future reference:
1. Full spin and orbital invariance U(1)C Ä SU(2)S Ä SU(2)O is realized for JP ¼ 0, JX ¼ J,andU0 ¼U� J (note: notU0 ¼U� 2J). This actually applies to an arbitrary number oforbitals and yields the Hamiltonian equation (Equation 12) introduced by Dworin &Narath (45) in the context of magnetic impurities:
12
�U � J
2
��bN � 2�2 þ JT
!2 ¼ 12
�U � 3J
2
��bN � 2�2 � JS
!2 þ 2J. 27:
2. Setting JP ¼ 0, JX ¼ J, and U0 ¼ U, we obtain a Hamiltonian that still implements the
essence of Hund’s rule physics while maintaining a partial O(2) orbital symmetry (it
commutes with T!2
and Tz). This Hamiltonian was introduced by Caroli et al. (47) (see
also References 81 and 229) and reads
U2
� bN � 2�2 þ J
�T!2 � T2
z
�. 28:
DISCLOSURE STATEMENT
The authors are not aware of any affiliations, memberships, funding, or financial holdings thatmight be perceived as affecting the objectivity of this review.
ACKNOWLEDGMENTS
We are grateful to M. Aichhorn, H. Alloul, S. Biermann, M. Capone, M. Casula, M. Ferrero,A. Fujimori, E. Gull, P. Hansmann, K. Haule, M. Imada, G. Kotliar, J. Kune�s, A. Liebsch,C. Martins, I. Mazin, A.J. Millis, M. Randeria, G. Sangiovanni, G. Sawatzky, Y. Tokura, A. Toth,D. van der Marel, L. Vaugier, P. Werner, and R. �Zitko for very useful discussions and remarks.This work was supported by the Partner University Fund, the Agence Nationale de la Recherche(grants ANR-09-RPDOC-019-01, ANR-2010-BLAN-040804, PNICTIDES), the Slovenian Re-search Agency (under contract J1-0747), the Swiss National Science FoundationMaNEP programand the JST-CREST program. Computer time was provided by IDRIS/GENCI under grant2011091393.
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Condensed Matter
Physics
Volume 4, 2013 Contents
Why I Haven’t RetiredTheodore H. Geballe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Quantum Control over Single Spins in DiamondV.V. Dobrovitski, G.D. Fuchs, A.L. Falk, C. Santori,and D.D. Awschalom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Prospects for Spin-Based Quantum Computing in Quantum DotsChristoph Kloeffel and Daniel Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Quantum Interfaces Between Atomic and Solid-State SystemsNikos Daniilidis and Hartmut Häffner . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Search for Majorana Fermions in SuperconductorsC.W.J. Beenakker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Strong Correlations from Hund’s CouplingAntoine Georges, Luca de’ Medici, and Jernej Mravlje . . . . . . . . . . . . . . . 137
Bridging Lattice-Scale Physics and Continuum Field Theory withQuantum Monte Carlo SimulationsRibhu K. Kaul, Roger G. Melko, and Anders W. Sandvik . . . . . . . . . . . . 179
Colloidal Particles: Crystals, Glasses, and GelsPeter J. Lu ( ) and David A. Weitz . . . . . . . . . . . . . . . . . . . . . . . 217
Fluctuations, Linear Response, and Currents in Out-of-Equilibrium SystemsS. Ciliberto, R. Gomez-Solano, and A. Petrosyan . . . . . . . . . . . . . . . . . . . 235
Glass Transition Thermodynamics and KineticsFrank H. Stillinger and Pablo G. Debenedetti . . . . . . . . . . . . . . . . . . . . . 263
Statistical Mechanics of Modularity and Horizontal Gene TransferMichael W. Deem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
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Physics of Cardiac ArrhythmogenesisAlain Karma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Statistical Physics of T-Cell Development and Pathogen SpecificityAndrej Ko�smrlj, Mehran Kardar, and Arup K. Chakraborty . . . . . . . . . . 339
Errata
Anonline log of corrections toAnnual Review of CondensedMatter Physics articlesmay be found at http://conmatphys.annualreviews.org/errata.shtml
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