J-freezing and Hund’s rules in spin-orbit-coupled multiorbital Hubbard models

Aaram J. Kim,1 Harald O. Jeschke,1 Philipp Werner,2 and Roser Valentı1

1Institut fur Theoretische Physik, Goethe-Universitat Frankfurt,Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany

2Department of Physics, University of Fribourg, Chemin du Musee 3, 1700 Fribourg, Switzerland

We investigate the phase diagram of the spin-orbit-coupled three orbital Hubbard model at ar-bitrary filling by means of dynamical mean-field theory combined with continuous-time quantumMonte Carlo. We find that the spin-freezing crossover occurring in the metallic phase of the non-relativistic multiorbital Hubbard model can be generalized to a J-freezing crossover, with J = L+S,in the spin-orbit-coupled case. In the J-frozen regime the correlated electrons exhibit a non-trivialflavor selectivity and energy dependence. Furthermore, in the regions near n = 2 and n = 4 themetallic states are qualitatively different from each other, which reflects the atomic Hund’s thirdrule. Finally, we explore the appearance of magnetic order from exciton condensation at n = 4 anddiscuss the relevance of our results for real materials.

PACS numbers: 71.10.Hf, 71.15.Rf, 71.30.+h, 75.25.Dk

Introduction. In 4d and 5d transition metal oxides theinterplay and competition between kinetic energy, spin-orbit coupling (SOC) and correlation effects results inseveral interesting phenomena, such as spin-orbit assistedMott transitions [1, 2, 4, 5, 7, 8], unconventional super-conductivity [9, 10], topological phases [11], exciton con-densation [10, 12, 13], or exotic magnetic orders [14, 15].Transition metal oxides involving 4d and 5d electronsshow diverse structures like the Ruddlesden-Popper se-ries [1, 9], double perovskite, [14–16] two-dimensionalhoneycomb geometry [3–8] or pyrochlore lattices [17]. Inan octahedral environment, as in most of the 4d and 5dmaterials mentioned above, the five d orbitals are splitinto low energy t2g and higher energy eg levels. TheSOC further splits the low energy t2g levels into a so-called j = 1/2 doublet and j = 3/2 quadruplet. The en-ergy separation between the j = 1/2 and j = 3/2 bandsis proportional to the strength of the SOC. Existing ab-initio density functional theory calculations [17, 18] sug-gest that in some materials a multiorbital description in-cluding both the j = 1/2 and j = 3/2 subbands shouldbe considered.

Most theoretical studies of 4d and 5d systems have fo-cused on material-specific models with fixed electronicfilling. Here we follow a different strategy and explorethe possible states that emerge from a multiband Hub-bard model with spin-orbit coupling at arbitrary filling.This allows us to investigate unexplored regions in pa-rameter space which may exhibit interesting phenomena.Specifically, by performing a systematic analysis of thelocal J moment susceptibility (J = L + S) as a functionof Coulomb repulsion U , Hund’s coupling JH, spin-orbitcoupling λ and filling n, we identify Mott-Hubbard in-sulating phases and complex metallic states. We find aJ-freezing crossover between a Fermi liquid (FL) and anon-Fermi liquid (NFL) phase where the latter shows adistinct flavor selectivity that originates from the SOC.In addition, we detect near filling n = 2 a highly fluctu-

ating metallic phase with properties reminiscent of theatomic Hund’s third rule. Finally, we investigate dopingeffects on the excitonic magnetism at n = 4.Method. We consider a three-orbital Hubbard model

with spin-orbit coupling. The model Hamiltonian con-sists of three terms,

H = Ht +Hλ +HU , (1)

where Ht, Hλ, and HU denote the electron hopping,spin-orbit coupling, and local Coulomb interaction terms,respectively. We assume that the hopping part ofthe Hamiltonian corresponds to degenerate semi-circulardensities of states (DOS), ρ0(ω) = (2/πD)

√1− (ω/D)2,

whose half-bandwidth D is set to unity. Hλ is con-structed by projecting the SOC term of d orbitals ontothe t2g subspace,

Hλ = λ∑

αβσσ′

c†iασ〈ασ|Pt2gLdPt2g · S|βσ′〉ciβσ′ , (2)

where Pt2g is the projection operator. ciασ (c†iασ) denotesthe annihilation (creation) operator of a spin σ electronat site i and orbital α. The angular momentum operatorwithin the t2g subspace can be represented by an effectiveL = 1 angular momentum operator with an extra minussign [14].

The local Coulomb interaction Hamiltonian is writtenin Kanamori form [19] including the spin-flip and pair-hopping terms as:

HU = U∑

i,α

niα↑niα↓ +∑

i,α<α′

σσ′

(U ′ − JHδσσ′)niασniα′σ′

−JH∑

i,α<α′

(c†iα↑c†iα′↓ciα′↑ciα↓ + h.c.)

+JH∑

i,α<α′

(c†iα↑c†iα↓ciα′↓ciα′↑ + h.c.) . (3)

arX

iv:1

607.

0519

6v1

[co

nd-m

at.s

tr-e

l] 1

8 Ju

l 201

6

2

(a) λ /D = 0, JH /U = 0.15

0 1 2 3 4 5 6

n

0

1

2

3

4

5

U /D

(b) λ /D = 0.25, JH /U = 0.15

0 1 2 3 4 5 6

n

0

1

2

3

4

5

U /D

MI

FL FL

NFLNFL

J-freezing

(c) λ /D = 0.25, JH /U = 0.25

0 1 2 3 4 5 6

n

0

1

2

3

4

5

U /D

0

2

4

6

8

∆χloc

FIG. 1. Dynamic contribution to the local susceptibility, ∆χloc in the (U/D, n) phase diagram for (a) λ/D = 0.0, JH/U = 0.15,(b) λ/D = 0.25, JH/U = 0.15, (c) λ/D = 0.25, JH/U = 0.25, and T/D = 0.03. For the definition of this quantity, see the maintext. Cross symbols mark the maximum values of ∆χloc corresponding to the J-freezing crossover points.

Here, U is the on-site Coulomb interaction and JH de-notes the Hund’s coupling. U ′ is set to U − 2JH to makethe interaction rotationally invariant in orbital space.

We employ the dynamical mean-field theory(DMFT) [20] to solve the model Hamiltonian Eq. (1)and to investigate it in a broad parameter space. SinceDMFT is a nonperturbative technique within the localapproximation, we can access metallic and insulatingphases on the same footing. In addition, the dynamicalfluctuations encoded in the DMFT solution containvaluable information on the degree of moment correla-tions and the corresponding susceptibility. We will usethe local J moment susceptibility as a central quantityto investigate the phase diagram.

As an impurity solver, we adopt the continuous-timequantum Monte Carlo method (CTQMC) in the hy-bridization expansion variant [21, 22]. For the singleparticle basis of the CTQMC calculation, we choose therelativistic j effective basis (j = 1/2, j = 3/2) whichis an eigenbasis of the SOC Hamiltonian. It was previ-ously reported that the j effective basis reduces the signproblem of the CTQMC simulation [23]. Typically, weuse 108 ∼ 109 Monte Carlo steps. For symmetry brokenphases, we consider the off-diagonal hybridization func-tions.

Results. A strong Coulomb interaction localizes elec-trons and can lead to the formation of local moments.The freezing of these local moments is signaled by a slowdecay, and eventual saturation, of the dynamical corre-lation function 〈Jz(τ)Jz(0)〉 on the imaginary-time axis.Hence, the local susceptibility, defined as

χloc =

∫ β

0

dτ 〈Jz(τ)Jz(0)〉 , (4)

allows us to investigate the formation and freezing of lo-cal moments. In addition, we define the dynamical con-tribution to the local susceptibility by eliminating thelong-term memory of the correlation function from the

original χloc [24]:

∆χloc =

∫ β

0

dτ(〈Jz(τ)Jz(0)〉 − 〈Jz(β/2)Jz(0)〉

). (5)

As the system evolves from an itinerant to a localizedphase, ∆χloc exhibits a maximum in the intermediateCoulomb interaction regime [25]; both, (i) the enhancedcorrelations compared to the noninteracting limit and (ii)the larger fluctuations compared to the localized limitlead to the maximum in ∆χloc. The location of the∆χloc maxima in the phase diagram can be viewed asthe boundary of the local moment regime and has beenused to define the ‘spin-freezing crossover line’ in the non-spin-orbit coupled system [24, 26]. However, since spinis not a good quantum number in the spin-orbit-coupledsystem, we introduce the total moment J = S + L togeneralize the ‘spin-freezing’ to a ‘J-freezing’ crossover.

In the following, we discuss the paramagnetic phasediagram of Eq. (1) obtained with DMFT(CTQMC) as afunction of U , JH, λ and n. Figures 1 (a-c) show contourplots of ∆χloc in the interaction vs. filling plane for threedifferent parameter sets of λ and JH. Since SOC breaksparticle-hole symmetry, Fig. 1 (b), (c) are not symmet-ric about the half-filling axis, n = 3. Plots for χloc areshown in Ref. 25. The Mott insulating phase (black linesin Fig. 1) which we identify as the region where the spec-tral function vanishes at the Fermi-level (not shown) andwhere ∆χloc is smallest, appears at each commensuratefilling. Nonetheless, compared to the system withoutSOC (Fig. 1 (a)), the change of the critical interactionstrength Uc shows a complex behavior depending on thefilling and λ. We can quantitatively analyze the change ofUc using the Mott-Hubbard criterion, according to whicha Mott transition occurs when the atomic charge gap be-comes comparable to the average kinetic energy:

∆ch(n,Uc, JH, λ) ≡ Uc + δ∆ch(n, JH, λ) = W (n, JH, λ).(6)

3

0

1

2

0 1 2 3 4 5

n=1

n=2

n=3

n=4

n=5

⟨Jz2 ⟩

U / D

FIG. 2. Size of the local Jz-moments as a function of in-teraction strength U/D for λ/D = 0.25, JH/U = 0.15 andT/D = 0.03 at various commensurate fillings. The param-eter set is the same as in Fig. 1(b). Solid (Open) symbolscorrespond to the metallic (insulating) solutions. The arrowsrepresent the corresponding values from the Hund’s rule.

∆ch is the charge gap of the local Hamiltonian [25], andW (n, JH, λ) is the average kinetic energy. Here, n is in-teger for commensurate Mott insulators. Since SOC re-duces the degeneracy of the atomic ground states, Wis basically a decreasing function of λ except for n = 3where the ground state degeneracy is not changed by in-troducing SOC. By diagonalizing the local Hamiltonian,we observe that δ∆ch is an increasing function of λ forn = 1, 2, and 4, but a decreasing function for n = 3 and5 [25]. Altogether, for n = 1, 2, and 4, the two terms con-tributing to Uc = W−δ∆ch cooperate to reduce Uc as wealso observe in our DMFT results. A smaller Uc at n = 4compared to n = 2 is consistent with the Mott-Hubbardcriterion. In contrast, for n = 5 the two contributionsto Uc compete and it is hard to predict the behavior ofUc from this criterion. We can anticipate based on theDMFT results that the reduction of the kinetic energydominates the slight decrease of the atomic gap. Finally,at n = 3 there is an unchanged degeneracy and δ∆ch

decreases due to SOC implying a slight increase of Uc

(compare Fig. 1 (a) and (b) and see Ref. 25).

The effect of the Hund’s coupling can be seen by com-paring Figs. 1 (b) and (c). Away from half-filling, Uc

increases with JH but at half-filling it slightly decreases,which is consistent with the behavior of δ∆ch [25].

We now concentrate on the metallic regions. In thespin-orbit-coupled multiorbital system the dynamic con-tribution to the susceptibility is larger below half-fillingcompared to the particle-hole transformed state (red areain Fig. 1 (b) and (c)). Especially near n = 2 we find ametallic phase with very large ∆χloc implying a statewith highly fluctuating J-moments. A recent study [24]has shown that in the case of a multiorbital Hubbardmodel without spin-orbit coupling, such highly fluctu-ating moments can induce s-wave spin-triplet supercon-ductivity along the spin-freezing line. The effect of SOC

on this superconductivity will be an interesting futureresearch topic.

The enhanced susceptibility (Ref. 25) and dynamicalcontribution to the susceptibility (Fig. 1 (b) and (c))below half-filling can be explained by Hund’s third rulewhose origin is the spin-orbit coupling [27, 28]. Follow-ing Hund’s third rule, in the atomic limit the alignmentbetween L and S depends on whether the filling is be-low or above half-filling. In our calculation, L and S arealigned in the same direction below half-filling, while theyare anti-aligned above half-filling. Therefore, the size ofthe total J-moment is larger at fillings below n = 3 aswe increase the interaction strength and further local-ize the electrons. Figure 2 shows the evolution of 〈J2

z 〉as a function of Coulomb interaction strength for fivecommensurate fillings and parameter values as chosen inFig. 1 (b). In the intermediate and strong interactionregion, U/D & 2, an enhanced value of the J-moment isfound at n = 2 and 1 compared to the cases n = 4 and 5,respectively. In the strong correlation (Mott insulating)regime, the alignment of the spin, orbital, and J-momentis consistent with the atomic results according to Hund’srules [25]. The large J-moment at n = 2, explained by theHund third’s rule, results in a highly fluctuating metallicstate and large χloc even at moderate U values.

Inside the J-freezing region (denoted by crosses inFig. 1 (b)), we observe a non-Fermi liquid (NFL) behav-ior of the metallic state. In order to explore this statewe show in Figs. 3 (a) and (b) the imaginary part of theself-energy on the Matsubara frequency axis across the J-freezing crossover line for the same parameter values asin Fig. 1 (b) and various fillings. In the low frequency re-gion, Im Σ(iωn) can be expressed in the form −Γ−Cωαn .As we cross the J-freezing line, (region between n ' 2 andn ' 4 for U = 3) Γ changes from zero to a finite valueindicating a Fermi-liquid (FL) to NFL crossover. Nearthe J-freezing line, a small Γ value with a non-integerexponent α is found.

These two characteristic properties of the FL to NFLcrossover are reminiscent of the spin-freezing crossoverobserved in the model without SOC [26]. However, dueto SOC, the self-energy Im Σ(iωn) of the j = 1/2 electronis different from that of the j = 3/2 electrons (denotedby solid and open symbols in Fig. 3 (a), (b) respectively).At low frequency, the difference between j = 1/2 and 3/2is enhanced in the NFL phase compared to the FL phase.

A remarkable finding is that there exists an intersectionbetween the two self-energies from the different j bandsin the NFL phase (see shadings in Fig. 3 (a) and (b)).This intersection implies that the scattering rate near theFermi-level, ImΣ(ω ∼ 0), and the total scattering rate,∫∞−∞ dω ImΣ(ω) have different relative magnitudes for

the j = 1/2 and 3/2 electrons. For example, for n = 3.5,the j = 3/2 electrons have a larger value of Γ with largerscattering rate at the Fermi-level, while they exhibit asmaller high energy coefficient of the 1/(iωn) tail, imply-

4

−ωn

−ωn½

j = 1/2

j = 3/2

−1

0spin orbit coupling

(a)Im

Σ(i

ωn)

n=1.5 n=2.0 n=2.5

crystal field splitting

(c)α =1

α =2,3

j = 1/2

j = 3/2

−1

0

0 1 2

(b)

ImΣ

(iωn)

ωn

0 1 2ωn

α =1

α =2,3

0 1 2

(d)

ωn

n=3.5 n=4.0 n=4.5

FIG. 3. Imaginary part of the self-energy on the Matsub-ara axis for the system with (a,b) spin-orbit coupling (SOC)(c,d) crystal-field (CF) splitting. For (c,d), the CF Hamilto-nian, HCF = ∆CF

∑σ n1σ is introduced instead ofHSOC. The

strength of the SOC and the CF are chosen to produce thesame noninteracting DOS: λ/D = 0.25 and ∆CF/D = 0.375.U/D = 3.0, JH/U = 0.15, and T/D = 0.015. Solid (open)symbols in (a,b) denote the j = 1/2 (3/2) results. Solid(open) symbols in (c,d) correspond to α = 1 (α = 2, 3). Forj = 3/2 in (a,b) the average over mj = ±1/2,±3/2 is shown.In (c,d), we plot the average of α = 2, 3 and spin. The shad-ings in (a,b) highlight the intersections between the differentself-energies. The dashed (dotted) lines correspond to −ωn(−ω0.5

n ) as a guide for the low frequency scaling.

ing a smaller total scattering rate. Such a behavior isnot observed in the Hubbard model with ordinary crys-tal field (CF) splitting (no SOC) as shown in Figs. 3 (c)and (d) [29]. We suggest that the basis transformationand corresponding modification of the interaction, es-pecially of the Hund’s coupling, are the origin of thisphenomenon. This implies that the interplay betweenspin-orbit coupling effects and electronic correlation can-not be fully captured by an effective crystal-field split-ting description. We call this phenomenon spin-orbit-correlation induced flavor selectivity.

Note that the frozen J-moment and the NFL behaviorare characteristic features of multiorbital systems withlarge composite moments. Within the J-freezing region,even the j = 1/2 electrons show NFL behavior, and thesingle-band description for j = 1/2 is not valid anymore.Accordingly, the J-freezing crossover line delimits the re-gion of validity of the single-band description.

Besides the paramagnetic phase, we also investigatethe excitonic magnetism (EM) near n = 4 [10, 12, 13, 30].To access such a symmetry broken phase, we introducethe off-diagonal components of the Green function anddefine the order parameter of the exciton condensed

phase as ∆j′m′

jm = 〈c†jmcj′m′〉, where j′ 6= j. The magnetic

0

0.4AFM (n = 4)

(a) λ /D = 0.25JH /U = 0.15

∆1/2,m3/2,m

−0.05

0

0.05(b)

Mj,m

(b)

(c)

0 1 2 3 4 5 6 7 8U /D

3.8

4

4.2

n

0

0.4

∆1/2,m3/2,m

EMMI

J−freezing

0

0.3

(d)

FM (n = 4.2)

∆1/2,m3/2,m

λ /D = 0.1λ /D = 0.15λ /D = 0.2λ /D = 0.25

−0.6

0

0.6

0 1 2 3 4 5 6 7 8

λ /D = 0.1JH /U = 0.25

(e)Mj,m

U /D

j,m = 1/2,1/2j,m = 3/2,1/2j,m = 3/2,3/2

FIG. 4. (a) Excitonic order parameter and (b) magneticcomponents as a function of U/D at T/D = 0.33 for n =4.0, λ/D = 0.25 and JH/U = 0.15. (c) Density plot for theAFM excitonic order parameter. Here, EM represents theexcitonic magnetism. The black bar and yellow line indicatethe boundary of the (symmetric) metal-insulator (MI) and J-freezing regime, respectively. (d) Excitonic order parameterand (e) magnetic components as a function of U/D at T/D =0.33 for n = 4.2, JH/U = 0.25 and various λ/D values.

components are defined as Mj,m = 〈nj,+m〉 − 〈nj,−m〉.We find two types of excitonic magnetism: Antiferro-magnetism (AFM) and ferromagnetism (FM) at differ-ent fillings. At n = 4 an AFM excitonic state ap-pears at intermediate interaction strength [10, 30–32].The corresponding region is located around the metal-insulator transition point of the paramagnetic calcula-tions, Uc/D ∼ 3.5. Figures 4 (a) and (b) show that

AFM (Mj,m 6= 0) and excitonic order (∆1/2,m3/2,m 6= 0) ap-

pear simultaneously.

Upon electron doping, the AFM state is rapidly sup-pressed and eventually vanishes around n ∼ 4.2, whichis shown in Fig. 4 (c). For n = 4.2 with a large JH/Uvalue and small λ value, an excitonic FM state emergesin the strong interaction region (Fig. 4 (d) and (e)). InFig. 4 (d) we show that the critical interaction strengthof the onset of the FM state is an increasing function ofspin-orbit coupling.

Conclusions. We have explored the paramagneticphase diagram of the spin-orbit-coupled three-orbitalHubbard model at general filling. We found a general-ized J-freezing crossover as a function of U , JH, λ and nwhich exhibits a strong particle-hole asymmetry and wehave detected a highly fluctuating metallic phase nearn = 2 which is the effect of Hund’s third rule on theitinerant phase. Across the J-freezing line, a FL-to-NFLcrossover appears with a peculiar flavor selectivity in theNFL phase. This is a unique feature of SOC, which isnot present in models with ordinary crystal-field split-

5

ting. We expect that hole-doping of materials with d5

filling like iridates or rhodates will shift the systems to-ward the J-freezing line. Near n = 4, we observe exci-tonic magnetism with both AFM and FM order whichis consistent with a recent mean-field study [10]. Uponelectron doping, AFM at n = 4 is suppressed and FMemerges with enhanced Hund’s coupling. These resultsoffer new routes for finding exotic phases by doping 4dand 5d based materials.

Acknowledgements We thank Ying Li, Steffen Backes,Steve Winter, Ryui Kaneko, Jan Kunes, and Alexan-der I. Lichtenstein for helpful discussions. This researchwas supported by the Deutsche Forschungsgemeinschaftthrough FOR1346. The computations were performed atthe University of Frankfurt.

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Supplemental Information: J-freezing and Hund’s rules in spin-orbit-coupledmultiorbital Hubbard models

Aaram J. Kim,1 Harald O. Jeschke,1 Philipp Werner,2 and Roser Valentı1

1Institut fur Theoretische Physik, Goethe-Universitat Frankfurt,Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany

2Department of Physics, University of Fribourg, Chemin du Musee 3, 1700 Fribourg, Switzerland

I. Jz correlation function and local susceptibility

In this section, we illustrate the correlation functions which are the basic tool for investigating the phase diagramof the three-band Hubbard model with spin-orbit coupling. Figure S1 presents the local correlation function of thetotal moment Jz and the corresponding susceptibility for two different fillings, n = 2 and 4. Figure 1 in the maintext and Fig. S2 are constructed based on these correlation functions. The instantaneous value of the correlationfunction, 〈J2

z 〉, is determined by the size of the local moment and the right most value, 〈Jz(β/2)Jz〉, estimates thelong-term memory for a given temperature scale. The local susceptibility χloc (Eq. (4) in the main text) and itsdynamic contribution ∆χloc (Eq. (5) in the main text) are graphically illustrated in Fig. S1 (a).

In the n = 2 case, the correlation function and the susceptibility show a monotonic behavior. As strong Coulombinteractions localize the electrons, both the size of the local moment and its long-term memory increase simultaneously.The increase in both quantities contributes to the susceptibility. However, the dynamic contribution shows a peakat an intermediate interaction value (Fig. S1 (c)). Compared to the noninteracting limit, the enhanced correlationsand slow decay of the moment result in a larger value of ∆χloc, but this quantity again vanishes in the fully localized

n = 2

(a)

⟨Jz(τ

)J

z(0

)⟩

τ

0.0

0.5

1.0

1.5

2.0

2.5

0 4 8 12 16

χloc

∆χloc

n = 4

(b)

⟨Jz(τ

)J

z(0

)⟩

τ

U /D = 0U /D = 1U /D = 2U /D = 3U /D = 4U /D = 5

0.0

0.5

1.0

1.5

0 4 8 12 16

(c)

Uc

χloc

∆χloc

U /D

χloc∆χloc

0

20

40

60

80

0 1 2 3 4 5 0

4

8(c)

Uc

(d)

EM

Uc

χloc

∆χloc

U /D

0

2

4

6

8

0 1 2 3 4 5 0

3

6

Figure S1: (a), (b) Dynamical correlation function of the total moment Jz for various U/D values. (c,d) Thesusceptibility and its dynamical contribution as a function of U/D. (c), (d) The vertical dashed lines mark the

critical interaction strength Uc of the metal-insulator-transition. The green shading in (d) represents the excitonicAFM region. The left and right columns show the n = 2 and n = 4 results, respectively. λ/D = 0.25, JH/U = 0.15,

and T/D = 0.03 for all panels. The color scheme is fixed within the same row. χloc and ∆χloc are graphicallyrepresented in panel (a).

2

(a) λ /D = 0, JH /U = 0.15

0 1 2 3 4 5 6

n

0

1

2

3

4

5

U /D

(b) λ /D = 0.25, JH /U = 0.15

0 1 2 3 4 5 6

n

0

1

2

3

4

5

U /D

MI

FL FL

NFLNFL

J-freezing

(c) λ /D = 0.25, JH /U = 0.25

0 1 2 3 4 5 6

n

0

1

2

3

4

5

U /D

0

20

40

60

χloc

Figure S2: Local susceptibility in the (U/D, n) phase diagram for (a) λ/D = 0.0, JH/U = 0.15, (b) λ/D = 0.25,JH/U = 0.15, (c) λ/D = 0.25, JH/U = 0.25, and T/D = 0.03. Cross symbols mark the maximum values of ∆χloc

(compare Figs. S1 (c) and (d)) which may be used to define the J-freezing crossover points.

limit. Hence, a peak in ∆χloc can be naturally expected as a function of U .On the other hand, the behavior of the local susceptibility is nonmonotonic in the n = 4 case. The long-term memory

of the correlation function and the susceptibility show a peak structure in the intermediate interaction regime. Thestrong suppression of the susceptibility in the localized phase, which may be attributed to the nonmagnetic characterof the Van-Vleck-type ground state (vanishing J-moments) leads to this peak structure. The form of the ground stateis shown in Table SII. In other words, the small peak in χloc is the result of a competition between J-freezing andquenching of the local moment. If symmetry breaking is allowed, the antiferromagnetic and excitonic order extendinto the region with enhanced local spin susceptibility. In this sense, we may regard the fluctuating local moments inthis crossover regime as the hosting background for the symmetry-broken states.

The very different behaviors of the susceptibility in the n = 2 and 4 cases are illustrated in Fig. S2. Comparedto the non-spin-orbit-coupled system, Fig. S2 (a), the enhancement at n = 2 and the suppression at n = 4 of thelocal susceptibility in and near the localized phase are clearly evident in Fig. S2 (b). These strong effects persist intothe intermediate coupling regime and for doped systems, as shown in Figs. S2 (b) and (c). The susceptibility of theitinerant phase with n = 2 is much larger than for n = 4.

II. Local Hamiltonian

The local Hamiltonian not only describes the physical properties of the fully localized limit but also affects theproperties of the correlated itinerant phases. In our model, the local Hamiltonian is composed of two terms: the SOCand the Kanamori-type Coulomb interaction,

Hloc = Hλ +Hint . (1)

Using the total electron number N and the total moment Jz as quantum numbers, we diagonalized Hloc. Tables SIand SII summarize the properties of the ground states for each N . In Table SI, the numbers in parenthesis are theground state degeneracies of the non-spin-orbit-coupled Hamiltonian and µ is the chemical potential. As we turn onthe SOC, the degeneracy is reduced except for N = 3, which means that the average kinetic energy is reduced, asdiscussed in the main text. Based on the ground state energy, we calculate the charge gap, defined by

∆ch(N) = (Eg(N + 1)− Eg(N))− (Eg(N)− Eg(N − 1)) , (2)

where Eg(N) is the ground state energy of the local Hamiltonian for filling N . The results are summarized inTable SIII, and Fig. S3 represents δ∆ch(N, JH, λ) ≡ ∆ch − U . The summary of the Hund’s rules which apply to thelocal Hamiltonian is also presented in table SIV.

Figures S3 (b) through (f) illustrate the effect of the SOC on the charge gap, which turns out to be different fordifferent N . In the regime of interest, finite JH and relatively small λ, δ∆ch is an increasing function of λ for N = 1,2 and 4, but a decreasing function for N = 3 and 5. Based on the information of the degeneracy and charge gap, wediscuss the qualitative behavior of the critical interaction strength Uc in the main text.

3

N Degeneracy Ground State Energy (Eg)0 1 (1) 01 4 (6) −λ

2− µ

2 5 (9) 14

(4U − 8JH − λ−

√16J2

H + 8JHλ+ 9λ2

)− 2µ

3 4 (4) 3U + f(JH, λ) − 3µ

4 1 (9) 12

(12U − 21JH − λ−

√25J2

H + 10JHλ+ 9λ2

)− 4µ

5 2 (6) 10U − 20JH − λ− 5µ6 1 (1) 15U − 30JH − 6µ

Table SI: Ground state energy and degeneracy of the local Hamiltonian for different total particle numbers. Thenumbers in the parentheses show the ground state degeneracies when λ = 0.

N Jz Ground State

1

+3/2

∣∣∣⟩

+1/2

∣∣∣⟩

-1/2

∣∣∣⟩

-3/2

∣∣∣⟩

2

+2 α

∣∣∣⟩− β

∣∣∣⟩

+1 α

∣∣∣⟩− β

∣∣∣⟩− γ

∣∣∣⟩

0 α(∣∣∣

⟩+

∣∣∣⟩)

+ β(∣∣∣

⟩−∣∣∣

⟩)

-1 α

∣∣∣⟩

+ β

∣∣∣⟩

+ γ

∣∣∣⟩

-2 α

∣∣∣⟩

+ β

∣∣∣⟩

3

+3/2 α

∣∣∣⟩− β

∣∣∣⟩− γ

∣∣∣⟩− δ

∣∣∣⟩

+1/2 α

∣∣∣⟩− β

∣∣∣⟩− γ

∣∣∣⟩

+ δ(∣∣∣

⟩+

∣∣∣⟩)

-1/2 α

∣∣∣⟩− β

∣∣∣⟩

+ γ

∣∣∣⟩

+ δ(∣∣∣

⟩−∣∣∣

⟩)

-3/2 α

∣∣∣⟩

+ β

∣∣∣⟩

+ γ

∣∣∣⟩

+ δ

∣∣∣⟩

4 0 α

∣∣∣⟩

+ β(∣∣∣

⟩+

∣∣∣⟩)

5+1/2

∣∣∣⟩

-1/2

∣∣∣⟩

Table SII: Ground state for a given sector of the local Hamitonian. In our notation, the upper (lower) levelrepresents the j = 1/2 (3/2) states and the lower left (right) level corresponds to mj = ±1/2 (mj = ±3/2). Full(empty) circles mark the positive (negative) mj electron. All coefficients can be chosen to be real and duplicated

symbols in different sectors have nothing to do with each other.

N Charge Gap (∆ch)

1 U − 2JH + 3λ4

− 14

√16J2

H + 8JHλ+ 9λ2

2 f(JH, λ) + U + 4JH + 12

√16J2

H + 8JHλ+ 9λ2

3 U − 25JH2

− 3λ4

− 14

√16J2

H + 8JHλ+ 9λ2 − 12

√25J2

H + 10JHλ+ 9λ2 − 2f(JH, λ)

4 f(JH, λ) + U + JH +√

25J2H + 10JHλ+ 9λ2

5 U − JH2

− λ2− 1

2

√25J2

H + 10JHλ+ 9λ2

Table SIII: Charge gap for different total particle numbers. Note that ∆ch can be expressed as U + δ∆ch(N, JH, λ).We do not specify f(JH, λ) because it is difficult to express it in a simple form. Numerical values are used to obtain

Fig. S3.

4

0

2

4

6

8

10

1 2 3 4 5

(a)

Uc /D

n

(b) N = 1

0 0.5 1JH

0

0.5

1

λ

-3

-2

-1

0

δ∆ch

(c) N = 2

0 0.5 1JH

0

0.5

1

λ

-3

-2

-1

0

δ∆ch

(d) N = 3

0 0.5 1JH

0

0.5

1

λ

-1

0

1

2

δ∆ch

(e) N = 4

0 0.5 1JH

0

0.5

1

λ

-3

-2

-1

0

1

2

δ∆ch

(f) N = 5

0 0.5 1JH

0

0.5

1

λ

-5

-4

-3

-2

-1

0

δ∆ch

Figure S3: (a) Critical interaction strength Uc as a function of electron filling. From left to right within a group ofthree bars, the bars represents the Uc for (λ, JH/U) = (0, 0.15), (0.25, 0.15), and (0.25, 0.25), respectively. (b)-(f)Density plot of δ∆ch as a function of JH and λ for different total particle number N . δ∆ch(N, JH, λ) is defined as

∆ch − U .

n S L J1 1/2 1 3/22 1 1 23 3/2 0 3/24 1 1 05 1/2 1 1/2

Table SIV: Summary of Hund’s rules including the third law.