Interplay between spin-orbit coupling and van Hove singularity in the Hund’s metallicity ofSr2RuO4

Hyeong Jun Lee,1, 2, 3 Choong H. Kim,2, 3, ∗ and Ara Go1, †

1Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea2Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Republic of Korea

3Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea(Dated: February 13, 2020)

We investigate the dynamical properties of Sr2RuO4 at zero and very low temperature using density func-tional theory plus dynamical mean-field theory with an exact diagonalization solver. By considering rotationallyinvariant local interaction, we examine how Hunds coupling and spin-orbit coupling affect the correlated natureof the system. In the absence of Hund’s coupling, the system shows a Fermi liquid behavior over the entirerange of temperatures we consider. We confirm that the Fermi liquid persists at zero temperature even withnonzero Hund’s coupling; however, at sufficient temperatures Hund’s coupling significantly reduces the Fermiliquid regime and the system evolves into a typical Hund’s metal. At the bare electronic occupancy of Sr2RuO4

(t42g), a stronger Hund’s metallicity accompanies a larger long-time correlator. Remarkably, electron dopingfurther destabilizes the Fermi liquid even though the long-time correlator and magnetic fluctuations decreaseupon doping. This suppression of the Fermi liquid is driven by the van Hove singularity above the Fermi levelin Sr2RuO4, combined with an enhanced Van Vleck susceptibility by spin-orbit coupling. Such findings pointto the important role that electronic structure plays in the behavior of Hund’s metals, in addition to magneticfluctuations.

Strong electronic correlations create numerous fascinatingemergent phenomena that are hardly understood by simplesingle-particle pictures. The transition metal compounds withpartially filled d-orbitals have been a captivating playgroundto investigate correlation effects, including high-Tc supercon-ductivity, metal-insulator transitions, and so on [1]. In caseswhere local electronic interaction dominates such systems, theessential ingredients to scale the correlations are the HubbardU and Hund’s coupling JH [2]. The on-site Coulomb repul-sion U governs the Mott physics, yielding an optimal num-ber of electrons in the system and confining the electrons toeach site with an increasing energy cost for doubly occupiedorbitals. On the other hand, Hund’s coupling affects inter-orbital exchange and controls the electronic distribution overconfigurations with a given number of electrons.

The recent discovery of Hund’s metals promotes the im-portance of Hund’s coupling in correlated multi-orbital sys-tems [2–4]. While Hund’s coupling can appear to reducecorrelations by suppressing the Mott phase, it also enhancesanother type of correlation that exhibits various character-istic behaviors—for example, large renormalization massesdepending on filling [5, 6], spin (or J) freezing [7, 8], andnon-Fermi liquid [7]. Furthermore, a strong Hund’s couplingin multi-orbital systems induces significant orbital-selectivebehaviors [9–13]. As the name implies, the aformentionedcharacteristics of Hund’s metals originate from local corre-lation effects. Interplay between local correlation and elec-tronic structure often develops exotic phenomena, such as theunconventional superconductivity in iron-based superconduc-tors [14].

Strontium ruthenate, Sr2RuO4, is a well-known correlatedmetal [15–28] that undergoes a crossover from Hund’s metalto Fermi liquid as the temperature drops [29–31]. It exhibitsunconventional superconductivity below 1.5 K [32–36], the

precise mechanism of which is still under debate [37, 38]. Re-cent experiments observing that Tc is enhanced by strain [38–43] have puzzled the relation between the van Hove singu-larity (VHS) and superconductivity. A good starting point toclarify the effect of the VHS, then, is to investigate how thenormal state of Sr2RuO4 reacts when the VHS approaches theFermi level [23, 44–47].

Pioneering studies have adequately explained the normalstate properties of Sr2RuO4 via three-band models [48–61]employing dynamical mean-field theory (DMFT) in combi-nation with density functional theory (DFT). Research hasshown that Hund’s coupling causes orbital-dependent correla-tions [15–17] and bad metallic behaviors at nonzero tempera-tures, which are prototypical features of Hund’s metals. An-other important local interaction in Sr2RuO4 is the spin-orbitcoupling (SOC) that lifts the degeneracy between jeff = 1/2and 3/2 (where jeff is the effective total angular momen-tum [62]) [63] and also modifies the Fermi surface [28, 53–56, 64, 65]. Because of the extreme computational costs aswell as the fermionic sign problem though, the nonperturba-tive nature of electronic correlations at nonzero but very lowtemperature has not yet been explored.

In this Letter, we study the ground state and very low tem-perature physics of Sr2RuO4 via the DFT+DMFT method [66,67] with an exact diagonalization solver [68]. We include therotationally invariant Slater–Kanamori interaction [69] with aSOC of λSOC = 0.1 eV on top of the tight-binding Hamil-tonian [70] of DFT-based maximally localized Wannier or-bitals [71] obtained from the Vienna Ab-initio SimulationPackage [72]. We construct a comprehensive phase diagramto clarify the effect of Hund’s coupling at various tempera-tures in the presence of SOC, and further investigate how thesystem evolves across the VHS by applying electron doping.We report a strong suppression of the Fermi liquid behavior

arX

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l] 1

2 Fe

b 20

20

2

(a)

(FL)

(b)

FIG. 1. (a) Power exponent α of the imaginary part of the Matsubaraself-energy and (b) long-time correlator χS

zz(τ = β/2) in the T -JHplane. The dashed line in (a) denotes the crossover between Fermiliquid and Hund’s metal regimes, and the yellow star marks a realisticvalue of JH for Sr2RuO4.

by Hund’s coupling, where the suppression is reinforced bythe VHS upon doping. We used three (six, considering spin)correlated and nine (eighteen) bath orbitals to satisfy the min-imum requirement to describe multi-orbital physics [73–76].

The layered structure of Sr2RuO4 induces a strong orbitalselectivity with the aid of Hund’s coupling [50]. Based on theself-energy, the dxy orbital is more strongly correlated in com-parison to the other two t2g orbitals (see Supplementary In-formation for more details [70]). Since the dxy orbital experi-ences a drastic change in correlations, its self-energy Σxy(iω)is a good measure for identifying whether it follows Fermi liq-uid behavior or not [70]. We present the power exponent of theimaginary part of the self-energies in Fig. 1(a) as Hund’s cou-pling and temperature vary. We obtain power exponent α byfitting the self-energies to ImΣ(iωn) ∼ ωαn with the two low-est values of ωn. The linear power of the self-energy (markedin blue in Fig. 1(a)) with respect to the Matsubara frequency,i.e. ImΣxy(iω) ∼ ω, indicates that the system is in the Fermiliquid phase. This phase is stable over the entire range of tem-perature that we can access when Hund’s coupling is zero. Wedefine the Fermi liquid temperature TFL for a given JH as thetemperature whose corresponding value of α equals the α atT = 25 K for JH = 0.4 eV, where T = 25 K is based on ear-lier experimental works [29–31]. To present how the Fermiliquid regime evolves as a function of JH , we mark TFL witha dashed line in Fig. 1(a). TFL decreases once Hund’s cou-pling is turned on, but the Fermi liquid ground state persistsup to JH = 0.7 eV. At JH = 0.4 eV, which is believed to bea typical Hund’s coupling strength in Sr2RuO4 [50, 55], thelinear power seems to hold below 10 K in our result. Thisscale is comparable with previous experimental results fromresistivity curves [29–31].

Deviation from Fermi liquid behavior has been extensivelystudied with three-band models, where the long-lived localmagnetic moment may promote the scattering rate of elec-trons [7, 75, 77, 78]. Development of the long-lived paramag-

(a) (b)

FIG. 2. Doping dependence around the van Hove singularity. (a)Power exponent of the imaginary part of the Matsubara self-energyand (b) long-time correlator χS

zz(τ = β/2) in the T -nel plane. In-set: Projected density of states on t2g for the non-interacting case.Vertical dotted lines denote the Fermi level when nel = 4 and 4.35around the VHS of the dxy band.

netic moment (or so-called moment freezing) can be observedin the related magnetic susceptibility as a non-vanishing long-time correlator. The long-time correlator is defined as

χSzz(τ = β/2) = 〈Sz(β/2)Sz〉, (1)

where β is the inverse temperature, and Sz is the z-componentof the total angular momentum operator. To examine the ori-gin of the power reduction in Sr2RuO4, we compute the mag-netic susceptibility as displayed in Fig. 1(b). We observe aclear similarity of the T - and JH -dependencies between thetwo quantities in Fig. 1 (a) and (b). When the exponent in-dicates Fermi liquid, the long-time correlator is vanishinglysmall. We note here that the mere existence of the long-livedmoment does not necessarily lead to time-reversal symmetrybreaking; rather, the long-lived moment coexists with its time-reversal partner and the system remains paramagnetic.

We have observed that Hund’s coupling suppresses theFermi liquid regime at t42g , showing consistent behavior withthe long-time correlator. This power exponent deviation andTFL reduction in Sr2RuO4 are reminiscent of the JH -inducedspin-freezing crossover in the three-band model at t42g [7, 8].However, there is a prominent difference upon electron dop-ing due to the peculiar electronic structure of Sr2RuO4. Pre-vious model calculations for the Bethe lattice showed thatFermi liquid behavior is enhanced by electron doping on topof t42g [7, 8], while earlier DFT+DMFT works focused on theHund’s metallicity of bare Sr2RuO4 [50–61]. In Sr2RuO4,though, there is a nontrivial doping dependence of the powerexponent. In Fig. 2, we compare the power exponent fromΣxy and the long-time correlator for electron-doped cases.Unlike the t42g case shown in Fig. 1, the two quantities re-veal a quantitatively distinct behavior for nel > 4 (where nel

is electron occupancy). Here, TFL becomes minimal aroundnel ∼ 4.1, while the long-time correlator monotonically de-

3

FIG. 3. Spectral density of (a) the Fermi surface and its projectiononto the (b) t2g- and (c) jeff-basis states. (d) Dispersion along differ-ent angles θ between the ΓM and ΓX lines depicted in (a). All dataare obtained with U = 2.5 eV, JH = 0.4 eV, and λSOC = 0.1 eV at T= 23 K.

creases in the regime shown in Fig. 2. This can be attributedto the electronic structure of Sr2RuO4 that has a VHS slightlyabove the Fermi level in the absence of doping [79, 80]. Thissupports that a mechanism separate from spin-freezing leadsthe power deviation. The VHS strongly suppresses hybridiza-tion and amplifies the correlation effect delivered to the self-energy. At realistic parameters of Sr2RuO4, the lowest TFL

is achieved between nel = 4 and the doping where the VHScrosses the Fermi level. The lowest TFL moves to a highernel if SOC is turned off [70]; this originates from the VHSand the enhanced magnetic fluctuations by SOC, which willbe discussed in more detail later in this Letter.

To locate the realistic parameters of Sr2RuO4, we computethe spectral functions on the Fermi level and compare themto experiments. We present the Fermi surface and low en-ergy hole excitation spectra of Sr2RuO4 in Fig. 3, with pa-rameters U = 2.5 eV and JH = 0.4 eV. This reproducesthe Fermi surface geometry previously obtained from ARPESspectra [18–25, 28], which is consistent with earlier theoreti-cal works [28, 53, 55]. The strong correlation effects have anintimate connection with the Fermi surface spectra. The ef-fective masses from our results also satisfy m∗γ > m∗β > m∗α[81] in good agreement with experimental results [17, 23, 82]and theoretical works [50, 51, 54, 55]. Orbital selectivity nearthe Fermi level brings us the pocket-dependent scattering ofthree sheets, α, β, and γ, as marked in Fig. 3(a).

Another indispensable ingredient to explain the spectralproperties here is the SOC [55, 63]. While SOC is known tobe essential to produce the three pockets on the Fermi surfaceas discussed in previous studies [64, 65], the role of SOC indescribing low-lying excitation has yet to receive much focus.

(b)

(a)

FIG. 4. Spin susceptibilities as a function of temperature in thepresence or absence of SOC of (a) long-time correlator χS

zz(β/2),and (b) static spin susceptibility χS

zz and its Van Vleck contributionχS;VVzz .

As SOC introduces a mixing between t2g orbitals, the strongorbital dependence of the self-energy appears controversial tounderpin the importance of SOC. In spite of this, here we findthat the enhanced SOC affects the character of the Fermi sur-face pocket from the orbital projection of the spectra.

We project the spectral weights of the Fermi level inFig. 3(a) onto the t2g and spin-orbit coupled (jeff ) basis statesin Fig. 3(b) and (c), respectively. The two 1D-like spectra as-sociated with yz and zx are discontinuous near the ΓX linesdue to orbital mixing by SOC. Interestingly, the Fermi pocketsare well-defined on the spin-orbit eigenstates in the realisticparameter regime. The clear decomposition by the jeff basis isnot only limited to the Fermi level but also extends to an en-ergy window of a few hundred meV. In Fig. 3(d), we presentspectral weights as a function of energy along a few differentpaths between ΓM and ΓX (marked in Fig. 3(a)), which aredirectly related to a recent ARPES experiment [28]. The scat-tering of the γ pocket is the largest because of the strong xycomponents in (jeff , jz) = ( 3

2 ,± 12 ). The strong jeff character

remains by the enhanced SOC from correlation effects. Whilethe bare SOC is approximately 0.1 eV on Ru, local interactionboosts the splitting induced by SOC [83]. The effective SOCis thus twice as large as the bare value [53], which is com-parable to the Hund’s coupling of realistic parameters in ourcalculation. This indicates that the low energy excitation nearthe Fermi level should be described based on jeff rather thanon t2g degrees of freedom.

Spin-orbit coupling is also vital to understand the magneticresponse of this system. We compute the static susceptibility

4

χSzz(ω = 0) as

χSzz(ω) =βδ(ω)∑

a,b

e−βEa |〈a|Sz|b〉|2δ(Ea − Eb)

+∑

a,b

e−βEa |〈a|Sz|b〉|2(

1

ω + ωba− 1

ω − ωba

),

(2)

where summations run over the energy eigenstates, Ea is theeigenvalue of the ath energy eigenstate, and ωba = Eb − Ea.The first term in Eq. (2) reflects the contribution from thelong-lived moment, which is stated as the Curie term, whereasthe second term expresses the Van Vleck (VV) component.Figure 4(a) displays the long-time correlator that governs theCurie term in the susceptibility. The Curie term is gener-ated by energetically degenerate eigenstates and directly af-fected by the long-lived magnetic moment. SOC lifts thedegeneracy of the eigenstates and is supposed to reduce theCurie term and the long-time correlator in t42g . This resultsin a reduction of the long-time correlator in the Hund’s metalregime (T>TFL), as shown in Fig. 4(a). On the other hand,the degeneracy-lifting introduced by the SOC has an oppo-site effect on the VV term. The VV contribution comes fromthe angular-momentum-mediated virtual excitations betweenthe eigenstates. The newly contributing excitations betweenthe SOC-lifted multiplets increases the VV term, as presentedin Fig. 4(b). In the range of T we consider, susceptibilityis boosted by SOC due to the prominent VV contribution.We note that the VV contribution dominates ∆χ = χ(ω =0) − βχ(τ = β/2), which measures the strength of fluctua-tions [84]. Hence, the enhanced VV at low temperature showsthe significance of SOC in the correlated nature of Sr2RuO4.Combined with the VHS, spin-orbit coupling makes the sys-tem reach its lowest TFL at smaller nel than expected whenbased solely on the VHS.

The temperature dependence of the two contributions ofEq. (2) shows a sharp contrast. As T increases, the long-timecorrelator rises while the VV counterpart decreases. Overall,the enhancement of the total susceptibility by SOC decreasesas a function of T . Based on the envelope, SOC appears tosuppress the susceptibility at higher temperatures. This wouldbe consistent with a previous study that reported a reducedspin susceptibility by SOC at T & 300 K [59].

In summary, we performed DFT+DMFT calculations withan exact diagonalization solver on Sr2RuO4. Considering therotationally invariant Slater–Kanamori interaction as well asspin-orbit coupling, we constructed a comprehensive phasediagram on the T − JH plane by extracting power exponentsof the self-energy. The ground state is a Fermi liquid, andthe system enters the Hund’s metal regime as the temperatureincreases, with Hund’s coupling suppressing the Fermi liquidtemperature. We identified the particular parameter set thatreproduces experimentally observed spectral functions. Theground state of Sr2RuO4 turned out to be very close to theHund’s metal regime in the phase diagram, reflecting its corre-

lated nature. We then applied electron doping to study how theVHS affects the correlated metal, and found that the VHS ex-tends the Hund’s metal regime, enhancing the scattering rateof the electrons, whereas the long-time correlator weakens asthe system is doped.

We also investigated the role of SOC by comparing two in-dependent calculations, with and without SOC included in theDMFT loop. When SOC is included, the Fermi surface andlow lying excitations are clearly decomposed in the jeff basis,indicating the fundamental importance of SOC in the low en-ergy excitation of this system. Furthermore, SOC revives theorbital degrees of freedom to affect the magnetic responses.More specifically, it reduces the long-time correlator and am-plifies the VV contribution in the susceptibility. Combiningthese two effects, SOC increases susceptibility at sufficientlylow temperature and reinforces the Hund’s metallicity by en-hancing magnetic fluctuations near nel = 4.

Our results call for further studies in various directions. Weobserved that VV susceptibility with SOC outperforms theCurie part as the temperature decreases; this reveals genuineeffects of SOC in correlated multi-orbital systems. By ana-lyzing the VHS and static susceptibility, we also showed theimportance that the electronic structure has on the correlationeffects in Sr2RuO4. Investigation into the interplay betweenthe VHS and superconductivity would be another promisingdirection for future works. From a technical perspective, itwould be interesting to test the role of additional bath orbitalsby employing adaptively trunctated Hilbert spaces [85–87] inhighly correlated regimes where Hund’s coupling is large.

We thank Andrew J. Millis and Philipp Werner for insight-ful comments and discussion. This work was supported bythe Institute for Basic Science under Grants No. IBS-R024-D1 (HJL and AG) and IBS-R009-D1 (CHK).

∗ chkim82@snu.ac.kr† arago@ibs.re.kr

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Supplemental Material for“Fermi liquid behavior and van Hove singularity in Sr2RuO4”

Hyeong Jun Lee,1, 2, 3 Choong H. Kim,2, 3, ∗ and Ara Go1, †

1Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea2Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Republic of Korea

3Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea

Tight-binding Hamiltonian

We obtain the tight-binding (TB) Hamiltonian HTB =∑i,j

∑µ,β t

ijµνc†iµcjν on the maximally localized Wannier functions

(MLWF) from the density functional theory (DFT) calculation with the PBEsol functional. The indices µ and ν denote the t2gorbitals. Some parameters between nearest neighbors and next-nearest neighbors are obtained as t±1,0,0xy,xy = −0.388, t±1,0,0yz,yz =−0.325, t0,±1,0yz,yz = −0.048, t±1,±1,0xy,xy = −0.130, t±1,±1,0yz,yz = −0.017, and t±1,±1,0xy,yz = 0.010. We introduce the spin-orbitcoupling (SOC), λSOC

∑i li · si (where i is index for electrons on a site), on the TB Hamiltonian. From fitting the band

structure of Wannier to the DFT bands having the SOC, the strength of the SOC, λSOC, is set to 0.1 eV (as shown in Fig. S1).We perform the dynamical mean-field theory (DMFT) calculation based on top of this TB Hamiltonian with jeff-basis states.

We also perform the LDA calculations and obtain its MLWF and TB Hamiltonian. It has slightly reduced bandwidths com-pared to that of the PBEsol. This gives only minor correction on the magnitude of self-energies for fixed U and JH of ourinterest in the DMFT calculation. We confirm that this correction does not affect our main result significantly. It is expected tohave a small shift of the critical Hund’s coupling strength associated with the resulting correation effects.

FIG. S1. (a) DFT and MLWF bands of Sr2RuO4 along the high-symmetry lines. (b) Density of states of MLWF bands on t2g .

Orbital-dependent correlations

From the DFT band structure and density of states, the effective model of Sr2RuO4 can be described by two degenerate dyzand dzx bands and the other dxy band. This difference between the bands enables Sr2RuO4 to have the orbital selectivity, whichis one of remarkable aspect of correlations effects driven by the Hund’s coupling. Figure S2(a) shows the orbital-depenentcorrelations in the self-energy on t2g . The imaginary part of the self-energy of the dxy is more enhanced than that of the dyz/dzx near ω = 0. This difference exists noticeably in low-energy excitations and maintains up to 0.5 eV.

The Matsubara-frequency self-energy near ω = 0 is associated with renormalized mass and renormalization factor whenits imaginary part follows linear (or Fermi liquid) behavior in ωn. From the T=0 data, we compute and compare the renor-malized masses m∗ on t2g and jeff defined as m∗ = (1− ∂ωImΣ(iω)|ω→0)

−1. We obtain (m∗( 32 ,

12 ),m∗

( 12 ,

12 ),m∗

( 32 ,

32 )

) '(6.03, 5.58, 4.11), and (m∗xy,m

∗yz,m

∗zx) ' (7.37, 4.21, 4.21) in Sr2RuO4 .

arX

iv:2

002.

0482

5v1

[co

nd-m

at.s

tr-e

l] 1

2 Fe

b 20

20

2

(b) (a) (c)

FIG. S2. (a) Imaginary part of Matsubara self-energy projected on the t2g-basis in Sr2RuO4 at T ' 10K. Inset shows the log-log plot withfitting lines showing different power exponents - 0.75 for dxy and 0.91 for dyz/dzx. Power exponent α of the dyz compoenent (b) in the T -JHplane and (c) in the T -n plane.

Comparison to the case in the absence of SOC

(b) (a) (c)

FIG. S3. Power exponent α of the imaginary part of the Matsubara self-energy (a) of the dxy-component and (b) of the dyz-component. (c)long-time correlator χS

zz(τ = β/2) in the T -n plane.