Signatures of Mottness and Hundness in archetypal correlated metals

Physical properties of multi-orbital materials depend not only on the strength of the effective interactions among the valence electrons but also on their type. Strong correlations are caused by either Mott physics that captures the Coulomb repulsion among charges, or Hund physics that aligns the spins in different orbitals. We identify four energy scales marking the onset and the completion of screening in orbital and spin channels. The differences in these scales, which are manifest in the temperature dependence of the local spectrum and of the charge, spin and orbital susceptibilities, provide clear signatures distinguishing Mott and Hund physics. We illustrate these concepts with realistic studies of two archetypal strongly correlated materials, and corroborate the generality of our conclusions with a model Hamiltonian study.

Many realistic correlated materials are multi-orbital systems with strong on-site atomic-like interactions described by Coulomb repulsion and Hund's coupling. The role of Coulomb repulsion in inducing the correlation effects has been extensively studied [1], and that of Hund's coupling was less discussed. Hund's coupling has a contrasting effect on the effective Coulomb repulsion depending on the occupancy in the correlated shell [2,3]. And Hund's coupling strongly affects the energy scale characterizing the screening of local moments in multi-orbital models [4,5]. We refer to Ref. [6] for a review of these two effects. The importance of Hund's coupling in realistic materials starts to be appreciated in investigations on iron-based superconductors and ruthenates [7][8][9], which are then coined "Hund's metals". However defining signatures that identify the origin of correlation effects, Mott physics or Hund's physics, are sparse. Theoretical studies highlights some features of "Hund's metals", such as the low coherence scale, the large charge fluctuation and the separation of spin and orbit degree of freedom [6][7][8][9][10][11][12][13], however, whether these features can be used to distinguish a Hund's metal from a metal of Mott type is not known.
In this work, we point out that Mott physics and Hund's physics manifest in different temperature dependence of correlated spectra and of several local quantities. This study is built on the successes of density functional theory plus dynamical mean-field theory (DFT+DMFT) [14][15][16] in describing the available experimental measurements in two archetypal materials, V 2 O 3 [17][18][19][20][21][22] and Sr 2 RuO 4 [9,23,24]. We find that a clear signature of Mottness in V 2 O 3 in the high temperature local spectra of correlated orbitals as a two-peak structure with a pseudogap at the Fermi level, while in Sr 2 RuO 4 the correlated spectra has only a single incoherent peak up to very high temperature. As the temperature decreases, the coherent resonance in V 2 O 3 emerges from the pseudogap regime, and in Sr 2 RuO 4 it is built from large broad hump of spectral weight. The local charge susceptibility and local charge fluctuation are suppressed significantly along with the disappearance of resonance in V 2 O 3 , while in Sr 2 RuO 4 it is only slightly reduced and retains a large value up to high temperature. In V 2 O 3 both the local spin and orbital susceptibility exhibit a Curie behavior at the same temperature T M at which the coherence resonance disappears. Thus the onset of screening of the spin degree of freedom T onset spin coincides with the onset of screening of the orbital degree of freedom T onset orb in Mott system V 2 O 3 . In Sr 2 RuO 4 the local spin and orbit susceptibility exhibit Curie behavior at much higher temperature than those in V 2 O 3 , and a spin-orbital separation is seen in their characteristic temperatures T onset orb T onset spin . We further identify that the orbital degree of freedom is completely screened at a temperature T cmp orb . In both materials it is much larger than the corresponding Fermi liquid temperature T F L , which is presumably the scale T cmp spin at which the spin degree of freedom is fully screened. Therefore the completion of screening features by a spin-orbital separation in both materials. The entropy of the correlated atom in both materials exhibits a plateau of ln 9, a large value expected for a high spin (S = 1) state with large contribution from orbital degreedom of freedom. In V 2 O 3 the plateau is reached at the same temperature scale T M .
We summarize briefly the basic facts of these two materials. V 2 O 3 , a paramagnetic metal at ambient conditions, is proximate to an isostructural Mott transition that can be induced by slightly Cr-doping, and a temperaturedriven magnetic transition [25]. It exhibits a Fermi liquid behavior at low temperature when antiferromagnetism is quenched by doping or pressure [25]. Sr 2 RuO 4 , on the other hand, is a paramagnetic metal far away from MIT [26]. As temperature decreases it shows a Fermi liquid behavior and eventually becomes superconducting at very low temperature [27]. Despite the very different distance to a Mott insulating state, both materials have large specific heat coefficients in their Fermi liquid states [25,27]. In both materials the observed Fermi liquid scales are extremely , is featured by a suppression at a characteristic temperature TM = 1000K (indicated by arrow) in V2O3 (a), while it evolves smoothly in Sr2RuO4 (b). As temperature decreases, in V2O3 the coherent resonance of both e π g and a1g orbitals emerges from the pseudogap regime with low density of states between two incoherent peaks (c,d), and in Sr2RuO4 the coherent resonance of both d xz/yz and dxy orbitals emerges from a single broad incoherent peak with large finite density of states at the Fermi level (e,f).
low (around 25K [25,28]), much smaller than the bare band energy or interaction parameters (order of eV). Pronounced quasiparticle peaks are observed in both materials using photoemission spectroscopy [29][30][31][32], and large values of mass renormalization are seen in Sr 2 RuO 4 in various measurements [33][34][35]. Notably the local physics on V/Ru sites are similar, with nominally two electrons/holes in three t 2g orbitals. Due to crystal field of surrounding oxygen, the t 2g orbitals of V are split into e π g orbitals with two-fold degeneracy and a 1g orbital, while those of Ru are split into xz/yz orbitals with two-fold degeneracy and xy orbital. Two electrons (holes) in three orbitals favors a spin-triplet S = 1 atomic state because of Hund's coupling in both V 2 O 3 [17,[36][37][38] and Sr 2 RuO 4 [9].
The Mottness in a Mott insulating state manifests as a significant gap in the spectra that separates into the socalled "Hubbard" bands. To induce metallicity from a Mott state is to build coherent resonance from a regime where the density of states is small, either from the center of the gap or the edges of Hubbard bands. Vice versa, a gap (or pseudogap) between incoherent spectra is restored starting from a correlated metal of Mott-type when the coherent resonance is destroyed. This is known in the context of Metal insulator transition (MIT) induced by doping or tuning the ratio of the interaction versus the bandwidth [1]. It is also true when the coherence resonance is destroyed gradually by increasing temperature. For example, in the model study of doped Mott insulator at infinite dimensions, the spectra at high temperature exhibits a two-peak structure, a reminiscent of Mott gap, as the "resilient quasiparticles" disappear [39]. We hypothesize that a gap/pseudogap regime appearing in the local spectra when the coherent resonance are destropyed is a defining signature of Mottness in general situations and test it in the metallic phase of V 2 O 3 .
We compute the spectra of the relevant correlated orbitals in V 2 O 3 and Sr 2 RuO 4 up to high temperature. We focus first on the density of states at the Fermi level, taking D(iω 0 ) = − 1 π G(iω 0 ) (ω 0 is the first matsubara frequency, G is the local Green's function computed) as an estimation. The temperature dependence of D(iω 0 ) for e π g and a 1g orbitals in V 2 O 3 is depicted in Fig. 1(a). The results show that both orbitals share a characteristic temperature T M = 1000K: D(iω 0 ) is fairly flat at temperature above T M which implies an approximate rigid picture of the spectra. Below T M D(iω 0 ) gradually acquires a larger magnitude in both orbitals as temperature is lowered,, signaling the forming of coherent resonance. Interestingly it increases monotonically in e π g orbital, but in a 1g orbital it increases and then decreases a little. Thus at low temperature the density of states at the Fermi level is dominated by e π g character. We emphasize that the evolution of D(iω 0 ) is smooth and a first-order Metal-insulator transition is not involved.
The estimated density of states D(iω 0 ) of d xz/yz and d xy orbitals in Sr 2 RuO 4 exhibit very different temperature dependence, as depicted in Fig. 1(b). For both orbitals they decrease continuously as temperature increases, showing no signs of flattening thus no characteristic temperature as found in V 2 O 3 . The decay of D(iω 0 ) is very slow, even at extremely high temperature it retains a value larger than the value of V 2 O 3 above T M .
We now look at the real frequency spectra D(ω) = − 1 π G(ω). This is obtained by analytically continuing the computed Matsubara self energy and then computing the local Green's function. The results of V 2 O 3 is depicted in Fig. 1(c)(d). At low temperature both orbitals show a coherent resonance forming by quasiparticle bands. In e π g orbital the resonance is peaked at the Fermi level while in a 1g orbital it is slightly above the Fermi level. As The computed local charge susceptibility χ chg (iω = 0) (a) and local charge fluctuation ∆N 2 (b) of V2O3 and Sr2RuO4. Both χ chg (iω = 0) and ∆N 2 in Hund's system Sr2RuO4 are large and weakly temperature-dependent, and those in V2O3 are much smaller and strongly temperature-dependent. The arrows indicate that in V2O3 the minimum of the local charge susceptibility and fluctuation occurs at the same temperature scale TM determined from the local spectra evolution.
temperature is increased, the magnitude of the coherence peak in both orbitals decreases gradually, and the missing weight is transferred to two broad hump peaked around -1eV and 2eV. Eventually a pseudogap opens in the spectra of both orbitals. The temperature evolution of zero frequency density of states in both orbitals is consistent with the D(iω 0 ) discussed above. The characteristic temperature for the onset of coherence resonance forming is roughly consistent with T M = 1000K determined above as well. In addition the a 1g quasiparticle peak slightly moves away from the Fermi level and reduces its density of states at the Fermi level when the temperature is lowered, which explains the the nonmonotonic behavior of the corresponding D(iω 0 ) in Fig. 1(a). In Sr 2 RuO 4 the slow decay of the density of states near the Fermi level is clear from the real frequency spectra D(ω) = − 1 π G(ω), as shown in Fig. 1(e)(f). At low temperature, the spectra of both d xz/yz and d xy are similar to their corresponding DFT values with a renormalized band-width. The d xz/yz spectra has one-dimensional character with two peaks corresponding to the quasiparticle band edges (Fig. 1(e)) and the d xy orbital has two-dimensional character and shows a pronounced peak of Van Hove singularity. When temperature increases, the sharp peaks in the spectra are broadened gradually, and the overall spectra evolves into a single broad bump. Only a small fraction of spectral weight is transferred from a 1eV range around the Fermi level to higher frequency. The bump seems quite rigid at high temperature, that except a shift in its position its shape is almost unchanged when the temperature is above 2300K. This is very different from that in V 2 O 3 which shows a two-peak structure at high temperature.
Therefore from the temperature dependence of the local spectra we can distinguish the Mott physics in V 2 O 3 and the Hund's physics in Sr 2 RuO 4 . The coherent resonance of V 2 O 3 emerges from a pseudogap regime with very low density of states between incoherent spectra at high temperature. This is consistent with the widely held belief that Mott physics governs V 2 O 3 . And Mott physics is governed by a single characteristic temperature scale T M which indicates the onset of formation of coherence resonance. This is in contrast with Sr 2 RuO 4 which is featured by a single incoherent peak that retains a large value at the Fermi level at very high temperature. The two roads for forming coherent Fermi liquid at low temperature is one of the main results of this work.
The static local charge susceptibility, defined as χ chg (iω Fig. 2(a). χ chg (0) of Sr 2 RuO 4 is large in value and exhibts a weak temperature dependence with a reduction only about 10% even at the highest temperature studied. On the other hand, χ chg (0) in V 2 O 3 is much smaller than that in Sr 2 RuO 4 , and it has a significant temperature dependence with a characteristic temperature 1000K, same as T M determined above. As the temperature increases from low temperature, χ chg (0) decreases quickly and the reduction reaches around 50% at T M , and then it becomes almost flat. Similar temperature is seen in the temperature dependence of the local charge fluctuation ∆N 2 = N 2 d − N 2 d , as shown in Fig. 2(b): large charge fluctuation with fairly weak temperature dependence is seen in Sr 2 RuO 4 while in V 2 O 3 the charge fluctuation is much weaker and features a suppression at T M . The suppression of the local charge fluctuation accompanied by the gap/pseudogap opening is seen as well in the single-band doped Mott insulator [39]. Now we turn to the static local spin and orbit susceptibility that are defined as χ spin = β 0 S z (τ )S z (0) dτ and χ orb = results are depicted in Fig. 3(a)(b). In V 2 O 3 both spin and orbital susceptibility exhibit Curie behavior ( Fig. 3(a) upper panel), that is, T χ spin and T χ orb approximately a constant at high temperature. Notably the Curie behavior starts at the same characteristic temperature, T M = 1000K, determined above from the local spectra evolution. Thus the spin and orbital degree of freedom start to be screened simultaneously with the formation of coherence resonance in the prototype Mott system V 2 O 3 , T onset spin = T onset orb = T M . In Sr 2 RuO 4 ( Fig. 3(b) upper panel), the Curie behavior appears in the spin susceptibility at around T onset spin 2300K, a scale much higher than that in V 2 O 3 . The orbital susceptibility does not seem to show a well-defined Curie behavior at the highest temperature studied (T onset orb ≥ 6000K), but T χ orb seems approaching gradually to a constant. This is evidence of spin-orbit separation in Sr 2 RuO 4 : the screening of the orbital degree of freedom starts at much high temperature than that of the spin degree of freedom T onset orb T onset spin . In Hund's metal the spin-orbital separation has been pointed out in numerical studies of the frequency dependence of local self energy and susceptibilities [10,12] and in analytical estimate of the Kondo scale [11]. Here our results reveal that it exists in the temperature domain. The onset of the Curie behavior in the spin/orbit susceptibility, or the onset of screening of spin/orbital degree of freedom, is very different in V 2 O 3 and Sr 2 RuO 4 , which is another signature that distinguishes "Mottness" and "Hundness".
It is interesting to note that the full screening of spin and orbital susceptibility appear at very different temperature scale in both V 2 O 3 and Sr 2 RuO 4 . The orbital susceptibility approaches more or less constant at low temperature in both materials. This characteristic temperature is about T cmp orb 300K in V 2 O 3 (Fig. 3(a) lower panel), which is much smaller than that in Sr 2 RuO 4 T cmp orb 1100K (Fig. 3(b) lower panel). One the other hand, the spin susceptibility increases with decreasing temperature, and is not fully screened even at the lowest temperature studied in both materials. This is consistent with the experimental observations that in both materials T F L are as low as about 25K, and T F L provides an estimation for T cmp spin at which the spin degree of freedom is fully screened. The low scale of T cmp spin is understandable since in both materials a large spin S = 1 formed by two electrons (holes) in the t 2g shell due to Hund's coupling, is difficult to be fully screened [6,7]. Therefore the orbital degree of freedom is fully screened at a much higher temperature than the spin degree of freedom T cmp orb T cmp spin . We emphasize that, the spin-orbital separation in the completion of screening is not unique to "Hund's metal" but also seen in multi-orbital Mott material. . A plateau about ln(9) is seem in both materials, starting at about 1000K in V2O3 and about 2000K in Sr2RuO4, as indicated by solid arrows. As temperature decreases, the entropy crosses ln (3) continuously, at about 400K in V2O3 and about 750K in Sr2RuO4, as indicated by unfilled arrows. The pentagon denotes an estimation of the entropy in V2O3 from experimental measurements [25]. The blue line with diamond indicates a Fermi liquid approximation of the electronic entropy of Sr2RuO4 S = γT , taking the specific heat coefficient γ = 38mJ/molK 2 at low T [41].
We note that our computed spin susceptibility of Sr 2 RuO 4 is consistent with early result in a narrower temperature range [40]. Finally we calculate the impurity entropy and the results are depicted in Fig. 4. In both V 2 O 3 and Sr 2 RuO 4 , as the temperature increases, the impurity entropy increases first quickly, and then reaches a plateau starting at about T M = 1000K in V 2 O 3 and around T = 2000K in Sr 2 RuO 4 . Notably the value of the plateau is approximately ln 9 for both materials. This indicates that both spin and orbital degree of freedom contributes significantly to the entropy in both materials, and the most relevant atomic states likely have large spin S = 1 and large effective orbital angular momentum L = 1 as discussed in models [6]. It is therefore not surprising that in V 2 O 3 , the entropy plateau is reached at the same characteristic temperature T M = 1000K, where the Curie behavior appears in the spin and orbital susceptibility thus the spin and orbital degree of freedom are fully active. In Sr 2 RuO 4 the entropy reaches the plateau at much larger temperature than that in V 2 O 3 , and the characteristic temperature T = 2000K is close to the temperature at which the Curie behavior appears in the spin susceptibility T onset spin . We note at this temperature the orbital degree of freedom is close to fully active although it does not exhibit a Curie behavior explicitly as shown by the T χ orb (0) (Fig. 3(b)), therefore the entropy contribution due to orbital degree of freedom is large. In addition the large charge fluctuation in Sr 2 RuO 4 may contribute to the entropy accumulation as well. As temperature decreases, the entropy decreases continuously and crosses a value ln(3) expected for an unscreened S = 1 atomic state. We note that the crossing occurs at a temperature when the orbital degree of freedom is (almost) fully screened: in V 2 O 3 it occurs at about 400K, and in Sr 2 RuO 4 it occur at about 750K (Fig. 4), and both temperatures are comparable to the screening scale T cmp orb in the corresponding materials. This is consistent with the observation that the orbital degree of freedom is fully screened at much higher temperature than the spin degree of freedom in both materials. The spin degree of freedom is responsible for the large entropy found in the extended temperature where the orbital degree of freedom is frozen. Overall, these results suggest that a strong correlation between the entropy accumulation and the unscreening of spin and orbital degree of freedom.
Both V 2 O 3 and Sr 2 RuO 4 have a large entropy at the lowest temperature considered, which highlights the strong correlated nature of these two materials. Interestingly the large values of entropy are found in experimental measurements. In V 2 O 3 the entropy change across the transition from a metallic state to the antiferromagnetic state at T = 150K is as large as 0.65k B [25]. Assuming that the electronic entropy of the ordered state is zero [42], this value provides an estimation of the entropy in metallic V 2 O 3 and fits very well in our computed impurity entropy (Fig. 4). In Sr 2 RuO 4 as a first order approximation, the electronic entropy of Sr 2 RuO 4 can be written as S = γT where γ is the specific heat coefficient in the Fermi liquid regime. The approximated entropy matches our computed result at about 100K. These agreements suggest that the large entropy in both materials are mainly due to the local correlated electrons. In connection with the evolution of the local spectra, we see that in a large temperature range beyond the Fermi liquid scale the large entropy is accompanied by coherent resonance, a feature of "resilient quasiparticles" [39].
We remark that the large total orbital angular momentum L = 1 highlights the importance of that the orbital degree of freedom in the electronic structure. Its role has been emphasized in Sr 2 RuO 4 [9,40], however is not much discussed in V 2 O 3 . The large orbital angular momentum in V 2 O 3 is a direct consequence that the a 1g orbital is partially filled and contributes to the atomic degree of freedom [17,22,43]. It is therefore very unlikely that the a 1g is effectively excluded by a correlation-enhanced crystal field splitting as suggested by several studies [19][20][21]. This is consistent with the conclusion drawn in recent angular-resolved photoemission spectroscopy measurement [44]. This finding shed light on the understanding of the nature of Mott transition in V 2 O 3 .
In conclusion, we revealed contrasting signatures of Mottness and Hundness in two archetypal materials V 2 O 3 and Sr 2 RuO 4 , in the coherence resonance forming, the temperature dependence of charge, spin, orbital susceptibility as well as impurity entropy. Mott physics and Hund's physics manifest in the process that the atomic degrees of freedom at high energies evolve towards low energies to form fermionic quasiparticles. We highlight the importance of recognizing the existence of four scales which characterize the onset and the completion of screening of the spin and the orbital degrees of freedom. In the Mott system V 2 O 3 , the onset of the screening of the spin and orbital degree of freedom coincide and is accompanied by the coherence resonance forming, while they occur in well separted stages in Sr 2 RuO 4 . These insights will be useful in interpreting experimental measurement on correlated metals, and identifying the origin of correlations.
Method: We use the all-electron DMFT method as implemented in Ref. [45] based on WIEN2k package [46] and the continuous-time quantum Monte-Carlo (CTQMC) impurity solver [47,48]. We use projectors within a large (20eV) energy window to construct local orbitals, thus the oxygen orbitals hybridizing with the d orbitals are explicitly included. With such a large energy window the resulting d orbitals are very localized. We treat t 2g dynamically with DMFT and all other states statically, and no states are eliminated in the calculations. The nominal "double counting" scheme with the form Σ DC = U (n imp − 1/2) − 1 2 J(n imp − 1) is used, where n imp is the nominal occupancy of d-orbitals. The onsite interactions in terms of Coulomb interaction U and Hund's coupling J are chosen to be (U, J) = (6.0, 0.8)eV for V in V 2 O 3 and (U, J) = (4.5, 1.0)eV for Ru in Sr 2 RuO 4 . The impurity entropy is computed with accuracy by integrating the impurity internal energy up to high temperature, following the formalism in Ref. [49]. Our DFT+DMFT setup was successful in describing the correlation effects in both materials [22,24]. We captured the phase diagram of V 2 O 3 with Mott-type MIT, and our computed electronic structure is consistent experimental measurements [22]. We captured the large mass enhancement and the electronic structure of Sr 2 RuO 4 which are in agreement with experimental measurements [24] and with other DFT+DMFT studies [9,23]. In addition our studies [22,24] captured the transport and optical properties of both materials in the experimental accessible temperature regime. These successes gave us confidence to carry out studies up to even higher temperature that not accessed by experimental measurements.
Work by X.