Signatures of spin-orbital states of {t_{2g}}^{2} system in the optical conductivity : The case of RVO_{3} (R=Y and La)
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Signatures of spin-orbital states of t g system in the optical conductivity: The case of R VO ( R =Y and La) Minjae Kim
1, 2, ∗ Centre de Physique Théorique, École Polytechnique,CNRS, Université Paris-Saclay, 91128 Palaiseau, France Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France (Dated: September 26, 2018)We investigate signatures of spin and orbital states of R VO ( R =Y and La) in the optical con-ductivity using density functional theory plus dynamical mean-field theory (DFT+DMFT). Fromthe assignment of multiplet state configurations to optical transitions, the DFT+DMFT reproducesexperimental temperature dependent evolutions of optical conductivity for both YVO and LaVO .We also show that the optical conductivity is a useful quantity to probe the evolution of the orbitalstate even in the absence of spin order. The result provides a reference to investigate spin andorbital states of t g vanadate systems which is an important issue for both fundamental physics onspin and orbital states and applications of vanadates by means of orbital state control. I. INTRODUCTION
Rare-earth vanadates, R VO ( R =rare-earth atom),are one of the most intensively discussed materials forthe nature of the interplay between the spin, orbital,and lattice degrees of freedom. Understanding thiscoupling between the spin, orbital, and lattice degreesof freedom in R VO is an important fundamental prob-lem, which has been a subject of debates regarding theorigin of spin and orbital ordering, electronic superex-change versus lattice distortion. Also, this spin-orbital-lattice coupling is an important ingredient in applicationsof R VO , such as in multiferroic materials and solarcells, by means of heterostructure engineering. Thelattice controlled spin state for R VO is a promising routeto multiferroicity, while the spin ordering driven longerlifetime of photodoped carriers for R VO is an importantsource to improve the efficiency of solar cells. YVO and LaVO are two representative materialsof the rare-earth vanadate family which exhibit differ-ent temperature ( T ) dependent evolutions of the crystalstructure. At high T , both YVO and LaVO haveorthorhombic structures of the P nma space group withthe a − a − c + type Glazer rotation of octahedrons. ForYVO , with cooling, the structural transition occurs at T =200 K to the monoclinic structure of the P /a spacegroup. For lower T , below 77 K, the crystal structureof this material turns into the P nma space group again.In contrast, LaVO has a larger size of the cation thanthat of YVO . Accordingly, the a − a − c + type rotationof LaVO is smaller than that of YVO . As a result, forLaVO , the structural transition from P nma to P /a occurs at T =140 K upon cooling, and there is no struc-tural transition below 140 K. The spin and orbital ordering of YVO and LaVO is related to the crystal structure for the given T . Fig-ure 1(a) shows the electronic configuration of these ma-terials that have two electrons in t g orbitals with theHund’s coupling induced high spin state. The xy orbitalhas an occupancy close to one because of its lower energy level with respect to that of xz and yz orbitals, whichis driven by the rotation and distortion of octahedron.Another electron has an orbital degrees of freedom be-tween xz and yz . Fig.1(b and c) shows dominant Jahn-Teller distortion patterns for P /a and P nma struc-tures. P /a structure has G type Jahn-Teller distortion(JTG), and P nma structure has C type Jahn-Teller dis-tortion (JTC). These patterns of JT distortion lift thedegeneracy on xz and yz orbitals inducing G type or-bital ordering (OOG) for P /a structure (JTG) and Ctype orbital ordering (OOC) for P nma structure (JTC).Fig.1(f and g) shows this structure driven spin and orbitalordering pattern for OOG (antiferro OO in all axis) andOOC (ferro OO in the c axis and antiferro OO in the ab plane). With the nominal configuration in Fig.1(a), thesingle electron in the xy orbital induces the antiferromag-netic (AFM) superexchange interaction between nearestneighbor atoms in the ab plane. The OOG induces fer-romagnetic (FM) superexchange interaction along the c axis from the Hund’s coupling induced lower energy su-perexchange path along the c axis. And, the OOC in-duces AFM superexchange interaction along the c axis,because of that the ferro OO along the c axis inducesthe superexchange energy gain in the case of AFM or-der along the c axis. Accordingly, JTG ( P /a ) andJTC ( P nma ) distortions induce C type AFM (AFMC)and G type AFM (AFMG), respectively, in agreementwith Goodenough-Kanamori-Anderson rules (GKA rules,Ref.15–17). These T -dependent spin orderings of YVO and LaVO are unambiguously determined by neutrondiffraction experiments. For the explanation of experimentally confirmed spinorderings for the given T , there have been debates onorbital states for R VO . Due to the small crystal-fieldinherent to t g orbitals, the quantum orbital fluctuationof xz/yz could be relevant for the T regime of spin or-dering, even in the presence of pictorial JT distortionsin Fig.1(b and c). Based on this small crystal-field, theresonant valance bond (RVB) state of xz and yz orbitalsis proposed as an orbital state of R VO . In this or-
Figure 1. (Color) (a) The t g electron configuration in R VO . (b) and (c) G type Jahn-Teller distortion (JTG) in P /a and C type Jahn-Teller distortion (JTC) in P nma ,respectively. Blue and red rhombus indicate distorted oc-tahedrons. (d) and (e) Multiplet configurations of opticaltransitions for α peak (optical gap : U − J ) and β peak (op-tical gap : U ) in (f) and (g) (see the text). (f) Spin-orbitalconfiguration in the JTG and C type antiferromagnetic or-der (AFMC). (g) Spin-orbital configuration in the JTC andG type AFM order (AFMG).Figure 2. (Color) Schematic partial density of states (PDOS)of R VO and its interpretation for optical transitions, α , β ,and O-V (O(2 p ) to V(3 d )). LHB and UHB present lower andupper Hubbard bands, respectively. O(2 p ) bands contribu-tions are also plotted. E F indicates the Fermi level. bital RVB, xz and yz orbitals of nearest neighbor atomson c axis have a singlet state, and this singlet makes theRVB along the c axis of the lattice. Accordingly, in agree-ment with the GKA rules, the FM superexchange inter-action emerges in the c axis and AFMC emerges. In the P /a structure, both the JTG and the orbital RVB in-duce AFMC. However, in LaVO at T ∼
150 K, which hasthe
P nma structure, the JTC in the structure induces theAFMG order, which is different from the orbital RVBinduced AFMC order. Both view points explain exper-iments such as the magnon spectrum of YVO . But,for LaVO , the continuous T -dependent evolution of spinand orbital states for T ∼
150 K still remains to be re-solved. The fingerprint of spin-orbital states of LaVO in the optical conductivity would resolve this puzzle, com-plementary to thermal properties such as electronic con-tributions to entropy. Optical conductivity is a powerful tool to probe phasetransitions in strongly correlated systems. Fingerprintsof multiplet states of the correlated shell in the op-tical conductivity provide information about the spinand orbital ordering. In R VO , there have been issueson the explanation of T -dependent evolution of opti-cal conductivities in experiments in terms of spin andorbital states. However, there is an ambiguity inthe quantitative assignment of optical peaks to multipletstates, which we explain in the Sec.III A. In addition,the T -dependent evolution of optical conductivity in thetheory to be compared with experiment is still lacking.In this paper, we compute T -dependent evolutions ofoptical conductivity of R VO ( R =Y and La) using den-sity functional theory plus dynamical mean-field theory(DFT+DMFT). The approach reproduces experi-mental T -dependent evolutions of optical conductivityfor both YVO and LaVO . We have shown that the T -dependent optical conductivity could detect spin andorbital orderings. This result could be achieved by thecorrect assignment of optical transitions to two peaks, α and β , as shown in Fig.1(d and e). Furthermore, by im-posing paramagentic (PM) state for given structures ofeach T in the phase diagram, we have shown the orbitalpolarization driven evolution of optical conductivity inthe absence of spin ordering. This result could be used asa reference for probing orbital states in the heterostruc-ture of R VO . Also, we have shown the difference in theoptical conductivity of LaVO at T ∼ II. METHODS
We calculate the optical conductivity of YVO andLaVO within DFT+DMFT using the full potential ofRef and the TRIQS library. We compute opticalconductivity using the Kubo formula with the bubble di-agram of the Green’s function in the DFT+DMFT.
We adopted experimental crystal structures of each rel-evant T for both YVO and LaVO . For YVO , (i)for high T (300-500 K), the experimental crystal struc-ture for T =297 K is used ( P nma ), (ii) for intermediate T (100K), the experimental crystal structure for T =100K is used ( P /a ), and (iii) for low T (50K), the exper-imental crystal structure for T =65 K is used ( P nma ). For LaVO , (i) for high T (150-500 K), the experimentalcrystal structure for T =150 K is used ( P nma ), and (ii)for low T (50-100K), the experimental crystal structurefor T =100 K is used ( P /a ). In the DFT part of thecomputation, the Wien2k package was used. The localdensity approximation (LDA) is used for the exchange-correlation functional. For projectors on the correlated
Figure 3. (Color) (a) and (b) Temperature ( T ) dependent optical conductivity σ ( ω ) of YVO for c axis and ab plane inDFT+DMFT. (c) and (d) T -dependent σ ( ω ) of YVO for c axis and ab plane of experiment, Reul et al., Ref.20. Solid lines in(a) and (b) are for the edge of O(2 p )-V(3 d ) optical transitions for each T . The peak, e , in (c) is the exciton peak of experiment,Reul et al., Ref.20 (see the text). (e) T -dependent phase diagram of YVO . JTC and JTG type distortions are dominant inthe P nma and the P /a structures, respectively. PM corresponds to paramagnetism. t g orbitals in DFT+DMFT, Wannier-like t g orbitalsare constructed out of Kohn-Sham bands within the en-ergy window [-1.1,1.1] eV with respect to the Fermi en-ergy. We use the full rotationally invariant Kanamoriinteraction as shown below, where L and S are angu-lar momentum and spin momentum operators of t g or-bitals, H int = ( U − J ) ˆ N ( ˆ N − − J ~S − J ~L . (see e.g.the algebra in Ref.29) For U and J parameters of theKanamori interaction, we used U =4.5 eV and J =0.5 eV.This parameter range is shown in Ref.2 to be relevantfor the description of experimental photoemission spec-trum (Ref.30 and 31) of YVO and LaVO from the t g low energy effective model of the DFT+DMFT. Onething should be noticed is that in Ref.2, only the density-density type interaction in the Kanamori interaction isused. Using the full rotationally invariant Kanamori in-teraction including the spin-flip and the pair-hoppingterms is essential to describe the correct splitting of α and β peaks in the optical conductivity which havedifferent multiplets in optical transitions as shown inFig.1(d and e). To solve the quantum impurity problemin the DMFT, we used the strong-coupling continuous-time Monte Carlo impurity solver as implemented inthe TRIQS library. III. RESULTS
In this section, we present our results for the opticalconductivity of YVO and LaVO . In Sec.III A, we as-signed main features of optical conductivity, α , β , andO(2 p )-V(3 d ) peaks, to the multiplet of final states of op-tical transitions, and also discussed their relation to mag-netic states. In Sec.III B, we presented the T -dependentevolution of optical conductivity and compared the re-sult with experiments. In Sec.III C, we presented the T -dependent the evolution of orbital states in the xz , yz ,and xy states. This T -dependent orbital state is alsocomputed by imposing PM state. Under this PM state,the T -dependent optical conductivity is discussed. InSec.III D, we presented evolution optical conductivity ofLaVO as function of T around 150K for various spinstates, and suggest possible fingerprints of the spin statein the optical conductivity for this T regime. A. Multiplet states and optical conductivity
Two main peaks, α and β , in the optical conductivitycan be assigned to the optical transitions having mul-tiplet configurations in Figure1(d and e). The α peakcorresponds to the inter-site electron transfer within theFM spin states, and the β peak corresponds to the inter-site electron transfer within the AFM spin states. Fig-ure2 presents schematic partial density of states (PDOS) Figure 4. (Color) (a) and (b) T -dependent σ ( ω ) of LaVO for c axis and ab plane in DFT+DMFT. (c) and (d) T -dependent σ ( ω ) of LaVO for c axis and ab plane of experiment, Miyasaka et al., Ref.21. Solid lines in (a) and (b) are for the edge ofO(2 p )-V(3 d ) optical transitions for each T . The peak e in the experiment, in (c) is interpreted as an exciton peak (see thetext). (e) T -dependent phase diagram of LaVO . of O(2 p ) and V(3 d ) bands and its interpretation for theoptical transitions, α , β , and O(2 p )-V(3 d ). Photon ener-gies for α and β peaks are U − J and U , respectively.Due to the incomplete orbital polarization of xz/yz andthe rotation of octahedron which breaks cubic symmetry,the inter-orbital optical transition is also possible. Thespectral weight from the optical transition, which has adouble occupancy in the same orbital for the final state,has the corresponding photon energy of U + 2 J . Thisspectral weight is buried in V(3 d )-O(2 p ) optical transi-tion as shown in Fig.2. Fig.3 shows that the peak po-sition of α and β in our DFT+DMFT results (Fig.3 aand b) is consistent with experiments (Fig.3 c and d) ofRef.20. The splitting between α and β peaks, 3 J ( ∼ . α and β peaks is correct. B. Temperature dependent evolution of opticalconductivity
Figure 3 presents the T -dependent evolution of the op-tical conductivity of YVO . From the change of T from500 K to 300 K, in the DFT+DMFT result, it is shownthat there are little changes in the optical conductivityin both ab plane and c axis. In this T range, the crystalstructure is P nma and the spin state is PM. From thechange of T from 300 K to 100 K, there is a structuraltransition from P nma to P /a , and accordingly, there is an onset of the AFMC ordering which is related to thetransition of octahedron distortion from JTC to JTG. Asshown in Fig.1(f), this AFMC ordering induces an en-hancement (suppression) of α ( β ) peak for c axis and theopposite trend for ab plane. In Fig.3, it is shown thatthe DFT+DMFT result is remarkably consistent withexperiment. With the change of T , from 100 K to ∼ P /a to P nma with the onset of the AFMG ordering and the transitionof distortion from JTG to JTC. In this state, all inter-sitespin configurations are AFM as shown in Fig1(g). As aresult, the optical conductivity becomes more isotropic,with a suppression of α peak and an enhancement of β peak for both c axis and ab plane, with respect to PMstate. This result in DFT+DMFT for 100 K to ∼
50 K isconsistent with the experiment. The overall evolutionof α peak heights in c axis in the present DFT+DMFTresult, (97, 281, and 105 Ω − cm − for 300, 100, and 50 K,respectively) is consistent with the experiment (198, 355,and 92 Ω − cm − for 300, 100, and 60 K, respectively). In the photon energy above 4 eV, the optical transition ofO(2 p )-V(3 d ) is activated. As a result, as shown in Fig.3,there is an upturn of optical conductivities, and the β peak is partially buried, consistent with experiments. There are several features in the experimental opticalconductivity which are not resolved in our DFT+DMFTcomputation. There is a sub-peak, e , below the α peak inthe experiment (Fig.3(c)). This peak is the exciton belowthe band gap as shown in Ref.20. The non-equilibriumevolution of the e peak and the α peak in the pump probe Figure 5. (Color) (a) and (b) T -dependent orbital density of YVO and LaVO , respectively. In the case of the P /a spacegroup, two sites ,1 and 2, are symmetrically non-equivalent. Dots are orbital densities in the PM states. Red (dashed) andblue (dotted) lines are orbital densities in the AFMC and AFMG states in Fig.3 and Fig.4, which are averaged per site. Thedeviation from the averaged value in AFM states is less than 0.077. (c) and (d) T dependent evolution of crystal-field for YVO and LaVO , respectively, for crystal structures used in the present DFT+DMFT computation. The lowest crystal-fieldenergy level is set as a zero energy for each structures.Figure 6. (Color) (a) The maximum and the minimum oforbital density deviations within all non-equivalent site fromthe site averaged orbital density, for each spin state for YVO .Orbital which has the maximum or the minimum are pre-sented for each cases. (b) Same as (a) for LaVO . It is shownthat the onset of magnetism results in the reduction of thedeviation for both YVO and LaVO for P /a structure. experiment of YVO in Ref.35 shows that this subpeakbelow the α is indeed the exciton below the band gap.This assignment of peaks is different from previous re-ports in R VO of Ref.21 and 22, which argue that the e and the α peaks in our viewpoint are α and β peaks inFig.1(d and e) and Fig.2, respectively. This is the reasonwhy previous optical conductivity data from DFT is dif-ferent from experiments. The emergence of the e peakis from the condensation of the exciton below the bandgap which is induced by non-local correlations, inter-preted as a condensation of the charge carrier state alongthe ferromagnetic chain in the AFMC state. PresentDFT+DMFT computation includes local dynamical elec-tronic correlations by the non-perturbative manner andspatial static electronic correlations from the exchange-correlation functional in the LDA. As a result, the e peak from the non-local correlation is not described inthe present results.Figure 4 presents the T -dependent evolution of opticalconductivity for LaVO . For T range from 500 K to 300K, in the DFT+DMFT result, it is shown that there is asmall change in the optical conductivity in both ab planeand c axis, similar to the case of YVO . With changesof T from 300 K to 50 K, the structural transition from P nma to P /a occurs. Accordingly, there is an onsetof the AFMC ordering which is related to the transitionfrom JTC to JTG. Similar to the case of YVO for 300 Kto 100 K, this AFMC ordering induces an enhancement of α peak heights for c axis and the opposite trend in the ab plane. As shown in Fig.4, the DFT+DMFT result for 300K to 50 K is consistent with the experiment for c axis. We suggest that the e peak in Fig.4(c) is the exciton peaksimilar to the YVO . For the optical conductivity of ab plane, differently from the DFT+DMFT result, there is asmall evolution of optical conductivity in the experiment,which is remained to be resolved. The overall evolutionof α peak heights in c axis in the present DFT+DMFTresult, (120, and 357 Ω − cm − for 300, and 50 K, respec-tively) is consistent with the experiment (330, and 570Ω − cm − for 293, and 10 K, respectively). There are two important differences in the optical con-ductivity for YVO and LaVO . Firstly, due to thesmaller rotation of the octahedron in LaVO , which giverise to the enhanced O(2 p )-V(3 d ) hybridization, the op-tical transition of O(2 p )-V(3 d ) is activated at the lowerphoton energy (around 4 eV) in LaVO with respect tothat in YVO (above 4 eV). As a result, as shown in Fig.4,the β peak center is buried from the O(2 p )-V(3 d ) peakin LaVO , consistent with experiments. This difference
Figure 7. (Color) (a) and (b) T -dependent σ ( ω ) of YVO for c axis and ab plane in DFT+DMFT with the constraint of PMstate. (c) and (d) T -dependent σ ( ω ) of LaVO for c axis and ab plane in DFT+DMFT with the constraint of PM state. Solidlines are for the edge of O(2 p )-V(3 d ) optical transitions. of the energy of the O(2 p )-V(3 d ) transition explains thepresence of the T dependent evolution of the β peak inYVO and the absence of the T dependent evolution ofthe β peak in LaVO . Therefore, we suggest to anal-yse (i) T -dependent evolution of the α peak to resolvespin-orbital structure in the case of bulk LaVO and (ii)the appearance of the β peak in the heterostructure ofLaVO in the case of compressive strain to confirm thereduced O(2 p )-V(3 d ) hybridization from the enhancedrotation of octahedron. Secondly, there is a differencein the hight of the e peaks. The height of e peaks are179 and 760 Ω − cm − for YVO and LaVO , respec-tively. We suggest that this larger height of the e peak inLaVO is due to the smaller octahedron rotation whichresults in the larger stabilization of the kinetic energy ofthe exciton for LaVO . C. Orbital states dependence of opticalconductivity
Figure 5 presents the T -dependent evolution of orbitaldensity and crystal-field of xz , yz , and xy orbital in the t g manifold for YVO and LaVO . None of orbitalsare fully polarized due to the low symmetry crystal-fieldcontribution from the rotation of the octahedrons andthe covalent bonding with cations. For T range of 300-500 K, in Fig.5(a) and 5(b), there isa small evolution of orbital polarization which is consis-tent with Ref.2 This result explains the small evolutionof the optical conductivity in the T range of 300-500 K,as shown in Fig.3 and 4. Fig.5(c) and (d) shows T depen-dent crystal-field levels in the crystal structures used inpresent results. The P nma structure has a single type ofcrystal-field level for all sub-lattice of V (site-uniform),and the P /a structure has two types of crystal-field level for sub-lattice of V (site-non-uniform). In the PMstate of this T range, the site-uniform crystal-field split-ting, is not effective to induce the T -dependent evolutionof orbital polarization.Figure 6 shows that in the AFMC state, the orbital po-larization of xz and yz is more uniform than that of thePM state for P /a structure. Also, as shown in Fig.5(a),especially for the P nma structure of YVO for T =50 K,the orbital polarization in the AFMG state is much largerthan that of the PM state. These results suggest that JTdistortions from the structural transition induce a finiteorbital polarization, and the superexchange interactionmodifies this orbital polarization with the onset of AFMspin ordering. The size of the induced magnetic momentfor each site is 1.97 µ B for AFMG state. This large mag-netic moment induced exchange field to orbitals resultsin the large orbital polarization in the AFMG state withrespect to that in the PM state as shown in Fig.5(a).This result shows that both JT distortion and superex-change interaction contribute to the orbital polarization.This result is in line with the inelastic x-ray scatteringexperiment on YVO which shows that the orbital ex-citation has contributions from both superexchange andcrystal-field. Fig. 6 shows the maximum and the minimum of or-bital density deviations within all site from the site aver-aged orbital density, for each spin states, for YVO andLaVO . It is shown that the onset of magnetic orderingresults in the reduction of the site dependence of the or-bital polarization in both YVO (Fig.6(a)) and LaVO (Fig.6(b)). As shown in Fig.5(c) and (d), the crystal-field in the structure of P /a space group is non-uniformfor sites. On the other hand, the magnetic moment formagnetic states of all case is uniform for sites. The sitedependent deviation of magnetic moment size is smallerthan 0.01 µ B . As a result, the site-uniform orbital polar-ization from the site-uniform spin exchange interaction isinduced in all AFM state. This site-uniform orbital statein AFM states is different from the crystal-field drivensite-non-uniform orbital polarization, which is inducedfor PM states of the P /a space group.In P /a structures for YVO and LaVO , the orbitalpolarization for xz and yz is larger in the case of LaVO for PM spin state. This result is due to the nature ofthe JT distortion in the P /a structure. In Ref.9, itwas shown that both JTC and JTG distortions exist for P /a structures of YVO and LaVO . In the case ofLaVO , JTG type distortion is much larger than JTCtype distortion. On the one hand, YVO has also largerJTG type distortion than that of JTC, but the differenceis smaller than that of LaVO . As a result, the orbitalpolarization is larger for LaVO with respect to that ofYVO for P /a structure in PM state, consistent withRef.2, as shown in Fig.5(a and b).Figure 7 presents the T -dependent evolution of opticalconductivity in DFT+DMFT with the constraint of PMstate. With the constraint of PM state, Kramers doublet,spin up and spin down components, are set to equal, andorbital differentiation is allowed with constraint from thegiven symmetry of crystal structure. Even in the con-straint of PM, due to the superexchange energy gain ac-cording to the GKA rules, antiferro OO and ferro OOenhances FM ( α peak) and AFM ( β peak) spin correla-tions, respectively. As a result, the orbital polarizationin the PM state also contributes to the evolution of op-tical conductivity. For the structural transition from the P nma to the P /a , 300K to 100K of T , LaVO shows alarge evolution of optical conductivity, while YVO showsa small variation. The larger JTG type orbital polariza-tion in the P /a structure of LaVO induces the AFMCtype spin correlation. On the other hand, the optical con-ductivity of YVO for P /a structure shows small vari-ations because of the small difference in the magnitudeof JTG and JTC type distortions. Thus, the trend, suchthat α ( β ) peak is much enhanced (suppressed) for c axisand the opposite change occurs for ab plane, is larger forLaVO . This trend is consistent with the larger orbitalpolarization in LaVO with respect to that in YVO for P /a structures in PM state as shown in Fig.5 (a) and(b).For YVO , in Fig.5(a), it is shown that in the P nma structure of T =50K for PM state, the orbital occupancyof the xy orbital is close to 1. And also, there is un-quenched orbital fluctuation of xz and yz for the PMstate. This result provides an explanation of the opticalconductivities of T =50K for PM state in Fig.7(a and b).With the constraint of ferro OO in c axis (OOC) fromthe JTC distortion of P nma structure, the orbital stateof xz and yz in the apical direction has same phase. As aresult, there is a strong suppression of optical transitionfor c axis due to the blocking of the charge transfer for c axis. On the other hand, the orbital fluctuation of xz and yz is in different phase for ab plane due to the symmetryof P nma , which enhances in-plane FM correlation. As a result, the height of α peak is enhanced for ab plane.These results suggest that the optical conductivity is auseful quantity to probe the orbital state of R VO , whichis complimentary to the resonant x-ray diffraction exper-iment. We suggest to compare heights of the α peakfor consistent analysis of spin-orbital states from opticalconductivity of R VO systems, because, in the case ofLaVO , the β peak is buried in the O(2 p )-V(3 d ) peak. D. Spin state and optical conductivity of LaVO for T str 150 K. At this T , there is a debate on the spinordering. The structural distortion of JTC in the P nma space group above T str induces the AFMG order-ing. On the other hand, the superexchange driven orbitalRVB state induces AFMC. We compute the optical con-ductivity of AFMG and PM states at 150 K in the struc-ture of the P nma space group. For this T , the AFMCstate is not stabilized, which implies that the descriptionof the free energy in the present level of approximationin the DFT+DMFT does not have a local minimum forthe AFMC state for the structure of T =150 K. The re-sult is compared with results for the AFMC state at 100K ( P /a structure) and the PM state at 300 K ( P nma structure). It is clearly shown that in the case that theAFMG ordering emerges for T str < T < T N , the α peak issuppressed for c axis with respect to the case for theAFMC state. This trend from the AFMG state is appar-ently different from the anisotropic optical conductivityfor the AFMC state. We suggest that the measurementof the evolution of optical conductivity in the regime of T str < T < T N would be useful for resolving the issue onspin ordering between the JTC structure driven AFMG Figure 8. (Color) (a) and (b) T -dependent σ ( ω ) of LaVO for c axis and ab plane in DFT+DMFT including σ ( ω ) at T =150K ( P nma space group) for PM and AFMG states. TheAFMC state is not stabilized at T =150 K. (c) T -dependentphase diagram of LaVO around 150 K. Solid lines are for theedge of O(2 p )-V(3 d ) optical transitions. and the electronic superexchange induced orbital RVBdriven AFMC. IV. DISCUSSION We have shown that the T -depedent evolution of op-tical conductivty of YVO and LaVO has signatures ofspin and orbital states in these materials. DFT+DMFTreproduces experimental results from the correct assign-ment of multiplet states to peaks in the optical conduc-tivity. Two types of magnetic state, AFMG and AFMC,could be resolved by the optical conductivity. Further-more, we have shown that there is a fingerprint of the T -dependent evolution of the orbital polarization in theoptical conductivity even in the absence of the spin or- der. This result provides a reference to probe orbitalstates in the heterostructure of R VO . The clear dif-ference between AFMG and AFMC states indicates thatthe optical conductivity is useful for resolving issues onthe spin ordering of LaVO for the regime of T str < T < T N ,which depends on its origin between the electronic su-perexchange induced orbital RVB and the JT crystal-field induced OO. 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