PHYSICAL REVIEW B 86, 125103 (2012)

Resonant inelastic x-ray scattering in a Mott insulator

Nandan Pakhira,1 J. K. Freericks,1 and A. M. Shvaika2

1Department of Physics, Georgetown University, Washington, DC 20057, USA2Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Street, 79011 Lviv, Ukraine

(Received 20 December 2011; published 5 September 2012)

We calculate the resonant inelastic x-ray scattering (RIXS) response in a Mott insulator, which is describedby the Falicov-Kimball model. The model can be solved exactly within the single site dynamical mean-fieldtheory (DMFT) approximation and the RIXS response can also be calculated accurately up to a local backgroundcorrection. We find that on resonance the RIXS response is greatly enhanced. The response systematically evolvesfrom a single-peak structure, arising due to relaxation processes within the lower Hubbard band, to a two-peakstructure, arising due to relaxation processes within the upper Hubbard band and across the Mott gap into thelower Hubbard band. This occurs as we vary the incident photon frequency to allow excitations from the lowerHubbard band to the upper Hubbard band. The charge transfer excitations are found to disperse monotonically aswe go from the center of the Brillouin zone towards the zone corner. These correlation-induced features have beenobserved by Hasan et al. [Science 288, 1811 (2000)] and many other experimentalists in RIXS measurements overvarious transition-metal oxide compounds. They are found to be robust and survive even for large Auger lifetimebroadening effects that can mask the many-body effects by smearing out spectral features. As a comparison, wealso calculate the dynamic structure factor for this model, which is proportional to the nonresonant part of theresponse, and does not show these specific signatures.

DOI: 10.1103/PhysRevB.86.125103 PACS number(s): 71.10.Fd, 71.27.+a, 74.72.−h, 78.70.Ck

I. INTRODUCTION

Resonant inelastic x-ray scattering is essentially a deepcore level spectroscopic method that is increasingly becom-ing an essential technique in understanding the complexelectronic dynamics of a wide class of novel materials likecuprates, manganites, and various other transition-metal oxidecompounds. In the RIXS process, a highly energetic x-rayphoton (with energy ∼1–10 keV) excites a deep core levelelectron into the unoccupied states of the conduction band. Theexcited electron then undergoes inelastic scattering processeswith various intrinsic excitations present in the system andfinally, a conduction band electron fills up the core-hole andemits a photon with relatively lower energy. So, this is atwo-photon inelastic process with no core-hole present inthe final state. The transferred energy and momentum to theintrinsic excitations of the system as well as the change inthe polarization of the scattered photon can provide importantinformation regarding these excitations. Also, RIXS being aresonant technique, the incident photon energy can be chosento coincide with, and hence resonate with, certain intrinsicx-ray atomic transitions which in effect can greatly enhancethe inelastic scattering cross section. This enhancement allowsRIXS to be used as a probe for charge, magnetic, and orbitaldegrees of freedom on selective atomic sites.

RIXS has several advantages over other spectroscopictechniques like angle resolved photo-emission spectroscopy(ARPES) and neutron scattering. First, in ARPES, the incidentphoton knocks out an electron from the system and hencecan only probe the occupied states in a system, whereasRIXS, being a high-energy process, can excite a systeminto unoccupied intermediate states (like the upper Hubbardband in a Mott insulator) and hence can be used as aprobe for understanding complex electron dynamics in thosestrongly correlated intermediate states. Inverse photoemissiontechniques, in which an electron is injected into the system,

can also access the unoccupied states of a system. But thismethod will charge the system and so far no momentumresolved inverse photoemission spectroscopy with sufficientenergy resolution has been developed. Second, the scatteringphase space, i.e., the range of energies and momenta thatcan be transferred in the RIXS process, is much larger thanother available photon scattering techniques involving visibleor infrared light. As a result, RIXS can probe low-energyexcitations over a wider range of the Brillouin zone and mostimportantly it can be used to probe all three directions ofthe Brillouin zone, and hence can be used even for materialsthat are intrinsically three dimensional in nature. On theother hand, ARPES, because of the in plane momentumconservation, is widely used for materials which are inherentlytwo dimensional in nature. For three-dimensional materials,the analysis of ARPES spectra is much more complicated (theperpendicular component of momentum is integrated over inthe ARPES spectra). Another advantage is that the electron-photon interaction is much stronger than the electron-neutroninteraction (which arises through the magnetic dipole-dipoleinteraction). As a result, RIXS can be used on small volumesamples, thin films, surfaces, and nano-objects in addition tobulk single crystal or powder samples. Besides these, RIXSis polarization dependent and hence can be used to probemagnetic excitations and also it can be used as a probefor certain specific elements or orbitals in a system. Themain disadvantage of RIXS over other techniques is that itrequires substantially large incident photon flux in order tohave comparable or better energy and momentum resolution.But recent progress in RIXS instrumentation has dramaticallyimproved upon this situation and RIXS is beginning to becomean important probe for condensed matter physics.

Over the past decade or so, RIXS measurements havebeen performed over large classes of transition-metal oxidecompounds like cuprates, manganites, iridates, etc. (seeRef. 1 for a detailed review). Most notably, RIXS

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NANDAN PAKHIRA, J. K. FREERICKS, AND A. M. SHVAIKA PHYSICAL REVIEW B 86, 125103 (2012)

measurements have been performed at the Cu K

edge2–11 over a large class of cuprates like the un-doped La2CuO4,3,7 Nd2CuO4,2,4 Sr2CuO2Cl2,2 Ca2CuO2Cl2,5

quasi-one-dimensional cuprates like SrCuO2,6,8 Sr2CuO3,6

hole-doped cuprates La2−xSrxCuO4,11 YBa2Cu3O7−δ ,9 andelectron-doped cuprates Nd1.85Ce0.15CuO4.10 Also, RIXShas been performed at the Cu L3 edge12–14 over vari-ous undoped cuprates CuO,12 Sr2CuO2Cl2,12 La2CuO4,12,14

and doped systems Bi2Sr2CaCu2O8+δ , Nd2−xCexCuO4,12

La2−xSrxCuO4.12–14 Besides these, RIXS measurements at theMn K edge in the orbitally ordered manganite LaMnO3,15 atthe Mn L2,3 absorption edge in MnO,16 at the Ni L3-edgein NiO,17 and at the Ir L3 edge in the 5d Mott insulatorSr2IrO4

18,19 have also been reported. A common feature ofthese materials is that all of them are either Mott insulatorsor doped Mott insulators and have interesting magneticground states. RIXS measurements on these materials haveprobed energy and momentum resolved features of chargetransfer excitations,5,6,19 dd excitations12,13,16 (arising due totransitions between crystal field split d orbitals), orbitons15 inorbitally ordered systems and even magnetic excitations likemagnons14 and bimagnons.11

Theoretical approaches in understanding the RIXS responseare mainly either based on exact diagonalization of modelHamiltonians over finite but small clusters5,20–23(typically witha large Auger broadening put in by hand) or based on a single-particle approach24–30 that includes realistic band structureeffects. The correlation effects are treated perturbativelyunder the random phase approximation24–28 (RPA) or undera self-consistent renormalization29,30 (SCR) approach and theeffect of scattering from the core hole in the first-order Bornapproximation or multiple scattering approximation. The exactdiagonalization method treats the strong correlation effects ex-actly but because of the exponentially growing basis problem,this method is limited to small size clusters and small numberof orbitals and hence has limited momentum resolution. Also,the effects of the core hole in this approach as well as in theSCR-based calculations are taken either through an input core-hole lifetime, arising due to Auger and fluorescence effects,which broadens the intermediate states or under the ultrashortcore-hole lifetime (UCL) approximation31,32 that is found tobe perturbatively exact for small as well as large core-holepotentials. But the effect of the core-hole lifetime arising solelydue to intrinsic strong correlation effects in a Mott insulatoron the RIXS response has not been addressed so far and inthis work we use the Falicov-Kimball33 (FK) model to addressthis issue. The main motivation in choosing the FK model isthat the FK model is one of the simplest models of stronglycorrelated electron systems that can be exactly solved34,35

under the single-site dynamical mean-field theory36,37 (DMFT)approximation and most notably shows a Mott insulatingground state for large interaction strength between the itinerantand the static electrons. Also, the fully renormalized two-particle dynamic charge correlation function involving theitinerant species as well as the finite temperature core-holepropagator in this model can be calculated exactly.

One might ask how this solution for RIXS in the Falicov-Kimball model compares with the solution in the Hubbardmodel. Since the exact solution for the Hubbard model isnot known, we can only speculate here. But because the

predominant property of the Mott phase (above the magneticordering temperature) is its gap, we expect that much of theresults in this regime will be generic to the two models.This reasoning has shown to be true when comparing Ramanscattering calculated in the Falicov-Kimball model to thatcalculated in the Hubbard model,38,39 where in the insulatingphase much of the behavior was quite similar, with detailsvarying in the temperature dependence.

In the following sections, we calculate the RIXS responsein the limit of large core-hole energy using only two ap-proximations: (1) we neglect some momentum independentbackground contributions and (2) we calculate the chargevertex for the core-hole–band-electron exchange processes(see below) under the Hartree-Fock approximation. Theorganization of the paper is as follows. In Sec. II, we provide abrief mathematical formulation for the calculation of the RIXScross section followed by Sec. III where we show a moredetailed calculation for the RIXS response in the FK model.In Sec. IV, we show our results for a half-filled Mott insulatorfollowed by Sec. V where we discuss the core-hole lifetimebroadening effects on the RIXS response and in Sec. VI weshow some results for the case of particle-hole asymmetricMott insulator. Finally, in Sec. VII we conclude.

II. MATHEMATICAL FORMULATION OF RIXS

Our starting point is the familiar electron-photon interactionHamiltonian1

Hint =N∑

i=1

[e

mA(ri ,t) · pi + eh

2mσ i · ∇ × A(ri ,t)

+ e2

2mA2(ri ,t) − e2h

4m2c2σ i · ∂A(ri ,t)

∂t× A(ri ,t)

](1)

for a system of N electrons. A(r,t) is the vector potential forthe external electromagnetic field and can be expanded in aplane-wave basis as

A(r,t) =∑k,ε

√A0[εakεe

i(k·r−ωt) + ε∗a†−kεe

−i(k·r−ωt)], (2)

where A0 = h2Vε0ωk

, V is the volume of the system, andε is the polarization of the light, and we have fixed the gaugeby choosing ∇ · A(r,t) = 0 in Eq. (1).

In the RIXS process, an incident x-ray photon withmomentum ki , energy ωi , and polarization εi is scatteredto a final state described by momentum kf , energy ωf , andpolarization εf . Fermi’s golden rule to second order in Hint

gives the scattering cross section for this process:

W = 2π

h

∑F,I

∣∣∣∣∣〈F|Hint|I〉 +∑

n

〈F|Hint|n〉〈n|Hint|I〉EI − εn

∣∣∣∣∣2

× ρ(εi)δ (EF − EI) , (3)

where |I〉 ≡ |i〉 ⊗ |ki ,ωi〉, |F〉 ≡ |f 〉 ⊗ |kf ,ωf 〉, and |n〉 arethe direct product states for the initial, final and the interme-diate states of the systems, respectively (both electronic andphoton states are present in the initial and final states, whilethe intermediate states are just the electronic states), and EI,EF, and εn are the corresponding energies, respectively. It

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RESONANT INELASTIC X-RAY SCATTERING IN A MOTT . . . PHYSICAL REVIEW B 86, 125103 (2012)

is interesting to mention that the intermediate state |n〉 hasa core hole, while the initial and final states, |I〉 and |F〉,have no core hole. Also, at finite temperature, the systemis in a mixed state, so we sum over all possible initialconfigurations weighted by the appropriate Boltzman factor,ρ(εi). The first-order amplitude is in general dominant overthe second-order contribution except near resonance when theincident photon energy is nearly equal to a specific atomictransition in a material, i.e., ωi ≈ εn − εi . At resonance, thesecond-order term becomes overwhelmingly large comparedto the first-order term and hence the second-order term causesresonant scattering while the first-order term gives rise tononresonant scattering.

The diamagnetic term proportional to A2 as well as thespin-orbit coupling term proportional to σ · (∂A/∂t) × A inEq. (1) contribute to the first-order amplitude. The latter issmaller than the former by a factor of ωi(f )/mc2 1 andalso, at resonance, the contribution from the diamagnetic termis negligibly small compared to the resonant term and hencetheir contributions will be neglected. So, then the resonant partof the second-order amplitude at zero temperature is givenby1,40,41

e2h√

ωkiωkf

Vε0

∑n

[〈f |Dkf

|n〉〈n|D†−ki

|i〉εn − εi − ωi

+ 〈f |D†−ki

|n〉〈n|Dkf|i〉

εn − εi + ωf

], (4)

where

Dk = 1

imωk

N∑i=1

eik·ri

(ε · pi + ih

2σ i · k × ε

)(5)

is the relevant transition operator for the RIXS cross section.The first term in Eq. (5) causes nonmagnetic scattering. Thesecond term, arising from the spin-orbit coupling term in Hint,causes magnetic scattering which, for typical incident photonenergy (∼1–10 keV) and the localized core levels involvedin a RIXS process, is about 100 times smaller than the non-magnetic term1 and hence will also be neglected. Finally, undersuch circumstances, we assume the dipole limit for the RIXSprocess and the transition operator is then given by

D = ε · D with D = 1

imωk

N∑i=1

pi . (6)

The expression for the transition operator is model specific,and in the following section, we will explicitly show it for theFalicov-Kimball model.

III. RIXS RESPONSE IN THE FALICOV-KIMBALL MODEL

The Falicov-Kimball33 model involves the interaction ofmobile conduction electrons with static localized electrons. Itwas originally developed for rare-earth compounds, and hencethe particles were denoted as d electrons for the conductionelectrons and f electrons for the localized electrons. One canapply it in an approximate way to transition metal compoundswhere the labels d and f should no longer be associated withthe corresponding atomic orbitals. The Hamiltonian for the

Falicov-Kimball model (in the hole representation) includingthe interaction with a core hole is given by

H = − t∗

2√

d

∑〈ij〉

d†i dj −

∑i

μndi +∑

i

(Ef − μ)nf i

+∑

i

(Eh − μ)nhi +∑

i

Undinf i +∑

i

Qdndinhi

+∑

i

Qf nf inhi, (7)

where ndi = d†i di , nf i = f

†i fi , and nhi = h

†i hi are the occu-

pation number operators for the d-hole, f -hole, and core-holestate at a given site i, respectively. t∗/2

√d is the nearest-

neighbor hopping amplitude of the itinerant d hole on a d-dimensional hypercubic lattice and μ is the common chemicalpotential. U is the onsite Coulomb interaction between theitinerant d and the static f holes, Qd > 0 and Qf > 0 arethe Coulomb interactions between the core hole, the d and f

holes, respectively, Ef is the site energy of the f state andEh ∼ 102–104 eV is the energy of the core-hole state. It isimportant to mention that at half-filling (〈nf 〉 = 〈nd〉 = 0.5),we choose μ = U/2 and Ef = 0, which corresponds tothe particle-hole symmetric case (in the restricted subspaceinvolving d and f operators). It is important to mention thatunder the hole-particle transformation d → d†, f → f † theinteractions between core-hole and band states transform asQd → −Qd , Qf → −Qf , i.e., they become attractive insteadof repulsive and the core-hole energy, Eh gets shifted toEh + Qd + Qf .

Under single-site DMFT the model reduces to an effectivesingle impurity problem,34,35 described by the local Hamilto-nian

Hloc = Undnf + Qdndnh + Qf nf nh − μnd

+ (Ef − μ)nf + (Eh − μ)nh (8)

together with an effective bath to which the d holes hop in andout. The equilibrium density matrix for the single-impurityproblem in DMFT is then given by

ρ = e−βHloc

Z Tc exp

[−i

∫c

dt ′∫

c

dt ′′d†(t ′)λ(t ′,t ′′)d(t ′′)]

, (9)

where the time ordering and integration are performed over theKadanoff-Baym-Keldysh contour (see Fig. 1) and β = 1/kBT

is the inverse temperature. Here, the dynamical mean fieldλc(t ′,t ′′) and the chemical potential μ are taken from the equi-librium solution of the conduction electron problem withoutthe core hole, arising under the single-site DMFT approxima-tion. This in effect implies that we are treating the creation ofthe core hole under the sudden approximation instead of a fullnonequilibrium treatment of the core-hole propagator.

The equilibrium impurity problem arising under the DMFTapproximation can be solved exactly in this case and the locald-hole propagator, Gloc

d (ω), is given by

Glocd (ω) = w0

ω+ + μ − λ(ω+)+ w1

ω+ + μ − λ(ω+) − U, (10)

where ω± = ω ± iδ (δ > 0) and w0 and w1 are the prob-abilities for finding a given site unoccupied and occupied

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NANDAN PAKHIRA, J. K. FREERICKS, AND A. M. SHVAIKA PHYSICAL REVIEW B 86, 125103 (2012)

− iβ

0t

FIG. 1. The Kadanoff-Baym-Keldysh contour. The contour startsat time zero, moves forward along the real axis to time t then movesbackward along the real axis to time zero, and finally, downwardsalong the imaginary axis to time −iβ.

by an f hole, respectively. The momentum-dependent fullyrenormalized d-hole propagator is given by

Gd (q,ω) = 1

ω+ + μ − εq − �locd (ω+)

, (11)

where the local self-energy is related to the local propagatorthrough Dyson’s equation

�locd (ω+) = ω+ + μ − λ(ω+) − [

Glocd (ω+)

]−1. (12)

Similarly, the core-hole Green’s functions, G>h (t) =

−i〈h(t)h†(0)〉 and G<h (t) = i〈h†(0)h(t)〉, can also be

calculated42 at finite temperature by using either numerical in-tegration over the Kadanoff-Baym-Keldysh contour as shownin Fig. 1 or by the Wiener-Hopf sum equation approach.43,44

The angular brackets 〈 〉 denote a trace over all states weightedby density matrix in Eq. (9) and the operators are in theinteraction representation with respect to Hloc. Also, it isimportant to mention that for the calculation of the itinerant aswell as the core-hole propagators we use the d-dimensionalhypercubic lattice density of states (DOS) in the limit ofd → ∞ (DMFT approximation).

The interaction of the x-ray photon with the electronicsubsystem of matter can be represented by the diagrams shownin Fig. 2. The dipole operator31,32 is given by

D =∑

l

(e−iki .r l h†l dl + eikf .r l hld

†l + H.c.). (13)

The first two terms in Eq. (13) correspond to Figs. 2(a) and 2(b),while the Hermitian conjugate terms correspond to Figs. 2(c)and 2(d). We have explicitly shown the direct dependence ofthe core-hole propagator on the core-hole energy Eh, whichis typically much larger than the band energies and is ofthe order of the incident (ωi) and the scattered (ωf ) x-rayphoton energies, respectively. One can see that in the case oflarge photon and core-hole energies only the first two vertices[see Figs. 2(a) and 2(b)] contribute significantly whereas thecontribution from the remaining two [see Figs. 2(c) and 2(d)]are negligibly small because the hole propagators are evaluatedtoo far off the energy shell.

The calculation of the RIXS cross section involves theanalytic continuation of the four-particle correlation function

ωi

ωi

ωf

ωf

ω − Eh

ω − ωfω + ωi − Eh

ω

ω − Eh ω

ω − ωf − Ehω + ωi

(a) (b)

(c) (d)

FIG. 2. (Color online) RIXS interaction vertices. Wavy lines(blue) represent incident or scattered photons, the dashed lines (green)represent the core-hole propagator, and the solid lines (red) representthe propagator for the itinerant d holes. The labels indicate theenergies of the different particles. Note that the momentum and energyare conserved at each vertex.

χ(4)i,f,f,i(iνi,iνf ,iν ′

f ,iν ′i) from Matsubara frequencies

χ(4)i,f,f,i(iνi,iνf ,iν ′

f ,iν ′i)

F.T .−→ χ (4)(τ1,τ2,τ3,τ4)(14)

χ (4)(τ1,τ2,τ3,τ4) = −〈TτD(τ1)D(τ2)D(τ3)D(τ4)〉to real frequencies, which is a well defined but tediousprocedure.45,46 In the considered limit of large photon andcore-hole energies, the bare-loop contribution to the amplitudefor the RIXS process is represented by the two diagramsshown in Fig. 3. The contribution of the top diagram tothe four-particle correlation function, χ

(4)i,f,f,i(iνi,iνf ,iν ′

f ,iν ′i),

evaluated on the imaginary axis, is equal to

− 1

β

∑m

χdd0 (iωm − iνi + iνf ,iωm|q)

×χhh0 (iωm + iνf ,iωm + iν ′

f ), (15)

whereas the contribution of the bottom diagram toχ

(4)i,f,f,i(iνi,iνf ,iν ′

f ,iν ′i) is equal to

− 1

β

∑m

χhh0 (iωm + iνi − iνf ,iωm)

×χdd0 (iωm − iνf ,iωm − iν ′

f |0). (16)

Here, we have introduced the bare charge susceptibilities

χdd0 (iωm + iν,iωm|q) = − 1

N

∑k

Gd (k + q,iωm + iν)

× Gd (k,iωm), (17)

χhh0

(iωh

m + iν,iωhm

) = −Gh

(iωh

m + iν)Gh

(iωh

m

), (18)

and iωhm ≡ iωm − Eh. Since the core-hole propagator is local,

the bottom diagram in Fig. 3 does not depend on the photonwave vector and hence can only contribute to momentum-independent background effects as evident in Eq. (16). Thefirst diagram does depend on the transferred momentumq = ki − kf . Since in the present study, we are interested inthe energy and wave-vector dependence of the RIXS response,

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RESONANT INELASTIC X-RAY SCATTERING IN A MOTT . . . PHYSICAL REVIEW B 86, 125103 (2012)

ωk

ω + ωfω + ωf

ω − Ωk − q

ω − ωf

k − kf

ω − ωf

k − kf

ω

ω + Ω

j l

j l

j l

j l

Gdd(q, ω)

Ghh(ω)

ωi,ki ωi,ki

ωi,ki ωi,ki

ωf ,kf ωf ,kf

ωf ,kf ωf ,kf

FIG. 3. (Color online) Bare loop contribution to the four-particlecorrelation function corresponding to the direct RIXS process. Thebottom diagram gives a momentum-independent contribution (seetext) to the RIXS process and hence will be neglected in thiscalculation.

we neglect all such momentum-independent contributions.The technical reason behind neglecting these terms is thatthe momentum independent contributions are the local onesand hence they include all types of many body scatteringprocesses. Hence they can only be derived from the solutionof the single-impurity problem for the four-particle correlationfunction χ (4) (which involves multiparticle vertices and manymore complications) and at this moment we do not have welldeveloped approach for this even in the simplified Falicov-Kimball model.

Analytic continuation to real frequencies of the barediagram in Fig. 3(a) gives the following contribution to theRIXS cross section:

− 1

2π2

∫ +∞

−∞dω [f (ω) − f (ω + �)] χhh

0 (ω− + ωi,ω+ + ωi)

×Re[χdd

0 (ω−,ω− + �|q) − χdd0 (ω+,ω− + �|q)

], (19)

where f (ω) = 1/[exp(βω) + 1] is the Fermi function and

χhh0 (ω− + ωi,ω

+ + ωi) = −∣∣Gh

(ω+ + ωh

i

)∣∣2. (20)

Here, ωhi,f = ωi,f − Eh is the incident and scattered photon

energy measured with respect to the core-hole energy, Eh.Next, we introduce the renormalization of the bare charge

susceptibilities through inclusion of charge vertices. In thesimplest case, this can be done by inserting the two-particlecharge vertex either in between the two d-hole propagators(dd channel) or between the two core-hole propagators (hh

channel). Physically speaking, the incident x-ray photoncreates a core-hole–d-electron pair. The excited d-electron

FIG. 4. (Color online) Diagrams for the renormalized four-particle correlation function χ (4). Panel (a) total direct resonantscattering, (b) full exchange resonant scattering, and (c) partialexchange resonant scattering processes. Explicit Eh dependence ofthe core-hole propagator is not shown.

undergoes various inelastic scattering processes with othercharge excitations in the correlated d band as well as withthe electron-hole pairs created due to the presence of thecore-hole potential. As a result the total scattering amplitudecontains contributions from both direct and indirect scatteringprocesses.1,32 The total scattering cross section, as shown inEq. (3), is the square of the total scattering amplitude andcontains four contributions. The corresponding contributionsto the four-particle correlation function χ (4) are representedby different rows in Fig. 4. The first row [see Fig. 4(a)]corresponds to the pure direct Coulomb processes which donot involve core-hole–band-electron scattering. The secondrow [see Fig. 4(b)] corresponds to full exchange scatteringprocesses and involves dynamical screening of the core-holepotential. The last two rows [see Fig. 4(c)] describe mixedscattering arising due to the quantum mechanical interferenceof the direct and indirect scattering processes, which we callpartial exchange.

In real experiments, the RIXS processes31,32 can happeneither through a direct process, as in the case of the L2,3-edge2p → 3d RIXS in which the core electron is excited to anunoccupied state of the correlated valence band (d band)or through an indirect process, as in the case of K-edge1s → 4p RIXS in which the excited core-electron goes intoan uncorrelated 4p band several electron volts above the Fermilevel. In the present case, the excited core-electron goes to thecorrelated d band and the dipole selective transition 2p → 3d

or 3p → 4d is consistent with the involvement of a 2p or 3p

core hole and hence our study is related to the direct RIXSprocesses like L-edge or M-edge RIXS.

The sum of the two diagrams in Fig. 4(a) correspondsto the replacement in Eq. (19) of the bare charge suscepti-bility χdd

0 (iωm + iν,iωm|q) by the fully renormalized charge

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NANDAN PAKHIRA, J. K. FREERICKS, AND A. M. SHVAIKA PHYSICAL REVIEW B 86, 125103 (2012)

susceptibility χdd (iωm + iν,iωm|q), which in the case of theFalicov-Kimball model is given by

[χdd (iωm + iν,iωm|q)]−1 = [χdd

0 (iωm + iν,iωm|q)]−1

+�(iωm + iν,iωm), (21)

where the irreducible charge vertex47–49 is

�(iωm + iν,iωm) = 1

T

�(iωm) − �(iωm + iν)

G(iωm) − G(iωm + iν). (22)

The sum of the two diagrams in Fig. 4(a) corresponds to thedirect scattering contribution to the RIXS process and is givenby

− 1

2π2

∫ +∞

−∞dω[f (ω) − f (ω + �)]

∣∣Gh

(ω+ + ωh

i

)∣∣2

×Re[χdd (ω−,ω− + �|q) − χdd (ω+,ω− + �|q)

]. (23)

The contribution of the diagram in Fig. 4(b), which welabel as the full exchange scattering process to the four-particlecorrelation function, χ (4)

i,f,f,i(iνi,iνf ,iν ′f ,iν ′

i), on the imaginaryaxis is given by

Q2d�(iνi,iνf ,iνi − iνf )�(iν ′

i ,iν′f ,iνi − iνf )

×χdd (iνi − iνf ,q). (24)

In this equation,

χdd (iν,q) = 1

β

∑m

χdd (iωm + iν,iωm|q)

= i

2π

∫ +∞

−∞dω[χdd (ω+,ω + iν|q)

−χdd (ω−,ω + iν|q) + χdd (ω − iν,ω+|q)

−χdd (ω − iν,ω−|q)]f (ω) (25)

is the dynamical charge susceptibility and � satisfies

�(iνi,iνf ,iνi − iνf ) = − 1

β

∑m

Gh(iωm + iνi − Eh)

×Gd (iωm)Gh(iωm + iνf − Eh). (26)

It is important to mention that in the derivation of the fullexchange scattering process contribution to RIXS in Eq. (24),we have approximated the core-hole–d-hole charge vertexby the core-hole–d-hole interaction Qd under the Hartree-Fock approximation, which is exact to leading order. Afteranalytic continuation to real frequencies, the correspondingcontribution to the RIXS cross section is equal to

Q2d

πIm[�(ωi + iδ,ωf + iδ,� + iδ)

×�(ωi − iδ,ωf − iδ,� + iδ)χdd (� + iδ,q)], (27)

where �(ωi ± iδ,ωf ± iδ,� + iδ) is given by

1

π

∫ +∞

−∞dω f (ω)

[Gh(ω± + ωh

i )Gh

(ω± + ωh

f

)ImGd (ω+)

+Gd (ω∓ − ωi)Gh(ω∓ − � − Eh)ImGh(ω+ − Eh)

+Gd (ω∓ − ωf )Gh(ω± + � − Eh)ImGh(ω+ − Eh)]. (28)

For large core-hole energy, Eh, we need to keep only the firstterm that has a small difference of energies μ + ωi,f − Eh andcan safely neglect the other two terms containing large differ-ences in energies (μ − Eh and μ + � − Eh, respectively). Wethen obtain the contribution from the full exchange processesas

Q2d

π|�(ωi + iδ,ωf + iδ)|2Imχdd (� + iδ|q), (29)

where

�(ωi + iδ,ωf + iδ) = 1

π

∫ +∞

−∞dωf (ω)Gh

(ω+ + ωh

i

)×Gh

(ω+ + ωh

f

)ImGd (ω+), (30)

Imχdd (� + iδ,q) = 1

2π

∫ +∞

−∞dω [f (ω) − f (ω + �)]

× Re[χdd (ω−,ω− + �|q)

−χdd (ω+,ω− + �|q)]. (31)

Apart from the direct and full exchange resonant scatteringprocesses, we also have processes, termed as the partialexchange processes, as shown in Fig. 4(c). The sum oftheir contributions to the four-particle correlation function,χ

(4)i,f,f,i(iνi,iνf ,iν ′

f ,iν ′i), on the imaginary axis is equal to

−Qd

1

β

∑m

χdd (iωm,iωm + iνi − iνf |q)

× [Gh(iωm + iν ′i − Eh)�(iνi,iνf ,iνi − iνf )

+Gh(iωm + iνi − Eh)�(iν ′i ,iν

′f ,iν ′

i − iν ′f )]. (32)

After analytic continuation to real frequencies, we obtain thefollowing partial exchange contribution to the RIXS response:

−Qd

π2Re

{�(ωi + iδ,ωf + iδ)

∫ +∞

−∞dω [f (ω) − f (ω + �)]

×Gh(ω− + ωi − Eh)Re[χdd (ω−,ω− + �|q)

−χdd (ω+,ω− + �|q)]

}. (33)

It is interesting to mention that for the L-edge RIXS process,which we consider in the present study, pure direct resonantscattering processes are overwhelmingly dominant over theresonant exchange (both full and partial) processes. Besidesthe diagrams considered above, we could also have consideredother contributions to the charge vertex between the two core-hole propagators which produce diagrams like the parquetdiagram in Fig. 5. From a simple power counting argument,we can show that the contribution of such diagrams goes atleast as an inverse power of the dimension of the lattice, d. So,in the limit of d → ∞, they all have vanishing contributionsexcept in the case when they are all local (and we neglect allsuch momentum-independent contributions).

The nonresonant part of the RIXS response is found tobe related to the density-density correlation function. To beprecise, the nonresonant part is proportional to the dynamicalstructure factor,50S(q,�), which is given by

S(q,�) = − 1

π[1 + nB(�,T )] Imχdd (� + iδ|q), (34)

125103-6

RESONANT INELASTIC X-RAY SCATTERING IN A MOTT . . . PHYSICAL REVIEW B 86, 125103 (2012)

FIG. 5. (Color online) Parquet diagrams with vanishing contribu-tion to the RIXS response in the limit of infinite dimension.

where nB(ω,T ) = 1/[exp(βω) − 1] is the Bose distribu-tion function and χdd (� + iδ|q) is the dynamic chargesusceptibility48[see Eq. (31)] of the system.

IV. RIXS RESPONSE FOR THE HALF-FILLEDMOTT INSULATOR

As has been already stated, the Falicov-Kimball model athalf-filling (〈nd〉 = 〈nf 〉 = 0.5, μ = U/2, and Ef = 0) on ahypercubic lattice shows a Mott insulating ground state forU > Uc = √

2. We choose U = 2.0, Qd = Qf = 2.5, andT = 0.1 in units of effective hopping amplitude t∗ and fromhere onwards we choose t∗ = 1. This choice of U gives a small“gap” (�gap � 0.25 in units of t∗) Mott insulator. Note that forthe d → ∞ hypercubic lattice DOS there is no true gap asthere is an exponentially small DOS inside the gap.

In Fig. 6, we show the spectral function for the itinerantspecies, Ad (ω), as well as for the core-hole, Ah(ω − Eh).Ad (ω) clearly shows a gap (Mott gap) at the Fermi level(ω = 0) while Ah(ω − Eh) also shows a gap at some otherfrequency (at ω = Eh − 1.2 in this example) and the originof this gap is related to the same strong correlation effectsthat gives rise to the Mott gap in the itinerant species spectralfunction. Surprisingly, the gap structure is quite different forthe core hole, which is arising due to the asymmetry in the

-4 -3 -2 -1 0 1 2 3 4 5 6

ω/t*

0

0.2

0.4

0.6

0.8

1

A(ω

)

Ad(ω)Ah(ω-Eh)

U = 2.0Qd = 2.5Qf = 2.5

T = 0.1

(ω - Eh)/t*

FIG. 6. (Color online) Spectral function for the d hole, Ad (ω),and the core hole, Ah(ω − Eh). For Ah(ω − Eh), the frequency ismeasured with respect to the core-hole energy, Eh, as shown in thetop horizontal axis.

Green’s function for large Eh. Ah(ω − Eh) is dominated bya broad feature, arising from the projection of G>(t) ontothe nh = 0,nf = 0 configuration, along with a very sharppeak arising from the projection onto the nh = 0,nf = 1configuration in the final state.42

From the knowledge of the itinerant electron prop-agator, Gd (q,ω), core-hole propagator, Gh(ω − Eh), andthe fully renormalized two-particle charge susceptibility,χdd (iωm,iωm + iν|q), we can calculate the RIXS responseeither as a function of transferred energy (�) for a givenfixed incident photon energy, ωi (measured with respect tothe core-hole energy Eh), or as a function of ωi for a givenfixed transferred energy, �, for various transferred momentaq of the photon. It is interesting to mention that in the limitd → ∞ the momentum on the hypercubic lattice only entersthrough the dimensionless parameter,51

X(q) = limd→∞

1

d

d∑i=1

cos(qi). (35)

So, −1 � X � 1 and X = 1 and X = −1 corresponds tothe center, (0, . . . ,0) (� point), and the corner, (π, . . . ,π )(M point), of the Brillouin zone of a d-dimensional hyper-cubic lattice, respectively. It is convenient to think of thisparametrization as corresponding to RIXS scattering in thediagonal 〈1 · · · 1〉 direction.

First, in Fig. 7, we plot the resonant part of the RIXSresponse as a function of the transferred energy, �, for varioustransferred momenta, X, for three different incident photonenergies, ωi = −0.5, 0.5, 1.5. For ωi = −0.5, the coreelectron, excited by the incident x-ray photon, is injected intothe lower Hubbard band and the inelastic relaxation processescan happen only within the lower Hubbard band that is evidentin the single-peak structure in the RIXS response in Fig. 7(a).At the M point, the peak is large and well defined but as we gotowards the middle of the Brillouin zone (X = 0.0), the peakgets broadened and the position of the peak does not dispersesignificantly. Finally, as we approach the zone center, theposition of the peak disperses significantly and moves towardslower energy, also at the same time the peak gets more andmore well defined though the integrated spectral intensityunder the peak gradually diminishes and eventually goes tozero at the center of the Brillouin zone (X = 1). This is relatedto the vanishing of the uniform charge susceptibility (due tothe fact that we exactly include the screening dynamics for theconduction electrons and long-wavelength charge excitationsare fully screened).

As we increase the incident photon energy above the Mottgap, we start to excite the system into the upper Hubbardband and in Fig. 7(b), we show a characteristic responsewhen the incident photon, with energy ωi = 0.5, excites acore electron into the bottom of the upper Hubbard band.Near the zone corner the response still shows a single-peakstructure albeit shifted by the insulating gap, but as we gotowards the zone center, the response at low energy developsa very narrow secondary peak separated from the broad mainpeak by the Mott gap. The low-energy peak arises due torelaxation processes within the upper Hubbard band and isnondispersive in nature, whereas the high-energy peak arisesdue to relaxation processes across the Mott gap into the lower

125103-7

NANDAN PAKHIRA, J. K. FREERICKS, AND A. M. SHVAIKA PHYSICAL REVIEW B 86, 125103 (2012)

-1-0.8

-0.6-0.4

-0.20

0.2 0.4

0.6 0.8

0 0.5

1 1.5

2 2.5

3

0.1

0.2

0.3

0.4

0.5

RIX

S re

spon

se (

arb.

uni

ts)

Transferred momentum (X) Transferred energy (Ω)/t*

(a)

RIX

S re

spon

se (

arb.

uni

ts)

fω

ωihE

0

-1-0.8

-0.6-0.4

-0.20

0.2 0.4

0.6 0.8

0 0.5

1 1.5

2 2.5

3 3.5

3

6

9

12

15

RIX

S re

spon

se (

arb.

uni

ts)

Transferred momentum (X) Transferred Energy (Ω)/t*

(b)

RIX

S re

spon

se (

arb.

uni

ts)

Eh

ω

ω i

0

f

-1-0.8

-0.6-0.4

-0.20

0.2 0.4

0.6 0.8

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

5 10 15 20 25 30 35 40 45

RIX

S r

esp

on

se (

arb

. un

its)

Transferred momentum (X)Transferred energy (Ω)/t

*

(c)

RIX

S r

esp

on

se (

arb

. un

its)

ω i

ωf

Eh

0

FIG. 7. (Color online) RIXS response as a function of transferredenergy (�) for various fixed transferred momenta (X) with threedifferent incident photon energies, ωi . (a) The incident photon energyis at the lower Hubbard band, in (b), at the bottom of the upperHubbard band, and in (c), it is at the middle of the upper Hubbardband. The inset of each plot schematically depicts the possibleRIXS relaxation processes. The chosen parameters are U = 2.0,Qd = Qf = 2.5, and T = 0.1. Note, the signal vanishes exactly atX = 1 due to screening.

Hubbard band and is found to be dispersive over the Brillouinzone. With further increase in the incident photon energy toωi = 1.5 the low-energy peak and the spectral weight under it,as shown in Fig. 7(c), grows significantly and is visible for allmomenta along the 〈1 · · · 1〉 direction. However, the intensityof this low-energy peak shows nonmonotonic behavior: it

0 1

2 3

4 5

6 7-0.5

0 0.5

1 1.5

2 2.5

3

5

10

15

20

25

30

35

40

45

RIX

S re

spon

se (

arb.

uni

ts)

Transferred Energy (Ω)/t * Incident Energy (ω i)/t*

(a)

RIX

S re

spon

se (

arb.

uni

ts)

0 1

2 3

4 5

6 7 0

0.5 1

1.5 2

2.5 3

5

10

15

20

25

30

35

RIX

S re

spon

se (

arb.

uni

ts)

Transferred Energy (Ω)/t * Incident Energy (ωi)/t*

(b)

RIX

S re

spon

se (

arb.

uni

ts)

0

1

2

3

4

5-0.5 0

0.5 1

1.5 2

2.5

3

6

9

12

15

RIX

S re

spon

se (

arb.

uni

ts)

Transferred Energy(Ω)/t * Incident Energy(ω i)/t*

(c)

RIX

S re

spon

se (

arb.

uni

ts)

FIG. 8. (Color online) RIXS response as a function of transferredenergy (�) for various fixed incident photon energies, ωi , varyingfrom the bottom of the lower Hubbard band to the top of the upperHubbard band for three different momenta, X: (a) zone corner (Mpoint) X = −1, (b) somewhere in the middle of the zone X = 0, and(c) near the zone center X = 0.9. Inset of each plot shows the positionof each X inside the first Brillouin zone. All other parameters are thesame as in Fig. 7.

first increases up to X = 0 and then starts to decrease andeventually vanishes at the zone center, X = 1. On the otherhand, the intensity of the high-energy peak monotonicallydecreases as well as disperses to lower energies as we gofrom the zone corner towards the zone center and eventuallyvanishes at the center of the Brillouin zone.

125103-8

RESONANT INELASTIC X-RAY SCATTERING IN A MOTT . . . PHYSICAL REVIEW B 86, 125103 (2012)

0

3

6

9

12X = -1.0X = -0.5X = 0.0X = 0.5X = 0.9

0

10

20

30

RIXS

resp

onse

(arb

. uni

ts)

X = -1.0X = -0.5X = 0.0X = 0.5X = 0.9

0 0.5 1 1.5 2 2.5 3 3.5 4Transferred energy (Ω)/t

*

0

10

20

30

40 X = -1.0X = -0.5X = 0.0X = 0.5X = 0.9

0 1 2 3 4 5 60

5

10

15

20X = -1.0X = -0.5X = 0.0X = 0.5X = 0.9

0

8

16

24X = -1.0X = -0.5X = 0.0X = 0.5X = 0.9

0

10

20

30

40X = -1.0X = -0.5X = 0.0X = 0.5X = 0.9

ωi = 0.5

(a)

(b)

ωi = 1.0

(c)

ωi = 1.5

ωi = 2.0

ωi = 2.5

(d)

(e)

(f)

ωi = 3.0

FIG. 9. (Color online) Systematic evolution of the RIXS responseas a function of transferred energy (�) for six incident photon energiesfrom the bottom of the upper Hubbard band: ωi = 0.5 (a) to the topof the upper Hubbard band ωi = 3.0 (f). The dispersive nature ofthe high-energy peak (Mott gap excitation) is noticeable. All otherparameters are the same as in Fig. 7.

In Figs. 8(a)–8(c), we show the systematic evolution of theRIXS response for three transferred momenta X = −1, 0, and0.9, respectively. As we vary the incident photon energy, theenergy of the excited core electron varies from the bottom ofthe lower Hubbard band to the top of the upper Hubbard bandand the RIXS response evolves from a single-peak structure toa two-peak structure. Finally, when the excited core electrongoes into states beyond the edges of the Hubbard bands theresponse vanishes quickly due to an exponential reductionof the density of states that in effect drastically reduces thephase space for inelastic scattering. Also, the overall responsedecreases as we go towards the zone center.

In Fig. 9, we show a more detailed and systematic evolutionof the RIXS response with varying energy of the incidentphoton, which excites the core electron into the upper Hubbardband. As we can clearly see, the low-energy peak, which arisesdue to relaxation processes within the upper Hubbard band,does not disperse over the Brillouin zone (except its intensityvaries), whereas the high-energy peak (which arises due torelaxation processes across the Mott gap) shows significantdispersion over the Brillouin zone. Similar features like thetwo-peak structure and the dispersive nature of the high-energypeak have been observed in the RIXS measurements on aMott insulator Ca2CuO2Cl2 by Hassan et al.5 and have beenattributed to strong correlation effects.

It is interesting to mention that at zero temperature in thehalf-filled Mott-insulating ground state, the chemical potentialas well as the Fermi level lies within the Mott gap, whichresults in a completely filled lower Hubbard band (LHB)and a completely empty upper Hubbard band (UHB). As aresult, when the incident photon energy is within the LHB,the RIXS response will vanish due to the unavailability of anyunoccupied state in the LHB to which the core electron can

be excited and when the incident photon energy is within theupper Hubbard band the RIXS response will have a single-peakstructure arising due to charge transfer excitations across thegap. At finite temperature, some of the states near the top ofthe LHB get thermally excited across the gap and occupy thebottom of the UHB. So, at finite temperature, if the incidentphoton excites a core-electron into the unoccupied states ofthe LHB, the RIXS response will have a single-peak structurecorresponding to relaxation processes involving LHB states.If the excited core-electron goes into the unoccupied states ofthe UHB then it can undergo relaxation processes either withthe thermally excited UHB electrons occupying the bottomof the UHB giving rise to a low-energy nondispersive peak orthrough charge transfer excitations across the gap giving riseto a dispersive high-energy peak. Also, the whole structuregets shifted to higher energy with increasing incident photonenergy due to the fact that with increasing incident photonenergy the transferred energy must also increase in order tohave resonant scattering from the thermally excited states,which predominantly occupy the bottom of the UHB and thetop of the LHB. In the case of a large gap Mott insulator, as willbe shown in a following section (see Sec. VI), the intrabandrelaxation processes from the thermally excited states arenegligibly small compared to the excitations across theMott gap.

Finally, in Figs. 10(a)–10(c), we plot the RIXS response asa function of the incident photon energy, ωi , for various fixedtransferred photon energies, �, for three transferred momentaX = −1, 0, and 0.9, respectively. For small �, the RIXSresponse shows a two-peak structure in ωi , which correspondsto the relaxation processes within the individual bands (upperand lower Hubbard bands). As we increase � an additionalpeak develops between the two peaks. This peak correspondsto the interband relaxation processes across the Mott gap andgrows very rapidly with increasing �, while the other twopeaks decrease in intensity until we are finally left with a lonepeak. Hence, we can infer that in a Mott insulator, interbandrelaxation processes across the Mott gap are dominant overintraband relaxation processes. Also, as we go from the zonecorner to the zone center, the intensity of the peaks decreasesas in Fig. 7.

We also have calculated the dynamical structure factor,S(q,�), which is proportional to the nonresonant part of theRIXS response.48 In Fig. 11, we plot S(q,�) for the small-gapinsulator at T = 0.1. Near the zone corner, S(q,�) has a broadcharge transfer peak but as we go towards the zone center,a secondary peak develops near � = 0 and the integratedspectral weight under the charge transfer peak decreases.Finally, exactly at the zone center (X = 1), the charge transferpeak completely vanishes while the peak around � = 0 turnsinto a δ function, which again arises due to the vanishingof the uniform charge susceptibility at finite frequency. Thisbehavior is quite different from the resonant response wherethe two-peak structure is most prominent near the zone cornerand also the charge transfer peak in S(q,�) is much smallerand much less dispersive than the high-energy peak observed inthe resonant response and most importantly the position of thepeak cannot be identified with any particular x-ray transitionprocess.

125103-9

NANDAN PAKHIRA, J. K. FREERICKS, AND A. M. SHVAIKA PHYSICAL REVIEW B 86, 125103 (2012)

0.5 1

1.5 2

2.5 3

3.5 4

4.5-0.5 0

0.5 1

1.5 2

2.5 3

3.5 0 5

10 15 20 25 30 35 40 45

RIX

S re

spon

se (

arb.

uni

ts)

Transferred Energy (Ω)/t *Incident Energy (ω i)/

t*

(a)

RIX

S re

spon

se (

arb.

uni

ts)

0.5 1

1.5 2

2.5 3

3.5 4-0.5

0 0.5

1 1.5

2 2.5

3 3.5

0 5

10 15 20 25 30 35 40

RIX

S re

spon

se (

arb.

uni

ts)

Transferred Energy (Ω)/t *Incident energy (ω i)/

t*

(b)

RIX

S re

spon

se (

arb.

uni

ts)

0.25 0.5

0.75 1 1.25

1.5 1.75

2 2.25

2.5 2.75

3-1-0.5

0 0.5

1 1.5

2 2.5

3 0 2 4 6 8

10 12 14 16 18 20

RIX

S re

spon

se (

arb.

uni

ts)

Transferred Energy (Ω)/t *Incident E

nergy (ω i)/t*

(c)

RIX

S re

spon

se (

arb.

uni

ts)

FIG. 10. (Color online) RIXS response as a function of incidentenergy, ωi , for various fixed transferred energies, �, for threecharacteristic transferred momenta, X: (a) zone corner (M point)X = −1, (b) in the middle of the zone X = 0, and (c) near the zonecenter X = 0.9. The inset of each plot shows the position of each X

inside the first Brillouin zone. All other parameters are the same asin Fig. 7.

V. CORE-HOLE BROADENING EFFECTS

In the preceding section, we have not included anyadditional core-hole lifetime broadening effects that can arisedue to various nonradiative Auger and fluorescence effects andare important in the transition metal RIXS processes (we onlyincluded the intrinsic many-body effects in determining thecore-hole lifetime). In our calculation, we can easily includesuch effects by simply making the core-hole energy Eh com-plex, i.e., by making the transformation Eh → Eh − i� intothe retarded Green’s function, Gr

h(t) = �(t)[G>h (t) − G<

h (t)],

-1 0 1 2 3 4 5 6 7 8Ω

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

S(q

,Ω)

X = -1.0X = -0.9X = -0.8X = -0.7X = -0.6X = -0.5X = -0.4X = -0.3X = -0.2X = -0.1X = 0.0X = 0.1X = 0.2X = 0.3X = 0.4X = 0.5X = 0.6X = 0.7X = 0.8X = 0.9X = 1.0

FIG. 11. (Color online) Dynamical structure factor, S(q,�), forU = 2.0 and T = 0.1. Note, it vanishes for X = 1 due to screening.

and Eh → Eh + i� into the advanced Green’s function,Ga

h(t) = �(−t)[G<h (t) − G>

h (t)], respectively. Here, � = h/τ

with τ being the core-hole lifetime and for most materials,� ∼ 100–400 meV (1–3 in units of t∗).

While this may seem like an ad hoc procedure, microscopiccalculation of Auger lifetimes in condensed matter systems isbeyond the scope of this work, and this broadening effect on thecore hole spectra function is important to determine whetherall of the features calculated above survive when the core holespectral function is further broadened. Hence it is an importantelement in any analysis.

First, in Fig. 12, we show the systematic evolution of thecore-hole spectral function, Ah(ω − Eh), with additional core-hole broadening effects parameterized by �. With increasing�, the height of the sharp peak in Ah(ω − Eh) reduces, whileits width significantly broadens but the asymmetrical natureof the structure is largely retained. Also, the Mott gap in the� = 0.0 case (near ω − Eh ∼ −1.0) is replaced by a dip inthe spectral function and the tail of Ah(ω − Eh) increases withincreasing �.

-4 -3 -2 -1 0 1 2 3 4

(ω - Eh)/t*

0

0.4

0.8

1.2

1.6

2

Ah(ω

-Eh)

Γ = 0.0Γ = 0.1Γ = 0.2Γ = 0.3

U = 2.0Qd = 2.5

Qf = 2.5

T = 0.1

FIG. 12. (Color online) Core-hole spectral function, Ah(ω − Eh),evolution with core-hole broadening parameter, �. All other param-eters are same as in Fig. 7.

125103-10

RESONANT INELASTIC X-RAY SCATTERING IN A MOTT . . . PHYSICAL REVIEW B 86, 125103 (2012)

0

5

10

15

20

25

30 X = -1.0X = -0.8X = -0.6X = -0.4X = -0.2X = 0.0X = 0.2X = 0.4X = 0.6X = 0.7X = 0.8

6 7Transferred energy (ω)/t

*

0.03

0.06

0.09

0.12

RIXS

resp

onse

(arb

. uni

ts)

X = -1.0X = -0.8X = -0.6X = -0.4X = -0.2X = 0.0X = 0.2X = 0.4X = 0.6X = 0.8

0

0.01

0.02

0.03

0.04

0.05

0.06X = -1.0X = -0.8X = -0.6X = -0.4X = -0.2X = 0.0X = 0.2X = 0.4X = 0.6X = 0.8

0 11 2 3 4 5 2 3 4 5 6 70

0.01

0.02

0.03

X = -1.0X = -0.8X = -0.6X = -0.4X = -0.2X = 0.0X = 0.2X = 0.4X = 0.6X = 0.8

0 0.2 0.4 0.6 0.8 1Ω/t

0

1

2

3×10R

IXS

res

po

nse

Γ = 0.0

Γ = 0.1

Γ = 0.2

Γ = 0.3

(a)

(b)

(c)

(d)

FIG. 13. (Color online) Evolution of the RIXS response as afunction of the transferred energy, �, and various fixed transferredmomenta, X, for incident photon energy ωi = 2.5 with variouscore-hole lifetime broadening parameters, �. (a) Response for � = 0and the inset of panel (a) shows the quasielastic peak in the blownup far-infrared region. (b)–(d) Responses for broadening parametersmuch smaller (� = 0.1), comparable (� = 0.2), and larger (� = 0.3)than the Mott insulating gap �gap ∼ 0.25. All other parameters arethe same as in Fig. 7. For all the four sets, calculations have beenperformed from X = −1.0 (top most curve) to X = 0.9 (bottom mostcurve) with a step size of 0.1 for the momentum variable X.

In Figs. 13(a)–13(d), we show a detailed evolution ofthe RIXS response with various broadening parameters, �

(measured in units of t∗). We choose four characteristicparameters � = 0.0, 0.1, 0.2, and 0.3 corresponding to nobroadening, much smaller, comparable, and larger broadeningcompared to the intrinsic Mott-insulating gap in the system,respectively. All other parameters chosen are the same as in theprevious case. In Fig. 13 (a), we plot the RIXS response withoutany additional core-hole broadening effect and the responseshows a clear two-peak structure with the peaks well separatedby the Mott gap. The low-energy peak is nondispersive but thehigh-energy peak is highly dispersive in momenta and alsochanges its shape significantly as we go from the zone cornerto the zone center. Also, interestingly, close to the zone centera quasielastic peak develops in the far-infrared region. Theintensity as well as sharpness of this peak increases as we gotowards the zone center.

From the detailed analysis of the weights of the differentscattering processes presented in Fig. 4 for the total RIXSresponse, we find that the main two-peak structure and itsrather strong momentum dependence originates from the directscattering processes described by Eq. (24) [see Fig. 4(a)].Contribution from the full exchange (indirect) processes inEq. (29) [see Fig. 4(b)] is much smaller and are mainlyresponsible for the quasielastic peak, which corresponds tothe dynamical structure factor, S(q,�) [see Fig. 11].

The presence of additional core-hole lifetime broadeningeffects strongly suppresses the direct contribution. We nolonger see a clear gap structure in Figs. 13(b)–13(d) but thetwo-peak structure is still clearly evident for all momenta. Also

the high-energy peak still remains dispersive throughout theBrillouin zone, while the low-energy peak is more or lessnondispersive in nature as in the case with no additionalcore-hole broadening. The height of the quasielastic peakclose to the zone center slightly decreases with increasing �,which is related to the weak dependence of the � prefactor inEq. (29) on �, while its width and dispersive features remainssimilar to the � = 0.0 case. However, in a real experiment,finite resolution and the resolution broadened tail of thehuge elastic peak will mask such quasielastic features whichwill then not be observable. So, the presence of additionalcore-hole broadening effects can significantly modify theoverall response, but the most important qualitative features,like the two-peak structure and the dispersive features of thesepeaks remain similar.

VI. RESPONSE AWAY FROM HALF-FILLING

Finally, we consider a Mott insulator at arbitrary filling toexamine the breaking of particle-hole symmetry in the RIXSresponse. We choose U = 4.0, Qd = Qf = 5.0, 〈nf 〉 = 0.25,〈nd〉 = 0.75, and T = 0.1. This choice of parameters givesa large-gap Mott insulator (�gap ∼ 1.8 in units of t∗). InFig. 14(a), we show the d-hole spectral function, Ad (ω), as wellas the core-hole spectral function, Ah(ω − Eh), with (� = 1.0)and without (� = 0.0) core-hole broadening effects. Ad (ω) isdominated by two asymmetrical peaks separated by a largegap at the Fermi level. Ah(ω − Eh) [without any additionalcore-hole broadening effect (� = 0.0)] shows two very closelyspaced sharp peaks on top of a broad feature but with theinclusion of large core-hole broadening (� = 1.0), the wholestructure gets drastically modified and Ah(ω − Eh) resemblesa broad nearly symmetrical single peak.

In Fig. 14(b), we show the RIXS response as a functionof the transferred energy, �, for various fixed transferredmomenta, X, for � = 0.0. The response in this case isoverwhelmingly dominated by a huge peak arising due torelaxation processes across the Mott gap into the LHB, whilethe intraband (within the UHB) relaxation processes as shownin the inset of Fig. 14(b) are much weaker than the interbandprocesses. As already has been mentioned in Sec. IV, ina large-gap Mott insulator, the density of thermally excitedstates, which occupy the bottom of the UHB, is extremelysmall and hence cannot provide any significant relaxation tothe core electrons excited to the UHB. The high-energy peak,just as in the case of particle-hole symmetric half-filled case,shows significant dispersion with transferred momentum—thepeak disperses outwards in energy as we go from the zonecenter towards the zone corner. Finally, we study the RIXSresponse in the presence of finite core-hole broadening. InFig. 14(c), we show results for a typical broadening (of theorder of the Mott gap) � = 1.0. The first noticeable featureis the reemergence of the two-peak structure. This is mainlydue to a huge suppression of the sharp resonating peak inAh(ω − Eh), as can be observed in Fig. 14(a), which in effectdrastically reduces the resonant response across the Mott gapby six orders of magnitude. The high-energy peak still showssignificant dispersion across the entire Brillouin zone and

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NANDAN PAKHIRA, J. K. FREERICKS, AND A. M. SHVAIKA PHYSICAL REVIEW B 86, 125103 (2012)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

ω/t*

0

0.2

0.4

0.6

0.8

1

A(ω

)

Ad(ω)Ah(ω-Eh) , Γ = 0.0Ah(ω-Eh) , Γ = 1.0

U = 4.0Qd = 5.0Qf = 5.0

T = 0.1nf = 0.25

(a)

(ω - Eh)/t*

0 1 2 3 4 5 6 7

Transferred energy (Ω)/t*

0

1

2

3

4

5

6×103

RIX

S r

esp

on

se (

arb

. un

its)

X = -1.0X = -0.9X = -0.8X = -0.7X = -0.6X = -0.5X = -0.4X = -0.3X = -0.2X = -0.1X = 0.0X = 0.1X = 0.2X = 0.3X = 0.4X = 0.5X = 0.6X = 0.7X = 0.8X = 0.9

2 3 4 5 6 7 8 9Transferred energy (Ω)/t

*

0

0.2

0.4

0.6

0.8

1

RIX

S r

esp

on

se (

arb

. un

its)

X = -1.0X = -0.8X = -0.6X = -0.4X = -0.2X = 0.0X = 0.2X = 0.4X = 0.6X = 0.8X = 0.9

(b)

Γ = 0.0

0 1 2 3 4 5 6 7 8

Transferred energy (Ω)/t*

0.0

0.5

1.0

1.5

2.0

2.5

3×10-3

RIX

S r

esp

on

se (

arb

. un

its)

X = -1.0X = -0.9X = -0.8X = -0.7X = -0.6X = -0.5X = -0.4X = -0.3X = -0.2X = -0.1X = 0.0X = 0.1X = 0.2X = 0.3X = 0.4X = 0.5X = 0.6X = 0.7X = 0.8X = 0.9

0 0.1 0.2 0.3 0.4 0.5Ω/t

0

1

2

3

4×10-6

RIX

S r

esp

on

se (

arb

. un

its) X = -1.0

X = -0.9X = -0.8X = -0.7X = -0.6X = -0.5X = -0.4X = -0.3X = -0.2X = -0.1X = 0.0X = 0.1X = 0.2X = 0.3X = 0.4X = 0.5X = 0.6X = 0.7X = 0.8X = 0.9

Γ = 1.0

(c)

FIG. 14. (Color online) (a) d-hole spectral function, Ad (ω), aswell as the core-hole spectral function, Ah(ω − Eh), with (� = 1.0,blue line) and without (� = 0.0, green line) core-hole broadeningeffects in a particle-hole asymmetric large gap Mott insulator. For thecore-hole spectral function, energy is measured with respect to thecore-hole energy, Eh, i.e., ω → ω − Eh. (b) and (c) RIXS response asa function of the transferred energy, �, for various fixed transferredmomenta, X, without [� = 0.0 (b)] and with [� = 1.0 (c)] core-holebroadening effects for incident photon energy (measured with respectto Eh) ωi = 3.5. Magnified response in the inset of panel (b) clearlyshows a two-peak structure and the inset of panel (c) shows thequasielastic peak in the blown up far-infrared region. The parametersused for this plot are U = 4.0, Qd = Qf = 5.0, T = 0.1, and nf =0.25, nd = 0.75.

the low-energy peak also shows dispersive features. Also, asshown in the inset of Fig. 14(c), a very weak quasielastic peaksimilar to the half-filled case emerges due to the full exchange(indirect) processes.

VII. CONCLUSIONS

In conclusion, we have studied the RIXS response ina Mott insulator which is modeled by the Falicov-Kimballmodel. We have considered both the particle-hole symmetrichalf-filled case as well as the general particle-hole asymmetriccase. We find that when the incident photon energy is lyingwithin the upper Hubbard band, the resonant response showsa two-peak structure arising from the intraband (low-energypeak) and interband (high-energy peak) relaxation processes(as expected since the gap is larger but the temperature isthe same as before). The high-energy peak is found to bemuch larger and sometimes overwhelmingly larger (awayfrom half-filling case) than the low-energy peak and showsdispersive features throughout the entire Brillouin zone, whilethe low-energy peak remains more or less nondispersive. Suchdistinctive features have already been observed in a large classof transition metal K-edge RIXS responses in a wide class ofoxide materials and have been attributed to the nonlocal natureof the Mott gap excitations.

We also have considered moderately large core-hole broad-ening effects (due to finite Auger lifetime of the core hole) onthe RIXS response and we see that despite significant changein the RIXS response many interesting qualitative featureslike the two-peak structure and the dispersive nature of thehigh-energy peak remains more or less intact. The quasielasticfeature near the zone center, which originates from the purefull exchange (indirect) processes, becomes comparable to theother two peaks. However, this peak will be completely maskedby the resolution limited tail of the elastic peak (and cannot beobserved in any current experiments). For the half-filled case,we also have calculated the dynamical structure factor, S(q,�),which is proportional to the nonresonant part of the response.S(q,�) is either dominated by a very weakly dispersive chargetransfer peak when the transferred photon momentum, X, isnear the zone corner or by a narrow peak around � = 0,which corresponds to the quasielastic peak in RIXS fromthe full exchange (indirect) processes, when the momentumis close to the zone center. Exactly at the zone center S(q,�)vanishes for finite frequencies, while the narrow peak becomesa delta function peak. We believe, features like the two-peakstructure and the dispersive natures of the charge transfer peakare generic features of the RIXS response in a Mott insulatorand similar features are expected to be observed in calculationsbased on the more realistic Hubbard model.

ACKNOWLEDGMENTS

This work was supported by the US Department ofEnergy under grant No. DE-FG02-08ER46542 for the workat Georgetown and grants Nos. DE-FG02-08ER46540 andDE-SC0007091 for the collaboration. We would like tothank Tom Devereaux and Brian Moritz for many usefuldiscussions and a critical reading of this manuscript. Wewould also like to thank M. A. van Veenendaal, A. Bansil,

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RESONANT INELASTIC X-RAY SCATTERING IN A MOTT . . . PHYSICAL REVIEW B 86, 125103 (2012)

R. Markiewicz, J. Moreno, Z. Hussain, J. Rehr, andA. Sorini for useful discussions. J.K.F. also acknowledgessupport from the McDevitt bequest at Georgetown. All the

Feynman diagrams in this paper were drawn by using opensource software program JAXODRAW and the original referencehas been duly cited in Ref. 52.

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