Probing Competing and Intertwined Orders with Resonant Inelastic x-ray Scattering in the Hole-Doped Cuprates
PProbing Competing and Intertwined Orders with Resonant Inelastic x-ray Scatteringin the Hole-Doped Cuprates
David Benjamin, Israel Klich, and Eugene Demler Physics Department, Harvard University, Cambridge, Massachusetts, USA Department of Physics, University of Virginia, Charlottesville, VA, USA (Dated: September 27, 2018)We develop a formalism to study indirect resonant inelastic x-ray scattering (RIXS) in systemswith itinerant electrons, accounting for the attraction between valence electrons and the positively-charged core hole exactly, and apply this formalism to the hole-doped cuprate superconductors. Wefocus on the relationship between RIXS lineshapes and band structure, including broken symmetries.We show that RIXS is capable of distinguishing between competing order parameters, establishingit as a useful probe of the pseudogap phase.
PACS numbers: 78.70.Ck, 74.72.Gh
I. INTRODUCTION
The state of the underdoped cuprates above the su-perconducting transition temperature is an outstandingpuzzle in the field of high-temperature superconductivity.Many different types of order [1–11] have been hypothe-sized to coexist or compete with superconductivity andto explain pseudogap behavior in which the density ofstates is depleted around the Fermi energy. These po-tential phases are difficult to detect because, with theexception of charge density wave (CDW) and spin den-sity wave (SDW) in certain cuprates at specific dopings,they exhibit dynamic fluctuations and spatial inhomo-geneity [5, 12]. These fluctuations smear the signatures ofpotential order parameters and render them undetectableby quasistatic probes that effectively average over timescales longer than those of fluctuations. Resonant inelas-tic x-ray scattering (RIXS) can overcome this obstaclebecause its time scale is bounded by the finite lifetime(roughly (250 meV) − ≈ d -density wave [4]. We derive a formula for indirect RIXSin systems with itinerant electrons, accounting exactlyfor the core hole potential acting on valence electrons. II. THEORETICAL FORMALISM
In indirect RIXS a photon is absorbed and causes atransition of a core electron to an excited band above thevalence band, leaving behind a positively-charged corehole. In the cuprates, the most commonly-used transi-tion is 1 s → p . The excited band is generally weakly-interacting; in cuprates this occurs because it is derivedfrom a delocalized 4 p orbital. Thus the excited electrondoes very little in indirect RIXS except re-fill the corehole and emit a scattered photon. The interesting ac-tion is the effect of the core hole on the electrons inthe valence band, which in a conducting system is togenerate particle-hole “shake-up” pairs. When one mea-sures the energy difference ∆ ω and momentum difference∆ q between the incident and scattered photons, one ismeasuring the dispersion of shake-up pairs. The spectralweights of shake-up processes at different energy and mo-menta reveal the joint density of states of particles andholes, which in turn sheds light on quasiparticle disper-sions modified by different types of order.The intensity for incident x-rays of momentum q i andenergy ω i to be scattered into outgoing momentum q f = k i + ∆ k and energy ω f = ω i − ∆ ω is calculated from thefamiliar Kramers-Heisenberg formula [13] I ∝ (cid:88) f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n (cid:104) f | T | n (cid:105)(cid:104) n | T † | i (cid:105) ω i + E n − E i + i Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ( E f − E i − ∆ ω ) , (1)where | i ( n, f ) (cid:105) are the initial (intermediate, final) stateswith energies E i ( n,f ) , 1 / Γ is the lifetime of the inter-mediate state core hole, and T † = (cid:80) m e q i · R m s m p † m , T = (cid:80) m e − q i · R m s † m p m are transition operators. Thisnotation reflects the 1 s → p indirect RIXS in the high-T c cuprates. We assume momentum-independent dipolematrix elements and leave polarization dependence im-plicit [14]. The attractive potential due to the core holeis most easily accounted for by switching to the time do- a r X i v : . [ c ond - m a t . s t r- e l ] J u l main [15, 16], where we have I ∝ (cid:88) m,n e i ∆ q · ( R n − R m ) (cid:90) ∞−∞ ds (cid:90) ∞ dt (cid:90) ∞ dτe iω i ( t − τ ) − is ∆ ω − Γ( t + τ ) S mn , (2)where S mn = (cid:68) e iH τ p n e − iH n τ p † n e iH s p m e iH m t p † m e − iH ( t + s ) (cid:69) . (3)Eq. (3) is a Keldysh-like integral describing the historyof absorption and emission events separated by time evo-lution operators. Since the core hole is immobile and hasa constant energy that can be absorbed into ω i we areable to remove via H → H m ( n ) ≡ H + V m ( n ) , where H = H d + H p is the Hamiltonian of valence d electronsand the p band and V m is the potential due to the corehole at site m that acts on valence electrons. As the 4 p band is highly dispersive we assume that 4 p electrons donot interact with the core hole or the valence band; thisis the usual “spectator” approximation [17]. Therefore,the d and p bands are separable and the 4 p contributionto Eq. (3) reduces to a product of Green functions: S mn = G nnp ( τ ) G mmp ( − t ) × (cid:68) e iH d τ e − iH d,n τ e iH d s e iH d,m t e − iH ( t + s ) (cid:69) , (4)where G nnp ( t ) = (cid:104) | p n ( t ) p † n (0) | (cid:105) is an easily-calculatedsingle-particle quantity. We note that by Eq. (2) S mn is measurable as the Fourier transform of the intensity.Given the form of S mn as a Keldysh-like correlator it isintuitively clear that it may reveal the real-space struc-ture of H . We will discuss this point further below.Following a recent analysis of direct RIXS [18] wetreat the valence band as a system of non-interactingquasiparticles, which is valid when the quasiparticle life-time is long compared to the core hole lifetime 1 / Γ.In the cuprates Γ ≥
250 meV, which exceeds quasi-particle widths even quite far from the the Fermi sur-face. Many-body averages of products of exponentiatedquadratic operators such as in Eq. 4 have been dis-cussed in numerous works [19–23]. The standard for-mula (cid:10) e Z (cid:11) = det (cid:104) (1 − ˆ N ) + e z ˆ N (cid:105) , where uppercase ‘ Z ’denotes a quadratic many-body operator and lowercase‘ z ’ denotes its matrix elements Z = d † i z ij d j and ˆ N is theFermi occupation operator, gives S mn ( t, s, τ ) = G nnp ( τ ) G mmp ( − t ) det (cid:104)(cid:16) − ˆ N (cid:17) + e ih d τ e − ih d,n τ e ih d s e ih d,m t e − ih ( t + s ) ˆ N (cid:105) , (5)where ˆ N ≡ (cid:0) e H /k B T (cid:1) − and H d = d † i (cid:16) ˆ H d (cid:17) ij d j . Tohandle spin, we let m → ( m, σ ) represent a combinedsite and spin index and replace the basis { i } of Wannierorbitals with a spin-Wannier basis { ( i, σ ) } . The above determinant formula requires a quadractic Hamiltonianof the form H = d † i hd j . We map singlet pairing Hamilto-nians with terms of the form d †↑ d †↓ to the necessary form d †↑ d ↓ via a particle-hole transformation d †↓ ↔ d ↓ .The 4 p Green functions in Eq. (4) appear to complicatethe analysis of indirect RIXS but in fact simplify it byrestricting the number of particle-hole excitations causedby the core hole. Because the 4 p band is highly dispersivethe same-site Green function G nnp ( t ) decays very rapidly– it is unlikely that a 4 p electron created at site n willreturn except after very brief times. Therefore the timeintervals associated with t and τ are effectively truncatedmuch more than by the core hole lifetime alone. Thismakes numerical integration less computationally expen-sive, but more importantly dramatically reduces the con-tribution of processes in which the core hole potentialgenerates multiple particle-hole “shake-up” pairs. There-fore (see below) indirect RIXS spectra can be interpretedin terms of single shake-up pairs and are not dominatedby complicated processes involving multiple shake-ups. Acommon source of confusion is the assumption that shortintermediate state timescales t and τ imply poor energyresolution. However, the times t and τ are conjugate tothe incident photon energy ω i , and indeed spectra arevirtually featureless as a function of ω i . However, theenergy transfer ∆ ω is conjugate to the time s , duringwhich there is no core hole and no 4 p electron. Henceresolution of ∆ ω is limited only by instrumental resolu-tion. This preserves the dispersion information of ∆ ω vs.∆ k that is fundamental to RIXS. III. MAIN RESULTS
We are interested in whether indirect RIXS distin-guishes different types of short-range order in hole-dopedcuprates, particularly those that are hypothesized to ex-ist in the pseudogap phase of underdoped cuprates above T c . In order to exploit the determinantal formalism,which requires a quadratic Hamiltonian, we will treatthese orders as mean-field additions to the band struc-ture Hamiltonian H = (cid:88) k ,σ ε k d † k ,σ d k ,σ , (6)For concreteness we will use a single-band tight-binding dispersion ε k = − t (cos( k x ) + cos( k y )) − t cos( k x ) cos( k y ) − t (cos(2 k x ) + cos(2 k y )) − t (cos(2 k x ) cos( k y ) + cos( k x ) cos(2 k y )) with parametersfit to ARPES data: ( t , t , t , t ) = (126 , − , , . V m = − U c (cid:80) σ d † mσ d mσ for the core hole, with U c = 5 . H we add charge density wave(CDW), d -density wave, and antiferromagnetic (AF) FIG. 1: Indirect RIXS intensity (arbitrary units) versus momentum over entire Brillouin zone for underdoped Bi-2212 describedby Hamiltonians H (upper left) and mean-field Hamiltonians (clockwise from upper right): H + H CDW , H + H DDW , and H + H AF , with mean-field perturbations of amplitude V CDW,DDW,AF = 100 meV. Spectra are measured at energy transfer∆ ω = 100 meV and incident photon energy ω at the 4 p threshold. Here and elsewhere in this paper black denotes zero intensityand bright red denotes maximal intensity. orders: H CDW = (cid:88) k V CDW d † k + Q d k (7) H DDW = (cid:88) k V DDW ( k ) d † k + Q d k (8) H AF = (cid:88) k V AF (cid:104) d † k + Q , ↑ d k , ↑ − d † k + Q , ↓ d k , ↓ (cid:105) , (9)where the ordering wavevectors are Q = (2 π/ , Q = ( π, π ) (DDW), and Q = ( π, π ) (AF), and V DDW ( k ) = V DDW (cos k x − cos k y ). We restrict our at-tention here to period-4 commensurate CDW order, butthe qualitative features of the RIXS signal we present below are not specific to this wavevector. The form of H AF in Eq. (9) is that of an alternating sublattice mag-netization, which could occur in cuprates if residual lo-cal antiferromagnetic correlations persist after long-rangeantiferromagnetic order is destroyed by hole doping. Itis important to compare this form of AF to DDW be-cause they have the same ( π, π ) wavevector. Thus weare able to study whether indirect RIXS is sensitive notonly to the ordering wavevector but also to the form fac-tor V DDW ( k ). Another important distinction to note isthat the DDW is orthogonal to conventional charge order– while DDW is a form of translational symmetry break-ing, the charge density due to DDW does not exhibitsymmetry breaking. We will therefore be able to rejectany naive suspicion that indirect RIXS is only sensitiveto order parameters that accompany a density distortion.In direct RIXS experiments, and indeed in most spec-troscopic experiments it is customary to present datain the form of lineshapes, that is, intensity versus ∆ ω and ∆ k for momenta along some fixed cut in momen-tum space. This makes sense for presenting the disper-sion of collective modes. However, in indirect RIXS ofunderdoped cuprates the fundamental excitations are acontinuum of particle-hole pairs, the dispersion of whichis not inherently interesting. Rather, the indirect RIXSintensity measures the joint density of states of particlesand holes, which are useful in that they reflect the overallfermiology over the entire Brillouin zone. Thus, the mostnatural way to present indirect RIXS data is as plots ofintensity versus momenta over the entire Brillouin zonefor fixed ∆ ω .In Fig. 1 we examine the indirect RIXS spectrum in astate with no terms in H d other than the band structure H and compare it to states in which various mean-fieldorder parameters are added to H . The intensity in theunordered state corresponds closely to the joint densityof particle-hole pairs with total momentum ∆ k and totalenergy ∆ ω . Besides the peaknear zero momentu trans-fer, the dominant feature is the peak at ∆ k = ( π, π ) atenergies several hundred meV and less. This is due to thelarge density of states for both particles and holes nearantinodal regions ( π,
0) etc, which is caused by a saddlepoint in the dispersion. The peak does not occur exactlyat ( π, π ) because the Fermi surface does not cross antin-odal momenta (0 , π ) and ( π, π, − (0 , π ) = ( π, π ). In a non-interacting systemat half filling and only nearest-neighbor hopping thereare van Hove singularities at antinodal momenta and wewould expect the diamond-shaped Fermi surface to yielda RIXS maximum at ( π, π ) for small energy transfers.For the ordered systems we use order parameter ampli-tudes V DDW = 100 meV, V CDW = 100 meV, V AF = 100meV, which are typical energy scales for the pseudogap ofthe cuprates’ normal state. These yield distinct changesin the RIXS spectra that allow experiments to distin-guish them. Of particular interest is that the d -densitywave phase exhibits a clear signature distinct from otherphases, including the antiferromagnetic phase which alsohas a wavevector of ( π, π ). As seen in Fig. 1, indirectRIXS at small ∆ ω of systems with DDW and AF orderfollows a similar pattern to unordered systems describedby the band structure H : intensity maxima at zeromomentum and around ( π, π ), joined by arms runningalong the nodal directions ( k, k ). In the DDW system,the maximum near ( π, π ) is strengthened relative to themaximum near (0 , π, π ). As long as V AF is not extremely strong, the bulk of the Fermi sur-face is replaced by large oval-shaped pockets that overlap FIG. 2: Comparison of exact (Eqs. (2) and (5)) and approx-imate (Eq. (10) formulas for momentum-dependent indirectRIXS intensity of unordered system for doping p = 0 .
15, en-ergy transfer ∆ ω = 100 meV. the unreconstructed Fermi surface on one side and par-allel it on the other. From the point of view of the jointdensity of states of particle hole pairs, the effect is sim-ilar to a broadening of the Fermi surface. In contrast,the d -density wave has the same ( π, π ) wavevector, butthe form factor V DDW ( k ) vanishes in the nodal directionand is maximal in the antinodal direction. Hence theFermi surface reconstruction due to DDW order is re-stricted to the antinodal momenta near ( π,
0) and (0 , π ).This explains why the “arms” of the indirect RIXS in-tensity pattern are not broadened as in the system withAF order. The reason the intensity maximum at ( π, π )is strengthened is that the antinodal saddle point of thedispersion is buried inside the unreconstructed Fermi sur-face – there are antinodal holes, but only near -antinodalparticles. After reconstruction, there are more particlesavailable near the saddle point.Unlike the previous examples, CDW order stronglymodifies the intensity pattern of the orderless system.One distinct feature is the appearance of maxima near(0 , π/ Q CDW , which gen-erates low-energy particle hole pairs with total momen-tum Q CDW . Another obvious feature is the destructionof the intensity near ( π, π ), which was a maximum forthe unordered system as well as the DDW and AF sys-tems. We observe generally that a perturbation withwavevector Q does not destroy the density of states oflow energy particle-hole pairs with wavevector Q . (Ofcourse, the order yields a gap at energy scales of meVor tens of meV, which are much smaller than currentenergy scales measured by RIXS). Thus DDW and AForder are in a sense “compatible” with the RIXS spec-trum of the unordered system. Ordering at a differ-ent wavevector, on the other hand, can and does dras-tically change the particle-hole joint density of states.The clear qualitative difference between the CDW spec-trum and those of DDW and AF systems, along withthe maximum at Q CDW , are telltale signs of transla-tional symmetry breaking at a wavevector other than( π, π ). However, one can discern different orders evenmore clearly with a complementary measurement: Re-call from above that the Keldysh-like two-site correlator S mn can be measured as the Fourier transform of inten-sity. Specifically, the Fourier transform of I (∆ k , ω, ∆ ω )gives S (∆ r , ω, ∆ ω ) ≡ (cid:80) r m − r n =∆ r S mn ( ω, ∆ ω ). It is in-tuitively clear that S (∆ r ) ought to have a spatial struc-ture corresponding to that of H . In Fig. 3, we see thatthe Fourier transform of RIXS intensity exhibits the spa-tial periodicity of H , that is, a checkerboard pattern for DDW and AF orders and a stripe at ∆ x = 4 for period-4 CDW. (Even though our calculations were performedon 40 ×
40 systems, we show only a few near-neighborlattice sites in the Fourier-transformed spectra because S mn decays rapidly with separation between r m and r n ).Thus, each of the orders we have considered have notonly distinct patterns of indirect RIXS, but “smokinggun” signatures with simple interpretations. There are,of course, many proposed pseudogap order parametersother than the CDW, DDW, and AF that we have con-sidered here. However, these three examples demonstratethat one can easily obtain robust, falsifiable predictionsfor the indirect RIXS spectrum of any candidate order.Furthermore, these are among the most widely-proposedorders for the normal state of underdoped cuprates andit is encouraging that they exhibit such distinct patterns. IV. MODELLING RIXS WITH A SINGLEPARTICLE-HOLE PAIR
As mentioned above, our exact results ought to be well-approximated by considering only a single shake-up pairin indirect RIXS. It is straightforward to derive the RIXSintensity under this approximation for arbitrary bilinearHamiltonians H d containing any combination of mean-field and impurity potentials. We obtain I ∝ (cid:88) α,β (cid:12)(cid:12)(cid:12)(cid:12) ˜ V α,β ( k ) (cid:90) g ( ε ) dε ( ω − ( ε α − ε β − ε + i Γ)( ω − ε + i Γ) (cid:12)(cid:12)(cid:12)(cid:12) × n f ( ε α )(1 − n f ( ε β )) δ ( ε α − ε β − ∆ ω ) , (10)˜ V α,β ( k ) = (cid:88) m e i k · r m (cid:104) α | V m | β (cid:105) where | α (cid:105) and | β (cid:105) are single-particle eigenstates of H d , g ( ε ) is the 4 p density of states, and ˜ V ( k ) = (cid:80) m e i k · r m V m is the Fourier transform of the core hole potential opera-tor. We show in Fig. III that Eq. (10) gives results verysimilar to the exact formula Eq. (5), which we used togenerate the figures in this paper. In addition to mak-ing explicit the connection between RIXS and the jointparticle-hole density of states, Eq. (10) is also useful foranalyzing the dependence of the RIXS signal on the inci-dent photon energy ω . Here the resonance in ω is convo-luted with the broad and featureless 4 p density of states.The intensity is maximal when ω is in resonance to ex-cite the core electron to the 4 p band minimum, wherethe group velocity vanishes and the p electron is mostlikely to return to the core hole site. Other than thisfeature the resonant factor yields little structure. In par-ticular, the overall shape of the spectrum as a function of∆ k and ∆ ω is unchanged as ω varies (to the point that ω = 10 eV yields figures identical to those shown above,up to an overall scale), although the overall magnitude isstrongly ω -dependent. In experiments one should tune ω to maximize the intensity but varying ω yields no usefulinformation. V. ALTERNATIVE ANALYSIS OF RIXS DATA
It is possible that in real samples, with the combinedeffects of disorder, inhomogeneity, and non-resonant scat-tering, it may be desirable to increase the statisticalpower and robustness of measurements. One way to dothis is to define appropriate averaged variables. One nat-ural choice is simply to average over a range of ∆ ω : (cid:90) ∆ ω ∆ ω I (∆ k , ∆ ω ) d ∆ ω, (11)The most useful choice of the interval [∆ ω , ∆ ω ] willbe some range of small energy transfers with ∆ ω largeenough to avoid the elastic peak but with ∆ ω smallenough that the averaged quantity still reflects the redis-tribution of low-energy particles and holes due to Fermi FIG. 3: Fourier transforms of RIXS intensities from Fig. 1 exhibiting clear period-4 periodicity of the CDW system andcheckerboard periodicity in the DDW and AF systems.FIG. 4: Same as Fig. 1 but with ∆ ω integrated from 100 meV to 500 meV. Clockwise from top left: unordered state, CDW,AF, DDW. FIG. 5: Change in first moments integrated over 100 meV ≤ ∆ ω ≤
500 meV relative to unordered first moments as functionof momenta across the entire Brillouin zone. surface reconstruction. In Fig. 4 we plot this integratedintensity in the window 100 meV ≤ ∆ ω ≤
500 meV.The same qualitative distinctions between spectra thatappeared for fixed energy transfer ∆ ω = 100 meV arealso present in the integrated spectra, namely, concen-tration of intensity near ∆ k = ( π, π ) for DDW, increaseof intensity along the “arms” ∆ k = ( k, k ) , ≤ k ≤ π forAF order, and general diffuseness of intensity across theBrillouin zone for CDW. Another measure is the first mo-ment, that is, the averaged energy transfer ∆ ω weightedby the intensity:1st moment(∆ k ) = (cid:90) ∆ ωI (∆ k , ∆ ω ) d ∆ ω (12)This measures the shift of particle-hole pairs to higherenergies as gaps are opened at parts of the Fermi surface,and to lower energies as new parts of the Fermi surface arise. In Fig. 5 we show the distinct patterns in the firstmoments for DDW, CDW, and AF orders relative to theunordered state. Summary .– We presented a formalism for treating ex-act band structures and core hole potentials in indirectRIXS. We showed that indirect RIXS measures the jointdensity of states of particles and holes and is sensitive toperturbations to band structure due to the formation oflocal ordered states.
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