A Quantum Chemistry Plus Dynamical Mean Field Approach for Correlated Insulators: Application to La_{2}CuO_{4}
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b A Quantum Chemistry Plus Dynamical Mean-Field Approach for CorrelatedInsulators: Application to La CuO . M. S. Laad, L. Hozoi, and L. Craco
Max-Planck-Institut f¨ur Physik komplexer Systeme, 01187 Dresden, Germany (Dated: December 15, 2018)While the traditional local-density approximation (LDA) cannot describe Mott insulators, ab-initio determination of the Hubbard U , for example, limits LDA-plus dynamical mean field theory(DMFT) approaches. Here, we attempt to overcome these bottlenecks by achieving fusion of thequantum chemistry (QC) approach with DMFT. QC+DMFT supplants the LDA bandstructure byits QC counterpart as an input to DMFT. Using QC+DMFT, we show that undoped La CuO is a d -Mott insulator, and qualitatively discuss the circulating current- and incoherent metal phase, atsmall but finite hole doping. Very good quantitative agreement with experimental photoemission-and optical spectra constitutes strong support for efficacy of QC+DMFT. Our work thus opens anew avenue for truly ab-initio correlation-based approaches to describe correlated electronic systemsin general. PACS numbers: 71.28+d,71.30+h,72.10-d
High- T c superconductivity (HTSC) in quasi-two di-mensional (2 D ) cuprates is an outstanding, unsolvedproblem in modern condensed matter physics. Thesematerials are stochiometric Mott insulators (MI). Uponhole doping ( x ), a highly unusual, d -wave pseudogapped“metal” with strongly non-Fermi liquid (nFL) proper-ties smoothly evolves into the “strange metal” aroundoptimal doping ( x c ), where singular responses reminis-cent of a 1 D Luttinger liquid are observed [1]. At lower T , d -wave superconductivity, peaking around x c , is seen.Upon overdoping, d -SC rapidly disappears along with arapid crossover to a low- T FL metal. These unique ob-servations defy understanding in any FL picture, forcingone to search for non-FL alternatives.In reality, cuprates are charge-transfer (CT) “Mott”systems. ( i ) Strong particle-hole asymmetry is indeednecessary [2] for a quantitative description. LDA basedstudies [3] suggest that the maximum T maxc correlateswith t ′ /t . Moreover, t ′′ /t ′ is controlled by axial orbitals.( ii ) The role of these apical orbitals in cuprates is ill-understood; they may be relevant for the “hidden order”in the pseudo-gap (PG) phase, presumed to be of thecirculating current (CC) type [4, 5, 6]. If true , how doesthis constrain a minimal model for cuprates [4, 7, 8]?( iii ) A host of experiments [9, 10] clearly reveal the k -space differentiation of quasiparticle (QP) states indoped cuprates. The normal-state PG has the same d -wave symmetry as the SC at lower T . Quantumoscillation measurements show that small hole pocketsfor small x evolve, possibly via multiple electron- andhole-like sheets [11], into a full, Luttinger Fermi surface(FS) around optimal doping ( x opt ). ( iv ) Finally, near x opt , singular low energy responses suggest a branch cut,rather than a pole-like analytic structure of the one-particle Green’s function, G ( k , ω ), near E F [1]. Thisgoes hand-in-hand with FS reconstruction at x opt [12].Are these findings linked to a possible quantum critical point (QCP) at x ≃ x opt , as in f -band compounds [13]?If so, does the PG state have a “hidden” order? De-veloping an ab-initio formulation capable of reconciling( iii ) − ( iv ) above; i.e, the quantum oscillation (dHvA)data and angle-resolved photoemission (ARPES) disper-sion with one- and two particle dynamical responses is achallenge for theory, and has hitherto been studied withineffective models [14, 15]. If , however, the d -PG phaseindeed carries CC order, [4, 5, 6] these issues must bestudied using an extended Hubbard model involving both planar- and apical p − d states [3, 6].In light of ( i ) − ( iv ), inclusion of strong electroniccorrelations in a multi-band Hubbard model (with pla-nar and apical p − d states) is mandatory. Here, weextend earlier work [16] to compute the dynamical re-sponses by marrying the quantum-chemical (QC) bandstructure with multi-orbital DMFT. We “derive” an ef-fective model for cuprates that, by construction, is con-sistent with ( i ) − ( iii ) above. We show how both one-and two-particle spectra (PES and optics) in the CT-MI are quantitatively described by QC+DMFT. This is anovel theoretical route, hitherto unexplored, and avoidsthe “ U ” problem in LDA-based approaches [17], as de-tailed below.Earlier QC work is essentially a variational calcula-tion, using a multi-configurational wave-function (simi-lar to that used for the one-band Hubbard model [18],but including local and nearest neighbor (n.n) p − d and d − d excitations), and quantitatively captures the strongrenormalization of a doped carrier in a MI [16]. Shortrange electronic (AF spin) correlations anisotropicallyrenormalize the bare band structure, leading to k -spacedifferentiation, as derived before in the one-band con-text [19, 20], and in good quantitative agreement with both , the ARPES dispersion [21], and the Shubnikov-deHaas data [11, 22]. The nodal (N) QPs have predomi-nantly planar character, while the anti-nodal (AN) QPshave significant mixing of apical d z − p z states. How-ever, the QC work cannot compute dynamical excitationspectra. To address ( ii ) and ( iv ) above, the QC workneeds to be “married” with DMFT/cluster-DMFT calcu-lations. Cluster-DMFT successfully reproduces variousanomalous responses in cuprates, but within an effectivemodel framework. Here, we harmonize the successes ofthe QC method and DMFT-like approaches for the CT-MI phase in an ab-initio framework; the doped case willbe treated separately.Labelling the d x − y − p σ and d z − p z bands found inthe QC work by “1 ,
2” leads to an effective “two-band”Hubbard model, H = H + H + H mix , where H = P k ,a =1 , ,σ ǫ k ,a c † kaσ c kaσ + P i,a =1 , ∆ a n iaσ , H = U X i,a =1 , n ia ↑ n ia ↓ + U ′ X i,a = b n ia n ib − J X i,a = b S ia . S ib (1)and H mix = X k ,σ t m ( k )( c † k σ c k σ + h.c ) (2)where t m ( k ) = t m (cos k x -cos k y ) is the d -wave form factorassociated with intersite, inter-orbital d x − y − d z one-particle hopping. (∆ − ∆ ) = 1 .
15 eV and U ≃ . E N +1 + E N − − E N ≃ . J = 0 . U ′ ≃ U − J = 3 . t = 0 .
45 eV is chosen to be its “bare” value, while( t ′ , t ′′ , t m = 0 . , . , . H are renormalized (byshort-range static AF correlations) values taken from QCresults [16]. This is because DMFT mainly renormalizes t , but the farther-neighbor hoppings are renormalized bynon-local correlations beyond DMFT; we take the QC(static) renormalizations for these as an input into theDMFT machinery. Since QC already treats the effectof static correlations, [16], we do not include the staticHartree contribution in the DMFT treatment, therebyavoiding double counting of such terms. QC+DMFTnow treats the effects of local dynamical correlations andnon-local, static correlations on carrier dynamics in a sin-gle picture. The two dispersive bands are then givenas ǫ ( k ) = − t ( c x + c y ) − t ′ c x c y − t ′′ ( c x + c y ) and ǫ ( k ) = − t z ( c x + c y ) with t z << t, t ′ , t ′′ , t m . Here, c α =cos( k α ) with α = x, y, z . While similar values forthe bare hoppings are found in LDA approaches, an ad-vantage of QC is an ab-initio estimate of the Hubbard U within a correlated formulation, avoiding the problemsassociated with LDA in this context [17]. Also, in con-trast to LDA-based approaches, the QC work [16] gives,remarkably, a d -wave form-factor, and strong, local, d -shell correlations. Our two-band model is thus an ex-tended Anderson lattice model (EALM). Interestingly,Yin et al. [7] derive a similar two-band-like model from LDA+ U . In contrast to our QC results, however, with-out the d -wave hybridization. Within QC+DMFT, t m ( k )gives a d -CT-Mott insulator, as we show below.We solve H within DMFT using the multi-orbital it-erated perturbation theory (MO-IPT) as the impuritysolver [23]. Though not numerically “exact”, it hasmany advantages: (i) it gives quantitatively accurate re-sults for band-fillings upto half-filling [23] at arbitrary T , (ii) self-energies can be easily extracted, and (iii) itis numerically very effcient. More “exact” solvers ei-ther cannot reach low T of interest (QMC) [24] or areprohibitively costly in real , multi-orbital cases (NRG,D-DMRG) [25]. The relevant DMFT formalism has beendeveloped and used with very good success [23] for a va-riety of problems, and so we do not reproduce it here.The only input to the DMFT is the “free” density-of-states, given by ρ a ( ǫ ) = P k δ ( ω − ǫ a ( k )). We restrictourselves to the quantum paramagnetic phase, above the3 D Neel ordering temperature, T N . With a non-localhybridization, the Green function (GF) is a 2 × G ab ( k , ω ), (with a, b = 1 ,
2) in orbital space, as arethe local self-energies, Σ ab ( k , ω ) = Σ ab ( ω ). These equa-tions are similar to those appearing in the DMFT forthe EALM [26]. The diagonal GFs (and self-energies)yield the total many body spectral function, A ( k , ω ) =( − /π )Im[1 / ( ω − Σ a ( ω ) − ǫ a ( k ) − t m ( k ) G b ( k , ω ))]. The lo-cal DOS is just ρ a ( ω ) = P k A a ( k , ω ). Using the spectraltheorem, the off-diagonal spectral function, ρ ( k , ω ) =( − /π )Im G ( k , ω ), is easily seen to describe d -waveparticle-hole (excitonic) order (cf. d -wave form of t m ( k )).The CT-MI (found below with QC+DMFT) thus has d -wave p-h order, shown by the fact that h c † σ c σ i =( − /π ) P k R Im G ( k , ω ) dω = 0, as shown in the lowerinset of Fig 1 . Our two-band Hamiltonian will also yieldcirculating current (CC) order at finite doping concen-tration, x , as proposed by Varma [27]. Here, however,the apical p − d link is crucial for CC order with finite∆ = i h ( c † iσ c ,i + e x,y ,σ − h.c ) i = h T yi i 6 = 0 (see below).This can be readily seen in the large- U limit of our model,where a second-order in t/U expansion gives the “ex-change” part as [27, 28]: H = 2 U ′ − J X P i,j ( T (1) i,j + T (2) i,j ) (3)with P i,j = ( S i . S j − / T (1) i,j = ( t aa + t bb +2 t ab )( T zi T zj − /
4) + t aa t bb ( T + i T − j + h.c ) and T (2) i,j = t ab ( T + i T + j + T − i T − j )+( t ab ( t aa − t bb ))( T zi T xj + T xi T zj ), where T zi = ( n i − n i ) / , T + i = c † i c i , T − i = c † i c i . At mean-field level [27], (notice the difference in our T a s) this willyield a finite CC order, h T yi i 6 = 0 [27]. However, ourCC pattern involves both planar and apical states, i.e., itis closer to that proposed by Weber et al. [6], involvingthree oxygens on faces of the octahedra. While this sup-ports the view [27] that p − d interactions are importantin cuprates, our CC pattern is different in details. −6.0 −4.0 −2.0 0.0 2.0 4.0 ω (eV) ρ t o t a l ( ω ) U=0U=5eV −6.0 −4.0 −2.0 0.0 2.0 4.0 ω (eV) ρ σ ( ω ) −6.0 −4.0 −2.0 0.0 2.0 4.0 ω (eV) ρ σ ( ω ) −4.0 −2.0 0.0 2.0 ω (eV) −0.8−0.40.00.40.8 ρ ( ω ) FIG. 1: The “orbital resolved” and the total many-particledensity of states (DOS) for the two-orbital model at half-filling, h n i = 3. Notice the appearance of “orbital” dependentMott-Hubbard gaps, and the d -wave insulator, as explainedin the text. We now present our results. In Fig 1, we show the “un-perturbed” DOS (using QC) for our two-orbital modelas dash-line curves. Clearly, the planar states dominateat E F , which lies very close to a van-Hove singularity(vHs); the apical states lie much ( ≃ . U, U ′ , is readily manifest. TheCT-Mott gap equals ∆ MH = 1 . La CuO above T N .In Fig 2, we show this comparison. Quite remarkably,very good quantitative agreement with the PES spectrumupto − . ω ≥ . m ∗ /m b ≃ .
0, where m b is the LDA band mass. As a re-sult, we get ( t, t ′ , t ′′ ) ≃ (0 . , . , . renormalized dispersion quantitatively fits the dis-persion of the one-hole (“ZR-like”) states in ARPES [16] and the FS evolution with x , as E F shifts downward withhole doping. We show below that our QC+DMFT yieldsgood quantitative agreement with the optical conductiv-ity as well.The theoretical optical conductivity is compared withthe experimental data for La CuO above T N . In our −2.5 −2.0 −1.5 −1.0 −0.5 0.0 ω (eV) I n t e n s it y ( a r b . un it s ) PESU=5eV
FIG. 2: (Color Online) Comparison of the total one-hole spec-tral function obtained experimentally [29] (triangles) with theQC+DMFT spectrum with U = 5 . ≃ . two-band model, σ ( ω ) has two contributions: intrabandtransitions involving the planar- and apical states, and inter -band contributions involving transitions betwen thetwo bands. The usual DMFT equation for σ ( ω ) nowreflects both these processes [30], and reads σ ( ω ) = σ Z dǫρ ( ǫ ) Z dν f ( ω + ν ) − f ( ν ) ω ρ ǫ ( ω + ν ) ρ ǫ ( ν )(4)where ρ ǫ ( ω ) = P a =1 ρ ǫ a ( ω ) = ( − /π ) P a =1 Im [1 / ( ω + µ a − ǫ a − Σ a ( ω ))].Very good quantitative theory-experiment comparisonis clearly seen in Fig 3. Specifically, the relatively sharppeak-like structure at Ω = 2 . s ≃ . b ≃ . s : given that ∆ MH = 1 . . s must then be interpreted in terms of a quasi-continuum“excitonic” feature pulled down below the Mott gap. Inour two-band model, this arises directly from the inter -orbital transitions involving the planar- and apical bands.Thus, remarkably, QC+DMFT achieves very good quan-titative agreement with both , one- and two-particle dy-namical responses in the CT-MI phase, benchmarkingits efficacy.What do we expect at finite doping, x ? For small x ,we expect the lower-lying apical ( d z − p z ) band to re-main Mott-localized, as is generic in MO-Hubbard mod-els [20, 23], while the planar ZR band will selectivelymetallize. This would then be interpretable as an “or-bital selective” Mott transition (OSMT). One shouldthen expect nodal fermionic QPs to dominate the re-sponses at small x [20], also found in earlier QC work [16].This OSMT would thus realize the famed N-AN di-chotomy [9, 10] ubiquitous to cuprates. Eventually,around a critical doping, we expect the apical p − d bandto metallize as well, as E F shifts progressively downwardwith x . It is tempting to link the OSMT, where strongscattering between Mott localized (apical) and quasi-itinerant (planar) band carriers within DMFT would leadto low-energy infra-red singularities via the Anderson orthogonality catastrophe , to the doping driven avoided(pre-empted by d -SC) “quantum critical point” (QCP)around optimal doping, where low-energy singularitiesindeed dominate the “strange metal” [1, 4]. More workis needed to check this theoretically, and to see whetherthis coincides with a T = 0 melting of CC order [27].These issues will be addressed in detail in forthcomingwork. ω (eV) σ t o t a l ( a r b . un it s ) EXPU=5eV
FIG. 3: (Color Online) Comparison of the experimentaloptical conductivity (diamonds), taken from [31] with theQC+DMFT result (solid line) for La CuO . Very good quan-titative agreement, including description of the ≃ . below the charge-transfer-Mott gap, is clearly seen. In conclusion, we have proposed a new QC+DMFTmethod to compute the correlated electronic structure,along with the one- and two-particle spectra, for Mott(CT-Mott) insulators. Extending our earlier QC re-sults, where very good agreement with the dispersion ofone-hole “ZR-like” states and
FS was found, we haveshown how marrying QC with DMFT shows that, at“high”
T > T N , the CT-Mott insulator has d -wave or-der. The very good quantitative agreement with both PES and optical conductivity spectra in the insulating state of La CuO constitutes strong support for this con-clusion. Our QC+DMFT modelling thus reconciles theARPES and dHvA results with one- and two-particlespectral responses for the CT-MI phase. In light ofrecent ideas [1, 9, 10, 27], these findings serve as anexcellent starting point to study the physics of dopedcuprates in detail within an ab-initio correlated approach.QC+DMFT should also serve as a new theoretical toolwith wide application to other correlated systems of greatcurrent interest. Acknowledgements
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