Towards First-Principles Electronic Structure Calculations of Correlated Materials Using Dynamical Mean Field Theory (DMFT). Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University CMSN-Workshop on Predictive Capabilities for Strongly Correlated Systems UT November

Outline, Collaborators, References A. Poteryaev, A. Lichtenstein and G. Kotliar (preprint) (2003) S.Savrasov G. Kotliar and E. Abrahams, Nature 410,793 (2001). X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science, Vol300, 954 (2003) Funding: Basic Energy Sciences DOE.. DMFT and electronic structure calculations Case study 1: Ti2O3 Case study 2: Elemental Pu Conclusions: Future developments.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly Correlated Electrons Two limits of the electronic structure problem are well under control. Band limit, (LDA or GGA)+ GW, gives good spectra and total energy. Physical properties are accessible in perturbation theory in the screened Coulomb interactions Well separated atoms, in the presence of spin orbital long range order, expansion around the atomic limit, unrestricted HF, and LDA+U work well for ordered Mott insulators. Challenge ahead: materials that are not in either one of these regimes. Requires combination of many body theory and one electron theory.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly correlated systems are usually treated with model Hamiltonians Conceptually one wants to restrict the number of degrees of freedom by eliminating high energy degrees of freedom. In practice other methods (eg constrained LDA are used)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory Reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition. Instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992) DMFT

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT action and self consistency condition In the case of cluster is a matrix and is not the self energy, (but can be used to estimate the lattice self energy by projection ) In general tk is large matrix H[k], U is a matrix

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the DMFT equations Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: Effective Action point of view.R. Chitra and G. K Phys Rev. B (2000), (2001) S Savrasov and G. K. cond-matt Identify observable, A. Construct an exact functional of =a,  [a] which is stationary at the physical value of a. Example, density in DFT theory. (Fukuda et. al.) When a is local, it gives an exact mapping onto a local problem, defines a Weiss field. The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT. DMFT, build functionals of the LOCAL spectral function. Exact functionals exist. We also have good approximations! Extension to an ab initio method. Functional of greens function of electric field and electronic field, functional of the density and the local greens function.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT as an approximation to the exact functional of the Greens function, DMFT as a truncation of the BK functional of the full Greens function. Observable: Local Greens function G ii (  ). Exact functional  [G ii (  )  DMFT Approximation to the functional.(Muller Hartman 89)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT (II) DMFT U E dc

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMFT vs Nc=1

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 1d Hubbard U/t=4 exact diag 2+6.Capone Civelli and GK

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two roads for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Interfacing DMFT with band theory ROAD 1: Derive model Hamiltonians, solve by DMFT (or cluster extensions).  V.Anisimov A Poteryaev V.Korotin V.Anokin andG Kotliar J. Phys. Cond. Mat. 35, 7359 (1997).A.Lichtenstein and M Katsnelson PRB (1998).  ROAD 2: Define a functional of the density and of the local Greens function and extremize the functional to get coupled equations for the density and the spectral function and compute total energies. Kotliar, S.Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001).  G. Kotliar, S.Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001).

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT (I) The light, SP (or SPD) electrons are extended, well described by LDA. The heavy, D (or F) electrons are localized,treat by DMFT. LDA already contains an average interaction of the heavy electrons, substract this out by shifting the heavy level (double counting term). This defines H. The U matrix can be estimated from first principles (Gunnarson and Anisimov, McMahan et.al. Hybertsen et.al) of viewed as parameters. Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997).

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Application to Ti2O3

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Metal to insulator transition in Ti2O3 Isostructural to V2O3. All the qualitative physics of the high temperature part of the phase diagram of V2O3 can be understood within single site DMFT. Computations with a realistic density of states, and multiorbital impurity model (K. Held and D. Vollhardt ) substantial quantiative improvement. Is the same thing true in Ti2O3?

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ti2O3 V2O3 : Resistivities

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ti2O3 Structure

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Relevant Orbitals: Goodenough picture

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ti2O3 vs V2O3 As a function of temperature, there is no magnetic transition in Ti2O3, unlike V2O3 As a function of temperature, there is no structural change, unlike V2O3 which becomes monoclinic at low temperatures. In V2O3 the distance between the Vanadium pairs incrases as the temperature decreases. In Ti2O3 the distance between the Vanadium pairs decreases as one lowers the temperature. LTS 250 K, HTS 750 K.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Earlier work. Band Structure Calculations always produce a good metal. L.F. Mattheiss, J. Phys.: Condens. Matter 8, 5987 (1996) Unrestricted Hartree Fock calculations produce large antiferromagnetic gap. M. Cati, G. Sandrone, and R. Dovesi, Phys. Rev. B. f55, (1997).

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ti2O3 LDA-DOS LTS HTS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Methodology:1 and 2 site CDMFT Impurity solver. Multiband QMC. Derivation of the effective Hamiltonian. Massive downfolding with O Andersen’s new Nth order LMTOS. Coulomb interactions estimated using dielectric constant W=.5 ev. U on titanium 2 ev. J=.2 ev.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT fails. LTS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two-site CDMFT for beta=20, and beta=10 (T=500,1000) Poteryaev Lichtenstein and GK

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Important role played by the Coulomb nn repulsion.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Application to Plutonium

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Small amounts of Ga stabilize the  phase (A. Lawson LANL)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Elastic Deformations In most cubic materials the shear does not depend strongly on crystal orientation,fcc Al, c 44 /c’=1.2, in Pu C44/C’ ~ 7 largest shear anisotropy of any element. Uniform compression:  p=-B  V/V Volume conserving deformations : F/A=c 44  x/L F/A=c’  x/L

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT studies o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties. o Many studies (Freeman, Koelling 1972)APW methods o ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give o an equilibrium volume of the  phase  Is 30-35% lower than experiment o This is the largest discrepancy ever known in DFT based calculations.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT Studies LSDA predicts magnetic long range (Solovyev et.al.) Experimentally  Pu is not magnetic. If one treats the f electrons as part of the core LDA overestimates the volume by 30% DFT in GGA predicts correctly the volume of the  phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that  Pu is a weakly correlated system

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lda vs Exp Spectra

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Energy vs Volume [GGA+U=4 ev] EXPT: Bcc 14.7 Fcc 15.01

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS GGA+U spectra

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Other problems with LDA+U Predicts plutonium to be magnetic. Different theories of alpha and delta.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT - Technical details [spectra and energy] Atomic sphere approximation. Ignore crystal field splittings in the self energies. Fully relativistic non perturbative treatment of the spin orbit interactions. Impurity solver: interpolative scheme using slave bosons (low frequency ) and eqn of motion (high frequency).

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT- Phonon Spectra Full potential LMTO with two kappas. Linear response method in LMTO’s (S. Savrasov) Impurity solver: lowest order projection (Roth method) in the equations of motion.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: DMFT total energy vs Volume ( Savrasov Kotliar and Abrahams 2001)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu Spectra DMFT(Savrasov) EXP ( Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha and delta Pu

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha phase is also a correlated metal. It differs from delta in the relative weight of the resonance and the Hubbard band. Consistent with resistivity and specific heat measurements. Similar conclusions A. Mc Mahan K. Held and R. Scalettar, for the alpha to gamma transition in Cerium.

Summary LDA LDA+U DMFT Spectra Method E vs V

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon Spectra Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure. Phonon spectra reveals instablities, via soft modes. Phonon spectrum of Pu had not been measured. Short distance behavior of the elastic moduli.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = E i - E f Q = k i - k f

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Expt. Wong et. al.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Expts’ Wong et. al. Science (2003) Theory Dai et. al. Science 300, 953, (2003)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Shear anisotropy. Expt. vs Theory C’=(C11-C12)/2 = 4.78 GPa C’=3.9 GPa C44= GPa C44=33.0 GPa C44/C’ ~ 7 Largest shear anisotropy in any element! C44/C’ ~ 8.4

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The delta –epsilon transition The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase. What drives this phase transition? Having a functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon frequency (Thz ) vs q in epsilon Pu.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon entropy drives the epsilon delta phase transition Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta. At the phase transition the volume shrinks but the phonon entropy increases. Estimates of the phase transition following Drumont and Ackland et. al. PRB.65, (2002); (and neglecting electronic entropy). TC ~ 600 K.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonons epsilon

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion Develop new methods for treating realistic (system specific) strongly correlated electrons. The DMFT machinery is in a very primitive state. Study interesting materials science problems, develop some qualitative understanding of materials properties. Perform quantitative calculations. DMFT- in its current state of the art, allows us to do both.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion Kotliar, S.Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Cond-matt , S. Biermann F. Aeryasetiwan, A. Georgs PRL 2003] Serious bottle neck of current interface of DMFT and band theory: U as a frequency independent parameter. Solution: E-DMFT +GW. [G. Kotliar, S.Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Cond-matt , S. Biermann F. Aeryasetiwan, A. Georgs PRL 2003] Fully implemented at the level of model Hamiltonian [Ping Sun’s talk]. Needs to be carried over to electronic structure. Need further improvements of both electronic structure and many body tools. Illustrated compromises [Ti2O3 cluster, single site QMC +downfolding, Pu spectra and energy IPT+ ASA, Pu Phonons single site DMFT full potential+very primitive solver.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Careful comparison with experiments. What do we need to reproduce the softening of the 111 phonon ? Better solver at the single site level? Or cluster treatment of fcc Pu. Need further developments in linear response dynamics to accommodate better solvers.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A partial list of applications of DMFT to materials. Colossal Magneto-resistance LaSrMnO3 Double PerovskitesChattopadhyay:2001:PRB} LaSrTiO3 doping driven Mott transition Itinerant Magnetism: Iron Nickel Half Metals Pressure Driven Mott Transition V2O3 Presssure Driven Metal to Charge Transfer Insulator NiSeS Kappa Organics

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cerium : alpha to gamma transition Plutonium: delta and epsilon phase Mott insulators, phonons and spectra, NiO, MnO Bandwith control CaSrVO3, CaVO3 SrVO3 Heavy fermion without f eleLiV$_{2}$O$_{4}$ctrons Fullerines K$_{n}$C$_{60}$} Bechgaard Salts

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Biermann:2001 Quantum criticality of CeCuAu Si et.al. Heavy Fermion Insulators. Saso et.al. CrO$_2$. Laad et.al. FlNaV$_{2}$O$_{5}$ Fluctuating charge order Chattopadhyay:2001Magnetic Semiconductors Strongly Inhomogenous systems, surfaces and surface phase transitions.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Perfetti:2003, Liebsch:2003}. {Ruthenates} Sr$_{2}$RuO$_{4}$ Orbital differentiation. Ti2O3 Metal to insulator transition VO2 Metal to Insulator Transition.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Start with the TOE

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Rewrite the TOE as an electron boson problem.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Build effective action for the local greens functions of the fermion and Bose field r=R+  R unit cell vector  position within the unit cell. Ir>=|R,  Couple sources to

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Legendre transfor the sources, eliminating the field  Build exact functional of the correlation functionsW(r R,r’ R) and G (r R,r’ R)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS “Kohn Sham “ decomposition.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS (E)DMFT pproximation to Sum over all LOCAL 2PI graphs (integrations are restricted over the unit cell ) built with W and G Map into impurity model to generate G and W Go beyond this approximation by returning to many body theory and adding the first non local correction.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Sum of local 2PI graphs with local U matrix and local G

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Epsilon Plutonium.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lattice and cluster self energies

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Impurity cavity construction

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outer loop relax U E dc Impurity Solver SCC G,  G0G0 DMFT LDA+U Imp. Solver: Hartree-Fock

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT and LDA+U Static limit of the LDA+DMFT functional, with  atom  HF reduces to the LDA+U functional of Anisimov Andersen and Zaanen. Crude approximation. Reasonable in ordered Mott insulators. Total energy in DMFT can be approximated by LDA+U with an effective U. Extra screening processes in DMFT produce smaller Ueff. U LDA+U < U DMFT

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS C-DMFT: test in one dimension. (Bolech, Kancharla GK PRB 2002) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMFT vs Nc=1

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Very Partial list of application of realistic DMFT to materials QP bands in ruthenides: A. Liebsch et al (PRL 2000) N phase of Pu: S. Savrasov G. Kotliar and E. Abrahams (Nature 2001) MIT in V 2 O 3 : K. Held et al (PRL 2001) Magnetism of Fe, Ni: A. Lichtenstein M. Katsenelson and G. Kotliar et al PRL (2001) J-G transition in Ce: K. Held A. Mc Mahan R. Scalettar (PRL 2000); M. Zolfl T. et al PRL (2000). 3d doped Mott insulator La1-xSrxTiO3 Anisimov et.al 1997, Nekrasov et.al. 1999, Udovenko et.al 2002) ………………..

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT References Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsenelson PRB (1998). Reviews: Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001). Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001). A. Lichtenstein M. Katsnelson and G. Kotliar (2002)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction Introduce local orbitals,   R (r-R), and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for  (r) and G and performing a Legendre transformation,  (r),G(R,R)(i  )]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Self-Consistency loop DMFT U E

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on LDA+DMFT Static limit of the LDA+DMFT functional, with  =  HF reduces to LDA+U Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing. Luttinger theorem is obeyed. Functional formulation is essential for computations of total energies, opens the way to phonon calculations.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS References LDA+DMFT: V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, (1997). A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). S. Savrasov G.Kotliar funcional formulation for full self consistent implementation of a spectral density functional. Application to Pu S. Savrasov G. Kotliar and E. Abrahams (Nature 2001).

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS References Long range Coulomb interactios, E-DMFT. R. Chitra and G. Kotliar Combining E-DMFT and GW, GW-U, G. Kotliar and S. Savrasov Implementation of E-DMFT, GW at the model level. P Sun and G. Kotliar. Also S. Biermann et. al.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Put in the loop of dmft +lda and the functional And chitra. Put in the effective action perspective. Put in the coupling constant integration. Put in the cluster. Think of formula for simga-lattice.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition. Basic idea: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Master plan. 1) Fix titanite section by putting meet. 2) Fix plutonium section by transporting and putting meet. From Berkeley. 3) Put the ideology. Overview of how really DMFT is used. Models + non models. And within models two pictures. Including the effective action perspective. 1] Coupling constant integration formula for DMFT models. 4) Conclusion. EDMFT in r,r’ and non local corrections around it. Indirect evidence, Ping successes. Indirect evidence, from local GW, that it gives the U’s we need for DMFT……….

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Correlated electrons. Model Hamiltonians. DMFT-two perspectives-models and functionals. cavity.-mention cluster. How good the local approximation is. Functional perspective-effective action DMFT as an exact functional-DMFT as an approximation. Interface with electronic structure-Anisimov. Interace with a functional.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Chitra and GK ). Introduce local orbitals,   R (r-R)orbitals, and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for  (r) and G and performing a Legendre transformation,  (r),G(R,R)(i  )] Approximate functional using DMFT insights.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Model Hamiltonian. Exact functional of the local Greens function A +

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT for model Hamiltonians. Kohn Sham formulation. Introduce auxiliary field Exact “local self energy”

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS About XC functional. One can derive a coupling constant integration formulae (Harris Jones formula) for Generate approximations. The exact formalism generates the local Greens function and  ii is NOT the self energy. However one can use the approach as starting point for computing other quantities.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on functional construction Atoms as a reference point. Expansion in t. Locality does not necessarily mean a single point. Extension to clusters. Jii ---  Jii Ji i+  Aii ---  Ai i+   ii ---   i i+  Exact functional  Aii,Ai i+   he lattice self energy and other non local quantities extending beyond the cluster are OUTSIDE the formalism and need to be inferred.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on funct. construction. Construction of approximations in the cluster case requires care to maintain causality. One good approximate construction is the cellular DMFT: a) take a supercell of the desired range,b) c) obtain estimate of the lattice self energy by restoring translational symmetry. Many other cluster approximations (eg. DCA, the use of lattice self energy in self consitency condition, restrictions of BK functional, etc. exist). Causality and classical limit of these methods has recently been clarified [ G Biroli O Parcollet and GK]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition. Basic idea: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic applications of DMFT References: combinations of DMFT with band theory. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, (1997). A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). S. Savrasov and G.Kotliar and Abrahams funcional formulation for full self consistent Nature {410}, 793(2001). Reviews: Held et.al., Psi-k Newsletter \#{\bf 56} (April 2003), p. 65 Lichtenstein Katsnelson and Kotliar cond-mat/ :

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comparaison with the Hartree Fock static limit: LDA+U.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dependence on structure

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mapping onto impurity models. The local Greens function A, and the auxilliary quantity  can be computed from the solution of an impurity model that describes the effects of the rest of the lattice on the on a selected central site. One can arrive at the same concept via the cavity construction.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two roads for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions