Collaborators: G.Kotliar, S. Savrasov, V. Oudovenko Kristjan Haule, Physics Department and Center for Materials Theory Rutgers University Electronic Structure of Strongly Correlated Electron Materials: A Dynamical Mean Field Perspective.

Application of DMFT to real materials (Spectral density functional approach). Examples: –alpha to gamma transition in Ce, optics near the temperature driven Mott transition. –Mott transition in Americium under pressure Extensions of DMFT to clusters. Examples: –Superconducting state in t-J the model –Optical conductivity of the t-J model Overview

V2O3V2O3 Ni 2-x Se x  organics Universality of the Mott transition First order MIT Critical point Crossover: bad insulator to bad metal 1B HB model (DMFT):

Coherence incoherence crossover in the 1B HB model (DMFT) Phase diagram of the HM with partial frustration at half-filling M. Rozenberg et.al., Phys. Rev. Lett. 75, 105 (1995).

mapping fermionic bath Basic idea of DMFT: 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 of Spectral density functional approach: 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] Dynamical Mean Field Theory

Density functional theory Dynamical mean field theory: observable of interest is the electron density observable of interest is the local Green's function (on the lattice uniquely defined) fermionic bath mapping exact BK functional DMFT approximation DFT & DMFT from the unifying point of view

LDA+U corresponds to LDA+DMFT when impurity is solved in the Hartree Fock approximation observable of interest is the "local“ Green's functions (spectral function) Currently feasible approximations: LDA+DMFT and GW+DMFT: G. Kotliar et.al., cond-mat/ Spectral density functional theory basic idea: sum-up all local diagrams for electrons in correlated orbitals

f1 L=3,S=1/2 J=5/2 f6 L=3,S=3 J=0 Periodic table

Cerium

Ce overview volumesexp.LDALDA+U  28Å Å 3  34.4Å Å 3 Transition is 1.order ends with CP  isostructural phase transition ends in a critical point at (T=600K, P=2GPa)   (fcc) phase [ magnetic moment (Curie-Wiess law), large volume, stable high-T, low-p]   (fcc) phase [ loss of magnetic moment (Pauli-para), smaller volume, stable low-T, high-p] with large volume collapse  v/v  15 

LDA and LDA+U f DOS total DOS volumesexp.LDALDA+U  28Å Å 3  34.4Å Å 3 ferromagnetic

LDA+DMFT alpha DOS T K (exp)= K

LDA+DMFT gamma DOS T K (exp)=60-80K

Photoemission&experiment Fenomenological Landau approach: Kondo volume colapse (J.W. Allen, R.M. Martin, 1982) A. Mc Mahan K Held and R. Scalettar (2002) K. Haule V. Udovenko and GK. (2003)

Optical conductivity * + + K. Haule, et.al., Phys. Rev. Lett. 94, (2005) * J.W. van der Eb, A.B. Ku’zmenko, and D. van der Marel, Phys. Rev. Lett. 86, 3407 (2001)

Americium

"soft" phase f localized "hard" phase f bonding Mott Transition? f 6 -> L=3, S=3, J=0 A.Lindbaum, S. Heathman, K. Litfin, and Y. Méresse, Phys. Rev. B 63, (2001) J.-C. Griveau, J. Rebizant, G. H. Lander, and G.Kotliar Phys. Rev. Lett. 94, (2005)

Am within LDA+DMFT S. Y. Savrasov, K. Haule, and G. Kotliar Phys. Rev. Lett. 96, (2006) F (0) =4.5 eV F (2) =8.0 eV F (4) =5.4 eV F (6) =4.0 eV Large multiple effects:

Am within LDA+DMFT n f =6 Comparisson with experiment from J=0 to J=7/2 “Soft” phase very different from  Ce not in local moment regime since J=0 (no entropy) "Hard" phase similar to  Ce, Kondo physics due to hybridization, however, nf still far from Kondo regime n f =6.2 Different from Sm! Exp: J. R. Naegele, L. Manes, J. C. Spirlet, and W. Müller Phys. Rev. Lett. 52, (1984) Theory: S. Y. Savrasov, K. Haule, and G. Kotliar Phys. Rev. Lett. 96, (2006) V=V 0 Am I V=0.76V 0 Am III V=0.63V 0 Am IV

Beyond single site DMFT What is missing in DMFT? Momentum dependence of the self-energy m*/m=1/Z Various orders: d-waveSC,… Variation of Z, m*,  on the Fermi surface Non trivial insulator (frustrated magnets) Non-local interactions (spin-spin, long range Columb,correlated hopping..) Present in DMFT: Quantum time fluctuations Present in cluster DMFT: Quantum time fluctuations Spatially short range quantum fluctuations

The simplest model of high Tc’s t-J, PW Anderson Hubbard-Stratonovich ->(to keep some out-of-cluster quantum fluctuations) BK Functional, Exact cluster in k spacecluster in real space

t’=0 Phase diagram What can we learn from “small” Cluster-DMFT?

Insights into superconducting state (BCS/non-BCS)? J. E. Hirsch, Science, 295, 5563 (2226) BCS: upon pairing potential energy of electrons decreases, kinetic energy increases (cooper pairs propagate slower) Condensation energy is the difference non-BCS: kinetic energy decreases upon pairing (holes propagate easier in superconductor)

D van der Marel, Nature 425, (2003) cond-mat/ Optical conductivity overdoped optimally doped

Bi2212 ~1eV Weight bigger in SC, K decreases (non-BCS) Weight smaller in SC, K increases (BCS-like) Optical weight, plasma frequency D. van der Marel et.al., in preparation

Hubbard model Drude U t 2 /U t-J model J Drude no-U Experiments intraband interband transitions ~1eV Excitations into upper Hubbard band Kinetic energy in Hubbard model: Moving of holes Excitations between Hubbard bands Kinetic energy in t-J model Only moving of holes Hubbard versus t-J model

Phys Rev. B 72, (2005) cluster-DMFT, cond-mat/ Kinetic energy change Kinetic energy decreases Kinetic energy increases Exchange energy decreases and gives largest contribution to condensation energy cond-mat/

electrons gain energy due to exchange energy holes gain kinetic energy (move faster) underdoped electrons gain energy due to exchange energy hole loose kinetic energy (move slower) overdoped BCS like same as RVB (see P.W. Anderson Physica C, 341, 9 (2000), or slave boson mean field (P. Lee, Physica C, 317, 194 (1999) Kinetic energy upon condensation J J J J

Optics mass and plasma frequency Extended Drude model Within DMFT, optics mass is m*/m=1/Z and diverges at the Mott transition Plasma frequency vanishes as 1/Z (Drude shrinks as Kondo peak shrinks) In cluster-DMFT optics mass constant at low doping doping Plasma frequency vanishes because the active (coherent) part of the Fermi surface shrinks line: cluster DMFT (cond-mat ), symbols: Bi2212, Van der Marel (in preparation)

Optimal doping: Powerlaws cond-mat/ D. van der Marel et. al., Nature 425, 271 (2003).

Optimal doping: Coherence scale seems to vanish Tc underdoped overdoped optimally scattering at Tc

Local density of states of SC V shaped gap (d-wave) size of gap decreases with doping CDMFT-optimall doping PH symmetric STM study Cluster DMFT 1 Larger doping 6 Smaller doping

Pengcheng et.al., Science 284, (1999) YBa 2 Cu 3 O 6.6 (Tc=62.7K) 41meV resonance local susceptibility Resonance at 0.16t~48meV Most pronounced at optimal doping Second peak shifts with doping (at 0.38~120meV opt.d.) and changes below Tc – contribution to condensation energy

Pseudoparticle insight N=4,S=0,K=0 N=4,S=1,K=(  ) N=3,S=1/2,K=( ,0) N=2,S=0,K=0 A(  )  ’’ (  ) PH symmetry, Large 

LDA+DMFT can describe interplay of lattice and electronic structure near Mott transition. Gives physical connection between spectra, lattice structure, optics,.... –Allows to study the Mott transition in open and closed shell cases. –In both Ce and Am single site LDA+DMFT gives the zeroth order picture –Am: Rich physics, mixed valence under pressure. 2D models of high-Tc require cluster of sites. Some aspects of optimally doped, overdoped and slightly underdoped regime can be described with cluster DMFT on plaquette: –Evolution from kinetic energy saving to BCS kinetic energy cost mechanism –Optical mass approaches a constant at the Mott transition and plasma frequency vanishes –At optimal doping: Physical observables like optical conductivity and spin susceptibility show powerlaw behavior at intermediate frequencies, very large scattering rate – vanishing of coherence scale, PH symmetry is dynamically restored, 41meV resonance appears in spin response Conclusions

local in localized LMTO base Impurity problem (14x14): LDA Impurity solver DMFT SCC * * LDA+DMFT implementation

Carbone et.al., in preparation Comparison of spectral weight cluster DMFT / Bi2212 Spectral weight (kinetic energy) changes faster with T in overdoped system – larger coherence scale

Partial DOS 4f 5d 6s Z=0.33

More complicated f systems Hunds coupling is important when more than one electron in the correlated (f) orbital Spin orbit coupling is very small in Ce, while it become important in heavier elements The complicated atom embedded into fermionic bath (with crystal fileds) is a serious chalange so solve! Coulomb interaction is diagonal in the base of total LSJ -> LS base while the SO coupling is diagonal in the j-base -> jj base Eigenbase of the atom depends on the strength of the Hund's couling and strength of the spin-orbit interaction

Mott transition (B. Johansson, 1974): Kondo volume colapse (J.W. Allen, R.M. Martin, 1982): Classical theories Hubbard model Anderson (impurity) model changes and causes Mott tr. changes → chnange of T K bath either constant or taken from LDA and rescaled spd electrons pure spectators hybridization with spd electrons is crucial f electrons insulating f electrons in local moment regime (Lavagna, Lacroix and Cyrot, 1982) Fenomenological Landau approach:

LDA+DMFT ab initio calculation is self-consistently determined contains t ff and V fd hopping bath for AIM Kondo volume colapse model resembles DMFT picture: Solution of the Anderson impurity model → Kondo physics Difference: with DMFT the lattice problem is solved (and therefore Difference: with DMFT the lattice problem is solved (and therefore Δ must self- consistently determined) while in KVC Δ is calculated for a fictious impurity (and needs to be rescaled to fit exp.) In KVC scheme there is no feedback on spd bans, hence optics is not much affected.

An example Atomic physics of selected Actinides

Basov, cond-mat/ optics mass and plasma w

Two Site CDMFTin the 1D Hubbard model Two Site CDMFT in the 1D Hubbard model M.Capone M.Civelli V. Kancharla C.Castellani and Kotliar, PRB 69, (2004)

Slave particle diagrammatic impurity solvers NCA OCA TC A Luttinger Ward functional every atomic state represented with a unique pseudoparticle atomic eigenbase - full (atomic) base, where general AIM: ( )

SUNCA vs QMC two band Hubbard model, Bethe lattice, U=4D three band Hubbard model, Bethe lattice, U=5D, T=0.0625D three band Hubbard model, Bethe lattice, U=5D, T=0.0625D