Merging First-Principles and Model Approaches Ferdi Aryasetiawan Research Institute for Computational Sciences, AIST, Tsukuba, Ibaraki – Japan ISSP Collaborators: Antoine Georges and Silke Biermann (Ecole Polytechnique, France) Takashi Miyake and Rei Sakuma (RICS-AIST)

Outline Motivations: The need to go beyond one-particle picture in correlated materials Previous works: LDA+U, LDA+DMFT The GW approximation: success and difficulties Combining GW and DMFT: A first-principles scheme for correlated materials A simplified implementation: Application to ferromagnetic nickel First Part Second Part (if time permits) Constrained RPA: Calculating the Hubbard U from first-principles

Fujimori PRL 69, 1796 (1992) Difficult to treat within one-particle theory Spectral evolution as a function of U/bandwidth Georges et al Rev. Mod. Phys.1996 (Dynamical Mean-Field Theory) Can use LDA Can use LDA+U metal insulator

La/YTiO3 cubicorthorhombic

[1] Solovyev, Hamada, and Terakura, PRB 53, 7158 (1996) (LDA+U: correct magnetic structure of La/YTiO3) [2] Pavarini, Biermann, Poteryaev, Lichtenstein, Georges, and Andersen, PRL 92, (2004) (LDA+DMFT: consistent description of metal-insulator transition) These materials share similar electronic structure in LDA and they are all predicted to be metals Experimentally SrVO3 and CaVO3 are metals LaTiO3 and YTiO3 are insulators

Typical electronic structure of correlated materials: Partially filled narrow 3d or 4f band across the Fermi level By slight distortion or pressure the ratio of U/bandwidth changes and the materials can undergo, e.g., phase transitions (metal-insulator).  competition between kinetic energy (bandwidth) and U. The main action takes place here

D U U=0 U/D>1 Competition between kinetic energy and U (itineracy and localisation) satellite QP Lower Hubbard band Upper Hubbard band U/D>>1 When U is small it is preferable for the electrons to delocalise  metal When U is large it is costly for the electrons to hop  localised (Mott insulator) Electrons prefer to “spread” themselves to lower their kinetic energy. For intermediate U it is a mixture of localised and delocalised electrons.

D U U=0 U/D>1 Mapping to a Hubbard model satellite QP Lower Hubbard band Upper Hubbard band U/D>>1 More realistic models take into account the underlying one-electron band structure  LDA+U and LDA+DMFT

To treat systems with strong on-site correlations: Previous works: LDA+U and LDA+DMFT Ad-hoc double-counting term Adjustable U Anisimov et al, J. Phys. Condens. Matter 9, 7359 (1997) (LDA+U, LDA+DMFT) Lichtenstein and Katsnelson, PRB 57, 6884 (1998) (LDA+DMFT)

The Goal To construct a consistent theoretical scheme that can describe the electronic structure of correlated materials from first-principles. “Consistent” means that the scheme should be capable of describing the continuous transition from metal to insulator, i.e., it can describe both the itinerant and localised characters of the electrons. One of the main features of correlated electrons is that they have both itinerant and localised characters.

First-Principles Methods: Local Density Approximation (LDA) (ground states) GW method (excited states) Model Approaches: Exact diagonalisation (Lanczos) Quantum Monte Carlo (QMC) Dynamical Mean-Field Theory (DMFT) LDA+U and LDA+DMFT Can treat strong correlations but need adjustable parameters Insufficient to treat correlated materials Combine First-Principles and Model Approaches: GW+DMFT Electronic structure of correlated materials from first-principles spintronics electronic transport nanotech. optical devices Possible applications

The GW approximation Hartree-Fock approximation: GW approximation: v G = = = = = G W is a screened interaction Lars Hedin, Phys. Rev. 139, A796 (1965)

Screened exchange: from the poles of G Correlation hole: interaction between an electron and its screening hole Since W<v, it reduces the Hartree-Fock band gap. In addition we have another term arising from the poles of W

From Hybertsen and Louie, PRB34, 5390 (1986)

Assessing the GWA using the Polaron Hamiltonian: Exact solution:  Plasmons satellites

Quasi-particle and satellite energy Error in QP energy is smaller than error in satellite energy

 There is no spin dependence in W, only in G: Exchange is taken into account properly, but not correlation between the same spin.  Collective excitations only arise from RPA: (1-Pv)= 0 (plasmons). Satellites arising from local correlations (atomic multiplets) are probably not fully captured.  The Hubbard Hamiltonian cannot be transformed into a polaron-type Hamiltonian for which GW is good.  Usually start from a single Slater determinant: Difficult to treat systems that are inherently dominated by a few Slater determinants.  Self-consistent GW appears to give poor spectra (?) Some difficulties with the GWA

GW Band insulator Mott-Hubbard insulator Im   infinite Removes QP and transfers weight to Hubbard bands (QP life-time ~ 1/Im  ) Im   finite Shifts and broadens Quasi-Particle Band insulator vs Mott-Hubbard insulator U DMFT

Strongly correlated systems are problematic for LDA: LDA often predicts (anti-ferromagnetic) insulators to be metals. DMFT: Map the lattice to an impurity embedded in a “bath”. “bath” A. Georges et al, Rev. Mod. Phys. 68, 13 (1996) G. Kotliar and D. Vollhardt, Physics Today (March 2004) Dynamica l Even the GW approximation may not be sufficient. U A la Fukuyama

Effective dynamics for an impurity problem with the dynamical mean-field bath The Coulomb interaction is fully taken into account in one site (impurity), the rest of the sites (medium) is treated as an effective field

Dynamical Mean-Field Theory (DMFT) A. Georges et al, Rev. Mod. Phys. 68, 13 (1996) Self-consistency: Green’s function Self-energy Restore the lattice periodicity

Comparison between GW and DMFT DMFT (Dynamical Mean-Field Theory) GW  iGW Onsite, no k-dependenceFull k-dependence Onsite interactions are summed to all orders Approximate (RPA), good for long-range correlations Model Hamiltonian Adjustable parameters Real systems No parameters Can treat systems with strong onsite correlations (3d and 4f) Good for sp systems but has problems with 3d and 4f systems

Basic physical idea of GW+DMFT 0 R The onsite self-energy is taken to be  DMFT. The off-site self-energy is approximated by  GW. P. Sun and G. Kotliar, PRB 66, (2002) S. Biermann, F.A. and A. Georges, PRL 90, (2003)

General Framework Generalized Luttinger-Ward functional * * Almbladh, von Barth and van Leeuwen, Int. J. Mod. Phys. B 13, 535 (1999) Luttinger-Ward functional

Approximation for  wwww Conserving approximation

Self-Consistency Conditions The Hubbard U is screened within the impurity model such that the screened U is equal to the local W Additional condition: Determines U

Self-consistency loop Impurity: given Weiss field G and U Combine  GW and  imp Check self-consistency: G loc =G imp ? W loc =W imp ? New Weiss field G and U

Simplified GW+DMFT scheme Offsite self-energy: corrected by GW Onsite self-energy: corrected by DMFT Non-self-consistent GW + DMFT (static impurity model)

Test: Application to ferromagnetic nickel (Simplified GW+DMFT scheme) Correlation problems in nickel within LDA: LDAExp. 3d band width4 eV3 eV Exchange splitting0.6 eV0.3 eV Satellitenone6 eV GW 3.2 eV 0.6 eV Very weak (if any)

GW+DMFT (Experimental data from Buenemann et al, cond-mat/ ) majority minority majority minority exp LDA Nickel band structure Too large LDA bandwidth S. Biermann et al, PRL 90, (2003)

GW+DMFT density of states of nickel no 6 eV satellite in LDA Exchange splitting too large by 0.3 eV in LDA

Next target: From V. Eyert

metal insulator

From Miyake

Koethe et al

From Biermann et al, PRL 94, (2005) Non-local self-energy is important  a good test for GW+DMFT Single impurity DMFT does not open up a gap in M1

Preliminary GW result from Sakuma and Miyake O2p a_1g One-shot GW gives too small gap.  need DMFT?  self-consistency?

Weak satellites

The self-energy around the LDA energy is not linear.

Summary GW+DMFT scheme: -Potentially allows for ab-initio electronic structure calculations of correlated systems with partially filled localized orbitals and describes phase transitions. -Unlike LDA+DMFT, *The Hubbard U is determined self-consistently *Avoids double counting. Challenges Global self-consistency Solving the impurity problem with an energy-dependent U Treat all orbitals on equal footing