Electronic correlations in models and materials Jan Kuneš
Outline Dynamical-mean field theory Implementation (impurity problem) Single-band Hubbard model MnO under pressure moment collapse metal-insulator transition volume collapse Fe 2 O 3 AFM long-range order
Interacting electrons in periodic solid Hamiltonian = Kinetic energy + Crystal potential + Interaction
Hubbard model Hamiltonian = Hopping + Local interaction t U
Large dimension limit - classical Heisenberg model Cavity construction: Expansion in hybridization : Cumulant expansion: Scaling:
Hubbard model in d= How to construct non-trivial d= limit? E kin E int How to scale hopping? Metzner and Vollhardt, PRL 62, 324 (1998)
Dynamical mean-field theory Weak coupling expansion In d= limit only local contributions to the self-energy survive! Cumulant expansion (cavity construction) In d= limit tha action reduces to interacting site subject to time dependent external field equivalent to fermionic bath Georges and Kotliar, PRB 45, 6479 (1992)
DMFT Weiss molecular field
Dynamical Mean-Field Theory (DMFT) Single out a site from the lattice Replace the rest of the lattice by an effective medium Solve the impurity many-body problem Reconstruct lattice quantities A. Georges et al. RMP 68, 13 (1996) Physics Today (March 2004) Kotliar, Vollhardt
Hamiltonian formulation Impurity problem Action formulation
Impurity solver In U, Δ(ω) U Out Σ(ω) analytic (IPT, NCA) - diagrammatic expansions diagonalization (ED, NRG, DMRG) - Hamiltonian based QMC (Hirsch-Fye, CT-QMC) - action based
LDA Wannier projection: H LDA (k) constrained LDA: double counting : H = Σ k H 0 (k) + Σ R U ij n Ri n Rj DMFT G dd (iω n ) U ij H 0 (k)= H LDA (k)-e dc n d Find chem. potential Compute local propagator FP-LMTO, PW paramagnetic solution 8x8 (MnO), 38x38 (Fe 2 0 3 ) matrices on 3375 k-points Σ(iω n ) Construct impurity problem FT: ω n τ invft: τ ω n Compute Σ G(τ), U ij Solve impurity problem G dd (τ) Hirsch-Fye QMC CT-QMC hybridization expansion MaxEnt One- and two-particle quantities on real axis
DMFT (QMC) implementation QMC
Hubbard model: DMFT results crossover I critical point M 1st order transition Bulla et al. PRB 64, 045103 (2001)
Hubbard model: DMFT results I M Bulla et al. PRB 64, 045103 (2001)
Hubbard model: DMFT results I M Bulla et al. PRB 64, 045103 (2001)
Hubbard model: metal-insulator transition T=0 (NRG solver) Increasing U/W Bulla, PRL 83, 136 (1999)
Hubbard model: metal-insulator transition self-energy: with increasing U quasiparticles become heavier (m * ) in Mott insulator Σ(ω) develops pole inside the band in DMFT the gaps opens due to self-consistency Bulla et al. PRB 64, 045103 (2001)
Hubbard model: metal-insulator transition self-energy: with increasing U quasiparticles become heavier (m * ) in Mott insulator Σ(ω) develops pole inside the band in DMFT the gaps opens due to self-consistency Bulla et al. PRB 64, 045103 (2001)
Hubbard model: metal-insulator transition self-energy: with increasing U quasiparticles become heavier (m * ) in Mott insulator Σ(ω) develops pole inside the band in DMFT the gaps opens due to self-consistency Bulla et al. PRB 64, 045103 (2001)
Materials Models t U I M
MnO MnO FeO CoO Pressure NiO Mattila et al. PRL 98, 196404 (2007)
MnO experimental summary Mn 2+ O 2- => d 5 local configuration Conceptual phase diagram of MnO moment collapse insulator -> metal transition volume collapse structural transition C. S. Yoo et al., Phys. Rev. Lett. 94, 115502 (2005)
MnO - magnetic moment vs volume No change down to v c =0.68 Fluctuations increase dramatically in metallic phase at/below p c Orbital occupations follow the atomic scenario of J vs Δ cf competition M s2 = m z2 instantaneous moment M eff2 =T dτ m z (τ)m z (0) screened moment
MnO - magnetic moment vs volume No change down to v c =0.68 Fluctuations increase dramatically in metallic phase at/below p c Orbital occupations follow the atomic scenario of J vs Δ cf competition M s2 = m z2 instantaneous moment M eff2 =T dτ m z (τ)m z (0) screened moment
MnO - magnetic moment vs volume No change down to v c =0.68 At v c : t 2g gap closes Fluctuations increase dramatically in metallic phase at/below p c Orbital occupations follow the atomic scenario of J vs Δ cf competition Increasing J => shifts the transition
MnO - spectral density at ambient pressure PES/BIS (van Elp et al. PRB 44, 1530 (1991) ) vs Mn-3d spectral density: (U=6.9 ev, J=0.86 ev)
Pressure induced metallization
Pressure induced metallization
Pressure induced metallization
Pressure induced metallization
Pressure induced metallization
Pressure induced metallization Correlation effects weaker in LS
MnO - volume collapse E tot =E LDA +(E DMFT - E HF ) p=-de/dv
MnO - volume collapse E tot =E LDA +(E DMFT - E HF ) p=-de/dv p c
AFM order Fe 2 O 3 2320 K Fe-d total Fe-d spin resolved
AFM order Fe 2 O 3 1450 K Fe-d total Fe-d spin resolved
AFM order Fe 2 O 3 1160 K Fe-d total Fe-d spin resolved
AFM order Fe 2 O 3 580 K Fe-d total Fe-d spin resolved
Conclusions DMFT allows a systematic treatment of Hubbard model in the whole parameter range. LDA+DMFT has proved to be very useful for real materials, multi-band character leads to new physics JK, A. L. Lukoyanov, V. I. Anisimov, R. T. Scalettar, and W. E. Pickett, Nature Materials 7, 198 (2008) JK, Dm. M. Korotin, M. A. Korotin, V. I. Anisimov, P. Werner, Phys. Rev. Lett. 102, 146402 (2009)
Outlook Computation of susceptibilities: spin, charge, orbital, - investigation of ordering phenomena - dynamic susceptibilities for comparison with ineleastic spectroscopies 2 PhD positions available at Instutite of Physics, AS CR 1 PhD position available at Uni.. Augsburg
insulator -> metal transition resistance Patterson et al., Phys. Rev. B 69, 220101R (2004)