Spectral functions of Sr 2 IrO 4 From Cluster Dynamical Mean-Field - PowerPoint PPT Presentation

Spectral functions of Sr 2 IrO 4 From Cluster Dynamical Mean-Field Theory Benjamin Lenz IMPMC - Sorbonne Université, CNRS, MNHN, IRD benjamin.lenz@sorbonne-universite.fr HPC online lectures on Computational Materials Physics 2020-11-17, Paris/Singapore

om cluster DMFT • Sr 2 IrO 4 : A (not so) typical strongly correlated material (Cluster-) Dynamical Mean-Field Theory in a nutshell • Density functional theory and downfolding • Spectral functions from cluster-DMFT HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz model j e ff = 1/2 Outline • DMFT to the rescue: • Limitations of the

Sr 2 IrO 4 C. Martins et al., PRL 107, 266404 (2012) • Ir 4+ (5d 5 ) state: ⟶ extended 5d orbital R. Arita et al., PRL 108 , 086403 (2012) • Modest Coulomb interactions of 2eV (~bandwidth) C. Martins et al., O JPCM 29 , 263001 (2017) Sr Ir 001 110 3 mm HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

Sr 2 IrO 4 Ge et al. (2011) PRB 84 , 100402(R) C. Martins et al., PRL 107, 266404 (2012) • Ir 4+ (5d 5 ) state: ⟶ extended 5d orbital R. Arita et al., PRL 108 , 086403 (2012) • Modest Coulomb interactions of 2eV (~bandwidth) C. Martins et al., JPCM 29 , 263001 (2017) 0.075 Magnetization ( µ B /Ir) (1 0 17) Intenstiy • Interplay of Coulomb correlations and 0.050 B. J. Kim et al. (2008) spin-orbit coupling: PRL 101 , 076402 0.025 spin-orbit Mott insulator ( ) Δ = 0.25 eV 0.000 • Canted antiferromagnet below T N ~240K F. Ye et al. (2013) 0 100 200 300 PRL 87 , 140406(R) T (K) B. J. Kim et al. (2009) Science 323 , 1329 HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

Sr 2 IrO 4 PRB 84 , 100402(R) (a) Ge et al. (2011) Γ • Electron-doped (Sr 1-x La x ) 2 IrO 4 : PM metal down to lowest temperatures for x ≥ 0.04 • Sr 2 IrO 4 isostructural to superconducting oxides of La 2 CuO 4 family M x = 0.05 Angle-resolved photoemission spectroscopy ⟶ Fermi pockets emerge at sufficient doping X x = 0 x = 0.01 x = 0.05 ( π /2, π /2) ( π /2, π /2) ( π /2, π /2) (0,0) (0,0) (0,0) ( π ,0) ( π ,0) ( π ,0) /2) ( π , 0) Γ E B = 200 meV E B = 10 meV E B = E F 2.0 1.5 1.0 0.5 0.0 de la Torre et al. (2015) PRL 115 , 176402 Binding energy (eV) • Antinodal region shows depletion of spectral weight at Fermi level B. J. Kim et al. (2008) ⟶ Pseudogap phase (?) -0.4 -0.2 0.0 PRL 101 , 076402 E - E F (eV) HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

Sr 2 IrO 4 - Structure and Brillouin zone 11° � � � � � � � � � b* b* • Unit cell contains two layers in c-direction, which are shifted by (1/2, 0) in the a-b plane O k z =0 X=M • IrO 6 octahedra alternately tilted (anti)clockwise Sr a* M ⟶ two different configurations for the Ir atoms � = � ion � =X Ir ⟶ symmetry lowered from I4/ mmm to I4/ acd • Rotations of IrO 6 octahedra cause a doubling of the unit cell ⟶ halved first BZ, redefinition of high symmetry points 1 st Brillouin a* zone 2 nd Brillouin zone (1 st Brillouin zone undistorted) HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

Sr 2 IrO 4 - DFT and Downfolding • Ir 4+ (5d 5 ) state: extended d-orbitals • Strong spin-orbit coupling: ζ SO (Ir) = 0.40 eV j e ff = 1 2 , j e ff = 3 ⟶ t 2g states turn into states 2 DFT+SOC calculation, projection on states j e ff Energy (eV) j e ff = 1 j e ff = 3 2 | m j | = 3 C. Martins et al. | m j | = 1 Details of DFT calculation: J. of Phys. Cond. Mat. (2017) 2 2 2 HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

Sr 2 IrO 4 - DFT and Downfolding DFT+SOC calculation, projection on states j e ff upper Hubbard band UHB j eff = 1/2 Energy (eV) U LHB j eff = 1/2 lower Hubbard band � SO • j eff = 3/2 band Construct effective low-energy tight-binding model for j eff =1/2 manifold • Add electronic interactions between the j eff =1/2 states via a Hubbard U bands ⇣ ⌘ X X X c † H = − t ij i c j + h . c . + µ n i + U n i ↑ n i ↓ i,j i i ⟶ Solve low-energy model for U=1.1eV Scheme adapted from: B. J. Kim et al. Phys. Rev. Lett. (2008) HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

Dynamical Mean-Field Theory in a nutshell W. Metzner & D. Vollhardt, PRL 62 324 (1989) 
 Time A. Georges & G. Kotliar, PRB 45 6479 (1992) A. Georges et al., Rev. Mod. Phys. 68 6861 (1996) 
 G. Kotliar & D. Vollhardt, 
 + ¬ +R¬ +RA¬ 0 Physics Today (2004) G. Kotliar et al., Rev. Mod. Phys. 78 865 (2006) – + DMFT V n V n e– e– R A Electron reservoir Σ • Map many-body lattice problem to many-body local problem Rev. Mod. Phys. 68 6861 A. Georges et al. (1996) 
 • Approximate lattice self-energy by local self-energy : 
 Σ ( k , ω ) ≈ Σ imp ( ω ) Σ Σ Σ • Solve the local problem impurity solver 
 ⟶ (continuous-time quantum Monte-Carlo, exact diagonalisation 
 Σ imp iterative perturbation theory, numerical renormalization group,…) • Iterate until self-consistency : 
 Σ local lattice Green’s function = impurity Green’s function HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

Dynamical mean-field theory and beyond Local spectral function U ≫ 1 U = 0 Energy Success of DMFT: ∼ U Captures phase transition 
 Increase of e-e interaction strength U from metal to Mott insulator • Weakly correlated regime at small U: 
 LDOS Density of states resembles the band density of states PRL 70 6666 (1993) • Complex spectral function at intermediate U: 
 X.Y. Zhang et al. 
 Quasiparticle bands at low energies & Hubbard bands at high energy • Mott insulator at large U: 
 Two Hubbard bands, separated by a gap ~U How to include non-local fluctuations ? • Diagrammatic extensions of DMFT 
 G. Rohringer et al. (2018) 
 (D A, TRILEX, QUADRILEX, 
 Rev. Mod. Phys. 90 025003 
 Γ dual fermions, dual bosons, etc.) • Quantum cluster extensions of DMFT 
 T. Maier et al. (2005) 
 Rev. Mod. Phys. 77 1027 (DCA, Cellular DMFT, etc.) HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

Sr 2 IrO 4 - DFT and Downfolding DFT+SOC calculation, projection on states j e ff upper Hubbard band UHB j eff = 1/2 Energy (eV) U LHB j eff = 1/2 lower Hubbard band � SO • j eff = 3/2 band Construct effective low-energy tight-binding model for j eff =1/2 manifold • Add electronic interactions between the j eff =1/2 states via a Hubbard U bands ⇣ ⌘ X X X c † H = − t ij i c j + h . c . + µ n i + U n i ↑ n i ↓ i,j i i ⟶ Solve low-energy model for U=1.1eV Scheme adapted from: B. J. Kim et al. Phys. Rev. Lett. (2008) HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

Sr 2 IrO 4 - DFT and Downfolding (A) (B) DFT+SOC calculation, projection on states j e ff 1 1 upper Hubbard band 0 . 5 0 . 5 UHB Energy (eV) Energy (eV) j eff = 1/2 Energy (eV) 0 0 U − 0 . 5 − 0 . 5 LHB j eff = 1/2 lower Hubbard band − 1 − 1 � SO • j eff = 3/2 band Construct effective low-energy tight-binding model for j eff =1/2 manifold M X M X Γ Γ Γ Γ • Add electronic interactions between the j eff =1/2 states via a Hubbard U bands ⇣ ⌘ X X X c † H = − t ij i c j + h . c . + µ n i + U n i ↑ n i ↓ i,j i i ⟶ Solve low-energy model for U=1.1eV Scheme adapted from: B. J. Kim et al. Phys. Rev. Lett. (2008) HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

Undoped Sr 2 IrO 4 - Spectral Function C. Martins et al. (2011) Oriented-cluster DMFT applied to the half-filled j eff =1/2 band, filled j eff =3/2 bands from LDA+DMFT PRL 107 , 266404 � � � � � � • � Calculated spectral function � � Good agreement between theory and experiment • No matrix-element effects included in the calculated spectral function, but strong effect in ARPES measurements A. Louat, PhD thesis (2018) ⟶ Fourier component of symmetry lowering potential is weak • Identification of character of spectral density: 1 st BZ: dominated by j eff =1/2 band, j eff =3/2 contribution at -1.1eV along Γ -X 2nd BZ: dominated by j eff =3/2 band (mainly m j =1/2 band) ARPES ARPES 2 nd Brillouin zone 1 st Brillouin zone HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

� � � � � � � � � � � � Undoped Sr 2 IrO 4 - Spectral Function • Energy cuts of the spectral function in the paramagnetic phase • Good agreement between experiment and theory at both energies • Lowest energy excitations disperse up to -0.25eV at X and never cross the Fermi level • Spectrum similar to the antiferromagnetically ordered one (B) (A) C. Martins et al., Phys. Rev. Mat. 2 032001(R) (2018) HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz

Antiferromagnetic Phase Nie 2015 0 de la Torre 2015 Liu 2015 Cao 2016 • Y. F. Nie et al. PRL 114, 016401 − 0 . 5 (2015) • A. de la Torre et al. Energy (eV) PRL 115 , 176402 (2015) − 1 • Y. Liu et al. Sci. Rep. 5 , 13036 (2015) • Y. Cao et al. Nat. Comm. 7 , 11367 − 1 . 5 (2016) − 2 M X Γ Γ HPC online lectures - C-DMFT for Sr 2 IrO 4 - Benjamin Lenz B. Lenz et al., JPCM 31 293001 (2019)