Variational wave functions for multiband Hubbard models Federico - PowerPoint PPT Presentation

Variational wave functions for multiband Hubbard models Federico Becca CNR IOM-DEMOCRITOS and International School for Advanced Studies (SISSA) ICTP Trieste, November, 2017 C. de Franco (SISSA), L.F. Tocchio (Torino) R. Kaneko (Tokyo), R. Valenti (Frankfurt) Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 1 / 30

Variational wave functions for the Hubbard model 1 The Jastrow-Slater wave functions How to distinguish between metals and insulators Results for the two-band Hubbard model 2 The orbital-selective Mott transition on the square lattice Charge orders in organic charge-transfer salts Conclusions 3 Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 2 / 30

The one-band Hubbard Model X c † X H = − t i ,σ c j ,σ + h . c . + U n i , ↑ n i , ↓ i , j ,σ i The Hubbard model is the prototype for correlated electrons on the lattice NO exact solution in D > 1 • Does it give rise to (high-temperature) superconductivity? • Benchmark for several numerical methods (mostly in 2D): Several quantum Monte Carlo techniques (variational, diffusion, path integral) Density-matrix renormalization group and tensor networks (iPEPS) Dynamical mean-field therory and cluster extensions Embedding schemes (density-matrix embedding theory) Le Blanc et al. (Simons collaboration), PRX (2015) Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 3 / 30

Motivations and strategy GOAL: capture the ground state by variational wave functions We want to construct flexible variational states that may describe: Metals and superconductors Phases with charge and/or spin order, both metallic and insulating Mott insulators without any local order (Topologocal phases, including chiral spin liquids are also possible) We employ Jastrow-Slater wave functions and Monte Carlo sampling Non-interacting (Slater or BCS) determinant Long-range Jastrow factor Capello, Becca, Fabrizio, Sorella, and Tosatti, PRL (2005) Kaneko, Tocchio, Valenti, Becca, and Gros, PRB (2016) (Backflow correlations and Lanczos steps) Tocchio, Becca, Parola, and Sorella, PRB (2008) Tocchio, Becca, and Gros, PRB (2011) Becca and Sorella, PRL (2001) Cambridge University Press (November 2007) Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 4 / 30

Prehistory of correlated wave functions for Mott insulators e e −g −g Gutzwiller wave function i n i , ↑ n i , ↓ | Ψ 0 � | Φ g � = e − g P Gutzwiller, PRL (1963) Yokoyama and Shiba, JPSJ (1987) e −g It does not correlate empty and doubly occupied sites Metallic for g � = ∞ (any finite U / t ) Empty and doubly occupied sites play a crucial role for the conduction They must be correlated otherwise an electric field would induce a current Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 5 / 30

The Jastrow-Slater wave functions The low-energy properties reflect the long-distance behavior We must change the density-density correlations of | Ψ 0 � at large distance | Ψ � = J | Ψ 0 � ! ! − 1 − 1 X X J = exp v i , j n i n j = exp v q n − q n q 2 2 q i , j | Ψ 0 � is an uncorrelated determinant obtained from a non-interacting Hamiltonian: X t i , j c † X ∆ i , j c † i , ↑ c † H 0 = i ,σ c j ,σ + j , ↓ + h . c . i , j ,σ i , j For v i , i → ∞ (X ) f i , j c † i , ↑ c † | Ψ 0 � = exp | 0 � The RVB physics is recovered j , ↓ i , j Anderson, Science (1987) Find the optimal set of parameters v i , j , t i , j and ∆ i , j which minimizes the energy Sorella, PRB (2005) Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 6 / 30

Metal or insulator? Ansatz for the low-energy excitations Feynman, Phys. Rev. (1954) N q = � Ψ | n − q n q | Ψ � / � Ψ | Ψ � | Ψ q � = n q | Ψ � f -sum ≈ q 2 ∆ E q = � Ψ q | ( H − E 0 ) | Ψ q � = � Ψ | [ n − q , [ H , n q ]] | Ψ � rule � Ψ q | Ψ q � 2 N q N q N q ∼ | q | ⇒ ∆ E q → 0 ⇒ metal N q ∼ q 2 ⇒ ∆ E q is finite ⇒ insulator Example: 1D Hubbard model at half filling with U / t = 4 and 10 Long-range Jastrow WF Gutzwiller WF Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 7 / 30

Two-dimensional (paramagnetic) Hubbard model 1.0 20.0 Z k 0.2 | q | 2 v q D 0.8 15.0 : U/t=7, L=98 0.1 : U/t=8, L=98 : U/t=9, L=98 0.6 : U/t=10, L=98 10.0 : U/t=7, L=162 0.0 0.0 5.0 10.0 15.0 : U/t=8, L=162 U/t 0.4 : U/t=9, L=162 : U/t=10, L=162 : L=98 : U/t=7, L=242 5.0 : L=162 : U/t=8, L=242 : L=242 0.2 : U/t=9, L=242 : U/t=10, L=242 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 | q | 0.0 5.0 10.0 15.0 U/t N 0 |� Ψ N − 1 | c k ,σ | Ψ N �| 2 q ≈ 1 N q ≈ q Z k = 1+2 v q N 0 v q � Ψ N | Ψ N �� Ψ N − 1 | Ψ N − 1 � N 0 q is the uncorrelated structure factor | Ψ N − 1 � = J c k ,σ | Ψ 0 � 1 U / t � 8 . 5: v q ∼ | q | with Z k finite: FERMI LIQUID 1 U / t � 8 . 5: v q ∼ q 2 with vanishing Z k : MOTT INSULATOR AF parameter in the Slater determinant: AF order for U > 0 (BAND INSULATOR) Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 8 / 30

The two-band Hubbard model on the square lattice X t α c † H kin = − i ,α,σ c j ,α,σ + h.c. � i , j � ,α,σ X X H int = U n i ,α, ↑ n i ,α, ↓ + ( U − 2 J ) n i , 1 n i , 2 i ,α i X c † i , 1 ,σ c i , 1 ,σ ′ c † X c † i , 1 , ↑ c † H Hund = − J i , 2 ,σ ′ c i , 2 ,σ − J i , 1 , ↓ c i , 2 , ↑ c i , 2 , ↓ + h.c. i ,σ,σ ′ i Half-filling (2 electrons/site) Rotational symmetry of degenerate orbitals U ′ = U − 2 J −J J ′ = J J’ t 2 U Kanamori, Prog.Theor.Phys. (1963) 2U’ Small enough R = t 2 / t 1 ⇒ OSMI U’ t 2 t 1 R= / one orbital undergoes the MIT t 1 U while the other remains metallic Tocchio, Arrigoni, Sorella, and Becca, J. of Phys.: Cond. Matter (2016) Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 9 / 30

Ca 2 − x Sr x RuO 4 ruthenate: an orbital selective state? Maeno et al. , Nature (1994) Coexistence of spin-1/2 moments and metallicity (M-M phase) Possible explanation: presence of both localized and delocalized bands Anisimov, Nekrasov, Kondakov, Rice, and Sigrist, EPJB (2002) Several works that used dynamical mean-field theory and slave-particle approaches Liebsch, PRL (2003) Koga, Kawakami, Rice, and Sigrist, PRL (2004) Ferrero, Becca, Fabrizio, and Capone, PRB (2005) de Medici, Georges, and Biermann, PRB (2005) Arita and Held, PRB (2005) R¨ uegg, Indergand, Pilgram, and Sigrist, EPJB (2005) Inaba and Koga, PRB (2006) Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 10 / 30

The non-magnetic variational wave function | Ψ � = J | Ψ 0 � 0 1 @ − 1 X X v αβ J = exp i , j n i ,α n j ,β A 2 i , j αβ | Ψ 0 � is the ground state of a non-interacting Hamiltonian with Intra-orbital hopping t α (cos k x + cos k y ) − µ α } c † P k ,α,σ {− 2˜ k ,α,σ c k ,α,σ Intra-orbital singlet pairing with d -wave symmetry “ ” c † k ,α, ↑ c † P k ,α 2∆ α (cos k x − cos k y ) − k ,α, ↓ + c − k ,α, ↓ c k ,α, ↑ Inter-orbital triplet pairing (finite Hund’s coupling) “ ” c † i , 1 , ↑ c † i , 2 , ↓ − c † i , 2 , ↑ c † ∆ t P i , 1 , ↓ + c i , 2 , ↓ c i , 1 , ↑ − c i , 1 , ↓ c i , 2 , ↑ ⊥ i ˜ t 2 , ∆ α , ∆ t ⊥ and µ α are variational parameters to be optimized (˜ t 1 = 1) no further inter-orbital hopping t ⊥ can be stabilized in the wave function “ ” c † i , 1 ,σ c 1 , 2 ,σ + c † P t ⊥ i , 2 ,σ c i , 1 ,σ i ,σ Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 11 / 30

The phase diagram for decoupled bands X X t α c † H = − i ,α,σ c j ,α,σ + h.c. + U n i ,α, ↑ n i ,α, ↓ � i , j � ,α,σ i ,α U/ t 1 Insulator 9 0 < R = t 2 / t 1 < 1 7 OSMI U 1 5 t 1 = 7 . 5 ± 0 . 5 c 3 U 2 U 1 Metal t 1 = R c c t 1 1 0.1 0.3 0.5 0.7 0.9 t 2 /t 1 The two orbitals are decoupled and each one undergoes a MIT independently trivial OSMI Do they still have separated MIT when they are no longer decoupled? Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 12 / 30

The phase diagram for J = 0 X X X t α c † H = − i ,α,σ c j ,α,σ + h.c. + U n i ,α, ↑ n i ,α, ↓ + U n i , 1 n i , 2 � i , j � ,α,σ i ,α i Variational Monte Carlo DMFT 14 12 Mott Insulator 10 8 U/t 1 6 OSMI Metal 4 2 0 0 0.2 0.4 0.6 0.8 1 t 2 /t 1 Inaba and Koga, PRB (2006) see also: de Medici, Georges, and Biermann, PRB (2005) and Ferrero, Becca, Fabrizio, and Capone PRB (2005) The presence of the inter-band U favors a metallic phase Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 13 / 30

Metal-insulator transitions U=2 U=6 U=7 U=8 0.24 − q � ∼ q 2 for | q | → 0: N α ( q ) = � n α q n α 0.2 band α is insulating (gapped) 0.16 N α (q)/|q| N α ( q ) = � n α q n α − q � ∼ q for | q | → 0: 0.12 R=0.5 band α is metallic (gapless) 0.08 0.04 Three phases can be found: 0 Metal (e.g., U / t 1 = 6 , R = 0 . 5) 0 0.25 0.5 0.75 1 1.25 1.5 Mott (e.g., U / t 1 = 8 , R = 0 . 5) 0.24 U=4 U=5 U=7 U=8 OSMI (e.g., U / t 1 = 7 , R = 0 . 3) 0.2 0.16 N α (q)/|q| Small R : smooth metal-OSMI-Mott 0.12 R=0.3 transitions 0.08 0.04 Large R : first-order metal-Mott transition 0 0 0.25 0.5 0.75 1 1.25 1.5 |q|/ π Federico Becca (CNR and SISSA) Multiband Hubbard models ICTP 14 / 30