Spin and orbital freezing in unconventional superconductors Philipp Werner University of Fribourg Kyoto, November 2017
Spin and orbital freezing in unconventional superconductors In collaboration with: Shintaro Hoshino (Saitama) Hiroshi Shinaoka (Saitama) Karim Steiner (Fribourg) Kyoto, November 2017
Introduction Generic phase diagram of unconventional superconductors Superconducting dome next to a magnetically ordered phase Non-Fermi liquid metal above the superconducting dome Temperature bad metal magnetic order Fermi liquid pressure, doping, ... superconductivity
Method Georges and Kotliar, PRB (1992) Dynamical mean field theory DMFT: mapping to an impurity problem lattice model impurity model G latt ≡ G imp t t k Σ latt ≡ Σ imp Impurity solver: computes the Green’s function of the correlated site Bath parameters = “mean field”: optimized in such a way that the bath mimics the lattice environment
Method Werner et al., PRL (2006) CT -QMC solvers allow efficient simulation of multiorbital models � � H loc = − µn α , σ + Un α , ⇥ n α , ⇤ α , σ α � U ⌅ n α , σ n β , � σ + ( U ⌅ − J ) n α , σ n β , σ + α > β , σ J ( ψ † α , ⇤ ψ † β , ⇥ ψ β , ⇤ ψ α , ⇥ + ψ † β , ⇥ ψ † � β , ⇤ ψ α , ⇥ ψ α , ⇤ + h.c. ) − α ⇧ = β Relevant cases: 4 electrons in 3 orbitals: SrRu 2 O 4 3 electrons in 3 orbitals, J <0: A 3 C 60 6 electrons in 5 orbitals: Fe -pnictides
3-orbital model Werner, Gull, Troyer & Millis, PRL (2008) Phase diagram for U � = U − 2 J, J/U = 1 / 6 , β = 50 16 14 12 10 frozen U/t 8 Fermi liquid moment 6 4 glass transition Mott insulator ( β t=50) 2 0 0 0.5 1 1.5 2 2.5 3 n Metallic phase: “transition” from Fermi liquid to spin-glass Narrow crossover regime with self-energy Im Σ /t ∼ ( i ω n /t ) α , α ≈ 0 . 5
3-orbital model Werner, Gull, Troyer & Millis, PRL (2008) Fit self-energy by − Im Σ ( i ω n ) = C + A ( ω n ) α 1 0.8 intercept C, exponent α 0.6 0.4 exponent α ( β t=50) intercept C 0.2 exponent α ( β t=100) intercept C 0 0 2 4 6 8 10 U/t Square-root self-energy coincides with on-set of frozen moments
crossover spin-freezing Fermi-liquid spin-frozen 3-orbital model Hoshino & Werner, PRL (2015) Spin-freezing leads to a small “quasi-particle weight” z z ≈ 1 / (1 − Im Σ ( i ω 0 ) / ω 0 ) 5 1 (c) 4 0.8 3 0.6 2 0.4 Ising 1 0.2 rot. inv. 3 0 0 0 0.5 1 1.5 2 2.5 3 2.5 no quasi-particles in spin-frozen regime
3-orbital model Werner, Gull, Troyer & Millis, PRL (2008) Spin-spin and orbital-orbital correlation functions 0.25 n=1.21 n=1.75 0.2 n=2.23 <n 1 (0)n 2 ( τ )>, <S z (0)S z ( τ )> n=2.62 n=2.97 0.15 0.1 freezing of spin moments 0.05 0 -0.05 -0.1 0 5 10 15 20 25 τ t no freezing of orbital moments
3-orbital model Hoshino & Werner, PRL (2015) Consider the local susceptibility Z β χ loc = d τ h S z ( τ ) S z (0) i 0 and its dynamic contribution Z β ∆ χ loc = d τ [ h S z ( τ ) S z (0) i � h S z ( β / 2) S z (0) i ] 0 subtract the (frozen) long-time value
spin-frozen spin-freezing Fermi-liquid crossover 3-orbital model Hoshino & Werner, PRL (2015) Consider the local susceptibility and its dynamic contribution ∆ χ loc χ loc 50 5 1 (b) (c) 40 4 0.8 Ising Ising 30 3 0.6 Ising rot. inv. 20 2 0.4 10 1 0.2 0 3 0 0 0 0 0.5 1 1.5 2 2.5 Crossover regime is characterized by large local moment fluctuations
3-orbital model Werner, Gull, Troyer & Millis, PRL (2008) “quasi-particle weight” z from De’ Medici, Mravlje & Georges, PRL (2011) Hund coupling J : Strongly correlated metal far from the Mott transition
3-orbital model Werner, Gull, Troyer & Millis, PRL (2008) “quasi-particle weight” z from De’ Medici, Mravlje & Georges, PRL (2011) large local moment fluctuations Hund coupling J : Strongly correlated metal far from the Mott transition
Strontium Ruthenates Werner, Gull, Troyer & Millis, PRL (2008) A self-energy with frequency dependence implies an Σ ( ω ) ∼ ω 1 / 2 optical conductivity σ ( ω ) ∼ 1 / ω 1 / 2
Pnictides Strongly correlated despite moderate U 5.0 5.0 LDA total LDA p orbitals d spectral function d spectral function d spectral function 4.5 4.5 0.18 LDA d orbitals 4.0 4.0 0.12 3.5 3.5 LDA dynamic U 3.0 3.0 ρ ( ω ) (eV -1 ) ρ ( ω ) (eV -1 ) static U -1 incoherent metal state 0.06 2.5 2.5 resulting from Hund’s coupling ρ ω -16 -12 -8 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 -6 -4 -2 0 2 4 -6 -4 -2 0 2 4 ω (eV) ω (eV) ω Haule & Kotliar, NJP (2009)
Pnictides Strong doping and temperature dependence of electronic structure BaFe 2 As 2 : conventional FL metal in the underdoped regime non-FL properties near optimal doping incoherent metal in the overdoped regime Werner et al., Nat. Phys. (2012)
Pnictides Strong doping and temperature dependence of electronic structure
Long-range order Hoshino & Werner, PRL (2015) Identify ordering instabilities by divergent lattice susceptibilities Calculate local vertex from impurity problem Approximate vertex of the lattice problem by this local vertex Solve Bethe-Salpeter equation to obtain lattice susceptibility The following orders (staggered and uniform) are considered: diagonal orders: charge, spin, orbital, spin-orbital off-diagonal orders: orbital-singlet-spin-triplet SC, orbital-triplet-spin-singlet SC
[arb. unit] Long-range order Hoshino & Werner, PRL (2015) 3-orbital model, Ising interactions 2.5 2.5 (b) (c) FM FM 2 2 AFM near half-filling FM at large U away from 1.5 1.5 Spin-freezing Spin-freezing half-filling crossover crossover spin-triplet superconductivity AFM AFM 1 1 in the spin-freezing SC crossover region SC 0.5 0.5 Normal Normal 0 0 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5
[arb. unit] Long-range order Hoshino & Werner, PRL (2015) 3-orbital model, Ising interactions (lower temperature) 2.5 (c) FM FM 2 AFM near half-filling FM at large U away from 1.5 Spin-freezing half-filling crossover U spin-triplet superconductivity AFM AFM 1 in the spin-freezing crossover region SC 0.5 Normal 0 2 2.5 3 0 0.5 1 1.5 2 2.5 3 parameter regime relevant for Sr 2 RuO 4
Long-range order Hoshino & Werner, PRL (2015) T c dome and non-FL metal phase next to magnetic order (a) (b) 0.06 0.1 Normal Normal bad metal bad metal 0.08 0.04 0.06 Spin-freezing Spin-freezing crossover crossover 0.04 0.02 AFM FM 0.02 SC Fermi Fermi SC liquid liquid 0 0 1 1.5 2 2.5 3 0 0.5 1 1.5 2 Generic phasediagram of unconventional SC without QCP!
Long-range order Hoshino & Werner, PRL (2015) T c dome and non-FL metal phase next to magnetic order 0.01 spin-rotationally Ising limit invariant limit 0.008 Normal 0.006 0.004 SC 0.002 0 0 0.2 0.4 0.6 0.8 1 Need spin-anisotropy (SO coupling) for high T c probably relevant for: Sr 2 RuO 4 , UGe 2 , URhGe, UCoGe, ...
Long-range order Hoshino & Werner, PRL (2015) Pairing induced by local spin fluctuations Weak-coupling argument inspired by Inaba & Suga, PRL (2012) Effective interaction which includes bubble diagrams: ˜ U αγ χ γ ( q ) ˜ X U αβ ( q ) = U αβ − U γβ ( q ) γ Effective inter-orbital same-spin interaction U 1 " , 2 " (0) = U 0 − J − [2 UU 0 + ( U 0 − J ) 2 + U 0 2 ] χ loc ˜ in the weak-coupling regime: χ loc = ∆ χ loc
Crystal field splitting Hoshino & Werner, PRB (2016) Complicated phase diagrams, even in the two-orbital case High-spin/low-spin transitions Werner & Millis, PRL (2007) 10 Mott insulator level (spin triplet for J/U=0.25) high spin crossing 8 6 J/U=0 J/U=0.25 U/t 4 orbitally polarized 2 insulator low spin metal 0 0 0.5 1 1.5 2 2.5 3 3.5 ∆ /t
Crystal field splitting Hoshino & Werner, PRB (2016) Complicated phase diagrams, even in the two-orbital case High-spin/low-spin transitions Werner & Millis, PRL (2007) 10 Mott insulator level (spin triplet for J/U=0.25) high spin crossing 8 6 J/U=0 J/U=0.25 U/t excitonic order Kunes et al., PRB (2014) 4 orbitally polarized 2 insulator low spin metal 0 0 0.5 1 1.5 2 2.5 3 3.5 ∆ /t
Crystal field splitting Hoshino & Werner, PRB (2016) Complicated phase diagrams, even in the two-orbital case High-spin/low-spin transitions Werner & Millis, PRL (2007) 10 Mott insulator level (spin triplet for J/U=0.25) high spin crossing 8 6 J/U=0 J/U=0.25 U/t excitonic order Kunes et al., PRB (2014) 4 orbitally polarized spin freezing 2 insulator crossover low spin metal 0 0 0.5 1 1.5 2 2.5 3 3.5 ∆ /t
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