Spin freezing in unconventional superconductors Philipp Werner University of Fribourg Beijing, August 2018 Tuesday, August 21, 18
Spin freezing in unconventional superconductors In collaboration with: Shintaro Hoshino (Saitama) Hiroshi Shinaoka (Saitama) Aaram Kim (London) Naoto Tsuji (RIKEN) Beijing, August 2018 Tuesday, August 21, 18
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 Tuesday, August 21, 18
Introduction Connection between spin-freezing and Sachdev-Ye model Sachdev-Ye model as an effective model describing the spin- freezing crossover regime Behavior of out-of-time-order correlation functions Temperature peculiar non-FL exponents Σ ( ω ) ∼ √ ω bad metal magnetic order Fermi liquid pressure, doping, ... superconductivity Tuesday, August 21, 18
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 Tuesday, August 21, 18
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: Sr 2 RuO 4 3 electrons in 3 orbitals, J <0: A 3 C 60 6 electrons in 5 orbitals: Fe -pnictides Tuesday, August 21, 18
3-orbital model Werner, Gull, Troyer & Millis PRL 101, 166405 (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 Tuesday, August 21, 18
3-orbital model Werner, Gull, Troyer & Millis PRL 101, 166405 (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 Tuesday, August 21, 18
spin-freezing crossover Fermi-liquid spin-frozen 3-orbital model Hoshino & Werner PRL 115, 247001 (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 Tuesday, August 21, 18
3-orbital model Werner, Gull, Troyer & Millis PRL 101, 166405 (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 Tuesday, August 21, 18
3-orbital model Werner, Gull, Troyer & Millis PRL 101, 166405 (2008) Decay of spin correlations 5 n c from S ∼ 1 / τ 2 u/t=8 C 1/2 (T=0.02t)/C 1/2 (T=0.01t) Fermi liquid 4 3 C 1 / 2 ( β ) = h S z S z i ( τ = β / 2) ∼ 1 / τ 2 ∼ const frozen spins 1 0 1 1.5 2 2.5 3 n spin-freezing crossover Tuesday, August 21, 18
3-orbital model Hoshino & Werner PRL 115, 247001 (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 Tuesday, August 21, 18
crossover spin-freezing Fermi-liquid spin-frozen 3-orbital model Hoshino & Werner PRL 115, 247001 (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 Tuesday, August 21, 18
3-orbital model “quasi-particle weight” z from De’ Medici, Mravlje & Georges, PRL (2011) Hund coupling J : Strongly correlated metal far from the Mott transition Tuesday, August 21, 18
3-orbital model “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 Tuesday, August 21, 18
Strontium Ruthenates A self-energy with frequency dependence implies an Σ ( ω ) ∼ ω 1 / 2 optical conductivity σ ( ω ) ∼ 1 / ω 1 / 2 Tuesday, August 21, 18
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) Tuesday, August 21, 18
Pnictides Werner et al. Nature Phys. 8, 331 (2012) 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 Tuesday, August 21, 18
Pnictides Werner et al. Nature Phys. 8, 331 (2012) Strong doping and temperature dependence of electronic structure Tuesday, August 21, 18
Long-range order Hoshino & Werner PRL 115, 247001 (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 Tuesday, August 21, 18
[arb. unit] Long-range order Hoshino & Werner PRL 115, 247001 (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 Tuesday, August 21, 18
[arb. unit] Long-range order Hoshino & Werner PRL 115, 247001 (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 Tuesday, August 21, 18
Long-range order Hoshino & Werner PRL 115, 247001 (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! Tuesday, August 21, 18
Long-range order Hoshino & Werner PRL 115, 247001 (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, ... Tuesday, August 21, 18
Long-range order Hoshino & Werner PRL 115, 247001 (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 Tuesday, August 21, 18
Negative J and orbital freezing Steiner et al. PRB 94, 075107 (2016) 2-orbital model ( U =bandwidth=4) 0.10 paired MI 0.09 Mott Mott AOO AFM insulator 0.08 insulator 0.07 T 0.06 0.05 0.04 0.03 0.02 − 0.2 0 0.2 0.4 0.6 J spin-singlet SC spin-triplet SC Tuesday, August 21, 18
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