Spin and orbital freezing in unconventional superconductors Philipp - - PowerPoint PPT Presentation

Philipp Werner University of Fribourg

Spin and orbital freezing in unconventional superconductors

Kyoto, November 2017

In collaboration with: Shintaro Hoshino (Saitama) Hiroshi Shinaoka (Saitama) Karim Steiner (Fribourg)

Spin and orbital freezing in unconventional superconductors

Kyoto, November 2017

Generic phase diagram of unconventional superconductors Superconducting dome next to a magnetically ordered phase Non-Fermi liquid metal above the superconducting dome

Introduction

magnetic order superconductivity bad metal Fermi liquid pressure, doping, ... Temperature

Method

Dynamical mean field theory DMFT: mapping to an impurity problem 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

t

Σlatt ≡ Σimp Glatt ≡ Gimp

k

t

lattice model impurity model Georges and Kotliar, PRB (1992)

CT

• QMC solvers allow efficient simulation of multiorbital models

Relevant cases: 4 electrons in 3 orbitals: SrRu2O4 3 electrons in 3 orbitals, J<0: A3C60 6 electrons in 5 orbitals: Fe-pnictides Hloc = −

• α,σ

µnα,σ +

• α

Unα,⇥nα,⇤ +

• α>β,σ

U ⌅nα,σnβ,σ + (U ⌅ − J)nα,σnβ,σ −

• α⇧=β

J(ψ†

α,⇤ψ† β,⇥ψβ,⇤ψα,⇥ + ψ† β,⇥ψ† β,⇤ψα,⇥ψα,⇤ + h.c.)

Method

Werner et al., PRL (2006)

Phase diagram for Metallic phase: “transition” from Fermi liquid to spin-glass Narrow crossover regime with self-energy

3-orbital model

2 4 6 8 10 12 14 16 0.5 1 1.5 2 2.5 3 U/t n Fermi liquid frozen moment glass transition Mott insulator (βt=50)

ImΣ/t ∼ (iωn/t)α, α ≈ 0.5 U = U − 2J, J/U = 1/6, β = 50

Werner, Gull, Troyer & Millis, PRL (2008)

3-orbital model

Werner, Gull, Troyer & Millis, PRL (2008)

0.2 0.4 0.6 0.8 1 2 4 6 8 10 intercept C, exponent α U/t exponent α (βt=50) intercept C exponent α (βt=100) intercept C

−ImΣ(iωn) = C + A(ωn)α Fit self-energy by Square-root self-energy coincides with on-set of frozen moments

Spin-freezing leads to a small “quasi-particle weight” z

3-orbital model

Hoshino & Werner, PRL (2015)

(c)

spin-freezing crossover Fermi-liquid spin-frozen

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3

2.5 3 0 1 2 3 4 5 z ≈ 1/(1 − ImΣ(iω0)/ω0)

Ising

• rot. inv.

no quasi-particles in spin-frozen regime

Spin-spin and orbital-orbital correlation functions

3-orbital model

Werner, Gull, Troyer & Millis, PRL (2008)

• 0.1
• 0.05

0.05 0.1 0.15 0.2 0.25 5 10 15 20 25 <n1(0)n2(τ)>, <Sz(0)Sz(τ)> τt n=1.21 n=1.75 n=2.23 n=2.62 n=2.97 no freezing of orbital moments freezing of spin moments

Consider the local susceptibility and its dynamic contribution

3-orbital model

Hoshino & Werner, PRL (2015) subtract the (frozen) long-time value

χloc = Z β dτhSz(τ)Sz(0)i ∆χloc = Z β dτ[hSz(τ)Sz(0)i hSz(β/2)Sz(0)i ]

Consider the local susceptibility and its dynamic contribution Crossover regime is characterized by large local moment fluctuations

3-orbital model

Hoshino & Werner, PRL (2015)

(b) (c)

spin-freezing crossover Fermi-liquid spin-frozen

0.2 0.4 0.6 0.8 1

10 20 30 40 50 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 ∆χloc χloc

Ising Ising Ising

• rot. inv.

“quasi-particle weight” z Hund coupling J: Strongly correlated metal far from the Mott transition

3-orbital model

Werner, Gull, Troyer & Millis, PRL (2008) from De’ Medici, Mravlje & Georges, PRL (2011)

“quasi-particle weight” z Hund coupling J: Strongly correlated metal far from the Mott transition

3-orbital model

Werner, Gull, Troyer & Millis, PRL (2008) large local moment fluctuations from De’ Medici, Mravlje & Georges, PRL (2011)

Strontium Ruthenates

A self-energy with frequency dependence implies an

• ptical conductivity

Σ(ω) ∼ ω1/2 σ(ω) ∼ 1/ω1/2

Werner, Gull, Troyer & Millis, PRL (2008)

Strongly correlated despite moderate U

Pnictides

ρ(ω) (eV-1) ω (eV)

LDA d orbitals LDA p orbitals LDA total

• 6
• 4
• 2

2 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 ρ(ω) (eV-1) ω (eV) d spectral function

static U dynamic U LDA

• 6
• 4
• 2

2 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 d spectral function

0.06 0.12 0.18

• 16
• 12
• 8

ρ ω

• 1

ω d spectral function

Haule & Kotliar, NJP (2009) incoherent metal state resulting from Hund’s coupling

Strong doping and temperature dependence of electronic structure

Pnictides

BaFe2As2: conventional FL metal in the underdoped regime non-FL properties near

• ptimal doping

incoherent metal in the

• verdoped regime

Werner et al., Nat. Phys. (2012)

Strong doping and temperature dependence of electronic structure

Pnictides

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

• ff-diagonal orders:
• rbital-singlet-spin-triplet SC, orbital-triplet-spin-singlet SC

Long-range order

Hoshino & Werner, PRL (2015)

(b) (c)

AFM FM SC Normal AFM FM SC Normal

[arb. unit]

0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 3

Spin-freezing crossover Spin-freezing crossover

2.5 3

3-orbital model, Ising interactions

Long-range order

AFM near half-filling FM at large U away from half-filling spin-triplet superconductivity in the spin-freezing crossover region Hoshino & Werner, PRL (2015)

3-orbital model, Ising interactions (lower temperature)

(c)

AFM FM AFM FM SC Normal

[arb. unit]

0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 3 2 2.5 3

Spin-freezing crossover

Long-range order

U

AFM near half-filling FM at large U away from half-filling spin-triplet superconductivity in the spin-freezing crossover region parameter regime relevant for Sr2RuO4 Hoshino & Werner, PRL (2015)

Tc dome and non-FL metal phase next to magnetic order Generic phasediagram of unconventional SC without QCP!

Long-range order

(a) (b)

Spin-freezing crossover

AFM FM SC Normal

Spin-freezing crossover

SC Normal

Fermi liquid Fermi liquid

0.02 0.04 0.06 0.08 0.1 1 1.5 2 2.5 3 0.02 0.04 0.06 0.5 1 1.5 2

bad metal bad metal Hoshino & Werner, PRL (2015)

Tc dome and non-FL metal phase next to magnetic order Need spin-anisotropy (SO coupling) for high Tc probably relevant for: Sr2RuO4, UGe2, URhGe, UCoGe, ...

Long-range order

Hoshino & Werner, PRL (2015)

Ising limit spin-rotationally invariant limit

Normal SC

0.002 0.004 0.006 0.008 0.01 0.2 0.4 0.6 0.8 1

Pairing induced by local spin fluctuations Effective interaction which includes bubble diagrams: Effective inter-orbital same-spin interaction

Long-range order

Weak-coupling argument inspired by Inaba & Suga, PRL (2012)

˜ Uαβ(q) = Uαβ − X

γ

Uαγχγ(q) ˜ Uγβ(q) ˜ U1",2"(0) = U 0 − J − [2UU 0 + (U 0 − J)2 + U 02]χloc

in the weak-coupling regime: χloc = ∆χloc Hoshino & Werner, PRL (2015)

Complicated phase diagrams, even in the two-orbital case High-spin/low-spin transitions

Crystal field splitting

Werner & Millis, PRL (2007) Hoshino & Werner, PRB (2016) 2 4 6 8 10 0.5 1 1.5 2 2.5 3 3.5 U/t ∆/t J/U=0 J/U=0.25 Mott insulator (spin triplet for J/U=0.25) metal

• rbitally polarized

insulator level crossing low spin high spin

Complicated phase diagrams, even in the two-orbital case High-spin/low-spin transitions

Crystal field splitting

Werner & Millis, PRL (2007) Hoshino & Werner, PRB (2016) 2 4 6 8 10 0.5 1 1.5 2 2.5 3 3.5 U/t ∆/t J/U=0 J/U=0.25 Mott insulator (spin triplet for J/U=0.25) metal

• rbitally polarized

insulator level crossing excitonic order Kunes et al., PRB (2014) low spin high spin

Complicated phase diagrams, even in the two-orbital case High-spin/low-spin transitions

Crystal field splitting

Werner & Millis, PRL (2007) Hoshino & Werner, PRB (2016) 2 4 6 8 10 0.5 1 1.5 2 2.5 3 3.5 U/t ∆/t J/U=0 J/U=0.25 Mott insulator (spin triplet for J/U=0.25) metal

• rbitally polarized

insulator level crossing spin freezing crossover excitonic order Kunes et al., PRB (2014) low spin high spin

Complicated phase diagrams, even in the two-orbital case High-spin/low-spin transitions Excitonic (spin-orbital) order Exact mapping: Spin-orbital order and spin-triplet SC instabilities driven by fluctuating local moments half-filled model with ∆ > 0 → doped model with ∆ = 0 crystal field splitting ∆ → chemical potential shift µ spin-orbital order → spin-triplet SC

Crystal field splitting

Werner & Millis, PRL (2007) Kunes et al., PRB (2014) Hoshino & Werner, PRB (2016)

ci2σ → X

σ0

σx

σσ0c† i2σ0eiQ·Ri

Hoshino & Werner, PRB (2016)

2 4 6 8 10 12 0.8 1 1.2 1.4 1.6 1.8 2 U n FL

spin-freezing crossover

NFL

FM

Complicated phase diagrams, even in the two-orbital case High-spin/low-spin transitions Spin-orbital order (excitonic insulator phases)

Crystal field splitting

Werner & Millis, PRL (2007) Kunes et al., PRB (2014) Hoshino & Werner, PRB (2016)

1 2 3 4 5 6 0.5 1 1.5 2 2.5 3 3.5 U ∆ FL LSI HSMI

spin-freezing crossover

∆ > 0, n = 2 ∆ = 0, n < 2

AFM AFM SO tSC tSC AFM SO

2-orbital model (U=bandwidth=4)

Negative J and orbital freezing

spin-triplet SC spin-singlet SC

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 −0.2 0.2 0.4 0.6 T J MI

Mott insulator AFM AOO paired Mott insulator Steiner et al., PRB (2016)

2-orbital model (U=bandwidth=4) Mapping between J<0 and J>0:

Negative J and orbital freezing

Steiner et al., PRB (2016)

✓ di,1↓ di,2↑ ◆ − → ✓ 1 1 ◆ ✓ di,1↓ di,2↑ ◆ spin-singlet SC → spin-triplet SC antiferro OO → AFM ferro OO → FM

• rbital freezing

→ spin freezing

J<0: J>0:

Away from half-filling: SC dome peaks near orbital freezing line

Negative J and orbital freezing

line of maximum orbital fluctuations

0.02 0.03 0.04 −1.5 −1 −0.5 0.5 1 1.5 2 J T Metal SC SC’ Orbital Frozen Spin Frozen FOO AFM OF−crossover SF−crossover

Steiner et al., PRB (2016)

Away from half-filling: SC dome peaks near orbital freezing line

Negative J and orbital freezing

0.02 0.03 0.04 −1.5 −1 −0.5 0.5 1 1.5 2 J T Metal SC SC’ Orbital Frozen Spin Frozen FOO AFM OF−crossover SF−crossover

Steiner et al., PRB (2016)

1 2 3 4 5 6 −1.2 −1 −0.8 −0.6 −0.4 ∆χorbital J β = 30 β = 60 β = 90

T-dependence of orbital fluctuations 1.5 electrons in 2 orbitals

Orbital freezing seen in the decay of the (imaginary-time) orbital-

• rbital correlation function

fermi liquid metal:

• rbital-frozen metal:

Orbital freezing crossover line: maximum of orbital fluctuations

Negative J and orbital freezing

Steiner et al., PRB (2016)

ho(τ)o(0)i, o = n1 n2 ho(τ)o(0)i ⇠ 1/τ 2 (τ large)

Fermi liquid

• rbital-frozen

ho(τ)o(0)i ⇠ const > 0 ∆χorb ⌘ R β

0 dτ[ho(τ)o(0)i ho(β/2)o(0)i]

Orbital freezing seen in the decay of the (imaginary-time) orbital-

• rbital correlation function

fermi liquid metal:

• rbital-frozen metal:

Orbital freezing crossover line: maximum of orbital fluctuations Orbital fluctuations induce attractive interaction for on-site pairs Effective interaction which includes bubble diagrams:

Negative J and orbital freezing

Steiner et al., PRB (2016)

ho(τ)o(0)i, o = n1 n2 ho(τ)o(0)i ⇠ const > 0 ∆χorb ⌘ R β

0 dτ[ho(τ)o(0)i ho(β/2)o(0)i]

˜ Uαβ(q) = Uαβ − P

γ Uαγχγ(q) ˜

Uγβ(q) ⇒ ˜ U = U − 4U 0[U 0 + |J|]∆χorb + O(U 3) ho(τ)o(0)i ⇠ 1/τ 2 (τ large)

analogous to: Inaba & Suga, PRL (2012)

Hoshino & Werner (2016)

50 100 50 100 780 740 760 800 780 740 760 800

(c)

AFM SC PM PM MI SC JTM

(d)

0.005 0.01 0.015 0.02 0.025 0.03 0.5 1 1.5 2 0.01 0.015 0.02 0.025 0.03 CEP PM MI

• max. orbital

fluctuation

SOSM AFM SC CEP PM MI

• max. orbital

fluctuation

SOSM

Negative J and orbital freezing

=3

U/W SC dome peaks in the region of maximum

• rbital fluctuations

spontaneous symmetry breaking into an

• rbital selective Mott phase (“Jahn-Teller metal”)

Hoshino & Werner, PRL (2016) Fermi liquid metal orbital frozen metal Mott insulator

Half-filled 3-orbital model (A3C60)

Hoshino & Werner (2016)

Cuprates

Werner, Hoshino & Shinaoka, PRB (2016)

Unconventional SC in the spin-freezing regime Strontium ruthenates Uranium-based SC Pnictides CrAs ... Unconventional SC in the orbital-freezing regime Alkali-doped fullerides What about cuprates? Can spin-freezing play any role in a single-band 2D Hubbard model? naive answer: NO, correct answer: YES

Cuprates

Mapping to an effective two-orbital model: Slater-Kanamori interaction with nnn hopping translates into a crystal-field splitting

t 2t t’ G U J ~ U’ ~ U ~ δ basis transf. DMFT embedding

=3

c1 =

1 √ 2(d1 + d3)

c2 =

1 √ 2(d2 + d4)

f1 =

1 √ 2(d1 − d3)

f2 =

1 √ 2(d2 − d4)

=3

δ = 2t

=3

˜ U = ˜ U = ˜ J = U/2

=3

c

=3

f

=3

d

Werner, Hoshino & Shinaoka, PRB (2016)

Mapping to an effective two-orbital model: Slater-Kanamori interaction with nnn hopping translates into a crystal-field splitting

Hoshino & Werner (2016)

Cuprates

single site 2t G0,ij c f J ~ U’ ~ U ~ δ transf. DMFT embedding approx.

=3

c1 =

1 √ 2(d1 + d3)

c2 =

1 √ 2(d2 + d4)

f1 =

1 √ 2(d1 − d3)

f2 =

1 √ 2(d2 − d4)

=3

δ = 2t

=3

˜ U = ˜ U = ˜ J = U/2

Werner, Hoshino & Shinaoka, PRB (2016)

Phasediagram (1-site/2-orbital DMFT)

single site t 2t t’ G0,ij c f J ~ U’ ~ U ~ δ basis transf. DMFT embedding approx.

Hoshino & Werner (2016)

Cuprates

emerging (fluctuating) local moments = bad metal regime frozen moments =pseudo-gap phase

50 100 150 200 250 300 0.75 0.8 0.85 0.9 0.95 1 temperature (K) [W = 2 eV] filling a

• max. ∆χloc

f c δ=0 frozen moments (pseudo-gap) bad metal crossover antiferromagnetism

=3

c

=3

f

Werner, Hoshino & Shinaoka, PRB (2016)

single site t 2t t’ G0,ij U J ~ U’ ~ U ~ δ basis transf. DMFT embedding approx.

=3

c

=3

f Phasediagram (2-site/2-orbital cluster DMFT)

Hoshino & Werner (2016)

Cuprates

50 100 150 200 250 300 0.75 0.8 0.85 0.9 0.95 1 temperature (K) [W = 2 eV] filling b CDMFT, t’=0

• max. χstat

half-max. χstat frozen spins (pseudo-gap) bad metal crossover

emerging (fluctuating) local moments = bad metal regime frozen moments =pseudo-gap phase Werner, Hoshino & Shinaoka, PRB (2016)

Phasediagram (2-site/2-orbital cluster DMFT)

Hoshino & Werner (2016)

Cuprates

50 100 150 200 250 300 0.75 0.8 0.85 0.9 0.95 1 temperature (K) [W = 2 eV] filling b CDMFT, t’=0

• max. χstat

half-max. χstat frozen spins (pseudo-gap) bad metal crossover

emerging (fluctuating) local moments = bad metal regime frozen moments =pseudo-gap phase SC dome [4-site cluster DMFT, Maier et al, (2005)] induced by fluctuating local moments? Werner, Hoshino & Shinaoka, PRB (2016)

Hoshino & Werner (2016)

Cuprates

=3

(d†

1↑d† 2↓ − d† 1↓d† 2↑) − (d† 2↑d† 3↓ − d† 2↓d† 3↑)

+(d†

3↑d† 4↓ − d† 3↓d† 4↑) − (d† 4↑d† 1↓ − d† 4↓d† 1↑)

=3

− → 2(f †

1↑f † 2↓ − f † 1↓f † 2↑)

single site t 2t t’ G0,ij U J ~ U’ ~ U ~ δ basis transf. DMFT embedding approx.

=3

˜ U eff

(1,f,↑),(2,f,↓) = 2 ˜

U 3χ(f)

locχ(c) 12 + O( ˜

U 5)

=3

f

local spin fluctuations (needed because U’-J=0)

=3

c d-wave SC induced by local spin fluctuations Transformation of the d-wave order parameter: Effective attractive interaction: Leading contribution:

Werner, Hoshino & Shinaoka, PRB (2016)

Spin/orbital freezing as a universal phenomenon in unconventional superconductors Strontium ruthenates Uranium-based SC Pnictides Fulleride compounds Cuprates ... Pairing induced by local spin or orbital fluctuations Bad metal physics originates from fluctuating/frozen moments

Hoshino & Werner (2016)

Summary I

Experimental results for Cs3C60 and RbxCs3-xC60

Jahn-Teller metal

bad metal Potocnik et al., Sci. Rep. (2014) Zadik et al., Sci. Express (2015)

“coexistence of both localized and itinerant electrons”

3-band model of A3C60

Bandstructure 3 bands near Fermi level half-filling bandwidth ~ 0.4 eV, increasing correlations from A=K to Cs

Nomura et al. (2012)

3-band model of A3C60

Inverted Hund coupling U~1eV > bandwidth strongly correlated Extended molecular orbitals small bare J (~0.035 eV) Reduction of J by 0.05 eV due to Jahn-Teller phonons:

Nomura et al. (2012) Cs3C60

Jeff = JH(0) − Jph(0) ≈ −0.02 eV

3-band model of A3C60

Inverted Hund coupling Lowest energy atomic state has paired electrons (“seed” for SC) |J| small compared to bandwidth, but Ekin strongly reduced by large U: cooperation between correlation and phonon effects For superconductivity, pairs have to be mobile: important role of pair-hopping term U U − 2J > U U − 3J > U

Capone et al., Science (2002) Nomura et al., Science Expr. (2015)

Hoshino & Werner (2016)

50 100 50 100 780 740 760 800 780 740 760 800

(c)

AFM SC PM PM MI SC JTM

(d)

0.005 0.01 0.015 0.02 0.025 0.03 0.5 1 1.5 2 0.01 0.015 0.02 0.025 0.03 CEP PM MI

• max. orbital

fluctuation

SOSM AFM SC CEP PM MI

• max. orbital

fluctuation

SOSM

Jahn-Teller metal

=3

U/W SC dome peaks in the region of maximum

• rbital fluctuations

spontaneous symmetry breaking into an

• rbital selective Mott phase (“Jahn-Teller metal”)

Hoshino & Werner, PRL (2017) Fermi liquid metal orbital frozen metal Mott insulator

Half-filled 3-orbital model (A3C60)

(a) (b) (e) (f ) (c) (d) orbital-selective Mott state γ=1 γ=2 γ=3

site A site B

2 2.5 Disordered

• 0.25
• 0.2
• 0.15
• 0.1
• 0.05

0.5 1 1.5 2 0.5 1 1.5 2 SOSM SC

0.2 0.4 0.6 0.8 1 1.2 1.4

=1 =2 =3 Disordered

γ=1 γ=2 γ=3 Disordered

γ=1 γ=2 γ=3 Disordered

Hoshino & Werner (2016) Hoshino & Werner (2016)

Half-filled 3-orbital models with negative J exhibit a symmetry- broken phase characterized by a composite order parameter completely degenerate bands no ordinary orbital moment (all orbitals half-filled) but: orbital-dependent double-occupation coexistence of Mott insulating and metallic orbitals

Jahn-Teller metal

Hoshino & Werner, PRL (2017)

DMFT results for , density-density interaction

• rbital-dependent double occupation (a), kinetic energy (b), and

self-energy (c)

Hoshino & Werner (2016)

Jahn-Teller metal

J = −U/4

enhanced D in the (paired) Mott insulator paired Mott insulator

• rbital-selective Mott insulator

metal Hoshino & Werner, PRL (2017)

Define the time-dependent orbital moment Odd time/frequency component characterizes the SOSM state “diagonal order” version of odd-frequency superconductivity

Hoshino & Werner (2016)

Odd-frequency order

Gell-Mann matrix

• ribtal-dependent Ekin and double occupation

Hoshino & Werner, PRL (2017)

• rdinary orbital

moment (=0)

T 8

• dd =

X

γ

λ8

γγ(Kγ + 2UDγ) + terms depending on U 0, J

Berezinskii (1974), Kirkpatrick & Belitz (1991)

T η(τ) = X

iγγ0σ

hc†

iγσλη γγ0ciγ0σ(τ)i = T η even + T η

• ddτ + O(τ 2)

Summary II

A3C60: 3-band system with strong U and inverted J Tc dome: enhanced pairing in the orbital-freezing crossover region Analogous to unconventional superconductivity induced by spin- freezing in systems with J>0 Jahn-Teller metal: symmetry-broken state with a composite order parameter (orbital-dependent double occupation) Coexistence of 2 Mott insulating and 1 metallic orbital Diagonal-order analogue of odd-frequency superconductivity