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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

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Philipp Werner University of Fribourg

Spin freezing in unconventional superconductors

Beijing, August 2018

Tuesday, August 21, 18

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In collaboration with: Shintaro Hoshino (Saitama) Hiroshi Shinaoka (Saitama) Aaram Kim (London) Naoto Tsuji (RIKEN)

Spin freezing in unconventional superconductors

Beijing, August 2018

Tuesday, August 21, 18

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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 Fermi liquid pressure, doping, ... Temperature bad metal

Tuesday, August 21, 18

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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

Introduction

magnetic order superconductivity bad metal Fermi liquid pressure, doping, ... Temperature peculiar non-FL exponents

Σ(ω) ∼ √ω

Tuesday, August 21, 18

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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)

Tuesday, August 21, 18

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CT

  • QMC solvers allow efficient simulation of multiorbital models

Relevant cases: 4 electrons in 3 orbitals: Sr2RuO4 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)

Tuesday, August 21, 18

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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 101, 166405 (2008)

Tuesday, August 21, 18

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3-orbital model

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

Werner, Gull, Troyer & Millis PRL 101, 166405 (2008)

Tuesday, August 21, 18

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

3-orbital model

(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 Hoshino & Werner PRL 115, 247001 (2015)

Tuesday, August 21, 18

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Spin-spin and orbital-orbital correlation functions

3-orbital model

  • 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 Werner, Gull, Troyer & Millis PRL 101, 166405 (2008)

Tuesday, August 21, 18

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Decay of spin correlations

3-orbital model

spin-freezing crossover 1 2 3 4 5 1 1.5 2 2.5 3 C1/2(T=0.02t)/C1/2(T=0.01t) n nc from S u/t=8

C1/2(β) = hSzSzi(τ = β/2) ∼ 1/τ 2 ∼ 1/τ ∼ const

frozen spins Fermi liquid Werner, Gull, Troyer & Millis PRL 101, 166405 (2008)

Tuesday, August 21, 18

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Consider the local susceptibility and its dynamic contribution

3-orbital model

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 ]

Hoshino & Werner PRL 115, 247001 (2015)

Tuesday, August 21, 18

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

3-orbital model

(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.

Hoshino & Werner PRL 115, 247001 (2015)

Tuesday, August 21, 18

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

3-orbital model

from De’ Medici, Mravlje & Georges, PRL (2011)

Tuesday, August 21, 18

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

3-orbital model

large local moment fluctuations from De’ Medici, Mravlje & Georges, PRL (2011)

Tuesday, August 21, 18

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Strontium Ruthenates

A self-energy with frequency dependence implies an

  • ptical conductivity

Σ(ω) ∼ ω1/2 σ(ω) ∼ 1/ω1/2

Tuesday, August 21, 18

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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

Tuesday, August 21, 18

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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. Nature Phys. 8, 331 (2012)

Tuesday, August 21, 18

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Strong doping and temperature dependence of electronic structure

Pnictides

Werner et al. Nature Phys. 8, 331 (2012)

Tuesday, August 21, 18

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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 115, 247001 (2015)

Tuesday, August 21, 18

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(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 115, 247001 (2015)

Tuesday, August 21, 18

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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 115, 247001 (2015)

Tuesday, August 21, 18

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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 115, 247001 (2015)

Tuesday, August 21, 18

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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

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 Hoshino & Werner PRL 115, 247001 (2015)

Tuesday, August 21, 18

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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 115, 247001 (2015)

Tuesday, August 21, 18

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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 94, 075107 (2016)

Tuesday, August 21, 18

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

Negative J and orbital freezing

✓ 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: Steiner et al. PRB 94, 075107 (2016)

Tuesday, August 21, 18

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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 94, 075107 (2016)

Tuesday, August 21, 18

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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

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]

Steiner et al. PRB 94, 075107 (2016)

Tuesday, August 21, 18

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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 94, 075107 (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)

Tuesday, August 21, 18

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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 118, 177002 (2017) Fermi liquid metal orbital frozen metal Mott insulator

Half-filled 3-orbital model with J<0 (A3C60)

Tuesday, August 21, 18

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Hoshino & Werner (2016)

Cuprates

Unconventional SC in the spin-freezing regime Strontium ruthenates Uranium-based SC Pnictides CrAs, MnP ... 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

Werner, Hoshino & Shinaoka PRB 94, 245134 (2016)

Tuesday, August 21, 18

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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 94, 245134 (2016)

Tuesday, August 21, 18

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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 94, 245134 (2016)

Tuesday, August 21, 18

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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 94, 245134 (2016)

Tuesday, August 21, 18

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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 94, 245134 (2016)

Tuesday, August 21, 18

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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 94, 245134 (2016)

Tuesday, August 21, 18

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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 94, 245134 (2016)

Tuesday, August 21, 18

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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

Tuesday, August 21, 18

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Sachdev-Ye model Spin-S quantum Heisenberg model with infinite-range Gaussian- random interactions Fermionize spins, calculate saddle-point solution in the large-M limit

Hoshino & Werner (2016)

Connection to Sachdev-Ye model

H = 1 √ NM X

i>j

JijSi · Sj P(Jij) ∼ exp[−J2

ij/(2J2)]

N → ∞ : number of sites S : SU(M) spin operator (M large) G−1(iωn) = iωn − Σ(iωn), Σ(τ) = −J2G(τ)G(−τ)G(τ)

Sachdev & Ye, PRL (1993)

= ) G(iω) ⇠ 1/pωn, Σ(iωn) ⇠ ipωn, hSa

i (τ)Sa i (0)i ⇠ 1/τ

Tuesday, August 21, 18

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= ) G(iω) ⇠ 1/pωn, Σ(iωn) ⇠ ipωn, hSa

i (τ)Sa i (0)i ⇠ 1/τ

Sachdev-Ye model Same non-Fermi liquid exponents as in the spin-freezing crossover region

Hoshino & Werner (2016)

Connection to Sachdev-Ye model

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2

  • Im Σ/t

(ωn/t)0.5 n=2.62 n=2.35 n=1.99 n=1.79 n=1.60 1 2 3 4 5 1 1.5 2 2.5 3 C1/2(T=0.02t)/C1/2(T=0.01t) n nc from S u/t=8

∼ const ∼ 1/τ ∼ 1/τ 2

Werner et al., PRL (2008)

Tuesday, August 21, 18

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Sachdev-Ye model: recent extensions Sachdev-Ye-Kitaev (SYK) model: Fermionic version with Gaussian- random interaction tensor (same saddle point equations) Lattice of “SYK atoms”:

Hoshino & Werner (2016)

Connection to Sachdev-Ye model

Hlattice = X

r,r0,l

tr,r0c†

r0,lcr,l +

X

r

HSYK

r

…. …….. …. …. . . . . …. …. ………... . . . . . . .

i j k l

HSYK = 1 (2M)3/2 X

ijkl

Uijklc†

ic† jckcl,

M = number or orbitals

Chowdhury et al, arxiv:1801.06178

Tuesday, August 21, 18

slide-43
SLIDE 43

Hoshino & Werner (2016)

Connection to Sachdev-Ye model

Hlattice = X

r,r0,l

tr,r0c†

r0,lcr,l +

X

r

HSYK

r

b) scaling hole doping interaction strength AFM SC Fermi liquid moment frozen FM hole doping a) temperature AFM SC Fermi liquid moment frozen square root

HSYK = 1 (2M)3/2 X

ijkl

Uijklc†

ic† jckcl,

M = number or orbitals Sachdev-Ye model: recent extensions Sachdev-Ye-Kitaev (SYK) model: Fermionic version with Gaussian- random interaction tensor (same saddle point equations) Lattice of “SYK atoms”: high T: local physics dominates same as SYK low T: Fermi-liquid metal

Chowdhury et al, arxiv:1801.06178

Tuesday, August 21, 18

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SLIDE 44

Interaction tensor Gaussian-random interaction tensor is unphysical for a Slater-Kanamori interaction with M=2,3,5 orbitals Switch to density-density interactions and focus on inter-orbital terms ( terms)

Hoshino & Werner (2016)

Connection to Sachdev-Ye model

Uijkl P

α,β Uαβnαnβ

O(M 2)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

  • 2
  • 1

1 2 3 4 5 6 7 J=U/6 U’=U-2J a) P(Uijkl) Uijkl / J M=2 M=3 M=5 0.1 0.2 0.3 0.4 0.5

  • 2 -1 0 1 2 3 4 5 6 7

P(Uαβ) Uαβ / J

Werner, Kim & Hoshino arxiv:1805.04102

Tuesday, August 21, 18

slide-45
SLIDE 45

Interaction tensor Gaussian-random interaction tensor is unphysical for a Slater-Kanamori interaction with M=2,3,5 orbitals Switch to density-density interactions and focus on inter-orbital terms ( terms)

Hoshino & Werner (2016)

Connection to Sachdev-Ye model

Uijkl

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

  • 2
  • 1

1 2 3 4 5 6 7 J=U/6 U’=U-2J a) P(Uijkl) Uijkl / J M=2 M=3 M=5 0.1 0.2 0.3 0.4 0.5

  • 2 -1 0 1 2 3 4 5 6 7

P(Uαβ) Uαβ / J

P

α,β Uαβnαnβ

O(M 2)

0.5 1 P(Uαβ) b) J Uav Uαβ same spin

  • pposite spin

U’-J U’ large M

Werner, Kim & Hoshino arxiv:1805.04102

Tuesday, August 21, 18

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SLIDE 46

Interaction tensor Gaussian-random interaction tensor is unphysical Physically meaningful and much simpler model: Density-density interactions with random-bimodal distribution average interaction is the “Hubbard U” difference between the two interactions is the Hund coupling This model has the same saddle point equations as the SY(K) model

Hoshino & Werner (2016)

Connection to Sachdev-Ye model

0.5 1 P(Uαβ) b) J Uav Uαβ same spin

  • pposite spin

U’-J U’ large M

Werner, Kim & Hoshino arxiv:1805.04102

Tuesday, August 21, 18

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SLIDE 47

Important point: interaction vertex in the second order diagram is Hund coupling Consequence: no Hund coupling, no interesting non-FL properties Same equations as for the SYK lattice model same physics: non-FL properties ( ) at high T FL metal at low T

Hoshino & Werner (2016)

Connection to Sachdev-Ye model

Hartree diagram: monopole interaction 2nd order diagram: Hund coupling effect

Σ(iωn) ⇠ pωn, hSz(τ)Sz(0)i ⇠ 1/τ

Werner, Kim & Hoshino arxiv:1805.04102

Tuesday, August 21, 18

slide-48
SLIDE 48

Hoshino & Werner (2016)

Connection to Sachdev-Ye model

b) scaling hole doping interaction strength AFM SC Fermi liquid moment frozen FM hole doping a) temperature AFM SC Fermi liquid moment frozen square root

Interpretation of the generic DMFT phase diagram As filling increases, local moments appear due to effect of Hund coupling As these moments form, the “Kondo screening temperature” drops, resulting in a bad metal with frozen moments The SY equations describe the spin-freezing crossover regime characterized by fluctuating moments The SY equations also naturally explain the connection to superconductivity

Werner, Kim & Hoshino arxiv:1805.04102

Tuesday, August 21, 18

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SLIDE 49

Connnection to superconductivity Effective interaction which takes account of “polarization bubble”: From follows and thus

Hoshino & Werner (2016)

Connection to Sachdev-Ye model

G(iωn) ∼ 1/√ωn P(iωn) = − 1 √ 2π ˜ J log( ˜ J/ωn) ⇒ Ueff(ω → 0) → −∞ P(τ) ∼ 1/τ Ueff(iωn) = ˜ U + ˜ JP(iωn) ˜ J, P(τ) = G(τ)G(−τ)

Werner, Kim & Hoshino arxiv:1805.04102

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Out-of-time-order correlation functions probes chaotic nature of quantum systems Conjecture: universal bound on growth rate of OTOCs SYK model saturates this bound on chaos Question: nontrivial behavior of OTOCs in the spin-freezing crossover regime of multi-orbital Hubbard models?

Hoshino & Werner (2016)

Outlook

OTOC(t, t0) = hA(t)B(t0)A(t)B(t0)i OTOC(t, t0) = c0 + c1 exp[λ(t − t0)] + . . . , λ ≤ 2πβ

Tsuji & Werner in preparation Larkin & Ovchinnikov, JETP (1969) Maldacena, Shenker, Stanford, J. High Energy Phys. (2016)

Tuesday, August 21, 18

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Hoshino & Werner (2016)

Outlook

OTOC(t, t0) = hA(t)B(t0)A(t)B(t0)i

0.1 1 0.1 1 10

  • Im Σ

ωn nσ=0.167 0.242 0.282 0.304 0.330 0.358 0.390

0.1 1 0.5 1 1.5 2 2.5 3 normalized Re[<A(t)B(0),A(t)B(0)>] - <nσ>4 t 2.2 exp[-2.2t]

A = B = n1σ

√ωn

Out-of-time-order correlation functions Exponential decay of OTOC in the spin-freezing crossover regime Similar to ED results for finite-M SYK model

Fu & Sachdev, PRB (2016) Tsuji & Werner in preparation

Tuesday, August 21, 18

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Summary II

SY equations can be derived from M-orbital Hubbard model with bimodal-distributed density-density interactions SY equations describe the spin-freezing crossover regime and the superconductivity at low T Non-Fermi liquid behavior arises from Hund coupling Spin-OTOC exhibits exponential decay in the fluctuating moment regime (similar to finite-M SY model)

Spin-freezing: P . Werner, E. Gull, M. Troyer and A. Millis, PRL 101, 166405 (2008) Connection to superconductivity: S. Hoshino and P . Werner, PRL 115, 247001 (2015) Connection to A3C60: K. Steiner, S. Hoshino,

  • Y. Nomura and P

. Werner, PRB 94, 075107 (2016) Connection to cuprates: P . Werner, S. Hoshino and S. Shinaoka, PRB 94, 245134 (2016) Connection to Sachdev-Ye model: P . Werner, A. Kim and S. Hoshino, arxiv:1805.04102

Tuesday, August 21, 18