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Spatial Correlations and Superconductivity in Dynamical Mean-Field Theory Junya Otsuki Research Institute for Interdisciplinary Science, Okayama University 1 23-28 September, 2019

Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

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Page 1: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Spatial Correlations and Superconductivityin Dynamical Mean-Field Theory

Junya Otsuki

Research Institute for Interdisciplinary Science, Okayama University

1

23-28 September, 2019

Page 2: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Outline

1. Heavy Fermion Superconductivity

• Dual fermion approach: beyond DMFT

• Role of incoherent part on superconductivities

2. Strong-coupling formula for χ(q)

• Physically, easy to understand; Numerically, easy to compute

• Evaluation of Intersite interactions Jij in DMFT

2

Page 3: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Collaborators

Dual fermion approach

• Hartmut Hafermann (Huawei)

• Alexander Lichtenstein (U Hamburg)

Strong-Coupling formula

• Hiroshi Shinaoka (Saitama U)

• Kazuyoshi Yoshimi (ISSP, U Tokyo)

• Yusuke Nomura (RIKEN)

• Masayuki Ohzeki (Tohoku U)

Special Thanks

• Yoshio Kuramoto (KEK)

• Hiroaki Kusunose (Meiji U)

• Dieter Vollhardt (U Augsburg)3

Shinaoka

Nomura

Yoshimi

Ohzeki

Page 4: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Introduction

Heavy fermion systems

Itinerant and localized nature of f electrons

4

Page 5: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

f-electron materials

5

CeCu2Si2 Steglich 1979, Yuan et al. 2003

CePd2Si2, CeCoIn5, URu2Si2, UGe3, UCoGe, …

Magnetism (multipole ordering) Superconductivity

CeB6 Tayama et al 1997

AFQ

AFM

AFO

A lot of Ce, Pr, Nd, …, U compounds

Page 6: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Itinerant/Localized in d-electron systems

6

dd

U

wave particle

From Yoshida et al. 2003

Fermi arc

in high-Tc cuprates

Page 7: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Itinerant/Localized in f-electron systems

7

Fermi surface

CeB6

Similar to LaB6

Onuki et a. 1989

Harima, Kasuya 1989, 1996

LaRu2Si2

CeRu2Si2

Different from LaRu2Si2Yamagami, Hasegawa 1992

H. Aoki et al. 1993

Matsumoto et al. 2010

No 4f contribution

localized

4f electrons

participate in FS

itinerant

Ce3+ : 4f1, La3+ : 4f0

Page 8: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Itinerant/Localized in f-electron systems

8

Magnetic susceptibilityCeRu2Si2

Haen et al. 1987

waveparticle

Curie law

= localized moment

Page 9: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Itinerant/Localized in f-electron systems

9

high pressure low pressure

CeRh2Si2

f states

f1 states

f2 multiplets

DFT (GGA) DFT+DMFT (Hubbard-I)

U (Slater type)

Page 10: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

P

T

Heavy fermions = itinerant + localized natures

SC

10

CeCu2Si2 CeRu2Si2

f

p, d

Incoherent part

Localized momentquasiparticle weight z

1- z

Coherent part

Heavy fermions

p, d

f itinerant

f electrons

wave

particle

Heavy-fermion state:

eg.

99% of spectrum are incoherent

U/W

Page 11: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Dynamical Mean-Field Theory (DMFT)

11

Metzner, Vollhardt 1989

Georges, Kotliar 1992

Georges et al. 1996

hybridization with “bath”

U

Local correlations

exactly taken into account

Local approximationPlenary talk by Dieter Vollhardt

JO in summer school textbook 2016

Solution of the impurity Anderson model

by continuous-time QMC (CT-QMC)

Page 12: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Heavy Fermion Superconductivity

12

Page 13: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

AFM phase diagram in DMFT

13

nc=1

Superconductivity?

beyond DMFT

hypercubic in d=∞

Kondo lattice solved with DMFT + CT-QMC(CT-J) JO, Kusunose, Kuramoto, JPSJ 2009

Page 14: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Motivations

14

Localized

(incoherent)

Itinerant

(coherent)

Magnetism(long-range order)

Superconductivity

Both localized and itinerant nature should be taken into account cf. Phenomenology: ̀ duality model’ (Kuramoto, Miyake, 1990)

Page 15: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Motivations

15

Localized

(incoherent)

Itinerant

(coherent)

Magnetism(long-range order)

Symmetry breaking

induced by Kondo effect

Kondo resonance

Kondo singlet-CEF singlet order in PrFe4P12

Hoshino, JO, Kuramoto, 2011

Page 16: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Motivations

16

Localized

(incoherent)

Itinerant

(coherent)

Superconductivity

heavy-fermion SC

exotic pairing state

incoherent part

=localized moment

Page 17: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Differences between d- and f-electron SCs ?

17

d = 2

Page 18: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Expansion around DMFT

18

(1) Dynamical mean-field theory (DMFT)

Rubtsov, Katsnelson, Lichtenstein 2008

U

Heavy fermions

Mott insulator

Dynamical

pairing interactions

(2) Auxiliary fermion (dual fermion) lattice

“residual interactions”“quasiparticles”

Page 19: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Superconductivity

Impurity solver:

CT-HYB (Werner et al. 2006)

Improved vertex calculation

(Hafermann et al. 2012)

U/t=8, n=0.9, T/t=0.1

spin singlet spin triplet

A1g

A2g

[xy(x2-y2)]

B1g

[x2-y2]

B2g

[xy]

Eu [x]

Eu [y]

-use the power method

-multiply a phase factor, then

orbital projection

-plot even- and odd-freq. parts

Pauli principle is fulfilled(spin parity time-reversal)

How to solve

even-freq. odd-freq. even-freq. odd-freq.

19

Page 20: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Pairing instability

20

Hubbard model (square lattice) Kondo lattice (square lattice)

Competing d-wave and p-wave SC

JO, PRL 2015 JO, Hafermann, Lichtenstein, PRB 2014

Page 21: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Superconductivity in 2D Kondo lattice

21

Localized

(small Fermi surface)Itinerant

(large Fermi surface)

f

p, d qp weight z

1- z

JO, PRL 2015

Note:

Electron # in the system

does not change

2D Kondo lattice

Page 22: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Differences between d- and f-electron SC ?

22

d = 2

or

- Formation of large Fermi surface

- Dynamical qp interactions

Odd-frequency SCs:

CeCu2Si2, CeRhIn5, Fuseya, Kohno, Miyake, 2003

Thermodynamic stability, Solenov 2009, Kusunose et al., 2011

Page 23: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

χ(q) in DMFT

23

Page 24: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

DFT+DMFT calculation flow

24

DFT

DMFT

two-particle

beyond DMFT unconventional superconductivity

magnetism, multipole order

(Neutron scattering spectra)

single-particle excitations

(ARPES spectra)

energy band structure

Page 25: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Momentum-dependent susceptibilities in DMFT

25

Susceptibility matrix

local vertex

Bethe-Salpeter equations

= + Jarrell 1992

Georges et al 1996

calculated in the effective impurity model (very heavy!)

Page 26: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Decoupling

26

Similar approximation was addressed by…

- dual-boson approach: Stepanov et al. 2016

- diagrama analysis: F. Krien, arXiv:1901.02832

dynamical local interactions

Page 27: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

spin channel

Mathematical justification of the decoupling

27

Singular Value Decomposition (SVD)

charge channel

Page 28: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Strong-Coupling-Limit (SCL) formula

28

(ii) Local information

(i) Lattice information

in general

in the atomic limit

Page 29: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Numerical verification of the SCL formula

29

perfect agreement

even in the weak-coupling regime

Inverse susceptibility Phase diagram

BSESCL

Validity has been confirmed also

in a two-orbital model

Page 30: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Effective intersite interactions

30

strong-coupling limit

square-lattice Hubbard model

Page 31: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Effective intersite interactions in the atomic limit

31“Modern” derivation of effective intersite interactions

Hubbard model

kinetic exchange interaction

in the atomic limit

Periodic Anderson model

RKKY interaction

Kondo coupling

in the atomic limit

Page 32: Spatial Correlations and Superconductivity in …otsuki/pdf/2019-SCES.pdfIntroduction Heavy fermion systems Itinerant and localized nature of f electrons 4 f-electron materials 5 CeCu

Summary

32

Microscopic derivation of heavy-fermion superconductivity

Strong-coupling formula on χ(q) within DMFT

This new formula has advantages…

- physically, easy to understand

- Effective intersite interactions

- numerically, easy to compute

- Dual fermion approach: Spatial correlations beyond DMFT

- Itinerant-Localized crossovers can lead to an exotic pairing state