2024 Feb 20-22 @RIKEN Kobe

Relativistic corrections tostrongly correlated electron systems
Department of Physics, Saitama University
Shintaro Hoshino
Acknowledgement :Michi-To Suzuki (Tohoku Univ), Hiroaki Ikeda (Ritsumeikan Univ), Tatsuya Miki (D1,Saitama Univ),
H Kusunose (Meiji Univ), MJS Yang (Saitama Univ), S Michimura (Saitama Univ),
M Fukuda (ISSP), M Senami (Kyoto Univ), M Hirayama (U Tokyo), T Naito (RIKEN)

Outline
1. Introduction
2. Multipoles in strongly correlated electron systems
3. Spherical representation of Coulomb-Breit interaction:
Interaction among multipoles (Generalized Slater-Condon parameters)
4. SummaryRelativistic correction in condensed matter
Conventional
relativistic correction:
Spin-orbit coupling・Materials with large-atomic number elements:
ex. 5d electrons, f electrons: J-multiplet
・Transition metals including ３ｄ series, organic conductors:
ex. Dzyaloshinskii-Moriya interaction⇒ small energy scale but can be relevant

For elements with large atomic number, higher-order relativistic
corrections can be relevant for low-energy behaviors?

Three relativistic
corrections

A. Physical quantities (ex. electron chirality)

SH, M-T Suzuki, H Ikeda,
Phys. Rev. Lett. (2023)

B. One-body Hamiltonians (1/c3 corrections)SH, T. Miki, M-T Suzuki, H Ikeda,in preparation☞ C. Interactions (Breit interaction)

Today’s focus

SH, arXiv (2023)

Outline
1. Introduction
2. Multipoles in strongly correlated electron systems
3. Spherical representation of Coulomb-Breit interaction:
Interaction among multipoles (Generalized Slater-Condon parameters)
4. Summarystrongly correlated electron systems

Multipole in SCES

☞ Classification scheme for nearly-localized d- and f-electron’s degrees of freedom.
☞ Candidate order parameter for exotic ordered states.
☞ Cluster multipoles, bond orders, multipole expansion in k-space, …
KI Kugel and DI Khomskii, Sov. Phys. JETP Lett. (1972)
FJ Ohkawa, JPSJ (1983)
CeB6
R Shiina, H Shiba, P Thalmeier, JPSJ (1997)
Ce1-xLaxB6
Y Kuramoto, H Kusunose, JPSJ (2000)
NpO2
P Santini, G Amoretti, PRL (2000)
A Kiss and Y Kuramoto, JPSJ (2005)
PrRu4P12
T Takimoto, JPSJ (2005)
H Kusunose, JPSJ (2008)
K Haule and G Kotliar, Nat. Phys. (2009)
URu2Si2
H Ikeda, MT Suzuki, R Arita, T Takimoto, T Shibauchi, Y Matsuda, Nat. Phys. (2012)
MT Suzuki, T Koretsune, M Ochi, R Arita, PRB (2017)
S Hayami, M Yatsushiro, Y Yanagi, H, Kusunose, PRB (2018)
and more …

Y Kuramoto, PTP Suppl. (2008)

Electronic degrees of freedom☞ Spin degrees of freedom

𝟏𝟏𝒔𝒔 =𝟐𝟐

Operators
Charge (TR+)(in terms of multipole)
𝟏𝟏Spin (TR-)

𝟏𝟏

𝑺𝑺 = × = 𝟎𝟎 + 𝟏𝟏 (monopole + dipole)
𝟐𝟐
𝟐𝟐“rank”☞ Orbital degrees of freedom (ex. p electron)ℓ = 𝟏𝟏

Classification of operators (|𝑚𝑚⟩⟨𝑚𝑚′ |)Figs from H Kusunose & Y Kuramoto, KOTAI BUTSURI (2006)

𝑳𝑳 = 𝟏𝟏 × 𝟏𝟏 = 𝟎𝟎 + 𝟏𝟏 + 𝟐𝟐 (monopole + dipole + quadrupole)

Charge (TR+), Magnetic orbital (TR-), Electric orbital (TR+)

Even/Odd of rank corresponds to Even/Odd of time-reversal.

Spin-orbit coupled case
𝐽𝐽𝐽𝐽†
☞ Classification of 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 ∼ 𝑐𝑐𝑚𝑚𝑚𝑚
𝑐𝑐𝑚𝑚′ 𝜎𝜎′ with L-S coupled basis (𝑇𝑇𝐿𝐿𝐿𝐿 )(Spatial inversion is always even for a fixed ℓ)

Example 1: 𝐿𝐿 = 1 (TR=odd) and 𝑆𝑆 = 1 (TR=odd)

Rank 𝑝𝑝 = 2𝑞𝑞 or 2𝑞𝑞 + 1
SI: Spatial Inversion / TR: Time-reversal

𝐽𝐽 = 1 × 1 = 0+ + 1+ + 2+

Electric toroidal (ET) dipole (Odd-rank,TR=even)

Example 2: 𝐿𝐿 = 2 (TR=even) and 𝑆𝑆 = 1 (TR=odd)

𝐽𝐽 = 2 × 1 = 1− + 2− + 3−

Magnetic toroidal (MT) quadrupole (Even-rank,TR=odd)

Multipole operator

Examples of dipoles:
M dipole: 𝑳𝑳, 𝑺𝑺
ET dipole: 𝑮𝑮 = 𝑳𝑳 × 𝑺𝑺・Y Wang, H Weng, L Fu, X Dai, PRL (2017)
・H Kusunose, R Oiwa, S Hayami, JPSJ (2020)
・N Chikano, SH, H Shinaoka, PRB (2021)
・S Hayami, R Oiwa, H Kusunose, JPSJ (2022)
・SH, MT Suzuki, H Ikeda, PRL (2023)

Outline
1. Introduction
2. Multipoles in strongly correlated electron systems
3. Spherical representation of Coulomb-Breit interaction:
Interaction among multipoles (Generalized Slater-Condon parameters)
4. SummaryRelativistic correction to SCES: Motivation
Exotic behaviors in uranium-based materials:☞ Hidden order (URu2Si2)☞ Spin-triplet superconductor candidates (UBe13, UPt3, UGe2, URhGe, UCoGe, UTe2)
Why uranium (atomic number = 92) material is so special?
☞ Large atomic number  Relevance of relativistic corrections☞ Localized f orbital  Relevance of Coulomb interactions

Possible relevance of relativistic correction to Coulomb interaction
Most general
Hamiltonian(Coulomb gauge)

Coulomb-Breit interaction
Tracing out photons and keep leading-order term in 1/c expansion(Coulomb)(T Itoh, Rev. Mod. Phys. (1965): reduced Compton length)(Breit)Cf. in quantum chemistry / atomic physics contextEx.) E Eliav et al., Nucl. Phys. A (2015)
T Naito, R Akashi, H Liang, S Tsuneyuki, J. Phys. B (2020)[𝒓𝒓 = 𝒓𝒓1 − 𝒓𝒓2 , 𝐴𝐴1 = 𝐴𝐴(𝒓𝒓1 ) ]SCES: Slater-Condon parameters (F0,F2,…) describe
𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 effectively for conventional Coulomb part.
How about for relativistic corrections?

SH, arXiv (2023)

Stratonovich-Hubbard transformation
Effective Lagrangian (𝛼𝛼 = 1,2: transverse photons, 𝛼𝛼 = 0: “scalar photon”)

Spherical wave expansion of photons*Berestetskii-Lifshitz-Pitaevski, textbook*Igi-Kawai, textbook (Japanese)(𝜆𝜆 = 0,1)𝜆𝜆 = 1: electric photon: parity −1 𝐽𝐽
𝜆𝜆 = 0: magnetic photon : parity −1 𝐽𝐽+1
𝜆𝜆 = −1: scalar photon: parity −1 𝐽𝐽(1,0,−1)(Vector spherical harmonics 𝒀𝒀𝐽𝐽𝐽𝐽)SH, arXiv (2023)

Multipole representation
Angular momentum tensor (for electrons with fixed 𝑛𝑛, 𝑙𝑙)

J=even; SI=+; TR=+: Electric
J=odd; SI=+; TR=-: Magnetic
J=odd; SI=+; TR=+: Electric Toroidal
J=even; SI=+; TR=-: Magnetic Toroidal
magneticphoton

Interaction in spherical basis

electricphoton[Figs from N Papasimakis et al., Nat. Mater. (2016)]

By parity, magnetic photons couple to magnetic multipoles,
and electric photons couple to magnetic toroidal multipoles.

SH, arXiv (2023)

Generalized Slater-Condon parameters
Cf. No odd-rank multipoles inconventional Coulomb interaction
even in solids (cRPA)

S Iimura, M Hirayama SH, PRB Lett (2021)

Rough estimation:☞ Order of Breit itnerction
𝐻𝐻𝐵𝐵 ∼ 𝑍𝑍𝑍𝑍 2 𝐻𝐻𝐶𝐶

☞ Hund’s coupling for uranium:
𝐽𝐽𝐻𝐻 ∼ 0.5 eVElec. polarizationin Coulomb

Mag. multipoles- Mag. photons

Mag. Toroidal multipoles- Elec. photons(Elec. Toroidal multipoles appear in higher-order Coulomb)☞ Effective Nuclear charge forf electrons: 𝑍𝑍 ∼ 30

∴ 𝐻𝐻𝐵𝐵 ∼ 0.1 × 𝐽𝐽𝐻𝐻 𝑍𝑍𝑍𝑍 2 ∼ 100 K

Cf.
Heavy-electronstate: effective
band width
becomes 102 103 times smaller
than usual metal.

SummarySH, arXiv (2023)

Relativistic corrections to interactions in condensed matter☞ Possible relevance of Coulomb-Breit interactions in SCES.
☞ Atomic representation, i.e. generalized Slater-Condon parameters, is derived, which canbe combined with Hubbard model, Anderson lattice, LDA+U, LDA+DMFT, etc.
☞ Foundation for further exploring low-energy physics (ex. Amperean pairing).