奈良女子大学大学院集中講義

固体電子論特論Introduction to strongly-correlated material science: with a focus on molecular conductors

Hitoshi Seo (妹尾仁嗣)

RIKEN (理化学研究所)

レポート

・自分の研究（の計画でも）の紹介 A4 1枚程度・講義の感想。

オプション：以下の論文を1本読み、A4半分～1ページくらいのレジュメにまとめよ。1. H. Seo, H. Fukuyama, J. Phys. Soc. Jpn. 66, 1249 (1997) (mean-field)

2. H. Shimahara, J. Phys. Soc. Jpn. 58, 1735 (1989) (RPA)

3. M. Tsuchiizu, H. Yoshioka, H. Seo, 3人の共著論文であればどれでも (bosonization)

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固体中の電子の互いにクーロン相互作用のために避けあいながら量子

力学の法則に則って運動している。相互作用の効果が特に強い「強相

関電子系」では、モット転移、高温超伝導、近藤効果などの様々な異常

な振る舞いを示し活発に研究されている。この講義では、分子性導体を

中心に強相関電子系を理論的に記述するための枠組みとその基本的

性質を解説し、最近のトピックを紹介する。学部レベルの量子力学と統

計力学を前提として、初学者にも分かりやすく説明したい。

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0. イントロダクション

強相関電子系とは．遷移金属酸化物と分子性導体

1．多フェルミオン系の量子力学とバンド理論

1-1．多体系の量子力学の復習1-1-1. 行列表示1-1-2. フェルミオン系の第2量子化1-1-3. 補足：クーロン相互作用の期待値、場の演算子

1-2. 強束縛モデルとバンド理論1-2-1. 強束縛モデル1-2-2. 1次元鎖の例とバンド理論1-2-3. 2量化とパイエルス不安定性1-2-4. d-pモデルと軌道混成

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2．強相関系における金属絶縁体転移

2-1. バンド理論の破綻：モット絶縁体と電荷秩序2-1-1. ハバードモデルと反強磁性の平均場理論2-1-2. 拡張ハバードモデルと電荷秩序

2-2. 強結合からのアプローチと数値計算2-2-1. ハイゼンベルグモデルと擬スピン2-2-2. 数値計算手法とその結果の紹介（スライド）

3．超伝導

3-1. BCS理論3-1-1. 平均場理論3-1-2. 引力の起源3-1-3. BCS波動関数と変分法、および擬スピン表示

3-2. 非従来型超伝導3-2-1. 感受率と金属状態の不安定性3-2-2. 分子性導体における超伝導発現機構（スライド）

goals of this lecture

simple picture of band theory: what it can/cannot explain

what are Mott insulators, charge ordering

why do we study them

understand what (extended) Hubbard model is

general idea about what comes out from these models

connections to experimental systems = what is explained, what is not

band theory led to semiconductor technology. “success story of electron theory of solids”but…

Although the band theory of solids had been very successful in describingvarious electrical properties of materials, in 1937 Jan Hendrik de Boer andEvert Johannes Willem Verwey pointed out that a variety of transitionmetal oxides (TMOs) predicted to be conductors by band theory (becausethey have an odd number of electrons per unit cell) are insulators.[3] NevillMott and Rudolf Peierls then (also in 1937) predicted that this anomaly canbe explained by including interactions between electrons.[4]

Wikipedia : “Mott insulator”

carrier doping to Mott insulator → high-Tc superconductivity

layered perovskiteCuO2 plane

OCu

1986: discovery of high-Tc cuprates by Bednorz and Muller

Mott insulator http://for538.wmi.badw.de/projects/P4_crystal_growth

metal-insulator transition induced by doping

layered perovskiteCuO2 plane

OCu

1986: discovery of high-Tc cuprates by Bednorz and Muller

insulating state due to“strong correlation”

spin degree of freedom is left → magnetic order

Mott insulator http://for538.wmi.badw.de/projects/P4_crystal_growth

OCu

carrier doping in transition metal oxides (TMOs)

La

La2CuO4 : (La3+)2 Cu2+ (O2-)4

(3d)9

layered perovskite structure

La2-xSrxCuO4 :

(La3+)2-x(Sr2+)x Cu(2+x)+ (O2-)4

(3d)9-x

Sr

hole doping“filling control”

metal-insulator transition induced by pressure“band-width control”

metal vs insulator: how can we detect?

temperature dependence of electrical resistivity rr = R×A/l R: resistance from Ohm’s law V=RI

K. Kanoda, J. Phys. Soc. Jpn. 75 (2006) 051007

(Mott)insulator

metal(& SC at low-T)

metal-insulator transitions in TMOs

“Metal-insulator transitions”, M. Imada, A. Fujimori, Y. Tokura, Reviews of Modern Physics 70, 1049 (1998)

metal-insulator transitions in TMOs

A B:TM

O

3-D perovskite structuregeneral formula: ABO3

Mott insulator

metal

A site: Y3+ → Ca2+

B site: Ti3+ → Ti(3+d)+

(3d)1 → (3d)1-x

metal-insulator transition induced by doping

metal-insulator transitions in TMOs

metal-insulator transitionat a certain temperature Tc

metal-insulator transitions in TMOs

metal-insulator transition induced by magnetic fieldCMR manganites

molecular conductors

TM

ET

TM2X ET2X

Se

Se

Se

Se

CH3

CH3

H3C

H3C S

S

S

S

S

S

S

S

DCNQI

DCNQI2X

(TMTSF)

(BEDT-TTF)

“X”

“X”

“X”

spin-density-wave (SDW) in TMTSF2PF6

Review: Denis Jerome, Chemical Reviews 104 5565 (2004)

metal-insulator transitions in molecular conductors (1)

“Peierls instability”～ band theory

metal-insulator transition induced by pressure

“band-width controlled Mott transition”

K. Kanoda, J. Phys. Soc. Jpn. 75 (2006) 051007

Mottinsulator

metal(& SC at low-T)

metal-insulator transitions in molecular conductors (2)

k-(BEDT-TTF)2X

K. Kanoda, J. Phys. Soc. Jpn. 75 (2006) 051007

metal-insulator transitions in molecular conductors (2)

electron localizeson each dimer

“dimer Mott” insulators

k-(BEDT-TTF)2X

metal-insulator transitions in molecular conductors (3)

q-ET2RbZn(SCN)4 a-ET2I3

K.Bender et al., Mol. Cryst. Liq. Cryst. 108 (1984) 359N.Tajima et al. J. Phys. Soc. Jpn. 69 (2000) 543

H.Mori, S.Tanaka, T.Mori, Phys.Rev.B 57 (1998) 12023

Vij

“crystal of electrons”

strong correlation

charge ordering

metal-insulator transitions in molecular conductors (3)

effective model for molecular conductors

・ transfer integrals tij～ 0.1-0.3 eV

from Q. chem. / 1st principles band calc.

・ on-site Coulomb energy U～ 0.5-1 eV

・ inter-site Coulomb energy Vij ～ 0.2-0.5 eV

・ electron-phonon interaction

tij

extended Hubbard model based on molecular orbitals

＝

U

Vij

妹尾仁嗣､鹿野田一司､福山秀敏, 物理学会誌2004年11月号解説

H. Seo, C. Hotta, H. Fukuyama, Chemical Review 204 (2004), 5005

H. Seo, J. Merino, H. Yoshioka, M. Ogata, J. Phys. Soc. Jpn. 75 (2006), 051009

systematic study on molecular conductors

H = S tij ( cis† cjs + h.c. ) + U S ni↓ni↑ + S Vij ni nj

H. Kino, H. Fukuyama, J. Phys. Soc. Jpn. 64 (1995), 1877; 2726; 4523; 65 (1996), 2158

reviews

end of chapter 0

2. 強相関系における金属絶縁体転移

2-2-2 数値計算手法とその結果の紹介

☆ exact diagonalization of large matrix

・ power method

in number operator basis, state vector ～ 4N, matrix 4N×4N

what we want: ground state = eigenstate with largest absolute value

・ Lanczos methodwe don’t stock the Hamiltonian matrix, only vectorsdo recursive matrix * vector calculations matrix elements are calculated each time

=1019 for N=16

☆ variational Monte-Carlo method

cf) https://github.com/cmsi/MateriAppsLive/wiki/MaLiveTutorial

☆ DMFT = dynamical mean-field theory(outside the scope here)

theories on Mott transition: numerical

DMFTReview: Georges et alReviews of Modern Physics 68, 13 (1996)

paramagnetic insulator: reproduced.

Ground state phase diagram of 1/2-filled t-t’-U Hubbard model.

DMFT Kyung-Tremblay PRL 2006

theories on Mott transition: numerical

PIRG Morita et al JPSJ 2002 VMC Watanabe et al PRB 2008

VCA Laubach et al. PRB 2015

theories on charge ordering: numerical

Mila-Zotos, Europhysics Letters 1993Lanczos

Ejima et al, Europhysics Letters 2005DMRG (density matrix renormalization group)

Ground state phase diagram of 1-dimensional 1/4-filled t-U-V model.

theories on charge ordering: analytical

Yoshioka-Tsuchiizu-Suzumura JPSJ 2000

Bosonization

Yoshioka-Tsuchiizu-Seo JPSJ 2006

extension to q1D

Ground state phase diagram of 1-dimensional 1/4-filled t-U-V model.

theories on charge ordering: numerical

finite-T phase diagrams

Otsuka et al, JPSJ 2005, Quantum Monte-Carlo

Quasi-1D model coupled to classical phonons (like SSH model)

theories on charge ordering: numerical

ground state of 2-dimensional t-U-V model on square lattice

Hanasaki-Imada, JPSJ 2005 correlator projection method (extension of DMFT?)

Merino PRL 2007 DMFT

H = tpS ( cis† cjs + h.c. )

+ tcS ( cis† cjs + h.c. )

+ U S ni↓ni↑

+ Vp S ni nj + Vc S ni nj

tp

U

Vc

tcVp

H. Seo, J. Phys. Soc. Jpn. 69 (2000) 805

tp = -0.094 eV, tc = 0.021 eV (extended Huckel)

・mean-field calculations(U=0.7 eV, vary Vp ,Vc as parameters)

competing “stripe-type” CO patterns ～ frustration

q-ET2X

“vertical” “horizontal” “diagonal”

theories on charge ordering: numerical

3. 超伝導

3-2-2. 分子性導体における超伝導発現機構

high-Tc cuprates

TMTSF2PF6 k-(BEDT-TTF)2X

2-dimensional Hubbard model:

1/2-filling : Mott insulator, usually AF

+ doping → d-wave SC

Misawa-Imada, PRB 2014, VMC

2-dimensional Hubbard model:

1/2-filling : Mott insulator, usually AF

+ doping → d-wave SC

Misawa-Imada, PRB 2014, VMC

Quasi-1-dimensional Hubbard model

(tx >> ty >> tz) at 1/4-filling:

as increasing ty, SDW → SC

Shimahara, JPSJ 1986, RPA

Figure from H. Oike, Ph. D. thesis

superconductivity near Mott transition

similarities between high-Tc cuprates and organic SC?

T(K)hole

doping

electron

doping

AF

Mott ins.

SCSC

Mott insulator

doping rate

single-band Hubbard model

superconductivity near Mott transition

similarities between high-Tc cuprates and organic SC?

Paramagneticinsulator

Paramagneticmetal

Super--conductor

AF insulator

k-ET2X

Theory: Is SC reproduced

by 1/2-filled Hubbard model ??

K. Kanoda

Figure from H. Oike, Ph. D. thesis

Mott insulator

single-band Hubbard model

H = t S (cis†cｊs + h.c.) + t’ S (cis

†cｊs + h.c.)

+ U S ni↓ni↑

stable superconductivity (in theories) ??

anisotropic triangular lattice

𝒕𝒕𝒕′

𝑼

H. Morita, S. Watanabe, and M. Imada

JPSJ 71, 2109 (2002)

PIRG CDMFT VMC

B. Kyung and A.-M. S. Tremblay

PRL 97, 046402 (2006)

T. Watanabe et al.

PRB 77, 214505 (2008)

VCA

M. Laubach et al.

PRB 91, 245125 (2015)

AF (Mott insulator) → OK

SC → controversial (no SC phase in many works…)

cf) doped Hubbard model : stable SC phase

early works: d-wave superconductivity

H. Kino, H. Kontani, JPSJ 67, 3691 (1998); J. Schmalian, PRL 81, 4232 (1998);

H. Kondo, T. Moriya, JPSJ 67, 3695 (1998); K. Kuroki, H. Aoki, PRB 60, 3060 (1999)

K. Miyagawa et al., Chem. Rev. 104, 5635 (2004)

in experiments, ubiquitous superconductivity

k-ET

H. Kobayashi et al., Phys. Rev. B 56, R8526 (1997).

AF ?

l-(BETS)2GaBrzCl4-z

l-BETS

K. Kanoda, JPSJ 75, 051007 (2006)

• 4 molecules in a unit cell (4-band model)

• transfer integral:

• Coulomb interaction:

C. Hotta, PRB 82, 241104(R) (2010); M. Naka and S. Ishihara, JPSJ 79, 063707 (2010)

K. Kuroki et al., PRB 65, 100516(R) (2002): V=0

A. Sekine et al., PRB 87, 085133 (2013): V≠0

D. Guterding et al., PRB 94, 024515 (2016)

Strong coupling theory (effective Model)

Weak coupling theory (superconductivity)

Mean field approximation

H. Kino, H. Fukuyama, JPSJ 64 (1995) 2726; 65 (1996) 2158: V=0; H. Seo, JPSJ 69, 805 (2000): V≠0

let’ go back to the original 3/4-filled model

competition between 2 SC states

dxy (“high-Tc type”) vs extended-s+dx2-y2

H = S tij ( cis† cｊs + h.c. )

+ U S ni↓ni↑ + S Vij ni nj

• 4 molecules in a unit cell (4-band model)

• transfer integral:

• Coulomb interaction:

H = S tij ( cis† cｊs + h.c. )

+ U S ni↓ni↑ + S Vij ni nj

・Charge Jastrow factor :

・One-body part :(Slater determinant)

・Spin Jastrow factor :

up to 78th neighbor

(within ~ 4Rx radius)

Jastrow-type Trial Wave Function

variational Monte-Carlo method: ground state T=0 properties

stable superconducting phaseextended-s+dx2-y2 type symmetry

we need intersite V’s to stabilize it

3-fold CO-1(non-magnetic)

dimer-AFMott insulator

polar CO (magnetic)

1152 sites (24×24×2)VMC ground state T=0 phase diagram

paramagnetic

metal

H. Watanabe, H. Seo, S. Yunoki, J. Phys. Soc. Jpn. 86, 033703 (2017)

1st order Mott transition

(no SC along this line)

wide region where SC condensation energy is

finite along the border of charge instability

importance of charge correlation

3-fold CO-1(non-magnetic)

polar CO (magnetic)

1152 sites (24×24×2)VMC ground state T=0 phase diagram

paramagnetic

metal

H. Watanabe, H. Seo, S. Yunoki, J. Phys. Soc. Jpn. 86, 033703 (2017)

dimer-AFMott insulator