HΦ講習会@柏の葉キャンパス

HΦを利用した研究例の紹介
山地 洋平

東京大学大学院工学系物理工学専攻

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Proximity to quantum spin liquids
Na2IrO3: Mott insulator in vicinity of QSLSpin excitations by shifted Krylov subspace methods
Extension of Kitaevʼs spin liquid	Proximity toQuantum Spin LiquidQuantum Liquid
Typical examples of quantum liquids:
Liquid helium 3, metals
Landauʼs Fermi liquid Theory
L. D. Landau, Zh. Eksperim. I Teor. Fiz 30, 1058 (1956).

Spontaneous symmetry breaking (SSB) in Fermi liquid:Superﬂuidity, superconductivity, magnetic order

Spins in Mott insulators
R. Peierls, N. F. Mott (1937)

Example:
La2CuO4

Search for quantum spin liquid

P. W. Anderson, Mater. Res. Bull. 8, 153 (1973)

No universal basis for SSB in spins

Quantum Spin Liquid in Frustrated Magnets
RVB

P. W. Anderson, Mater. Res. Bull. 8, 153 (1973).
P. Fazekas & P. W. Anderson, Philos. Mag. 30, 423 (1974).

・A v.w.f. for S=1/2 Heisenberg model

Review: L. Balents, Nature 464, 199 (2010).

No spontaneous symmetry breakings at T =0
Fractionalization→・Entropy remaining at low temperatures (〜O(10-1-10-2J))・Continuum in spin excitations

Thermal Excitations

J. Nasu, M. Udagawa, & Y. Motome,
Phys. Rev. B 92, 115122 (2015)

・Heat capacity ofKitaevʼs spin liquid
10-4

10-3

10-2

T/4K

10-1

100

10

Y. Yamaji & T. Misawa, unpublished

0.6

・Typical candidates:
J1-J2 and kagome(32 site)

0.5
0.4
0.3

kagome(36 site)

0.2
0.1
00.01

0.1

1

10100

Spin Excitations

J. Knolle, D. L. Kovrizhin, J. T. Chalker, & R. Moessner,
Phys. Rev. Lett. 112, 207203 (2014).

・S(Q,ω) at T=0 ofKitaevʼs spin liquid・Recent candidate(but not likely):
YbMgGaO4

YbMgGaO4: Yb triangular latticeY. Shen, et al., Nature 540, 559 (2016).

Kitaev ModelKitaev Model
Exactly solvable

Kitaev, Annals Phys. 321, 2 (2006)

Gapless spin liquid:-No long-range order in 2D-Fermionic excitation (Majorana)

By a→N(TopGaple-N-Fe

An Example of Frustration in Magnets
Honeycomb lattice of S=1/2

An Example of Frustration in Magnets
Honeycomb lattice of S=1/2

At least in UHF level,
3x2N/2 degenerated states !

An Example of Frustration in Magnets
Honeycomb lattice of S=1/2

At least in UHF level,
3x2N/2 degenerated states !
No long range order @ T=0
Quantum spin liquids
Kitaev, Annals Phys. 321, 2 (2006)	

Anisotropic Exchange Couplingsin Honeycomb Iridates
A2IrO3(111)

Jeﬀ=1/2 doublet

J. Chaloupka, G. Jackeli, and G.Khaliullin,
Phys. Rev. Lett. 105, 027204 (2010)

Spin-orbit couplings →Spin and lattice are locked each other
Ideal octahedron(001) mirror symmetryRau-Kee 2014

C3 around (111) axis

d-p-d hopping + Hund →Kitaev K <0Ab Initio Spin Hamiltonian
Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, & M. Imada, Phys. Rev. Lett. 113, 107201 (2014).

bZ-bondY-bond X-bonda	Speciﬁc Heat of Na2IrO3 by TPQ

Y. Singh and P. Gegenwart,
PRB 82, 064412 (2010).

K. Mehlawat, A. Thamizhavel & Y. Singh,
Phys. Rev. B 95, 144406 (2017).

Spin Excitations	Simulating Spectroscopy Measurements
Linear response of ground state
Greenʼs function

Excitation spectrum

Example of perturbation & response:

Magnetization of spins under magnetic ﬁelds	Shifted Krylov Subspace Method for
Excitation Spectra
Greenʼs function
Liner equations←Solvable by Shifted Krylov subspace method
A. Frommer (1995, 2003)

T. Sogabe, T. Hoshi, S. L. Zhang, and T. Fujiwara, A numerical method for
calculating the Greenʼs function arising from electronic structure theory,
In Frontiers of Computational Science. pp.189-195, 2007.

→Stable and controlled convergence	Shifted Krylov Subspace Method for
Excitation Spectra-Shift invariance of Krylov subspace-Collinear residuals

A. Frommer, Computing 70, 87 (2003).

Seed switch

S. Yamamoto, T. Sogabe, T. Hoshi, S.-L. Zhang, & T. Fujiwara,
J. Phys. Soc. Jpn. 77, 114713 (2008).

Library Kω (released) by Dr. Kawamura (ISSP)

2-Norm of Residual Vector
Strong parameter dependence
in convergence of S(Q,ω) with sBiCG
1000
100

Kitaev model

Na2IrO3

101
0.1
0.01
0.001
0.00011e-05
0Kitaev
ab initio
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000	

Dynamical Spin Structure Factors(a)0.30.25zigzag

*0.2
0.15
0.10.05(b)0(b)

01020(c)1*0.80.60.40.2030051015204050

Application of HΦ:Exploration ofA “New” Spin LiquidExploration of A “New” Spin LiquidK-Γ modelcf.) Classical macroscopic degeneracy
I. Rousochatzakis & N. B. Parkins,
arXiv:1610.08463.

A. Catuneau, Y. Yamaji, G. Wachtel, H.-Y. Kee, & Y.-B. Kim,
arXiv:1701.07837.

Kim & H.-Y. Kee,
α-RuCl3 H.-S.
Phys. Rev. B 93, 155143 (2016).

α-RuCl3H.-S. Kim & H.-Y. Kee,Phys. Rev. B 93, 155143 (2016).

One-body: ab initio TBCoulomb: parameters

Extension of Kitaevʼs Spin LiquidKitaev-Γ model

A. Catuneau, Y. Yamaji, G. Wachtel, H.-Y. Kee, & Y.-B. Kim,
arXiv:1701.07837.
0.002

-0.20
-0.25
-0.30
-0.35

QSL
00.25

0.0
0.5

0.75

-0.002
1

QSL adiabatically connected to Kitaev limit

Heat Capacity

8642
00.01

0.1

110

Plateau of Entropy

10.8
0.6
0.4
0.2
00.01

0.1

110

Decomposition of Entropy
10.8
0.6
0.4
0.2
00.01

0.1

1A new plateau appears

101210864
200S(Q,ω) by Kω
-Local excitations
Short range
correlations-Continuum

1210864200

*

0.2
0.15
0.1
0.05

00-0.05-0.1-0.15

1210864
200

*

0.5

1

1.5

2

2.5

3

*

0.5

1

1.5

2

2.5

3

*

0.5

1

1.5

2

2.5

3

INS for α-RuCl3
A. Banerjee et al.,arXiv:1609.00103.

Relevance of K-Γ-J3 to α-RuCl3-Easy plane anisotropy-Transition from zigzag to forced FM at 〜10TJ. A. Sears, et al., Phys. Rev. B 91, 144420 (2015).
M. Majumder, et al., Phys. Rev. B 91, 180401 (2015).
R. D. Johnson, et al, Phys. Rev. B 92, 235119 (2015).
50

1400.8

300.6

200.4

100.2

000.20.4

0.6

0.8

10Energy unit 〜 10 meVHow to Simulate K-Γ-J3 Model by HΦ

Nearest neighbor
3rd neighborHow to Simulate K-Γ-J3 Model
model = "SpinGC”
method = "TPQ"
lattice = "Honeycomb”
a0w = 2a0l = 2a1w = 4
a1l = -2J0x = -0.80901699437
J0yz = 0.58778525229
J0zy = 0.58778525229
J1zx = 0.58778525229
J1y = -0.80901699437
J1xz = 0.58778525229
J2xy = 0.58778525229
J2yx = 0.58778525229
J2z = -0.80901699437
h = 0.07071067811
Gamma = -0.07071067811
2S=1

Add Exchange)
and Ising byExpert mode

Heat Capacity of K-Γ-J3 Model876

B=0.1, 28 samples
B=0.1, 56 samples
B=0.1, 112 samples
B=0.0, 28 samples
B=0.0, 56 samples
B=0.0, 112 samples

Error bars:
B=0.1, 112samples
Error bars:
B=0.0, 112samples

5432100.010.1110Typicality Approach
Finite-temperature pure state

M. Imada and M. Takahashi, J. Phys. Soc. Jpn. 55, 3354 (1986).
P. de Vries and H. De Raedt, Phys. Rev. B 47, 7929 (1993).
A. Hams and H. De Raedt, Phys. Rev. E 62, 4365 (2000).
S. Sugiura and A. Shimizu, Phys. Rev. Lett. 111, 010401 (2013).

Heat Capacity of K-Γ-J3 Model876

B=0.1, 28 samples
B=0.1, 56 samples
B=0.1, 112 samples
B=0.0, 28 samples
B=0.0, 56 samplesB=0.0, 112 samples

Error bars:
B=0.1, 112samples
Error bars:
B=0.0, 112samples

54
32
1
00.01

0.1

110

まとめHΦによる量子スピン液体近傍の熱励起とスピン励起
自発的対称性の破れのアリバイにかわる証拠探し-低温まで残るエントロピー-励起の局在性と連続スペクトル・Na2IrO3
低温まで残る比熱
局在的連続スペクトルからスピン波的励起へ
・キタエフ-Γ模型における拡がった量子スピン液体相
エントロピー・プラトーのクロスオーバー
頑健な局在的連続スペクトル
11

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

1

10

100

1000

00.01

0.1

110

Collaborators
Dr. Yusuke Nomura
Department of Applied Physics,
The University of Tokyo

Prof. Takafumi Suzuki
Graduate School of Engineering,
University of HyogoDr. Moyuru Kurita
NECMr. Takuto Yamada
Graduate School of Engineering,
University of Hyogo

Dr. Ryotaro Arita
CEMS, RIKEN

Prof. Sei-ichiro Suga
Graduate School of Engineering,
University of Hyogo

Prof. Masatoshi Imada
Department of Applied Physics,
The University of Tokyo

Prof. Naoki Kawashima
The Institute for Solid State Physics,
The University of Tokyo

Mr. Andrei Catuneau, Dr. Gideon Wachtel, Prof. Hae-Young,& Prof. Yong-Baek
from University of Tronto

Ackn

Dr. TaDr. KaDr. MiProf. S