数値対角化によるカゴメ格子反強磁性体の研究

量子スピン系研究会　　新潟　　 2013 年 3 月 3 － 4 日

中野博生 A, 坂井 徹 A,B

A 兵庫県立大 , B 原子力機構 SPring-8

Contents

• Introduction• S=1/2 kagome-lattice AF ・ Magnetization ramp ・ Spin gap issue ・ Other anomalies of magnetization curve• S=1 dstorted triangular-lattice AF　（昨日の中野さんの話）

2D frustrated systems

• Heisenberg antiferromagnets ji

ji SSJH,

Triangular lattice Kagome lattice

Classical ground state120 degree structure

Macroscopic degeneracy(a global plane is not fixed)

Spin liquid in frustrated systems

• S=1/2 distorted triangular-lattice AF

κ-(BEDT-TTF)2Cu2(CN)3

by Shimizu et al. 2003

• S=1 triangular-lattice AF

NiGaS (order ?)

by Nakatsuji et al. 2005

• S=1/2 kagome-lattice AF

herbersmithite, volbothite, vesignieithe

by Shores et al. 2005, Yoshida et al. 2009

S=1/2 Kagome Lattice AF

• Herbertsmithite ZnCu3(OH)6Cl2 impurities Shores et al. J. Am. Chem. Soc. 127 (2005) 13426

• Volborthite CuV2O7(OH)2 ・ 2H2O lattice distortion

Hiroi et al. J. Phys. Soc. Jpn. 70 (2001) 3377

• Vesignieite BaCu3V2O8(OH)2 　　 ideal ? Okamoto et al. J. Phys. Soc. Jpn. 78 (2009) 033701

MethodsFrustration

Kagome lattice

Triangular lattice

Pyrochlore lattice

Numerical approach

Numerical diagonalization

Quantum Monte Carlo

Density Matrix Renormalization Group

Exotic phenomena

(negative sign problem)

(not good for dimensions larger than one)

Magnetization process of S=1/2 kagome lattice AF

Hida: JPSJ 70 (2001) 3673 Honecker et al: JPCM 16(2004)S749

1/3 plateau ?

N=27 and 36

Not a plateau

Field derivative of magnetizationReexamination from the viewpoint of

as a function of

N=36N=36N=33N=30

Anomaly at m=1/3

H. Nakano and TS: JPSJ 79 (2010) 053707

Magnetization rampSki jump Jump ramp

Magnetization curve of Kagome lattice AF

Calculation of larger-size system

is to confirm the behavior of the magnetization rampin cases of larger system sizes on kagome lattice.

Magnetization process of N=39

We also compare the results of kagome lattice with the results of triangular lattice

typical magnetization plateau

Procedure

Lanczos

Computational costsN=39, total Sz=1/2

Dimension of subspace d = 68,923,264,410

Memory cost

Time cost

d * 8 Bytes * at least 3 vectors ～ 1.7TB

d * # of bonds * # of iterations

d increases exponentially with respect to N.

Parallelization with respect to d

One more vector required for MPI parallelization

Results for Rhombic Clusters

N=27N=36

N=39

Characteristics of the ramp appear clearly for N=39.

Triangular latticeN=39, 36, and 27 Rhombus

Typical magnetization plateau at M/Msat=1/3

Comparison of

Kagome Triangular

Clear difference at M/Msat=1/3

Ramp Plateau

Critical exponent

|m-mc|=|H-Hc|1/

=2 1D　 　 Affleck 1990, Tsvelik 1990, TS-Takahashi 1991　　　　　　　　D Katoh-Imada 1994

1/3 magnetization plateau

Hc1=Hc2 ?

Hc1 Hc2 H

m

Estimation of δcf. TS and M. Takahashi: PRB 57 (1998) R8091

Numerical diagonalization of rhombic clusters for N=12, 21, 27, 36, 39

Triangular lattice Kagome lattice

δ-=δ+=1 Conventional (2D) δ-=2 χ→∞ (1D like) δ+=1/2 χ=0

Hc1=Hc2 ? (Plateau vs Ramp)

Δ~k ⇒ Δ→1/N1/2 (N→∞) if gapless

Triangular lattice

Hc1 ≠ Hc2

1/3 plateau

Kagome lattice

Hc1 = Hc2

No plateau

Magnetization measurements

Summary 1We study magnetization process at 1/3 saturation magnetizationof Kagome and triangular lattice AF by numerical diagonalization.

Kagome lattice Magnetization “ramp” is established.

Triangular lattice Conventional plateau is confirmed.

References H. Nakano and TS: JPSJ 79 (2010) 053707TS and H. Nakano: PRB 83 (2011) 100405(R)

Anisotropic triangular latticeincluding kagome lattice

J1

J2

J2=J1 : triangular lattice ⇔ J2=0 : kagome lattice

Systematic series of clusters

Ns=9, 36 Ns=12, 27

Ns=21

Whether or not the edges of rhombic clusters are parallel

Analysis of exponent +

The boundary may exist around J2/J1 ～ 0.1.

J2=0

J2=0.12

J2=1

|m-mc| |H-H∝ c|1/ y=c1 - *x for N=9 and 36 y=c2 - *x for N=12 and 27

Exponent -

The unchanged behavior for J2/J1 > ～ 0.4Serious finite-size deviations in J2/J1 < ～ 0.2

The cluster of N=9 is too small.

N=12N=27N=36

Analysis of finite-size gaps (1/2)

J2=0

J2=0.12

J2=1

y=a + b1*x for N= 9 and 36y=a + b2*x for N=12 and 27

a: lower bound of the gap

Analysis of finite-size gaps (2/2)

Gapped region

Summary2 　We study magnetization process at 1/3 saturation magnetizationof Kagome and triangular lattice AF by numerical diagonalization.

Kagome lattice: Magnetization “ramp” is established.Triangular lattice: Conventional plateau is confirmed.

Study on an anisotropic triangular lattice including kagome lattice

Existence of a boundary between the ramp (kagome) and the plateau (triangular) J2/J1 ~ 0.1

Spin gap issue

Mendels & Bert: JPSJ 79 (2010) 011001

“The central question of whether the small spin-gap ＜～ J/20 survives or vanishes at the thermodynamic limit is still pending.”

No spin gap is observed experimentally.

mixture of even Ns and odd Ns

Theoretical examination

“it is impossible to distinguish between a gapless system and a system with a very small gap.”

P. Sindzingre and C. Lhuillier: EPL 88(2009) 27009

Numerical diagonalizations up to 36 sites

Calculations of larger systemsare required.

Computational costsN=42, total Sz=0

Dimension of subspace d = 538,257,874,440

Memory cost

Time cost

d * 8 Bytes * at least 3 vectors ～ 13TB

d * # of bonds * # of iterations

d increases exponentially with respect to N.

Parallelization with respect to d

4 vectors ～ 20TB

Δ= 0.14909214 cf. A. Laeuchli cond-mat/1103.1159

Demonstration of analysis

J2

J1 =J2/J1

=1: square lattice, LRO, gapless

=0: isolated dimers gapped

=0.52337(3): critical Matsumoto et al:PRB65(2001) 014407

Dimerized Square Lattice

Extrapolation plots=0.4: gapped=0.52337: critical, gapless=1: square lattice, LRO, gapless

/J1=A+Bexp(-CNs1/2) /J1=A+B/(Ns

1/2)Matsumoto et al:PRB65(2001) 014407 Linear dispersion

Classification of finite-size data

odd Ns

even Ns

rhombic

non-rhombic

Important to divide data into two groups of even Ns and odd Ns.

Not good to treat all the data together.

Analysis of our finite-size gaps

Two extrapolated results disagree from odd Ns and even Ns sequences. Feature of a gapless system

Summary 3

・ Kagome lattice AF has no spin gap.

Gapless spin liquid

H. Nakano and TS: JPSJ 79 (2010) 053707 (arXiv:1004.2528)TS and H. Nakano: PRB 83 (2011) 100405(R) (arXiv:1102.3486)H. Nakano and TS: JPSJ 80 (2011) 053704 (arXiv: 1103.5829)

2/3 of saturation magnetizationPlateau width W

Kagome Triangular

W=0.11 ±0.33 W=-0.25 ±0.09

No Plateau (gapless)

Critical exponentsKagome Triangular

δ-=δ+=1 δ-=δ+=2

2/3 magnetization

Kagome Triangular

H H

mm

2/3 2/3

Magnetization step No anomaly

JAEA Synchrotron Radiation Research Symposium

Magnetism in Quantum Beam ScienceDate: 11th (Mon.) - 13th (Wed.), March, 2013

Place: SPring-8 (Hyogo, Japan)

Scope: The symposium is devoted to magnetism in quantum spin systems, frustrated magnets, strongly correlated

electrons, high-temperature superconductors, multifunctional materials etc. We focus mainly on research using quantum beams: synchrotron X-rays, neutrons, muons, as well as some magnetic resonance techniques. Both experimental and theoretical work will be highlighted. The presentations consist ofinvited or contributed talks and posters.

Organizing Committee: T. Sakai (JAEA) K. Kakurai (JAEA) J. Mizuki (JAEA) Y. Katayama (JAEA) H. Konishi (JAEA) T. Inami (JAEA) K. Tsutsui (JAEA) H. Nojiri (Tohoku Univ.) H. Kageyama (Kyoto Univ.) T. Ziman (ILL) Contact: Toru Sakai (JAEA, SPring-8) E-mail: sakai@spring8.or.jpURL: http://cmt.spring8.or.jp/workshop/workshop-20130311.shtml

Invited Speakers:M. Boehm (ILL) S. Miyashita (U. Tokyo) S. Fujiyama (RIKEN)C. Detlefs (ESRF, France) H. Nakano (U. Hyogo) S. Fujimori (JAEA)S. Dunsiger (Tech. U. Munich) E. Rodriguez (U. Maryland) T. Watanuki (JAEA)Z. Hiroi (U. Tokyo) N. Shannon (OIST) Y. Sidis (CEA, Saclay)S. Ishihara (Tohoku U.) P. Steffens (ILL) H. Nojiri (Tohoku U.)Z. Islam (APS, USA)* H. Tanaka (TIT) S. Sebastian (Cambridge)T. Shimokawa (Kobe U) M. Nakamura (TIT)H. Kageyama (Kyoto U.) E. Torikai (Yamanashi U.)H. Kobayashi(U. Hyogo) A. Zheludev (ETH, Zurich)M. Matsuda (ORNL) K. Tsutui (JAEA)Ph. Mendels (U. Paris-Sud XI) T. Matsumura (Hiroshima U.)