数値対角化による カゴメ格子反強磁性体の研究
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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.)
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