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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. 数値対角化による カゴメ格子反強磁性体の研究. 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）.

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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

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  1. 量子スピン系研究会　　新潟　　2013年3月3－4日量子スピン系研究会　　新潟　　2013年3月3－4日 数値対角化によるカゴメ格子反強磁性体の研究 中野博生A, 坂井　徹A,B A兵庫県立大, B原子力機構SPring-8

  2. 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）

  3. 2D frustrated systems • Heisenberg antiferromagnets Triangular lattice Kagome lattice Classical ground state 120 degree structure Macroscopic degeneracy (a global plane is not fixed)

  4. 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

  5. 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・2H2Olattice distortion Hiroi et al. J. Phys. Soc. Jpn. 70 (2001) 3377 • Vesignieite BaCu3V2O8(OH)2ideal ? Okamoto et al. J. Phys. Soc. Jpn. 78 (2009) 033701

  6. Methods Frustration Exotic phenomena Kagome lattice Triangular lattice Pyrochlore lattice Numerical approach Numerical diagonalization Quantum Monte Carlo Density Matrix Renormalization Group (negative sign problem) (not good for dimensions larger than one)

  7. Magnetization process of S=1/2 kagome lattice AF Hida: JPSJ 70 (2001) 3673 Honecker et al: JPCM 16(2004)S749 N=27 and 36 1/3 plateau ?

  8. Not a plateau H. Nakano and TS: JPSJ 79 (2010) 053707 Reexamination from the viewpoint of Field derivative of magnetization as a function of Anomaly at m=1/3 N=36 N=36 N=33 N=30

  9. Magnetization ramp Jump ramp Ski jump Magnetization curve of Kagome lattice AF

  10. Calculation of larger-size system is to confirm the behavior of the magnetization ramp in 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

  11. Procedure Lanczos

  12. Computational costs Parallelization with respect to d N=39, total Sz=1/2 Dimension of subspace d = 68,923,264,410 Memory cost d * 8 Bytes * at least 3 vectors ～ 1.7TB One more vector required for MPI parallelization Time cost d * # of bonds * # of iterations d increases exponentially with respect to N.

  13. Results for Rhombic Clusters N=36 N=27 N=39 Characteristics of the ramp appear clearly for N=39.

  14. Triangular lattice N=39, 36, and 27 Rhombus Typical magnetization plateau at M/Msat=1/3

  15. Comparison of c Kagome Triangular Clear difference at M/Msat=1/3 Ramp Plateau

  16. Critical exponent |m-mc|=|H-Hc|1/d d=2 1D Affleck 1990, Tsvelik 1990, TS-Takahashi 1991 d=1 2D Katoh-Imada 1994 1/3 magnetization plateau m Hc1=Hc2 ? Hc1 Hc2 H

  17. Estimation of δcf. TS and M. Takahashi: PRB 57 (1998) R8091 Numerical diagonalization of rhombic clusters for N=12, 21, 27, 36, 39 Kagome lattice Triangular lattice δ-=2 χ→∞ (1D like) δ+=1/2 χ=0 δ-=δ+=1 Conventional (2D)

  18. Hc1=Hc2 ? (Plateau vs Ramp) Triangular lattice Δ~k ⇒ Δ→1/N1/2 (N→∞) if gapless Hc1 ≠ Hc2 1/3 plateau Kagome lattice Hc1 = Hc2 No plateau

  19. Magnetization measurements

  20. Summary1 We study magnetization process at 1/3 saturation magnetization of 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) 053707 TS and H. Nakano: PRB 83 (2011) 100405(R)

  21. Anisotropic triangular latticeincluding kagome lattice J1 J2 J2=J1 :triangular lattice ⇔ J2=0 : kagome lattice

  22. Systematic series of clusters Whether or not the edges of rhombic clusters are parallel Ns=9, 36 Ns=12, 27 Ns=21

  23. Analysis of exponent d+ y=c1 - d*x for N=9 and 36 y=c2 - d*x for N=12 and 27 |m-mc| ∝ |H-Hc|1/d J2=0 J2=1 J2=0.12 The boundary may exist around J2/J1 ～ 0.1.

  24. Exponent d- The cluster of N=9 is too small. N=12 N=27 N=36 The unchanged behavior for J2/J1 >～0.4 Serious finite-size deviations in J2/J1 <～0.2

  25. Analysis of finite-size gaps (1/2) y=a + b1*x for N= 9 and 36 y=a + b2*x for N=12 and 27 a: lower bound of the gap J2=1 J2=0 J2=0.12

  26. Analysis of finite-size gaps (2/2) Gapped region

  27. Summary2 We study magnetization process at 1/3 saturation magnetization of 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

  28. Spin gap issue No spin gap is observed experimentally. 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.” mixture of even Ns and odd Ns

  29. Calculations of larger systems are required. Theoretical examination P. Sindzingre and C. Lhuillier: EPL 88(2009) 27009 Numerical diagonalizations up to 36 sites “it is impossible to distinguish between a gapless system and a system with a very small gap.”

  30. Computational costs Parallelization with respect to d N=42, total Sz=0 Dimension of subspace d = 538,257,874,440 Δ= 0.14909214 cf. A. Laeuchli cond-mat/1103.1159 Memory cost d * 8 Bytes * at least 3 vectors ～ 13TB 4 vectors ～ 20TB Time cost d * # of bonds * # of iterations d increases exponentially with respect to N.

  31. Demonstration of analysis Dimerized Square Lattice J1 a=J2/J1 J2 a=1: square lattice, LRO, gapless a=0.52337(3): critical Matsumoto et al: PRB65(2001) 014407 a=0: isolated dimers gapped

  32. Extrapolation plots a=0.4: gapped a=0.52337: critical, gapless a=1: square lattice, LRO, gapless D/J1=A+Bexp(-CNs1/2) D/J1=A+B/(Ns1/2) Matsumoto et al: PRB65(2001) 014407 Linear dispersion

  33. Classification of finite-size data odd Ns rhombic non-rhombic even Ns Important to divide data into two groups of even Ns and odd Ns. Not good to treat all the data together.

  34. Analysis of our finite-size gaps Two extrapolated results disagree from odd Ns and even Ns sequences. Feature of a gapless system

  35. Summary3 ・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)

  36. 2/3 of saturation magnetization Plateau width W Kagome Triangular W=0.11 ±0.33 W=-0.25 ±0.09 No Plateau (gapless)

  37. Critical exponents Kagome Triangular δ-=δ+=1 δ-=δ+=2

  38. 2/3 magnetization Kagome Triangular m m 2/3 2/3 H H Magnetization step No anomaly

  39. JAEA Synchrotron Radiation Research SymposiumMagnetism in Quantum Beam Science Date: 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.jp URL: 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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