on Frustrated Magnets Toru SAKAI 1,2 , Hiroki Nakano 1 , Kiyomi - PowerPoint PPT Presentation

Computational Approach to Quantum Many-body Problems, ISSP University of Tokyo on July 16 - August 8, 2019 Numerical Diagonalization Study on Frustrated Magnets Toru SAKAI 1,2 , Hiroki Nakano 1 , Kiyomi Okamoto 1 , Takashi Tonegawa 3,4 1 Graduate School of Material Science University of Hyogo, Japan 2 National Institutes for Quantum and Radiological Science and Technology(QST) SPring-8, Japan 3 Professor Emeritus, Kobe University, Japan 4 Osaka Prefecture University, Japan K-computer SPring-8

Contents • Spin gap issue of the kagome-lattice AF • Plaquette singlet phase of the Shastry- Sutherland system • Magnetization plateau of S=2 AF chain with anisotropies

Spin gap issue of kagome-lattice AF Candidates of Quantum Spin Fluid 2D frustrated systems      • S=1/2 Heisenberg antiferromagnets H J S i S j , i j Triangular lattice Kagome lattice 120 degree LRO No (conventional) LRO

Kagome lattice Itiro Syôzi: Statistics of Kagomé Lattice, PTP 6 (1951)306 kagome Corner sharing triangles

S=1/2 Kagome Lattice AF • Herbertsmithite ZnCu 3 (OH) 6 Cl 2 impurities Shores et al. J. Am. Chem. Soc. 127 (2005) 13426 • Volborthite CuV 2 O 7 (OH) 2 ・ 2H 2 O lattice distortion Hiroi et al. J. Phys. Soc. Jpn. 70 (2001) 3377 • Vesignieite BaCu 3 V 2 O 8 (OH) 2 ideal ? Okamoto et al. J. Phys. Soc. Jpn. 78 (2009) 033701

Spin gap issue of kagome-lattice AF Gapped theories Valence Bond Crystal (VBC) MERA[Vidal] Z 2 Topological Spin Liquid [Sachdev (1992)] DMRG [White (2011)] Chiral Liquid [Messio et al. PRL 108 (2012) 207204] Gapless theories U(1) Dirac Spin Liquid[Ran et al. PRL 98 (2007) 117205] Variational method [ Iqbal, Poilblanc, Becca, PRB 89 (2014) 020407 ] DMRG [He et al. PRX 7 (2017) 031020]

Single crystal of herbertsmithite T. Han, S. Chu, Y. S. Lee: PRL 108 (2012) 157202 ZnCu 3 (OH) 6 Cl 2 Inelastic neutron scattering: Spin gap < J/70 Gapless M. Fu, T. Imai, T.-H. Han, Y. S. Lee: Science 350 (2015) 655 NMR : Gapped

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

Computational costs 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 . Parallelization with respect to d

Numerical diagonalizations of finite-size clusters up to N s =42 Important to divide data into two groups of odd N s even N s and odd N s . rhombic Not good to treat all the data together. non-rhombic even N s

Analysis of our finite-size gaps H. Nakano and TS: JPSJ 80 (2011) 053704 (arXiv: 1103.5829) D /J=A+Bexp(-CN s D /J=A+B/(N s 1/2 ) 1/2 ) Two extrapolated results disagree gapless is better ! from odd N s and even N s sequences.

Gapless or Gapped ？ Susceptibility analysis Field derivative of magnetization at M=0 as a function of

    ˆ     g z H J S S H S (g μ B =1) i j B j i , j j ↓ ↓ - HM E(M) M= Σ j S j z E(M)/N ~ ε(m) (N→∞) m=M/N E(M+1)-E(M) ~ [ ε’(m) + ε’’(m)/2N + ・・・ ]/S (E(M+1)-E(M))-(E(M)-E(M-1)) ~ ε’’(m)/NS 2 m=0 ↓ 2 Δ ~ ε’’(m)/NS 2 χ = dm/dh =S/ ε’’(m)→0 for Δ≠0 N →∞ =1/2ΔNS =1/ΔN

Demonstration of analysis Dimerized Square Lattice J 1 a = J 2 / J 1 J 2 a =1: square lattice, LRO, gapless a =0.52337(3): critical Matsumoto et al: PRB 65 (2001) 014407 a =0: isolated dimers gapped

Magnetization processes Gapless Gapped

Differential susceptibility vs. M Gapped Gapless

Size dependence of c at M=0 Gapless Gapped

Kagome-lattice Heisenberg AF

Kagome lattice AF Differential susceptibility vs. M N s =39 N s =36 N s =42

Size dependence of c at M=0 N s =30 N s =12 N s =36 N s =18 N s =24 N s =42 χ→ finite (N s →∞) ⇒ Gapless

Triangular lattice AF Size dependence of c N s =18 N s =30 N s =12 N s =36 N s =24 Consistent with gapless feature of triangular lattice AF

Conclusion • “Susceptibility analysis” confirmed that S=1/2 kagome-lattice AF is gapless, as well as S=1/2 triangular-lattice AF. • In order to confirm it, we should do the numerical diagonalization of larger-size clusters than 42 spins. TS and H. Nakano: Physica B 536 (2018) 85; arXiv:1801.04458

Plaquette singlet phase of the Shastry-Sutherland system • B. S. Shastry and B. Sutherland, Physica 108B , 1069 (1981): Exact dimer ground state • H. Kageyama et al. Phys. Rev. Lett. 82, 3168 (1999) ： Material SrCu2(BO3)2

Quantum phase transition 1 • S. Miyahara and K. Ueda, Phys. Rev. Lett. 82, 3701 (1999) J’/J < 0.69 : dimer J’/J >0.69 : Neel order

Quantum phase transition 2 • A. Koga and N. Kawakami: Phys. Rev .Lett. 84 (2000) 4461 Plaquette singlet phase

Experimental studies Zayed et al., Sakurai et al.: JPSJ 87 , 033701 (2018) Nature Physics, 13, 962 (2017) High-pressure and high-field ESR Neutron scattering measurement a = J’/J a = 0.64 plaquette Néel or dimer a c1 = 0.68 a c2 = 0.86 Pressure

Finite-size clusters N=40 N=36

Finite-size energy difference N=36 N=40 r=J’/J

Analysis of the finite-size gap S=1 S=0 N=40 N=36 Neel order Plaquette singlet

Analysis of the finite-size gap S=1 New gap phase S=0 N=40 N=36 Neel order Plaquette singlet

Correlation functions Neel N=36 N=40 New gap Plaquette Next-nearest neighbor correlation

Summary • Shastry-Sutherland model ： varring J’/J • N=36 、 40 Numerical diagonalization • Spin gap and spin correlation functions Possible new gap phase J’/J New gap Exact Neel Plaquette phase order dimer singlet 0.675 0.70 0.76

Magnetization plateau of S=2 AF chain Haldane gap Nobel prize 2016 Haldane conjecture (1983) Low-lying energy spectrum 1D quantum antiferromagnet S=1/2, 3/2, 5/2 … Half-odd-integer spins ⇒ Gapless at T=0 Integer spins S=1, 2, 3, … ⇒ Gap (Haldane gap) at T=0

Mechanism of Haldane gap Valence Bond Solid (VBS) Affleck-Kennedy-Lieb-Tasaki: Phys. Rev. Lett. 59 (1987)799 S=1 ⇒ triplet pair of two S=1/2 s singlet triplet S=1

Magnetization plateau Field-induced spin gap Oshikawa-Yamanaka-Affleck Phys. Rev. Lett. 78 (1997) 1984 Sakai-Takahashi: Phys. Rev. B 57 (1998) R3201 m Field-induced spin gap Field-induced spin liquid →AF order Spin gap 0 H

Mechanism of magnetization plateau Oshikawa-Yamanaka-Affleck Phys. Rev. Lett. 78 (1997) 1984 S=3/2 antiferromagnetic chain Field-induced Valence Bond Solid S=3/2 singlet

Necessary condition of plateau • Oshikawa-Yamanaka-Affleck ： PRL 78 (1997) 1984  m  ( ) integer Q S Q: Periodicity of ground state S: Spin quantum number of unit cell m: magnetization of unit cell • S=3/2 AF chain m=1/2 (1/3 of saturation) possible plateau

1/3 plateau of S=3/2 AF chain • TS and Takahashi: PRB 57 (1998)R3201   ˆ        z 2 z ( ) H J S S D S H S  1 j j j j j j j Single-ion anisotropy D +3/2 Sz=+1/2 ⇒ +3/2 gap +1/2 -1/2 -3/2 Phenomenological renormalization Dc ~ 0.9

Two mechanisms of 1/3 plateau Kitazawa and Okamoto: PRB 62 (2000) 940 • ハルデンギャップと同じＶＢＳ • Large-D タイプ

Level spectroscopy • Kitazawa and Okamoto: PRB 62 (2000) 940 １ / ３ plateau of S=3/2 AF chain + D No plateau Haldane Large-D 0 0.387J 0.943J D

S=2 AF chain Very small Haldane gap Numerical estimations ・ Wan-Qin-Yu PRB 60 (1999) 14529 : DMRG Δ= 0.0876 ± 0.0013 ・ Todo-KatoPRL 87 (2001) 047203 : QMC Δ= 0.08917 ± 0.00004 ・ Ueda-Kusakabe PRB 84 (2011) 054446 : DMRG Δ= 0.0891623 ± 0.0000009 Present work : H. Nakano and TS JPSJ 87 (2018) 105002 Numerical diagonalization (TBC N=20) Wynn’s epsilon algorithm Δ= 0.0890 ± 0.0007

Symmetry protected topological phase Pollmann-Turner-Berg-Oshikawa PRB 81 (2010) 064439 • SPT phase of S=2 AF chain + anisotropies Tonegawa-Okamoto-Nakano-Sakai-Nomura-Kaburagi JPSJ 80 (2011) 043001 Numerical diagonalization + Level spectroscopy Haldane phase Intermediate-D phase Large-D phase

SPT phase in magnetic field ? • S=2 AF chain Anisotropies λ : XXZ anisotropy D : single-ion anisotropy ½ magnetization plateau ?

S=2 AF chain Two mechanisms of 1/2 plateau • Haldane mechanism : SPT phase • Large-D mechanism

Level Spectroscopy Kitazawa-Okamoto method Loweset energy under PBC ： E PBC (M) Δ ２ ＝ [E(M+2)+E(M-2)-2E(M)]/2 Lowesr energy under TBC with P=+1 ： E TBC+ Lowesr energy under TBC with P=-1 ： E TBC- Lowest level ⇒ phase ： Δ ２ : gapless E TBC － ： plateau by Haldane mechanism (SPTP) E TBC+ ： plateau by Large-D mechanism

Level spectroscopy λ=1 Large-D plateau No plateau No plateau Large-D plateau

Level spectroscopy λ=2 Haldane plateau (SPT phase) Large-D plateau No plateau

Phase diagram

Magnetization curves