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

Numerical Diagonalization Study

• n Frustrated Magnets

Computational Approach to Quantum Many-body Problems, ISSP University of Tokyo

• n July 16 - August 8, 2019

Toru SAKAI1,2 , Hiroki Nakano1, Kiyomi Okamoto1, Takashi Tonegawa3,4

1Graduate School of Material Science

University of Hyogo, Japan

2National Institutes for Quantum and Radiological Science and

Technology(QST) SPring-8, Japan

3Professor Emeritus, Kobe University, Japan 4Osaka Prefecture University, Japan

SPring-8 K-computer

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

Candidates of Quantum Spin Fluid 2D frustrated systems

• S=1/2 Heisenberg antiferromagnets



 

j i j i S

S J H

,

 

Triangular lattice Kagome lattice 120 degree LRO No (conventional) LRO

Spin gap issue of kagome-lattice AF

Kagome lattice

Corner sharing triangles

kagome

Itiro Syôzi: Statistics of Kagomé Lattice, PTP 6 (1951)306

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

Spin gap issue of kagome-lattice AF

Gapped theories Valence Bond Crystal (VBC) MERA[Vidal] Z2 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

ZnCu3(OH)6Cl2

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

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)

Computational costs

N=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

Numerical diagonalizations

• f finite-size clusters up to Ns=42
• dd 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

• H. Nakano and TS: JPSJ 80 (2011) 053704 (arXiv: 1103.5829)

Two extrapolated results disagree from odd Ns and even Ns sequences.

gapless is better !

D/J=A+Bexp(-CNs

1/2)

D/J=A+B/(Ns

1/2)

Gapless or Gapped ？ Susceptibility analysis

Field derivative of magnetization

as a function of

at M=0

 

  

j z j B j j i i

S H S S J H  g  

,

ˆ

(gμB=1)

↓ E(M) ↓

• HM

M=ΣjSj

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)/NS2 m=0 ↓ 2 Δ ~ ε’’(m)/NS2 χ = dm/dh =S/ ε’’(m)→0 for Δ≠0 N →∞ =1/2ΔNS =1/ΔN

Demonstration of analysis

J2 J1 a=J2/J1

a=1: square lattice, LRO, gapless a=0: isolated dimers gapped a=0.52337(3): critical

Matsumoto et al: PRB65(2001) 014407

Dimerized Square Lattice

Magnetization processes

Gapless Gapped

Differential susceptibility vs. M

Gapless Gapped

Size dependence of c at M=0

Gapless Gapped

Kagome-lattice Heisenberg AF

Kagome lattice AF Differential susceptibility vs. M

Ns=42 Ns=36 Ns=39

Size dependence of c at M=0

χ→finite (Ns→∞) ⇒ Gapless

Ns=42 Ns=36 Ns=30 Ns=24 Ns=18 Ns=12

Triangular lattice AF Size dependence of c

Consistent with gapless feature of triangular lattice AF

Ns=36 Ns=30 Ns=24 Ns=18 Ns=12

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., Nature Physics, 13, 962 (2017)

ac1 = 0.68 ac2 = 0.86 a = J’/J

• r

Pressure a = 0.64 dimer Néel plaquette

High-pressure and high-field ESR Neutron scattering measurement

Sakurai et al.: JPSJ 87, 033701 (2018)

Finite-size clusters

N=36 N=40

Finite-size energy difference

r=J’/J

N=36 N=40

Analysis of the finite-size gap

Plaquette singlet Neel

• rder

N=36 N=40 S=1 S=0

Analysis of the finite-size gap

Plaquette singlet Neel

• rder

New gap phase

N=36 N=40 S=1 S=0

Correlation functions

N=36 Neel Plaquette New gap N=40 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

Exact dimer Plaquette singlet New gap phase Neel

• rder

J’/J

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 Half-odd-integer spins S=1/2, 3/2, 5/2 … ⇒ 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

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

integer ) (   m S Q

1/3 plateau of S=3/2 AF chain

• TS and Takahashi: PRB 57 (1998)R3201

  

   

 j z j j z j j j j

S H S D S S J H

2 1

) ( ˆ  

Single-ion anisotropy D

+3/2 +1/2

• 1/2
• 3/2

Sz=+1/2 ⇒ +3/2 gap

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：EPBC(M) Δ２＝[E(M+2)+E(M-2)-2E(M)]/2 Lowesr energy under TBC with P=+1：ETBC+ Lowesr energy under TBC with P=-1：ETBC- Lowest level ⇒ phase： Δ２: gapless ETBC－ ：plateau by Haldane mechanism (SPTP) ETBC+ ：plateau by Large-D mechanism

Level spectroscopy λ=1

No plateau Large-D plateau No plateau Large-D plateau

Level spectroscopy λ=2

No plateau Haldane plateau (SPT phase) Large-D plateau

Phase diagram

Magnetization curves

Summary

S=2 AF chain + anisotropies : λ and D ½ magnetization plateau ? Numerical diagonalization + Level spectroscopy λ=1 : Large-D plateau appears λ>1.55 : Haldane plateau (SPT phase) appears

TS, K. Okamoto , T. Tonegawa: to appear in PRB (arXiv: 1907.11931)