Outline I. Introduction: II. Pr 2 Ir 2 O 7 Like-charge attraction III.

Masafumi Udagawa Dept. of Physics, Gakushuin University Mar. 8, 16 @ in Gakushuin University Reference M. U., L. D. C. Jaubert, C. Castelnovo and R. Moessner, arxiv:1603.02872

Introduction

: Ising? Ising (J > 0) H = J i,j σ i σ j (σ j = ±1) = J 4 & (σ i + σ j + σ k ) 2 + Const. i k j (Wannier) S = 0.323k B /spin (NOT 0.338!)

: Dy 2 Ti 2 O 7, Ho 2 Ti 2 O 7 : 2-in 2-out (Ramirez) : 0.229k B /spin Pauling ( Bethe ) S Pauling = k B 2 log 3 2 0.203k B/spin

: Fe 3 O 4 Fe 3 O 4 A-site Fe 3+ B-site Fe 2+, Fe 3+ T c =120K (Verway) B-site spinel T c (1eV 10 4 K) - (Anderson) T c E/ S

: LiV 2 O 4 Li V O 2 4 Specific heat C J. Kondo et al. (1999) C. Urano et al. (2000) γ = C/T m* S = T C T dt γ T m* = 200me

Winding Number w = ( # of ) - ( # of )

Short summary : : : : /

Collaborators Dr. Ludovic D. C. Jaubert (OIST) Dr. Claudio Castelnovo (Cambridge) Prof. Roderich Moessner (MPI PKS Dresden)

Pr 2 Ir 2 O 7 : Ir: Itinerant site center of IrO octahedron 6 = Pr: Spin ice Fig.: Matsuhira (2008) A sub.: Pr spin ice B sub.: Ir e - Pr Ir

Pr 2 Ir 2 O 7 : Hall Anisotropy in σ xy (Machida 2007) 1e-04 8e-05 6e-05 [100] [111] Spin ice to Kagome ice crossover P triangle ~ 0.5 P triangle ~ 1.0 σ 4e-05 H 2e-05 H ~ 0 : spin ice : kagome ice Liquid-Gas crossover of monopoles 0-2e-05 0 1 2 3 4 H / J spin Monopole gas Monopole liquid M. U. and R. Moessner (2013). Hall : σ xy J y = σ xy E x

Pr 2 Ir 2 O 7 : Hall conductivity ( Ω 1 cm 1 ) -10-5 0 0.1 1 0.3 2 Temperature (K) B = 0 10 0.02 0.01 0.00 Magnetization µ Β per Pr atom ( ) : 0.3K < T < 2K B [111]: 7 Tesla 0: (M = 0) Machida (2010)

(ac ) : Dipolar (Jaubert et al.) Exp. (Snyder et al.) Arrhenius Matsuhira (2004), Snyder (2004), Jaubert (2010)

: RKKY ( ) 2-in 2-out

Dy 2 Ti 2 O 7, Ho 2 Ti 2 O 7 : µ 0 [ Si S j 4π rij 3 3(S i r ij )(S j r ij ) ] rij 5 i<j

Dy 2 Ti 2 O 7, Ho 2 Ti 2 O 7 : µ 0 [ Si S j 4π rij 3 3(S i r ij )(S j r ij ) ] rij 5 i<j = +1-1

Dy 2 Ti 2 O 7, Ho 2 Ti 2 O 7 : µ 0 [ Si S j 4π r 3 3(S i r ij )(S j r ij ) ] i<j ij rij 5 Q i Q j r i<j ij = +1-1 : dipolar interaction : Coulomb interaction

: non-contractible pair (ac ) : Dipolar (Jaubert et al.) Exp. (Snyder et al.) Arrhenius Matsuhira (2004), Snyder (2004), Jaubert (2010) H = µ i Q 2 i i<j Q i Q j r ij Castelnovo (2010)

J 1 -J 2 -J 3 spin ice model RKKY : but fast-decaying: r 3 sign-alternating H = J 1 S i S j + J 2 S i S j + J 3 S i S j n.n. 2nd. 3rd. = J 1 η i η j + J 2 η i η j + J 3 η i η j n.n. 2nd. 3rd. J3 J1 J2 η i = +1( 1), for S i out (in) for sublattice A sublattice A 2-in 2-out Q = 0 Tetrahedral Charge: Q ( p 1 ) H = 2 J Q 2 p J Q p Q q p,q for J 2 = J 3 = J H. Ishizuka & Y. Motome (2013) p 3-in 1-out 1-in 3-out Q = -2 Q = +2

N M = 2 N : Ω 1, Ω M d dt P (Ω j) = 1 [P (Ω i )W (Ω i Ω j ) P (Ω j )W (Ω j Ω i )] τ 0 i j W (Ω i Ω j ): Thermal bath W (Ω i Ω j ) = exp( βe(ω j )) exp( βe(ω i )) + exp( βe(ω j ))

Results

Results: (J 2 = J 3 = 0): T quench: T = 10 0 Monopole density monopole density 10 0 1 0.1 10-1 0.01 10-2 0.001 10-3 0.0001 10-4 10-5 1e-05-1 0 1 2 3 4 log(time) [MCstep] 10-1 10 0 10 1 10 2 10 3 10 4 Time Mean-field : ρ ρ 0 /(1 + 3 3 4 gρ 0t), with ρ 0 = 0.4869146729 (T = 10) d c.f. mean-field model: dt n + = d dt n = λn + n Castelnovo (2010)

Results: (J 2 = J 3 = 0): H quench: H = 100 0 [111] Monopole density 1 1.0 0.8 0.6 monopole density 0.4 0.2 0.0 0 10-3 10-2 10-1 10 0 10 1 10 2 10 3 10 4 log(time) [MCstep] 10-3 10-1 10 2 10 3 10-2 10 0 10 1 10 4 Time T T=0.1 = 0.1 T=0.01 0.01 Initial state Kagome Triangular Kagome

Results: J 1 J 2 J 3 model, (J 2 = J 3 = 0.1, T = 0.10): H quench Monopole density monopole density 1.0 0.8 0.6 0.4 0.2 0.0 0.8 0.6 0.4 0.2 10 0 10 5 10 10 10 15 10 20 0.0 time [MCstep] monopole M 1.0 0.0 10 5 10 10 10 15 10 20 10 0 Time Magnetization Normalized magnetization

Results: J 1 J 2 J 3 model, J = 0.10, T = 0.10 Monopole density monopole density 1.0 0.8 0.6 monopole density 0.4 0.2 0.2 0.0 10 0 Time J=-0.1, T=0.10 monopole magnetization tri magnetization (sat: 0.5) kag magnetization (sat: 0.5) 0 Magnetization (Triangular) Magnetization (Kagome) 0.0 10 5 10 10 10 15 10 20 0 5 10 0 10 5 time [MCstep] 10 10 10 15 10 20 time [MCstep] 1.0 0.8 0.6 Normalized magnetization 0.4 0.2 Magnetization ( 1 ) H = 2 + J Q 2 p + J Q p Q q p,q p Kagome Triangular Kagome

Results: J 1 J 2 J 3 model, J = 0.10, T = 0.10 Monopole density monopole density 1.0 0.8 0.6 monopole density 0.4 0.2 0.2 0.0 10 0 Time J=-0.1, T=0.10 monopole magnetization tri magnetization (sat: 0.5) kag magnetization (sat: 0.5) 0 Magnetization (Triangular) Magnetization (Kagome) 0.0 10 5 10 10 10 15 10 20 0 5 10 10 10 0 10 5 time [MCstep] 10 10 10 15 10 20 time [MCstep] 1.0 0.8 0.6 Normalized magnetization 0.4 0.2 Magnetization ( 1 ) H = 2 + J Q 2 p + J Q p Q q p,q p -

Results: J 1 J 2 J 3 model, J = 0.10, T = 0.10 Monopole density monopole density 1.0 0.8 0.6 monopole density 0.4 0.2 0.2 0.0 10 0 Time J=-0.1, T=0.10 monopole magnetization tri magnetization (sat: 0.5) kag magnetization (sat: 0.5) 0 Magnetization (Triangular) Magnetization (Kagome) 0.0 10 5 10 10 10 15 10 20 0 5 10 10 10 0 10 5 time [MCstep] 10 10 10 15 10 20 time [MCstep] 1.0 0.8 0.6 Normalized magnetization 0.4 0.2 Magnetization ( 1 ) H = 2 + J Q 2 p + J Q p Q q p,q p Exhaustion problem flip

Results: J 1 J 2 J 3 model, J = 0.10, T = 0.10 Monopole density monopole density 1.0 0.8 0.6 monopole density 0.4 0.2 0.2 0.0 10 0 Time J=-0.1, T=0.10 monopole magnetization tri magnetization (sat: 0.5) kag magnetization (sat: 0.5) 0 Magnetization (Triangular) Magnetization (Kagome) 0.0 10 5 10 10 10 15 10 20 0 5 10 10 10 0 10 5 time [MCstep] 10 10 10 15 10 20 time [MCstep] 1.0 0.8 0.6 Normalized magnetization 0.4 0.2 Magnetization ( 1 ) H = 2 + J Q 2 p + J Q p Q q p,q p 0 J < 0

Results: J 1 J 2 J 3 model, J > 0: H quench (J 2 = J 3 = J = 0.05, 0.10, 0.15, 0.20, J/T = 0.125) Monopole density 1.0 monopole density 1.0 0.8 0.6 0.6 0.4 0.4 0.2 0.2 1.0 0.8 0.6 0.4 0.2 Magnetization 0.0 0.0 0.0 10-4 10-2 10 0 10 2 10 4 10 6 10 10-4 10-2 10 0 time 10 2 [MCstep] 10 4 10 6 10 8 Time 0.0 J/T 1

Results: J 1 J 2 J 3 model, J > 0: H quench (J 2 = J 3 = J 0.25, J/T = 0.125) Monopole density 1.0 monopole density 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 1.0 0.6 0.4 0.2 0.0 0.0 10-4 10-2 10 0 10 2 10 4 10 6 10 10-4 10-2 10 0 time 10 2 [MCstep] 10 4 10 6 10 8 Time J = 0.25 0.24 0.225 0.20 0.8 0.0 Magnetization 1

Results: J 1 J 2 J 3 model, J 2 = J 3 = J > 0: H quench H = @J = 1/4 H = 1 4 ( 1 ) 2 J Q 2 p J Q p Q q p,q p Q 2 p 1 4 p p,q Q p Q q = :

Results: Experimental Implication T How to estimate?

Results: Experimental Implication [00k] 4 2 0-2 -4 T = 0.1 K -4-2 0 2 4 [hh0] T = 0.3 K -4-2 0 2 4 [hh0] T = 1 K -4-2 0 2 4 [hh0] 1.6 1.4 1.2 1 0.8 0.6 0.4 pinch point B µ (0)B ν (r) = 1 3x µ x ν r 2 δ µν 4πK r 5 S µν (q) 1 ( δ µν q ) µq ν K q 2

Summary: J 1 J 2 J 3