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つかさみやのじょう
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1 ( ) URL: mailto:hukusima@phys.c.u-tokyo.ac.jp December 2, 2003

2 Today s Contents Summary 2003/12/02 1

3 Cannella Mydosh(1972) Edwards Anderson(1975): Model Hamiltonian: EA model H( S) = ij J ij S i S j S i : m-component classical spin variable J ij : quenched disorder EA order parameter q µν = 1 mn i S iµ S iν (µ, ν = x, y, z) +? (1980 ) 2003/12/02 2

4 ? Bhatt Young, PRL 54, 924(1985) Ogielski Morgenstein, PRL 54, 928(1985). Ground state calculation (Domain Wall RG) : Banavar-Cieplak, PRL 48, 832 (1982) McMillan, PRB 31, 342(1985). Finite temperature Monte Carlo calculation: Olive-Young-Sherrington, PRB 34, 6341 (1986). Matsubara-Iyota-Inawashiro, PRL 67, 1458 (1991). Yoshino-Takayama, EuroLett 22, 631 (1993). (dynamics) = SG Experimental Spin-Glass Transition (Canonical spin glass: CuMn, AgMn) = isotropic Heisenberg model ) ( 2003/12/02 3

5 weak anisotropy effects: Olive-Young-Sherrington 86 Ising SG cf. d Ising LCD < 3 Heisenberg to Ising Ising SG ν η Ising SG (EA model) 1.8(2) 0.26(4) Fe 0.5 Mn 0.5 TiO Heisenberg SG: CuMn /12/02 4

6 Kawamura, PRL 68 (1992) STEP 1 STEP 2 broken symmetry preserve Z 2 Z 2 phase CG LRO SG LRO CG phase SG phase paramag. 2003/12/02 5

7 vs weak anisotropy picture chirality mechanism T/J (a) Paramagnetic phase Ising SG limit T/J (b) Paramagnetic phase T CG (D=0) Ising SG limit ~D 1/4 SG (CG) phase CG phase CG (SG) phase D/J T SG (D=0) D/J 2003/12/02 6

8 a finite temperature Chiral-Glass transition Kawamura, JPSJ 64, 26 (1995). Kawamura, PRL 80, 5421 (1998). Hukushima-Kawamura, PRE 61, R1008 (2000). Matsumoto-Hukushima-Takayama, PRB66, (2002). Imagawa-Kawamura, PRL87,207203(2001), JPSJ71,127(2002) - 1 step lack of self-averaging... SG vs aging rejuvenation-memory 2003/12/02 7

9 low temperature Chiral Glass phase (1) Order parameter distributon in the low T phase D Heisenberg SG T/J=0.10 L=16 L=12 L=10 L=8 L=6 3D Ising SG P(q χ ) q) P( CG oder parameter distribution function q χ SG order parameter q 1 STEP RSB Full STEP RSB A central peak at q χ = 0, in addition to side peakssuggests one step like RSG. 2003/12/02 8

10 Ising (1) (aging ) Rejuvenation(Chaos)-Memory Effect: K. Jonason et. al. PRL81 (1998) (AgMn) 1000 Fe 0.5 Mn 0.5 TiO 3 (a) 25 Ag(11 at% Mn) (b) χ (arb. units) χ χ ref T (K) χ (arb. units) χ χ ref T (K) P.E.Jönsson, et al cond-mat/ SG 2003/12/02 9

11 Ising (2) : R(t, t w ) = X(t,t w) k B T C(t,t w ) t w T eff = T/X, X = 1, T eff = T MC simulation (CuMn) (T/J)χ ZFC (t,tw) (T/J)χ D (q EA,(T/J)χ EA ) ~ C(t,tw) Correlation C(t',t) T eff 2T g Susceptibility χ(t',t) ~ ~ χ(t',t) ~ C(t',t) t'=100s t'=200s t'=500s t'=1000s t'=2000s t'=5000s t'=10000s 2003/12/02 10

12 SG!!...??? Matsubara-Shirakura-Endoh, PRB 64, (2001). Nakamura-Endoh, JPSJ 71, 2113 (2002). Lee and Young, Phys. Rev. Lett. 90 (2003) SG 2003/12/02 11

13 Lee Young s claim: Phys. Rev. Lett. 90 (2003) Single spin- and chiral-glass transition in vector spin glasses in three dimensions Correlation length : second moment method (L 12) ξ SG (T, L) = 1 2 sin(k min /2) ( χsg (k=0) χ SG (k min ) 1 ) 1/2 SG..., it would be feasible to extend these resutls to larger sizes by a major computational effort. CG 2003/12/02 12

14 Kawamura-Yonehara, J. Phys. A 36(2003) Mermin- Scaled SG correlation length ξ SG /L ξ L < 20 $&%(' $&%*) $,+,' $,-,' $/.0' ξ! #" 2003/12/02 13

15 SG ξ/l SG L= CG L= ξ SG /L ξ CG /L T/J T/J correction to scaling T CG > T SG L /12/02 14

16 (1) q (2) SG (L) = 2 4 * X µν 1 N X i! 2 +3 S (α) 5 iµ S(β) iν 2 CG 64 (L) = q (2) 1 3N X iµ χ (α) iµ χ(β) iµ 1 A ) SG q ( 10 1 T/J=0.22 T/J=0.20 T/J=0.19 T/J=0.18 T/J=0.17 T/J=0.15 L 2) CG q~( T/J=0.22 T/J=0.20 T/J=0.19 T/J=0.18 T/J=0.17 T/J=0.15 T = T CG T CG /J 0.19 q (2) SG q (2) CG L 2003/12/02 15

17 Spin-glass order parameter distribution function (diagonal part) diagonal compornent of the overlap q 1 N X (S (α) ix S(β) ix +S(α) iy S(β) iy +S(α) iz S(β) iz ) i P (q diag ) = δ(q q diag distribution function randomly frozen order Long range part q EA + trivial rotation O(3) diverging peak at ± 1 3 q EA Heisenberg SK model (Imagawa-Kawamura) Our result in 3D HSG diag) P(q T/J = 0.15 < T CG q diag Only single peak!! no SG ordering 2003/12/02 16

18 Summary SG T CG > T SG FDT context 2003/12/02 17

References: 3 June 21, 2002 K. Hukushima and H. Kawamura, Phys.Rev.E, 61, R1008 (2000). M. Matsumoto, K. Hukushima,

References: 3 June 21, 2002 K. Hukushima and H. Kawamura, Phys.Rev.E, 61, R1008 (2000). M. Matsumoto, K. Hukushima, References: 3 mailto:hukusima@issp.u-tokyo.ac.jp June 21, 2002 K. Hukushima and H. Kawamura, Phys.Rev.E, 61, R1008 (2000). M. Matsumoto, K. Hukushima, and H. Takayama, cond-mat/0204225. Typeset by FoilTEX