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

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1 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/ Typeset by FoilTEX

2 , ISSP, University of Tokyo 2002/06/21 1

3 Today s Contents ISSP, University of Tokyo 2002/06/21 2

4 Edwards-Anderson Model and Lower Critical Dimensions Lower Critical dimension dimensions (Mean-field model) m=1 (Ising) (SK model) m=2,3,.. (Vector) ISSP, University of Tokyo 2002/06/21 3

5 Edwards-Anderson Model and Lower Critical Dimensions Lower Critical dimension dimensions (Mean-field model) m=1 (Ising) (SK model) m=2,3,.. (Vector) Experimental Spin-Glass Transition = isotropic Heisenberg model Weak magnetic anisotropy RKKY or Long range interection Site disorder Chirality mechanism ISSP, University of Tokyo 2002/06/21 3

6 weak anisotropy effects: 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 Heisenberg SG Olive-Young-Sherrington 86 ISSP, University of Tokyo 2002/06/21 4

7 Kawamura 92 STEP 1 STEP 2 Z 2 Z 2 phase CG LRO SG LRO CG phase SG phase paramag. ISSP, University of Tokyo 2002/06/21 5

8 temperature 1/4 D Paramagnet Ising SG limit temperature SG(Ising) CG (SG) CG Paramagnet Anisotropy D Anisotropy D Isotropic limit ISSP, University of Tokyo 2002/06/21 6

9 STEP1: : Banavar-Cieplak, PRL 48, 832 (1982). McMillan, PRB 31, 342 (1985). MC Olive-Young-Sherrington, PRB 34, 6341 (1986). Matsubara-Iyota-Inawashiro, PRL 67, 1458 (1991). Yoshino-Takayama, EuroLett 22, 631 (1993). (dynamics) : Kawamura, PRL 68, 3785 (1992). ISSP, University of Tokyo 2002/06/21 7

10 STEP1: : Banavar-Cieplak, PRL 48, 832 (1982). McMillan, PRB 31, 342 (1985). MC Olive-Young-Sherrington, PRB 34, 6341 (1986). Matsubara-Iyota-Inawashiro, PRL 67, 1458 (1991). Yoshino-Takayama, EuroLett 22, 631 (1993). (dynamics) : Kawamura, PRL 68, 3785 (1992). Kawamura, JPSJ 64, 26 (1995). Kawamura, PRL 80, 5421 (1998). Hukushima-Kawamura, PRE 61, R1008 (2000). ISSP, University of Tokyo 2002/06/21 7

11 STEP1: : Banavar-Cieplak, PRL 48, 832 (1982). McMillan, PRB 31, 342 (1985). MC Olive-Young-Sherrington, PRB 34, 6341 (1986). Matsubara-Iyota-Inawashiro, PRL 67, 1458 (1991). Yoshino-Takayama, EuroLett 22, 631 (1993). (dynamics) : Kawamura, PRL 68, 3785 (1992). Kawamura, JPSJ 64, 26 (1995). Kawamura, PRL 80, 5421 (1998). Hukushima-Kawamura, PRE 61, R1008 (2000). Matsubara-Shirakura-Endoh, PRB 64, (2001). Nakamura-Endoh, cond-mat/ ISSP, University of Tokyo 2002/06/21 7

12 Model and Observables : i S(α) iµ S(β) iν Spin Sector q µν = 1 N 3 SG ( ) H = J ijsi S j + χ D µν SG = N ij Sµ i Sν µν q2 µν j ij ij C q = 1 Si N i (t w ) S i (t w + t) Chiral Sector: q χ = 1 3N iµ J ij : bimodal dist. χ(α) iµ S(β) iν D µν with χ ij : Uniform dist. iµ = S i ˆµ S i S i+ˆµ ( ) χ CG = N µν q2 χ C χ = 1 3N i χ i(t w ) χ i (t w + t) ISSP, University of Tokyo 2002/06/21 8

13 Spin sector Chiral sector w+t) Cq(tw;t T/J=0.20 D/J=0.00 t w =10 5 w+t) Cχ(tw;t L= T/J=0.20 D/J=0.00 t w =10 5 L= t [MCS] t [MCS] τ(t, L) T = T c τ(t c, L) L z ISSP, University of Tokyo 2002/06/21 9

14 R(t) R(t; T, L) = T/J=0.35 T/J=0.30 T/J=0.20 = Ratio S i (t 0 ) S i (t + t 0 ) r Si 2 (t 0 ) S i (t + t 0 L=16 t[mcs] chiral sector τ(t,l) T/J "!#$ SG CG CG ISSP, University of Tokyo 2002/06/21 10

15 τ(t, L) = L z f ( (T T C )L 1/ν) FSS FSS τq(t,l)l z T SG =0.0(1) ν=1.4(2) z=4.4(4) τχ(t,l)l z T CG =0.22(2) ν=1.1(3) z=4.1(4) slope ~ zν slope ~ zν T T SG L 1/ν consistent with T SG = 0 T T CG L 1/ν T CG /J 0.22 = ISSP, University of Tokyo 2002/06/21 11

16 :Statics squared order parameter q (2) SG (L) = 2 4 * X µν 1 N X i S (α) iµ S(β) iν! CG 64 (L) = q (2) 1 3N X iµ χ (α) iµ χ(β) iµ 1 A ) SG q ( 2) CG q ( T/J=0.22 T/J=0.21 T/J=0.20 T/J=0.18 T/J=0.15 L q (2) SG T = T CG q (2) = SG T CG T/J=0.22 T/J=0.21 T/J=0.20 T/J=0.18 T/J=0.15 L CG T CG /J 0.21 ISSP, University of Tokyo 2002/06/21 12

17 ( ) q 1 N X (S (1) ix S(2) ix + S(1) iy S(2) iy i P (q diag ) = δ(q q diag q EA O(3) + S(1) iz S(2) iz ) ± 1 3 q EA Toy model (Imawaga-Kawamura) diag) P(q ±J at T/J = 0.15 < T CG N=32 N=64 N=128 N=256 N=512 N=1024 P(q diag ) q diag q diag ISSP, University of Tokyo 2002/06/21 13

18 Simulations Experiments Ising SG Heisenberg SG Ising SG Heisenberg SG (EA model) (Chiral Glass) (Fe 0.5 Mn 0.5 TiO 3 ) CuMn & AgMn β (1) γ 4.1(5) (3) 2.2(1) ν 1.8(2) η 0.26(4) ( ) 1 step RSB Ising SG self-averaging ISSP, University of Tokyo 2002/06/21 14

19 STEP 2: D/J = 0.05 τq(t,l) τχ(t,l) T/J T/J D insensitive ISSP, University of Tokyo 2002/06/21 15

20 FSS τχ(t,l)l z T CG =0.24(2) D/J=0.05 T T CG L 1/ν slope ~ zν D SG CG SG Universality class CG Universality class ISSP, University of Tokyo 2002/06/21 16

21 : 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 ISSP, University of Tokyo 2002/06/21 17

22 ISSP, University of Tokyo 2002/06/21 18

( ) URL: December 2, 2003

( ) URL:   December 2, 2003 ( ) URL: http://dbs.c.u-tokyo.ac.jp/~fukushima mailto:hukusima@phys.c.u-tokyo.ac.jp December 2, 2003 Today s Contents Summary 2003/12/02 1 Cannella Mydosh(1972) Edwards Anderson(1975): Model Hamiltonian:

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