( ) 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: 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
? 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
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 3 1.7 0.35 Heisenberg SG: CuMn 1.4 0.4 2003/12/02 4
Kawamura, PRL 68 (1992) 3785. 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
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
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, 104404(2002). Imagawa-Kawamura, PRL87,207203(2001), JPSJ71,127(2002) - 1 step lack of self-averaging... SG vs aging rejuvenation-memory 2003/12/02 7
low temperature Chiral Glass phase (1) Order parameter distributon in the low T phase 0.045 0.04 0.035 0.03 3D Heisenberg SG T/J=0.10 L=16 L=12 L=10 L=8 L=6 3D Ising SG P(q χ ) 0.025 0.02 0.015 0.01 q) P( 0.005 0-0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 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
Ising (1) (aging ) Rejuvenation(Chaos)-Memory Effect: K. Jonason et. al. PRL81 (1998) 3243. (AgMn) 1000 Fe 0.5 Mn 0.5 TiO 3 (a) 25 Ag(11 at% Mn) (b) χ (arb. units) 800 600 400 200 χ χ ref 10 0 10 20 30 40 50 0 15 17 19 21 23 25 T (K) χ (arb. units) 20 15 10 5 χ χ ref 1 0 1 2 3 4 0 20 24 28 32 36 T (K) P.E.Jönsson, et al cond-mat/0307640 SG 2003/12/02 9
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 10 10 2 10 3 10 4 10 5 (q EA,(T/J)χ EA ) 0.5 0.0 0.2 0.4 0.6 ~ C(t,tw) Correlation C(t',t) T eff 2T g Susceptibility χ(t',t) ~ 1.0 0.9 0.8 0.7 0.6 ~ χ(t',t) 1.0 0.5 0.0 0.0 0.5 1.0 ~ C(t',t) t'=100s t'=200s t'=500s t'=1000s t'=2000s t'=5000s t'=10000s 2003/12/02 10
SG!!...??? Matsubara-Shirakura-Endoh, PRB 64, 09412 (2001). Nakamura-Endoh, JPSJ 71, 2113 (2002). Lee and Young, Phys. Rev. Lett. 90 (2003) 228203 SG 2003/12/02 11
Lee Young s claim: Phys. Rev. Lett. 90 (2003) 228203 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
Kawamura-Yonehara, J. Phys. A 36(2003) 10867 10880. Mermin- Scaled SG correlation length ξ SG /L ξ L < 20 $&%(' $&%*) $,+,' $,-,' $/.0' ξ! #" 2003/12/02 13
SG ξ/l SG L=20 16 12 8 CG L=20 16 12 8 ξ SG /L ξ CG /L T/J T/J correction to scaling T CG > T SG L 16 2003/12/02 14
(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) *0 @ 1 3N X iµ χ (α) iµ χ(β) iµ 1 A 2+3 7 5 10 1 2) 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~( 10 2 10 3 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
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
Summary SG T CG > T SG FDT context 2003/12/02 17