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Critical behaviour and dynamics of Ising
spin glasses Ian
Campbell LCVN Universite Montpellier
II Collaborations with - Helmut Katzgraber,
ETH Zurich and - Michel Pleimling, Erlangen
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Numerical Definition of the dynamical exponent
z(T) (Katzgraber and Campbell PRB 72 (2005)
014462) At Tc ceq(L) L2-h in
equilibrium and cne(t) t(2-h)/zc as
function of anneal time t following a quench to
Tc . So at Tc we can define an anneal time
dependent length scale L(t)
At1/zc Quite generally, if at any temperature
T the equilibrium SG susceptibility as a
function of sample size L is ceq(L,T), and the
non-equilibrium SG susceptibility after at
anneal time t after a quench to temperature T is
c ne(t,T) for a large sample of size Lb, then an
infinite sample size time dependent length scale
L(t) can be rigourously defined by writing the
implicit equation c ne(L(t),T) c eq(L,T)
as long as L(t) ltlt Lb. L(t,T) is about
twice the correlation length l(t) as one could
expect.
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From the measured values of L(t,T) we find
numerically that an equation having the same
functional form as at criticality L(t,T)
A(T)t1/z(T) with a temperature dependent
effective exponent z(T) and a weakly temperature
dependent prefactor A(T) gives an excellent
parametrization of ISG and GG data in each
system, not only at temperatures below T_c
(confirming the conclusions of Kisker et al 1996
and Parisi et al 1996) but also above T_c.
z(T) varies smoothly through SG transition
temperatures. Energy data can be analysed just
the same way as c data and give consistent
results for z(T).
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red c(L) black c(t) transposed to c(L(t))
using L(t)A(T)t1/z(T)
2d GG T0.173
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Energy-susceptibility relation
Red e(L)-e(inf) against c(L) Black
e(t)-e(inf) against c(t) (not a fit !)
2d GG T 0.173
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GG 2d
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GG 3d
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3d Gaussian ISG
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Conclusion 1 For Ising and vector SGs, the
dynamic exponent z(T) can be defined for the
entire range of temperatures, not only at Tc but
also well below and well above. zc is just one
particular value, z(TTc). z(T) varies smoothly
through spin glass transitions PRB 72
(2005) 014462
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Estimation of Tc in ISGs Standard method is to
use the Binder cumulant crossing point. Accurate
in 4d, but in 3d this is subject to large
corrections to scaling. Similardifficulties for
x(L)/L. Alternatively combine static and dynamic
measurements (1995) - c(L) L2-h in
equilibrium at Tc - c(t) t(2-h)/z) after
quench to Tc - C(t) t-(d-2h)/2z after a long
anneal at Tc The three measurements are
consistent at only one temperature, which must be
Tc, and from the values at Tc one gets h and
zc. Recent high temperature series results for
Tc in 4d ISGs are in excellent agreement with
numerical values using this method.
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Kurtosis ltJij4gt/ltJij2gt2 PRB 72
(2005) 092405
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Conclusion 2 The consistency method gives
accurate and reliable estimates for Tc
confirmed by high T series calculations. The
values of the associated exponents h and zc from
the same simulations can also be taken to be
accurate. PRB 72 (2005) 092405
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Dynamic exponents see Henke and Pleimling
2004 Definitions Anneal for a time s,
measure at time t (gts) At criticality TTc,
measure - Autocorrelation function decay after
quench (no anneal) C(t,s0) t-lc/zc
where -lc/zc d/zc - qc qc being
an independent dynamic exponent , the initial
slip. - Fluctuation-dissipation ratio
response R(t,s) dltSi(t)gt/dh(s) h0
(tgtgts) gives FD ratio X(t,s)
TR(t,s)/(dC(t,s)/ds)
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Autocorrelation relaxation after quench to Tc
3d ISG fluctuation-dissipation
ratio at Tc
Interaction distributions b Binomial g
Gaussian l Laplacian
Pleimling et al cond-mat/0506795 (cf Malte
Henkel)
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3d ISGs derivative of the autocorrelation
function decay without anneal (s0) at Tc
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3d ISGs fluctuation-dissipation ratios at Tc
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Conclusion 3 The dynamic exponents zc,
X(infinity), lc/zc (and the static exponent h)
are not universal in ISGs.
cond-mat/0506795
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What about n ? The consistency method gives
reliable values of Tc, h, and z. To determine the
exponent n needs data not at Tc but at T gt
Tc The critical divergences x T-Tc -n, c
T-Tc -g etc are generally quoted in
textbooks (and many papers) as x t -n c t
-g with the scaling variable t
(T-Tc)/Tc Is this the right scaling variable
? For c in canonical ferromagnets the range of
temperature over which the critical temperature
dependence remains a good approximation can be
considerably extended by writing c t -g
with t (T-Tc)/T as the scaling variable
(as proposed by Jean Soultie).
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Note that t 1 means T infinity, while t 1
means only T 2Tc i.e. with t as the scaling
variable, the range of critical behaviour c(t)
t -g extends as a good approximation to
infinite T, while with t as scaling variable the
critical region is very limited. Why ? Look at
high series work. In ferromagnets, the exact
high temperature series for c(b) is an expansion
in powers of b (1/T) c(b) 1 a1b a2b2
..... with the an determined exactly up to n
25 by combinatorial methods in 3d ( Butera and
Comi) and up to n325 in 2d (Nickel).

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A fundamental mathematical identity is
Darbouxs first theorem (1878) (1-x) -g
1 gx g(1g)/2x2 g(1g)(2g)/6x3
... i.e. 1 a1x a2x2 a3x3 ... with
an/an-1 ng-1/n Now t -g
(T-Tc)/T -g 1-b/bc -g (b
1/T) If c(b) is to behave as 1-b/bc -g over
the whole range of T, then the combinatorial
series of an must be 1, a1, a2, ... such that
rn an/a(n-1) (1(g-1)/n)/bc It turns out
that this works remarkably well down to quite
small n ( hence to high T) for the standard
Ising ferromagnets (2d, 3d sc, bcc, fcc....)
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So what about spin glasses ? The high T ISG
combinatorial series take the form (Daboul et
al) c(b) 1 a1b2 a2b4 a3b6 .......
( powers of b2 instead of b) So following
the same argument as for the ferromagnet
but replacing b by b2 , the appropriate scaling
form for the ISG c should be c(b)
(1-b2/bc2) -g (1-b2/bc2) gt 2(1-b/bc) at
T Tc so so critical scaling is OK With the
data available for the moment, this also works
well.
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From the same line of reasoning, other, more
exotic scaling variables are appropriate for
other parameters such as x and Cv, both for
ferromagnets and for spin glasses. e.g. for
ferromagnets you expect scalings x
b1/21 - b/bc-n and Cv b21 -
(b/bc)2-a and for ISGs x b1 -
(b/bc)2-n and Cv b41 - (b/bc)4-a One
consequence is that the finite size scaling rules
and all the literature values for n in ISGs need
to be carefully reconsidered the appropriate
analysis should lead to reliable values for
n. Values of h (essentially measured at TTc)
are not affected.
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Finite size scaling instead of
fL1/n (T-Tc) one should write
f(LT1/2)1/n (T-Tc)/T for a ferromagnet,
and f(LT)1/n(T2-Tc2)/T2 for a spin
glass. Oliviers question what happens if
Tc0 ?
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Conclusion 4 Reliable values for g and n can be
obtained using data from temperatures that are
much higher than Tc, if appropriate scaling
variables are used.
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Overall Conclusion For Spin Glasses - The
dynamical exponent z is well defined not only at
criticality but for a wide range of T, with no
anomaly at Tc. - One can obtain reliable and
accurate estimates for critical temperatures and
all the critical exponents with precision
limited only by the (considerable)
numerical effort which needs to be devoted to
each system. - Numerical Data as they stand are
not compatible with the canonical universality
rules.