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\newcommand{\mysemtitle}{Finite-dimensional spin glasses in the aging
regime}
\newcommand{\myname}{Federico Ricci-Tersenghi}
\newcommand{\myaffiliation}{ICTP, Trieste}
\newcommand{\mysemplace}{INFMeeting (Catania)}
\newcommand{\mysemdate}{June 14, 1999}

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\begin{document}
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% slide 0
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\begin{slide*}
\headl{\begin{tabular}{c}
\Cred{Finite-dimensional spin glasses}\\
\Cred{in the aging regime:}\\
\Cred{replica symmetry breaking}\\
\Cred{and ultrametricity}\\
\end{tabular}}
\headn{\Cblu{\myname}}
\heads{\Cblu{(\myaffiliation)}}
\vfill

\Cgre{In collaboration with:}\\
\Cbla{Silvio Franz (ICTP, Trieste)}\\
\Cbla{Enzo Marinari (Cagliari)}\\
\Cbla{Giorgio Parisi (Roma1)}\\
\Cbla{Juan Jesus Ruiz-Lorenzo (Madrid)}

\vfill

\headn{\COra{\mysemplace, \mysemdate}}

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}
One of the main features of \Cblu{spin glasses} is their very
\Cred{slow dynamics}: a real sample never reaches equilibrium on the
experimental (and maybe even geological!) time scales.

As a consequence, the result of any measure depends on the time the
sample spent in the frozen phase: the \Cgre{waiting time}.

\vfill

Typical experiment (or \Cblu{dynamical} simulation):
\bn
\item quench a system in the frozen phase at time \Cgre{$t=0$}
\item wait a time \Cgre{$t_w$} (with or without an external field)
\item start the measurements (the result will depend on \Cgre{$t_w$})
\en

{\scriptsize At the initial time in dynamical simulations, we quench
from \Cred{$T=\infty$} (random configuration) to \Cred{$T<T_c$}.}
\end{slide*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}
\bc
\Cred{Equilibrium simulations}
($L<\xi(t)$)\\
\Cred{to measure the ``statics''}\\
\vspace{0.2cm}
vs.\\
\vspace{0.2cm}
\Cblu{Off-equilibrium simulations}
($L>\xi(t)$)\\
\Cblu{to measure the ``dynamics''}\\
\ec
\vfill
Advantages of a dynamical study:
\bi
\item necessary to study a system which never reaches equilibrium
(like realistic spin glasses);
\item numerically: bigger volumes, faster simulations, no
thermalization times;
\item critical exponents via dynamical scaling (once we know $T_c$);
\item non self-averaging equilibrium quantities can be measured from
off-equilibrium self-averaging ones.
\ei
\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

\bd
C(t,t_w) \equiv \frac1N \sum_{i=1}^N s_i(t_w) s_i(t_w+t) \nn
\ed

\bc
\includegraphics[width=0.8\textwidth]{plot_corr.eps}
\ec

\begin{description}
\item[$t<t_w$:] \Cblu{``quasi-equilibrium''} regime, $C(t,t_w) \simeq
C_{\rm eq}(t)$;
\item[$t>t_w$:] \Cblu{aging} regime. If simple aging holds $C(t,t_w)
\propto {\cal C}(t/t_w)$.
\end{description}

\bd
q_{\scriptscriptstyle {\rm EA}} \equiv \lim_{t\fd \infty} \lim_{t_w\fd
\infty} C(t,t_w)
\ed

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}
\headl{\bt
\Cred{Fluctuation Dissipation}\\
\Cred{Theorem (FDT)}\\
\et}

We define
\ba
\Cblu{C(t,t') \equiv \la A(t) A(t') \ra} \nn \\
\Cblu{R(t,t') \equiv \left. \frac{\delta\la A(t) \ra}{\delta
\epsilon(t')} \right|_{\epsilon=0}} \nn
\ea
with $H' = H + \epsilon(t) A(t)$

Assuming time translation invariance (TTI)
\ba
C(t,t') = {\cal C}(t-t') \nn \\
R(t,t') = {\cal R}(t-t') \nn
\ea
we have
\headn{\Cblu{$R(t,t') = \beta\ \theta(t-t')\ \frac{\delta
C(t,t')}{\delta t'}$}}
\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}
\headl{\bt
\Cred{Off-equilibrium Fluctuation}\\
\Cred{Dissipation Relation (OFDR)}\\
\et}

In the aging regime the FDT can be modified introducing a
multiplicative \COra{violation factor $X$}, that for large times only
depends on the correlation.

\headn{\Cblu{$R(t,t') = \beta\ \COra{X[C(t,t')]}\ \theta(t-t')\ \frac{\delta
C(t,t')}{\delta t'}$}}

{\scriptsize In the simulations, we perturb the system switching on a
random magnetic field at time $t_w$: \Cblu{$h(t)=h_0\
\theta(t-t_w)$}.}
\ba
C(t,t') = \frac1N \sum_i \la s_i(t) s_i(t') \ra \nn \\
M_S[h](t) = \frac1N \sum_i \frac{\la h_i s_i(t) \ra}{h_0} = \nn \\
 = \int_{-\infty}^{t} \d t' R(t,t') h(t') \nn
\ea
\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

In the linear response regime (very low fields) we define
\bd
\Cblu{\chi(t,t_w) = \lim_{h_0 \fd 0} \frac{M_S[h](t)}{h_0}}
\ed
and with a little algebra we have

\Cblu{\Large
\bd
\chi(t,t_w) = \beta \int_{C(t,t_w)}^1 X(u) \d u
\ed
}

Defining
\bd
S(y) = \int_y^1 X(u) \d u
\ed
we can simplify the previous formula to
\bd
\Cblu{T\ \chi(t,t_w) = S(C(t,t_w))}
\ed

FDT: \COra{$X=1$} $\Rightarrow$ \Cblu{$T\ \chi(t,t_w) = 1-C(t,t_w)$}

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

\bc
\Cred{\underline{\large Link with the Statics}}
\ec

In the large times limit:

\CBro{$t \fd \infty$} and  \CBro{$t_w \fd \infty$}

such that \CBro{$C(t,t_w) = q$}

\headl{$\Cblu{X[C(t,t_w)]} \longrightarrow \Cred{x(q)}$}

where
\bd
\Cred{x(q) = \int_0^q \d q' P(q')}
\ed

Equivalently
\bd
\Cblu{S(y)} \longrightarrow \Cred{s(y) = \int_y^1 x(q) \d q}
\ed

{\scriptsize This link has been proved, under very general
assumptions, by Franz, M\`ezard, Parisi and Peliti\\
(Phys. Rev. Lett. {\bf 81} (1998) 1758).}
\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

\headn{\Cred{Classification}}

Three main kinds of dynamical response \Cblu{$S(C)$}, that correspond
two different shapes of the static \Cred{$P(q)$}.
\bc
\includegraphics[width=0.9\textwidth]{class.eps}
\ec

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

\headn{\Cred{Numerical Results}}

\bc
\underline{\Cblu{Edwards-Anderson (EA) model}}

D=3\ \ Volume=$64^3$\ \ $T=0.7\simeq \frac34 T_c$
\includegraphics[width=\textwidth]{qm.eps}
\ec

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

\bc
\Cblu{EA\ \ D=4}
\includegraphics[height=\textwidth, angle=270]{plot_mc.ps}
\ec

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

Check for the \Cred{linear response} to hold.
\bc
\includegraphics[height=0.8\textwidth, angle=270]{plot_lin_res.ps}
\ec

Check for the \Cred{large times} regime.
\bc
\includegraphics[height=0.8\textwidth, angle=270]{plot_due_tw_corr.ps}
\ec

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

\bc
\underline{\Cred{Parisi-Toulouse (PAT) hypothesis}}
\ec

\bd
\Cblu{q(\beta,x) = \tilde{q}(\beta x)}
\ed
{\scriptsize It is one more feature of the SK model which seems to
hold in the EA model.}
\bc
\includegraphics[width=0.72\textwidth]{pat.eps}
\includegraphics[width=0.72\textwidth]{scaled.eps}
\ec

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

As a main \Cred{check} we have applied the same method to models where
there is non RSB.

\Cblu{Diluted ferro: $400^3$, $T=0.9 T_c$, $t_w=10^3,10^4$}
\bc
\includegraphics[width=0.75\textwidth]{plot_C065_low.eps}
\ec

\Cblu{RFIM: $440^3$, $T=0.9 T_c$, $t_w=10^2,10^3$}
\bc
\includegraphics[width=0.75\textwidth]{plot_RFIM_low.eps}
\ec
\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

Three main \Cred{remarks}:
\bn

\item The method based on the OFDR does work.
\bi
\item {\scriptsize We have verified numerically all the analytical
assumptions.}
\item {\scriptsize It gives information on the off-equilibrium
dynamics (efective temperature, memory effects, ...) and on the right
equilibrium behavior, no matter of how complex the off-equilibrium
dynamics is.}
\item {\scriptsize Has many advantages with respect to equilibrium
simulations (large volumes, small number of samples, no thermalization
times, ...)}
\ei

\item The classification based on it has a physical meaning.

\item The same kind of measure can be done also experimentally on real
samples.
\en

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

\headl{\Cred{Ultrametricity}}

Given three spin configurations the ultrametric inequalities read
\bd
\Cblu{q_{12} \geq \min\{ q_{13},q_{23} \}} \quad , \quad
\Cblu{d_{12} \leq \max\{ d_{13},d_{23} \}}
\ed

\Cblu{Dynamically} the ultrametricity constrains the correlation
functions
\bd
\Cblu{C(t_1,t_3)= \min \{ C(t_1,t_2),C(t_2,t_3) \}}
\ed
\underline{\scriptsize Very hard to check numerically!}

We consider \COra{2 real replicas} and define the auto- and
cross-correlation function as
\ba
\Cblu{C(t,t') = \frac1N \sum_{i} S_i^1(t) S_i^1(t')} \nn \\
\Cblu{D(t,t') = \frac1N \sum_{i} S_i^1(t) S_i^2(t') \nn}
\ea

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

If we constrain the 2 replicas to have, at every time, a fixed overlap
\Cblu{$q_0\in[0,q_{\scriptscriptstyle {\rm EA}}]$}, then
ultrametricity implies
\bd
\Cblu{D(t,t')= \min \{ C(t,t'),q_0 \}}
\ed

\bc
\includegraphics[width=\textwidth]{area.eps}
\ec

We evolve simultaneously 2 replicas constrained to have an overlap
$q_0$ and we check if the data stay on the ultrametric bound.
\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

\bc
\Cblu{EA $3D$: $T=0.7\simeq \frac34 T_c$, $t_w=10^6,10^7$}
\includegraphics[width=\textwidth]{plot_sg.eps}
\ec

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

In a pure or diluted ferromagnet ultrametricity is not expected to
hold.

\bc
\Cblu{Ferro $2D$: $T=1.5$, $t_w=32,64,128$}\\
\Cblu{Diluted Ferro $3D$: $T=1.67$, $t_w=10^2,10^3,10^4$}
\includegraphics[width=\textwidth]{ferro_dilu.eps}
\ec

\end{slide*} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide*}

\COra{Zooming} in the region near the ultrametric bound and comparing
the EA model (where we expect ultrametricity) and the diluted
ferromagnetic (which is not ultrametric), we can see the different
behavior.

\bc
\includegraphics[width=\textwidth]{zoom.eps}
\ec

\end{slide*} 

\end{document}
