Dissipative quantum disordered models
Leticia F. Cugliandolo
This article reviews recent studies of mean-field and one dimensional quantum disordered spin systems coupled to different types of dissipative environments. The main issues discussed are: (i) The real-time dynamics in the glassy phase and how they compare to the behaviour of the same models in their classical limit. (ii) The phase transition separating the ordered -- glassy -- phase from the disordered phase that, for some long-range interactions, is of second order at high temperatures and of first order close to the quantum critical point (similarly to what has been observed in random dipolar magnets). (iii) The static properties of the Griffiths phase in random Ising chains. (iv) The dependence of all these properties on the environment. The analytic and numeric techniques used to derive these results are briefly mentioned.
santafe04-5.tex
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\begin{document}

\markboth{Leticia F. Cugliandolo}{Dissipative quantum disordered models}

\catchline{}{}{}{}{}

\title{Dissipative quantum disordered models}

\title{Dissipative quantum disordered models\footnote{Contribution to 
the 12th International
Conference on Recent Progress in Many-Body Theories. Santa Fe, New Mexico, 
23-27 August 2004}}
\author{Leticia F. Cugliandolo}
\address{Laboratoire de Physique Th\'eorique et Hautes Energies, Jussieu, 
France and 
\\
Laboratoire de Physique Th\'eorique, Ecole Normale 
Sup\'erieure, Paris, France
\\
leticia@lpt.ens.fr}
\maketitle 
\begin{history}
\received{13 September 2004}
%\revised{}
\end{history}


\begin{abstract}
This article reviews recent studies of mean-field and one dimensional
quantum disordered spin systems coupled to different types of
dissipative environments. The main issues discussed are: (i) The
real-time dynamics in the glassy phase and how they compare to the
behaviour of the same models in their classical limit.  (ii) The phase
transition separating the ordered -- glassy -- phase from the
disordered phase that, for some long-range
interactions, is of second order at high temperatures and of first
order close to the quantum critical point (similarly to what has been
observed in random dipolar magnets). (iii)
The static properties of the Griffiths phase in random Ising
chains. (iv) The dependence of all these properties on
the environment.  The analytic and numeric techniques
used to derive these results are briefly mentioned.
\end{abstract}

\keywords{Quantum spin models, disorder, glassiness} 

%\pagebreak


\section{Introduction}
\label{intro}

Glasses slowly evolve towards equilibrium though never reaching it in
observable times scales.  Scientific research in this area started
more than a century ago. The amount of experimental data gathered is
huge.  Systems that would reach equilibrium in observable time-scales,
such as weakly sheared complex liquids, can be driven out of
equilibrium by external perturbations and still evolve slowl.y Powders
stay in static metastable states unless externally tapped or sheared:
these non-equilibrium perturbations slowly drive them  towards more compact
configurations.  Even if {\it a priori} very different, these systems
share many dynamic properties.

In several cases of practical interest quantum effects play an
important role.  On the experimental side spin-glass phases have been
identified in many condensed matter systems at very low
temperature. Among them we can cite the bi-layer Kagome
system\cite{Kagome} S$_r$Cr$_s$Ga$_4$O$_{19}$, the polychlore
structure\cite{pyrochlore} Li$_x$Zn$_{1-x}$V$_2$O$_4$, the dipolar
magnet\cite{aeppli,aeppli-first} LiHo$_x$Y$_{1-x}$F$_4$ and the high
$T_c$ compound\cite{highTc} La$_{1-x}$Sr$_2$Cu$_2$O$_4$. Quantum
glassy phases exit also in electronic systems\cite{Zvi} 
and structural glasses\cite{Osheroff} such
as Mylar.  The driven case is also very
important in quantum systems, think of an electronic device driven by
an external current.

The non-equilibrium dynamics of classical glasses, 
weakly driven complex liquids
and granular matter have been rationalized within a theoretical
approach that is based on the solution of mean-field simple 
models.\cite{LesHouches}
How much of the classical glassy phenomenology survives at very low
temperatures where quantum effects are important is a question that
deserves careful theoretical and experimental analysis.  The
impossibility of simulating the real-time evolution of quantum systems
of moderate size enhances the importance of {\it solving} simple
mean-field or low dimensional models.

On the other hand, peculiar phenomena in quantum phase transitions
have been signaled analytically and experimentally in systems with and
without quenched disorder.\cite{Sachdev-book,Senthil}  For instance,
at low temperatures and intermediate dipole concentration, the
dipolar-coupled Ising magnet LiHo$_x$Y$_{1-x}$F$_4$ in a transverse
field exhibits a spin-glass-like phase.\cite{aeppli,aeppli-first} The
phase transition  is of second order at low transverse field but
becomes first order close to the quantum critical
point.\cite{aeppli-first}  Morever, the relaxation in the glassy phase
is extremely slow and has very strong memory effects.

Another hallmark of 
finite dimensional disordered quantum spin models are
Griffiths-McCoy
singularities, that lead to a  highly non trivial paramagnetic phase
and critical behaviour.

In this contribution we summarize the results of 
%work done in
%collaboration with G. Biroli, T. Giamarchi, D. R. Grempel, P. Le
%Doussal, G. Lozano, H. Lozza and C. A. da Silva Santos on the 
recent studies of
the statics, dynamics and critical properties
of mean-field and one dimensional quantum
disordered spin systems coupled to an environment.\cite{Culo-Culolo}
%\cite{Culo,Cugrsa1,Cugrsa2,Bicu,bath-spherical,bath-SU2,Culolo}.
We also briefly mention related studies on a driven mesoscopic
ring,\cite{Lili} dilute antiferromagnets,\cite{Malcolm} and a manifold
in an infinite dimensional quenched random potential.\cite{Pierre} In
Sect.~\ref{questions} we review the main questions that we attempted
to answer in these papers. In Sect.~\ref{models} we recall the
definition of the models that we studied. 
The basic techniques used to study classical glassy models with or
without disorder are well documented in the literature (the replica
trick, scaling arguments and droplet theories, the dynamic functional
method used to derive macroscopic equations from the microscopic
Langevin dynamics, functional renormalization, Montecarlo and
molecular dynamic numerical methods). On the contrary, the techniques
needed to deal with the statics and dynamics of quantum macroscopic
systems are much less known in general.  We briefly
mention the ones that we used to study 
dissipative disordered quantum models.\cite{Boulder}
Finally, in Sect.~\ref{perspectives} we list some
projects for future research.

\section{Questions and main results}
\label{questions}

%Let us give a short description of the questions asked to characterize
%the physical properties of quantum disordered systems as well as some
%of the answers found.

%\vspace{0.5cm}
%\noindent{\it 
\subsection{Effect of quantum fluctuations on glassiness}
%\vspace{0.5cm}

Since glasses are not expected to reach equilibrium in experimentally
accessible times, it is important to device a method to understand the
influence of quantum fluctuations on their trully nonequilibrium {\it
real-time} dynamics.  Intuitively, one expects quantum fluctuations to
only affect the short-time dynamics; however, they are also expected
to act as thermal fluctuations. It is then not clear {\it a priori}
whether quantum fluctuations would tend to destroy glassiness or
modify it drastically.

The usual methods of equilibrium quantum statistical mechanics are
inappropriate to describe this nonstationary situation. 
We presented a formalism suited to study the real-time dynamics of a
general nonlinear, possibly disordered, model in contact with a
bath that can also be applied to glassy models.\cite{Culo} 
The method is a combination of the Schwinger-Keldysh
or closed-time path technique to study real-time phenomena, with the
Feynman-Vernon approach to dissipation that consists in modelling the
coupling to the environment with an ensemble of quantum harmonic
oscillators.  As a particular case we studied the relaxation of the
spherical version of the quantum $p$ spin fully-connected disordered
model. We analyzed the relaxation of `random initial
conditions' in the limit of vanishing coupling strength taken after
the long waiting-time ($t_w$) limit.  The same technique was applied to
other quantum problems\cite{Lev}$^-$\cite{Premi}
%\cite{Lev,Kech,Bipa,Schmalian,Premi} 
and related studies appeared.\cite{other-quantum}$^-$\cite{Galitsky}
%\cite{other-quantum,Stamp,quantum-mct,noneq-quantum,Galitsky}.  
We later studied the effect of a strong coupling to the
environment\cite{bath-spherical} ({\it i.e.} $t_w\to\infty$
with $\alpha$ finite) as discussed below.

In the disordered phase the dynamics is fast and occurs in
equilibrium.  The correlation and linear response are stationary,
{\it i.e.} $C(t + t_w,t_w)=C(t)$
and $R(t+t_w,t_w)=R(t)$. They
both oscillate with a $\Gamma$-dependent 
frequency that also depends on 
the characteristics of the bath $(\alpha,s)$
if $\alpha$ is not taken to zero.  
Correlations and responses are linked by the quantum
fluctuation dissipation theorem ({\sc fdt}).  At high temperatures and
after a short transient the system decoheres and the dynamics becomes
classical ({\it e.g.} responses and correlations are related
by the classical {\sc fdt}).

In the ordered phase the glassy dynamics persists asymptotically if
the thermodynamic limit has been taken at the outset.  The dynamics of
glassy systems occurs out of equilibrium and the correlations and
responses loose time translation invariance.  If $t_w$ denotes the
time elapsed since a quench into the {\sc sg} phase, $C(t + t_w,t_w)$
and $R(t+t_w,t_w)$ depend on both $t$ and $t_w$. The order in which
the limits $t_w\to\infty$ and $t\to\infty$ are taken is very
important. For sufficiently long $t$ and $t_w$ but in the regime $t
\ll t_w$, the dynamics is stationary and the correlation reaches a
plateau $q_{\sc ea}$.  A few quantum oscillations exist at very short
$t$ (and arbitrarily large value of $t_w$) and they later disappear.
For times $t > t_w$, the system enters an {\it aging} regime where the
correlation function depends on $t_w$ explicitly.  In
this regime, the correlation vanishes at long times,
$\lim_{t \to \infty} C(t + t_w,t_w) = 0$, at a rate that depends on
$t_w$. The behaviour is thus qualitative similar to what observed
classically, even if the scaling laws are modified by the quantum
fluctuations. One checks that the relaxation of typical 
(highly energetical) initial conditions approaches a threshold 
level in phase space with a higher energy-density than the equilibrium 
one.

The comparison of responses and correlations in the ordered phase is
particularly interesting. The quantum {\sc fdt} is a complicated
integral relation between the correlation and the linear
response. This relation holds for $t\ll t_w$ when the correlation
decays to the plateau. In the second regime the quantum {\sc fdt} is
no longer verified, much as it happens in the classical problem. A
comparison of the integrated responses and the symmetrized correlation
in a parametric manner\cite{LesHouches} shows that the two quantities
are related by a {\it classical} {\sc fdt} with an effective
temperature\cite{Cukupe} $T_{\sc eff}$ that depends on the parameters
in the problem $(T, \Gamma)$ [and the characteristics of the bath,
$\alpha, s$].  We proved that $T_{\sc eff}>T$ and it is different from
zero even when the environment is at $T=0$.  $T_{eff}$ drives the
dynamics at late epochs and it makes the dynamics appear classical in
that two-time regime.  $T_{eff} > 0$ even
when the temperature of the bath is zero.  The generation of a
non-trivial $T_{eff}$ for the slow part of the decay gives
support to the ``decoherent'' effect observed in the decay of
correlations.

The similarity between the second decay in the classical and quantum
problem can be argued as follows. The responses decay rather
fast to zero when the time difference increases (though
integrated over a time-interval of very long length does not
vanish.\cite{Cuku}) The explicit dependence on $\hbar$ in the regime
of widely separated times comes from factors with higher powers of $R$
that vanish\cite{Culo}.  The effect of quantum fluctuations on the
slow time-difference regime is simply to renormalize\cite{Kech}
certain parameters in the equations that otherwise 
look classical.

Models with $p\geq 3$ interactions have different static and dynamic
phase transitions, with the latter surrounding a larger region of
the $(T, \Gamma)$ phase diagram.  The static and dynamic
ordered-disorder phase transition present a second-to-first order
transition.\cite{Niri,Cugrsa1} Close to the classical critical
temperature quantum effects are small and the phase transition is
discontinuous but of second order, as in the classical case.  There
are no discontinuities in the thermodynamic quantities but there is a
plateau that develops in the correlation function when the transition
is approached from the disordered side. This is the behaviour 
expected 
in classical glasses. Spin-glasses instead have continuous transitions
(without precursors).  Conversely, close to the quantum critical point
quantum fluctuations drive the transition first order
thermodynamically. Across the first order line the susceptibility is
discontinuous and shows hysteresis.  This is similar to what has been
observed in the dipolar-coupled Ising magnet LiHo$_x$Y$_{1-x}$F$_4$ in
a transverse field.\cite{aeppli-first}

We also adapted the {\it Ansatz} of marginal 
stability\cite{Cugrsa1,ams,Niri} to
identify the dynamic critical line that is consistent with the one
found using the Schwinger-Keldysh formalism. 
The analytic continuation of the imaginary-time dependent
correlation computed with the {\sc ams} in the absence of
the bath is identical to the {\it stationary} part of the
non-equilibrium correlation function ($C>q_{\sc ea}$) when one takes
the long-time limit first and the limit in which the coupling to the
bath goes to zero next.\cite{Cugrsa1,Pierre}  

In the classical case, the study of the Thouless-Anderson-Palmer ({\sc
tap}) free energy landscape has been very useful to understand the
behaviour of these systems.\cite{LesHouches} A {\sc tap} approach can
also be developed for quantum problems.\cite{Bicu} It helps
understanding the existence of a dynamic and a static critical line as
well as the change in nature of the transition close to the quantum
critical point.

%\vspace{0.5cm}
%\noindent{\it 
\subsection{Effect of the bath: decoherence and localization}
%\vspace{0.5cm}

The quantum systems mentioned in Sect.~\ref{intro}
are not totally isolated but in
contact with environments of different type. 


The low-energy physics of many tunneling systems is well described by
the spin-boson model in which the two equivalent degenerate states are
represented by the eigenstates $\sigma_z = \pm 1$ of an Ising
pseudo-spin. A transverse field coupled to $\sigma_x$ (say) represents
the tunneling matrix element. The coupling to the environment is given
in terms of its spectral density $I(\omega)\propto \alpha\,\omega^s$
for $\omega \ll \omega_c$, where $\alpha$ is a dimensionless coupling
constant and $\omega_c$ a high frequency cutoff.  The exponent $s$
characterizes different types of environment.  The Ohmic case ($s =
1)$ is quite generally encountered but superOhmic ($s>1$) and subOhmic
($s<1$) baths also occur in, {\it e.g.}, the Kondo effect
in unconventional hosts.

The coupling of quantum two-level systems ({\sc tls}) to a dissipative
environment has decisive effects on their dynamical properties. The
dilute case, in which interactions between the {\sc tls} can be
neglected, has been extensively
investigated.\cite{Standard,Leggett-review} This problem, is related
the $1d$ Ising model with inverse squared interactions and the
anisotropic Kondo model.  In the Ohmic case, at zero temperature,
there is a phase transition\cite{Bray-Moore} at $\alpha=1$.  For
$\alpha<1$ there is tunneling and two distinct regimes develop. If
$\alpha<1/2$ the system relaxes with damped coherent oscillations; in
the intermediate region $1/2<\alpha<1$ the system relaxes
incoherently.  For $\alpha>1$ quantum tunneling is suppressed and
$\langle \hat \sigma_z\rangle \neq 0$ signalling that the system
remains localized in the state in which it was prepared.  These
results also hold for sub-Ohmic baths while weakly damped oscillations
persist for super-Ohmic baths.  At finite temperatures (but low enough
such that thermal activation can be neglected), there is no
localization but the probability of finding the system in the state it
was prepared decreases slowly with time for $\alpha>\alpha^{\sc
c}$.

The effect of dissipation on the phase transition, critical behaviour,
ordered phase, localization and decoherence properties of macrocopic
interacting systems is only now starting to be
analyzed.\cite{bath-spherical,bath-SU2,Troyer}
In thermodynamic equilibrium, in the absence of the bath, the
interactions between the {\sc tls} lead to the appearance of an
ordered state at low enough temperature. If the
interactions are of random sign, as in the models we considered,
the latter will be a glassy state. In this phase
the symmetry between the states $\sigma_i^z = \pm 1$
 at any particular site is broken but there is no global
magnetization, $\sum_i \langle \hat \sigma_i^z \rangle = 0$.
The coupling to the bath also 
competes with the tunneling term. 
We thus expect the presence of noise to increase the
stability of the glassy state. The
consequences of this fact are
particularly interesting when there is 
localization at some $\alpha=\alpha^{\sc c}$:
% Consider a system of size $N$ with $\alpha > \alpha^{\sc crit}$ at $T=0$
%and suppose that we turn off the interactions between the {\sc tls}.
%The ground state of the system is then
%$2^N$-fold degenerate as each {\sc tls} can be in one of
%the states $\langle \hat \sigma_i^z \rangle = \pm \;\sigma_0$ (say)
%independently. If we
%now turn on an infinitesimal random interaction between the {\sc tls},
% this macroscopic degeneracy will be immediately lifted as the system
%will select among its
%$2^N$ degenerate configurations the one (or one among the ones)
%that minimizes the interaction
%energy. If we denote by $\tilde{J}$ the typical scale of the
%interactions and by $\alpha^{\sc crit}$ the localization threshold, 
a quantum critical point at ${J}=0$,
$\alpha=\alpha^{\sc c}$ separates the disordered and the ordered
state such that, for  $\alpha > \alpha^{\sc c}$, the glassy 
phase survives down to ${J}=0$.

A system of non-interacting localized {\sc tls} and a glassy state
{\it in equilibrium} are in some way similar
%: in both cases $\sum_i
%\langle \hat \sigma_i^z \rangle =0 $ and the presence of order is
%reflected by a non-vanishing value of the long-time limit of the
%correlation function, $q_{\sc ea} = \lim_{t \to \infty} N^{-1} \sum_i
%\langle \sigma_i^z(t) \sigma_i^z(0) \rangle $ (since we assume
%equilibration the correlation is stationary and the reference time can
%be taken to be zero).  
However, this resemblance is only superficial.
The details of the dynamics of the two systems are expected to be
quite different, with $C$ saturating at a finite value in the localized 
state, and $C$ decaying down to zero in the glassy phase.

  
%\vspace{0.5cm}
%\noindent{\it 
\subsection{Effect of the bath on interacting mean-field 
models} 
%\vspace{0.5cm}
 
The problem of a single {\sc tls} being a difficult one, that of an
infinite set of interacting {\sc tls} seems hardly solvable.
Therefore, as a first step, we focused on the effect of the reservoir
on the $p$-spin {\it spherical} model,\cite{bath-spherical} a problem
that we studied with the real-time approach and the replica Matsubara
technique. The position of the critical lines strongly depends on the
strength of the coupling to the bath and the type of bath (Ohmic,
subOhmic, superOhmic).  For a given type of bath, the ordered glassy
phase is favored by a stronger coupling.  The classical static and
dynamic critical temperatures 
remain unchanged by the coupling to the environment. The identity 
between the analytic continuation  of the 
imaginary-time correlation to real-time and the correlation 
in the Schwinger-Keldysh approach also holds if the strength of the 
bath is finite.

The spherical model localizes in the absence of interactions when
coupled to a subOhmic bath: $C(t+t_w,t_w)$ reaches, for any
waiting-time $t_w$ and long enough $t$, a plateau that it never
leaves.  When interactions are switched on localization disappears and
the system undergoes a phase transition towards a glassy phase.

Similar results were found for the $SU(N)$ random Heisenberg model in
the limit $N\to\infty$\cite{Bipa}, the $p=2$ spherical
model\cite{Premi} and the SU(2) $p$-spin model.\cite{bath-SU2}

%\vspace{0.5cm}
%\noindent{\it 
\subsection{Effect of disorder: Griffiths singularities}
%\vspace{0.5cm}

Griffiths singularities\cite{Griffiths}$^-$\cite{McCoy} 
in classical finite dimensional random systems
are so weak that their consequences have not been clearly observed neither
experimentally nor numerically. Instead, when quantum
fluctuations are introduced they are much stronger 
and rare regions completely determine the static and
dynamic behaviour in the Griffiths phase.
%In the $d+1$
%equivalent classical problem there is perfect correlation of disorder
%in the imaginary time direction, clusters appear as line defects in
%the path-integral representation of the statics of the quantum problem
%-- as opposed to point defects in purely classical systems -- and this
%enhances strongly the effect of the
%singularities\cite{Griffiths,Harris,McCoy}. 
%Dynamically, these regions
%have to evolve coherently and they 
%slow down the dynamics dramatically even in the paramagnet.

The isolated quantum random Ising chain  has been 
studied in great detail with a decimation technique.~\cite{dfisher}
%is equivalent to
%a $2d$ classical spin system that was studied analytically
%in\cite{McCoyWu,ShankarMurthy}.  The quantum model was later studied
%in great detail by D. S. Fisher\cite{dfisher} using an extension of
%the Dasgupta-Ma-Hu renormalization procedure\cite{Ma}.  The model
It undergoes a quantum phase transition from a {\sc pm} to a ferromagnet
for a special relation between the
distribution of exchanges and transverse fields. The quantum phase
transition is of second order with 
the correlation time scaling exponentially with the
spatial correlation length ({\it activated} scaling)
%and Griffiths-McCoy singularities are
%specially strong.  
%The dynamical exponent $z$ characterizing the
%relation between  the temporal and spatial correlation lengths at the phase
%transition varies continuously in the paramagnetic phase and {\it
%diverges} at the transition meaning that 
%There is .  This is to be confronted to the
%conventional power-law relation.
Within the renormalization group procedure this is a characteristic of 
an ``infinitely strong disorder'' fixed point.\cite{dfisher}
Within the {\sc pm} Griffiths phase 
%on the {\sc pm} side of the transition 
the distributions of local linear and non-linear
magnetic susceptibilities are large and typical and average values
are very different, with the latter being dominated by rare regions.
The behaviour is higher spatial dimensions is similar.
%In higher dimensions there are no exact results.  Thill and Huse
%developed droplet arguments to obtain a scaling description of the
%transition and Griffiths phase for the $d+1$ dimensional classical
%counterpart\cite{Thill-Huse}. 
%Later, Motrunich {\it et al}\cite{RGd} developed a
%mixed analytic-numeric approach that extends the decimation procedure
%to higher dimensions and concluded that the same `infinitely strong
%disorder' scenario, with important Griffiths effects, applies in $d>1$
%as well.

The analysis of Montecarlo simulations of the equivalent $d+1$ 
classical Ising model is quite tricky. Initially, 
it was claimed that there was conventional scaling in
$d=1$\cite{1dMC} as well as in $d>1$\cite{2dsim}
but  a more careful analysis of the numerical data confirmed the activated
scaling in both cases.\cite{1dfermions}

On the real-time dynamic side, there have been some studies of the
relaxation of special initial conditions of the isolated random Ising
chain at constant energy.\cite{IgloiRieger}


%\vspace{0.5cm}
\subsection{Disorder and dissipation: fate of Griffiths phase?}
%\vspace{0.5cm}

If the interactions between two-level systems placed in a finite
dimensional space are random one may wonder what is the effect of the
bath on the Griffiths phase. The answer to this question has been
debated over the last years.\cite{Antonio-etal}$^-$\cite{Miranda}
We addressed this problem using Montecarlo
simulations of the equivalent $2d$ classical system.~\cite{Culolo} 
Preliminary results for rather small systems ($N_x\leq 32$, $N_\tau\leq 256$)
show that an Ohmic bath favors the glassy phase. 
Our results are compatible with (but we do not prove) 
activated scaling at criticality, 
at least for small values of $\alpha$. 

\section{The models}
\label{models}

Disordered quantum spin-$\frac12$ models with two-body 
interactions are defined by 
\begin{equation}
H_S = - \sum_{ij} J_{ij} \hat \sigma^z_i \hat \sigma^z_j
+ \sum_i \Gamma_i \hat \sigma_i^x + \sum_i h_i \hat \sigma_i^z
\; .
\label{HS}
\end{equation}
%\begin{equation}
%[\hat \sigma^a_i,\sigma_j^b] = \delta_{ij} \epsilon_{abc} 
%\hat \sigma_i^c  
%\; 
%\end{equation}
%with $a,b,c=1,2,3$ associated to $x,y,z$, respectively and
$i=1,\dots, N$ labels the spins that lie on the
vertices of a cubic $d$ dimensional lattice
and are represented by Pauli matrices. The interaction strengths
$J_{ij}$ couple near-neighbours only and are chosen from a probability
distribution, $P(J)$. The average and variance are defined as
$[J_{ij}]=J_o$ and $[J_{ij}^2]=J^2/(2c)$, where $J_o$ and $J$
are $O(1)$ and $c=2d$ is the connectivity of the lattice.  The
next-to-last term is a coupling to a random quenched local transverse field
$\Gamma_i$. The last term is the coupling to a longitudinal field
that serves to compute local susceptibilities.

Several generalizations that render the model easier to treat
analytically are:

\noindent
-- {\it Fully-connected limit.}  One allows each spin to interact with
all others, $c\to N-1$.
%$d\to\infty$ limit in which the model is defined on the complete
%random graph.  The scaling of the variance, $[J_{ij}^2]\approx
%J^2/(2N)$ ensures a good thermodynamic limit, $N\to\infty$.

\noindent
-- {\it Multi-spin interactions.}
In the fully-connected case one can considers 
%$p$-spin interactions
%with Hamiltonian
\begin{equation}
H_S = - \sum_{i_1\dots i_p} J_{i_1\dots i_p} 
\hat \sigma^z_{i_1} \dots \hat \sigma^z_{i_p} 
+ \sum_i \Gamma_i \hat \sigma_i^x 
+ \sum_i h_i \hat \sigma_i^z
\; .
\end{equation} 
where the sum runs over all
$p$-uplets,\cite{pspin,Niri,Cugrsa1,bath-SU2} with $p$ an integer
parameter, $p \geq 2$.  The exchanges are random independent variables
with variance $p!{J}^2/(2 N^{p-1})$.  This model provides a mean-field
description of the structural glass transition and glassy physics that
is also intimately related to the mode-coupling
approach.\cite{LesHouches} In its dilute limit, with no geometry 
but finite connectivity on each site, this model is related
to the $K$-sat optimization problem.~\cite{Zecchina}

\noindent
-- {\it Spherical variables -- a particle in a random 
potential.}
One considers the spherical limit, 
$\sum_{i=1}^N \langle \hat \sigma_i^2\rangle =N$, in which 
the $\hat \sigma_i$ may be interpreted as the coordinates of a particle 
moving on an
$N$-dimensional sphere. A kinetic term,
$K = \sum_{i=1}^N \hat P_i^2/(2M)$, is then 
included in the Hamiltonian, 
%\begin{equation}
%\; ,
%\end{equation}
with $\hat P_i$ the conjugated momentum satisfying the commutation rules
%\begin{equation}
$[\hat P_i, \hat P_j] = 0$, 
$[\hat P_i, \hat \sigma_j] = - i \hbar \delta_{ij}  
$.
%\end{equation}
Other spherical models have been discussed.\cite{Theo-spherical}

The coupling to the environment is modelled by
%\begin{equation}
$H = H_S + H_B + H_I + H_{CT}$,
%\; ,
%\end{equation}
where $H_B$ is the Hamiltonian of the bath, $H_I$ represents the
interaction between the system and the bath and 
$H_{CT}=\sum_{l=1}^{\tilde{N}}(2 m_l \omega_{l}^2)^{-1} 
(\sum_{i=1}^{N} c_{i l} \hat{\sigma}_i^z )^2$ is a
counter-term that is usually added to eliminate an undesired mass
renormalization induced by the coupling.\cite{Standard}  
We assume that each spin
is coupled to its own set of $\tilde{N}/N$ independent harmonic
oscillators with $\tilde N$ the total number of them.  
%The bosonic
%Hamiltonian for the isolated reservoir is
%\begin{equation} 
%H_B=\sum_{l=1}^{\tilde{N}} \frac1{2m_l} \hat{p}_{l}^2 + 
%\sum_{l=1}^{\tilde{N}} \frac12 m_{l} \omega_{l}^2 \hat{x}_{l}^2 
%\; . 
%\end{equation} 
%The coordinates $\hat x_l$ and the momenta $\hat p_l$ satisfy
% canonical commutation relations.  
For simplicity we consider the
 bilinear coupling,
%\begin{equation} 
$H_{I} = -\sum_{i=1}^N  \hat{\sigma}_i^{z} \sum_{l=1}^{\tilde{N}} 
c_{i l} \hat{x}_{l} $.
%\end{equation} 
 For $p=2$ the fully-connected limit with two-body interactions 
models metallic spin-glasses.\cite{Kondo} 
%For $p > 3$ the model
%exhibits richer behavior.\cite{Niri,Cugrsa1,Bicu}
 
%\section{The techniques}
%\label{techniques}


%The statics of a disordered quantum mean-field model can be analyzed
%using the Matsubara approach to express the path-integral for the
%partition function, the replica trick to average over disorder and
%Parisi's replica symmetry breaking prescription to study the ordered
%and disordered phases.


%  the dynamics of the fully-connected models using two methods
%  especially designed to treat systems out of equilibrium.  One is
%  based on the Schwinger-Keldysh real-time approach to non-stationary
%  systems.  The second method is based on the {\it
%  Ansatz} of marginal stability ({\sc ams}) within the replica
%  analysis of the partition function.  In
%  Ref.\cite{Cugrsa1} the {\sc ams} was extensively applied to the
%  quantum spherical $p$-spin model in the absence of the bath. It was
%  shown that the position of the dynamic transition line predicted by
%  this method coincides precisely with that obtained using the
%  real-time approach. It was also shown that the time dependent
%  correlation function computed using the {\sc ams} in the absence of
%  the bath is identical to the {\it stationary} part of the
%  non-equilibrium correlation function ($C>q_{\sc ea}$) when one takes
%  the long-time limit first and the limit in which the coupling to the
%  bath goes to zero next.  The marginality condition imposed by the
%  {\it Ansatz} is intimately related to the fact that the correlation
%  will further decay from $q_{\sc ea}$ towards zero. (The details of
%  this second decay as, for instance, the two-time scaling are not
%  accessible with this method.) A localized solution with
%  $C(t+t_w,t_w)$ approaching, and never leaving, the plateau at
%  $q_{\sc ea}$ corresponds, in replica terms, to a stable replica
%  symmetry solution.  In\cite{bath-spherical,bath-SU2} 
% e extended the {\sc ams} to study
% the dynamics of models coupled
% to an environment.


\section{Perspectives}
\label{perspectives}

In this article we reviewed recent studies of insulating disordered magnets.

The principal merit of the fully-connected models is that they are
simple enough to be studied in detail. Yet, many of their properties
are generic and expected to hold at least qualitatively for more
realistic cases.\cite{LesHouches} The analysis of the fully-connected
models is by now quite complete, having applied the replica theory,
the real-time dynamics approach, and the investigation
of the {\sc tap} free-energy landscape. A problem that remains 
not fully developed though is the treatment of the relaxation of initial 
conditions that are correlated with disorder.~\cite{Cugrsa2}

Recently, much progress has been done in the study of the statics of
classical dilute spin models, that is to say, models defined on random
hyper graphs with finite connectivity. The statics of these models 
encode problems in combinatorial opimization such
as $K$-sat.\cite{Zecchina} An interesting mapping relates (isolated)
dilute quantum disordered spin systems to the dynamics of
special purpose algorithms used in combinatorial optimization -- such
as Walk-Sat.\cite{walk-sat} It would be interesting to study the
latter using tools developed for the former and {\it vise versa}.

The great challenge remains to understand the behaviour of glassy
systems with and without disorder in finite dimensions. In particular,
one could try to adapt the decimation 
technique\cite{dfisher} to study disordered spin$-\frac12$ models
in contact with an environment.
One can also envisage applications of the ideas described in this
paper to other quantum systems evolving out of equilibrium.  In this
respect, we have studied the generation of an effective temperature in
a conducting ring threaded by time-dependent magnetic field and
coupled to a reservoir;\cite{Lili} and we plan to analyse quantum dilute
antiferromagnets\cite{Malcolm} and the low-temperature
dynamics of the Bragg glass,\cite{Pierre} as well as other related physical 
problems.

\section*{Acknowledgements}

This contribution summarizes work
done in collaboration with G. Biroli,
D. Grempel, G. Lozano, H. Lozza and C. da Silva Santos. I
also wish to thank L. Arrachea, C. Chamon, T. Giamarchi, L. Ioffe,
M. Kennett, J. Kurchan, P. Le Doussal, R. Monasson, M. 
J. Rozenberg  and G. Semerjian
for collaborations on related 
problems.
I acknowledge financial support from an Ecos-Sud travel grant, an ACI
project,
%``Optimisation algorithms and quantum disordered
%systems", 
the J. S. Guggenheim Foundation and ICTP-Trieste, as well as
hospitality from the Universidad de Buenos Aires and Universidad
Nacional de La Plata, ICTP, and the KITP.


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\end{document}

[Wolff algorithm is good close to the critical line but 
it is bad deep in the pm (the size of the cluster is 
just one spin and Wolff looses time checking -- to 
discard -- all neighbours) and ferro (it flips the almost 
full lattice in each step, which is good, but it looses time 
in checking for neighbours -- that will be added with high 
probability) phases of the classical systems.] 


\subsection{Static properties}

The equilibrium density matrix reads
\begin{eqnarray}
\rho_{\sc tot}(s'',{x''_a}; s', {x'}_a) &=& \langle s'', x''_a | 
\, \hat \rho_{\sc tot} \; | s', x'_a\rangle
\nonumber \\
&=& 
\frac{1}{Z_{\sc tot}} \langle s'', x_a'' | 
\; \exp(-\beta \hat H_{\sc tot}) \; | s', x'_a\rangle
\; ,
\end{eqnarray}
where  $| s', x'_a\rangle$ represents the state of the system and 
the oscillators.
The partition function of the combined system for a particular
 realization of the bonds
\begin{equation} 
Z_{\sc tot}=\mbox{Tr}\; e^{- \beta H} 
\end{equation} 
involves a sum over all states of the system and bath.  
The expectation value of a generic observable $A$
that depends only on the state of the system reads
\begin{equation}
\langle A \rangle = 
\mbox{Tr} \; \left( A(\hat s) \hat \rho_{\sc tot} \right) 
\; 
\end{equation}  
where, again,  the trace should be computed over all the states of the system
and environment. If one constructs the reduced density operator 
$\mbox{Tr}_{\sc osc} \; \hat\rho_{\sc tot}$ 
that acts on the system's Hilbert space,
the expectation value of the observables of the system is given by
\begin{equation}
\langle A \rangle 
=
\mbox{Tr}_{\sc syst} \; \left( A(\hat s) \hat \rho_{\sc red} \right)
\end{equation}
with 
\begin{eqnarray}
\rho_{\sc red}(s'', s') = \mbox{Tr}_{\sc osc} 
\rho_{\sc tot}(s'',{x_a''}; s', {x'}_a)
\; .
\end{eqnarray}

The coupled system can be mapped onto a classical Ising spin model
using the Totter-Suzuki formalism.  This amounts to writing a
path-integral as a sum over spin and oscillator variables evaluated on
a discrete imaginary-time grid, $\tau_t=\beta\hbar t/N_\tau $, labeled
by the index $t=0,\dots,N_\tau-1$.  Periodic boundary conditions on
the discrete time-axis are imposed due to the fact that the evaluation
of the partition function and averages involve a trace. To recover the
correct representation of the trace the limit $N_\tau\to\infty$ should
be ultimately taken. The finite $N_\tau$ expression yields a sequence
of $N_\tau$-approximants to the asymptotic $N_\tau\to\infty$
formula. Moroever, the zero temperature limit of the quantum problem
is only captured when the length in the imaginary-time direction is
taken to infinity together with a vanishing spacing.


For an Ohmic bath with spectral density 
$I_{ij}(\omega) \equiv
\frac{\pi}{2}\sum_{l=1}^{\tilde{N}}\frac{c_{il} c_{jl}}{m_l \omega_{l}} 
\delta(\omega- \omega_{l})=
\delta_{ij} 2 \eta \hbar \omega \Theta(\omega_{max}-\omega)$,
with $\eta$ the friction constant, $\omega_{max}$  an ultra-violet cut-off 
and $\Theta(x)$ the Heaviside theta-function,
the reduced partition function reads
\begin{eqnarray} 
Z_{\sc red} &=& 
\mbox{Tr}_{\sc syst} \exp \left[\frac{\beta\hbar}{N_\tau} 
\sum_{t=0}^{N_\tau-1} 
H_S
 + 
\sum_{t=0}^{N_\tau-1} \sum_{i=1}^{N} \left( A+B\sigma_i^{t}\sigma_i^{(t+1)} 
\right) \right. 
\nonumber \\ 
& & \mbox{\hspace{2cm}}  \left. + 
2 \alpha \sum_{t\neq t'=0}^{N_\tau-1} \sum_{i=1}^{N}  
\left( \frac{\pi}{N_\tau} \right)^2
\frac{\sigma_i^{t} \sigma_i^{t'}}{\sin^2\left( \pi (t-t')/N_\tau\right)}
\right]
\end{eqnarray} 
where 
\begin{eqnarray} 
A &=& \frac{1}{2} \ln \left[ \sinh \left(\frac{\beta\hbar \Gamma}{N_\tau}  \right)  
\cosh \left(\frac{\beta\hbar \Gamma}{N_\tau}  \right) \right] 
\, , 
\\ 
B &=& \frac{1}{2} \ln \left[ \coth \left(\frac{\beta\hbar \Gamma}{N_\tau}  \right) 
\right] 
\; , 
\label{B-def}
\end{eqnarray} 
The trace represents the sum over all $2^{N \times N_\tau}$ distinct
classical Ising spin configurations, $\sigma_i^t=\pm 1$, for each
spin, $i=1,\dots, N$, evaluated at each time-slice, $t=0,\dots,N_\tau-1$.
To encompass all models in a single equation we kept the 
the Hamiltonian generic.

\subsubsection{Fully-connected}

The static properties are determined from the analysis of the
disordered averaged free-energy density, $f$,
\begin{equation}
-\beta N \, \overline f = \overline{ \ln Z_{\sc tot}}
\end{equation}
that is computed using the replica method to average over the random
quenched interactions:
\begin{equation}
-\beta N \; \overline{f}
=  \overline{\ln Z_{\sc tot}} =  \lim_{n \rightarrow 0} 
\frac{\overline{Z_{\sc tot}^n} -1}{n}
\; .
\label{eqreplica}
\end{equation}
After some standard manipulations, and up to some irrelevant factors,
we obtain
\begin{equation}
\overline{Z_{\sc tot}^n} =
\prod_{a,b=1}^{n} \prod_{t,t'=0}^{M-1} \int \mathcal{D}Q^{atbt'}
\mathcal{D}\Lambda^{atbt'}
\exp \left( -N P(\Lambda, Q) \right)
\label{single-spin}
\end{equation}
with
\begin{eqnarray}
&&
P(\Lambda, Q)=
\sum_{a,b=1}^{n} \sum_{t,t'=0}^{N_\tau-1}
\left(   \frac{\Lambda^{atbt'}}{N_\tau^2} Q^{atbt'} -
\frac{\beta^2 J^2}{4N_\tau^2} (Q^{atbt'})^{\bullet p} +
{\mathcal C}_{(t-t')}\delta^{ab}(1- Q^{atbt'})\right)
\nonumber \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
- \ln \mbox{Tr}_{\{\sigma^{at}\}} e^{H_{\sc eff}}
\; ,
\label{eqP}
\\
%\end{eqnarray}
%\begin{equation}
&&
H_{ eff} =
\sum_{a,b=1}^{n} \sum_{t,t'=0}^{N_\tau-1}
\frac{\Lambda^{atbt'}}{N_\tau^2} \sigma^{at} \sigma^{bt'} +
\sum_{a=1}^{n} \sum_{t=0}^{N_\tau-1}
\left(A +  B \sigma^{at} \sigma^{a(t+1)} \right)
\; .
\label{eqF}
\end{eqnarray}
where the bullet is used to distinguish the ordinary power from the
matrix power.  In the thermodynamic limit, $N\to\infty$, the integrals
in $[ Z^n]$ can be evaluated with the saddle point method at
the expense of exchanging the $N\to\infty$ and $n\to 0$ limits. The
disordered-averaged free-energy per spin is then
\begin{equation}
-\beta [f] = \lim_{n \rightarrow 0} \frac{P[\Lambda_0,Q_0]}{n}
\label{eqf}
\end{equation}
where
$\Lambda_0$ and $Q_0$ are such that,
\begin{equation}
\left.\frac{\delta P(Q,\Lambda)}{\delta Q} \right|_{Q_0,\Lambda_0}=0 \; ,
\,\,\,\,\,\,\,\,\,\,\,
\left.\frac{\delta P(Q,\Lambda)}{\delta \Lambda} \right|_{Q_0,\Lambda_0}=0
\; .
\end{equation}
Hereafter we omit subscripts in the saddle-point values $Q_0$ and
$\Lambda_0$.
The disorder-averaged entropy per spin is easily obtained from the
disorder-averaged free-energy density
and reads
\begin{eqnarray}
-\frac{\overline s}{k_B}  &=& \beta [f] +
\frac{1}{n} \sum_{a,b=1}^n \sum_{t,t'=0}^{N_\tau-1} \left[
\frac{\beta^2\hbar^2 J^2}{2 N_\tau^2}
 (Q^{atbt'})^{\bullet p}
-
\delta^{ab}(1- Q^{atbt'})
\frac{\partial {\mathcal C}_{(t-t')}}{\partial \beta}
\right]
\nonumber \\
& &
+ \frac{\beta\hbar \Gamma}{\sinh(2\beta \Gamma /N_\tau)}
\left[\cosh(2\beta\hbar \Gamma /N_\tau)-\frac{1}{n} \sum_{a=1}^n Q^{a(t+1)at}\right]
\; .
\label{eq:entropy}
\end{eqnarray}
Another physical observable of interest is
the magnetic susceptibility,
\begin{equation}
\chi = \left. \frac{\partial {\mathcal M}}{\partial h}\right|_{h=0}
= \frac{\beta}{M^2} \sum_{a,b=1}^n \sum_{t,t'=0}^{M-1} Q^{atbt'}
\label{eq:susc}
\end{equation}
where ${\mathcal M}=N^{-1} \sum_{i=1}^N \overline{\langle
\sigma_i^z\rangle} $ is the total disorder-averaged magnetization and
$h$ a longitudinal external magnetic field. 
In Eqs.~(\ref{eq:entropy}) and (\ref{eq:susc})
the right-hand-sides should be evaluated at the saddle-point
values.

The matrix elements $Q^{atbt'}$ are the order parameters of the
 model,
\begin{equation}
Q^{atbt'} =
\frac1{N} \sum_{i=1} \overline{\langle \sigma_i^{at} \sigma_i^{bt'}}
\rangle
\; .
\end{equation}
Because of the translational invariance in the Trotter time
 direction the diagonal terms in the replica indices
depend on the time-difference only,
\begin{equation}
Q^{atat'}=q_d(t-t')
\;.
\end{equation}
Notice that due to the periodic boundary condition $q_d(t)=q_d(M-t)$.
In addition, as $q_d(0)=1$, only
$(t-t')=1,2,\cdots,\mbox{int}\frac{M}{2}$ need to be considered.  The
off-diagonal elements in the replica indices, $Q^{atbt'}$ with $a\neq b$,
are $t$ and $t'$ independent as shown by Bray
and Moore\cite{Bray-Moore}.
In order to determine the different phases of the model, we consider
different {\it Ans\"atze} -- replica symmetric for the paramagnetic phase, 
stable one step replica symmetry breaking for the static spin-glass phase, 
marginally stable one step replica symmetry breaking to locate the dynamic 
phase transition -- that we shall not discuss here but can be found in 
\cite{bath-SU2}.


\begin{figure}
\begin{center}
\resizebox{12cm}{!}
{\includegraphics*[1.5in,4.7in][7in,7.7in]{dfp3static.ps}}
\resizebox{12cm}{!}
{\includegraphics*[1.5in,4.7in][7in,7.7in]{dfp10.ps}}
%\setcaptionwidth{13cm}
\caption{\footnotesize 
Top: Static phase diagram for the $p=3$ model obtained using {\sc dss}
technique for finite number of time slices $M$ and extrapolating the
data to $M \rightarrow \infty$.  The continuous line (dashed line)
indicates a second order (first order) phase transition. The critical
lines continue below the lowest value of $T$ for which we trust the
algorithm, $T\approx 0.25$, to reach a quantum critical point at
$T=0$. Bottom: comparison between the static and dynamic phase diagrams 
for the $p=10$ model. The solid line represent second order transitions
and the dashed lines first order ones. The set of curves with 
$T_s^{cl}\approx 0.6$ (thin lines) are the static transition lines for 
$\eta = 0$ (below) and $\eta=1$ (above); the set of curves with $T_s^{cl}\approx 0.82$ (bold lines) correspond to the dynamic transition for $\eta=0$
(below) and $\eta=1$ (above). 
}
\label{fig1}
\end{center}
\end{figure}

The partition function in (\ref{single-spin}) is computed by numerically performing the
{\it exact} sum, also called direct spin summation ({\sc dss})\cite{pspin},
over the $2^{N_\tau}$ possible discrete spin histories,
$\sigma_t =\pm 1$. Physical results are obtained
repeating this procedure for various values of $N_\tau$ and extrapolating
to $N_\tau\to\infty$. 

\subsubsection{Ising random chain}

ARREGLA R ESTO ***

To study the disordered spin chain we used MC simulations of the equivalent
$2d$ classical problem.  
The critical properties of the chain at the quantum critical 
point $(T=0,\Gamma_c)$ should be identical to the ones of the classical 
$2d$ problem of infinite size with layered disorder at its critical classical 
temperature $T_c$. This is based on the assumption that rescaling 
the coupling strengths should not change the critical properties.
Morever, if the critical properties are universal one can 
fix the imaginary time spacing to be $\beta\hbar/N_\tau=1$ 

In the absence of coupling to the bath, the classical $2d$ model has a
phase transition at $T_c$ given by\cite{McCoy}
\begin{equation}
2\beta_c B + \int_0^\infty d(\tilde J_{ij}) 
P(\tilde J_{ij}) \ln \tanh(\beta_c \tilde J_{ij})
=0
\; , 
\;\;\;\;\;\;\;
\mbox{with}
\;\;\;\;\;\;\;
\tilde J_{ij} = \frac{\beta}{M} J_{ij}
\; ,
\label{Tcl}
\end{equation}
with $B$ defined in Eq.~(\ref{B-def}).  Note that for {\it any}
distribution of disorder this equation determines the critical
trnasition temperature at $\alpha=0$.
Even if all pdf's of random interactions should lead to equivalent
results, the approach to the universal behaviour can be much slower in
some cases than others. This is made manifest in MC simulations by too
important finite size effects. The experience shows that a
`hierarchical' distribution is better.

All interactions are ferromagnetic. The horizontal links are quenched 
random variables taken from $P(J_{ij})$ and in the sketch take two values
represented by the solid and dashed thin lines. The value of the horizontal
link between sites $i$ and $i+1$ is the independent of the row
and the same for each column. 
The vertical links are: nearest-neighbour interactions of 
the same magnitude $B$ originating in the transverse field 
and long-range interactions due to the coupling to the bath (shown only 
on the first column to simplify the figure). The resulting 
nearest-neighbour interaction is drawn with dark solid line and the 
long-range terms are drawn with dashed-dotted lines. Note that due to 
periodic boundary conditions in the vertical direction the lines drawn 
exhaust all possibilities.

*** QUE PASO CON wmax ??? NO APARECE EN LA INTERACCION TEMPORAL??? NO ME 
ACUERDO ***


\subsubsection{Real-time dynamics}

In the typical experimental protocol used to study a system in
its glassy phase one ``quenches'' (infinitely rapid descent in
temperature from the disordered to the ordered phase)
or ``anneals'' (one tunes the external temperature with a finite cooling
rate) the sample at an initial time and subsequently follows its
evolution in time\cite{kurchan,bray,nordblad,Bowick}.
Once arrived at the final point in parameter
space, if no external forces are applied\cite{comment-forces},
the system relaxes; its energy density
decreases towards an asymptotic value. Hence, the system is
not isolated but in contact with a thermal environment that
acts as a source of dissipation.  The system plus environment is
``closed'' while the system alone is ``open''.  The nature of the
environment, {\it e.g.} whether it can be modeled by a  classical
or a quantum ensemble, will depend on the problem under study.
Right after a quench or a rapid annealing,
the system is not in equilibrium with the reservoir.
Whether it will eventually reach equilibrium in experimentally relevant
time scales depends on the nature of the system and its interaction
with the environment. Indeed, for glassy systems at low temperatures
the equilibration time scales becomes so long that it
crosses over the experimentally observable times.

The Schwinger-Keldysh formalism \cite{schwinger} 
allows one to analyse the 
real-time dynamics of a quantum system.
The starting point is the time dependent 
density operator
\begin{equation}
\hat \rho_{\sc tot}(t)
=
\exp\left(-\frac{i}{\hbar} \hat H_{\sc tot} t \right) \; 
\hat \rho_{\sc tot}(0) \;  
\exp\left(\frac{i}{\hbar} \hat H_{\sc tot} t \right)
\; .
\label{density-op}
\end{equation}
We have set the initial time $t_o$  to be $t_o=0$.
Introducing identities, an element of the time-dependent 
density matrix reads
\begin{eqnarray}
& & \rho(q'',{x_a''}; q', {x'}_a; t) 
=
\nonumber\\
& & 
\int_{-\infty}^\infty dQ dQ' dX_a dX'_a \; 
\langle q'',x''_a | \; 
\exp\left(-\frac{i}{\hbar} \hat H_{\sc tot} t\right) \; | Q, X_a \rangle
\;
\langle Q, X_a | \; \hat \rho_{\sc tot}(0) \; | Q', X'_a \rangle
\nonumber\\
& &
\;\;\;\;\;\;\;\;\;\;\;\;
\times
\langle Q',X'_a | \; \exp\left(\frac{i}{\hbar} \hat H_{\sc tot} t \right)\; 
| q', x_a' \rangle
\; .
\end{eqnarray}
The first  factor is the coordinate representation of 
the evolution operator $\exp(-i\hat H t/\hbar$ and it 
can be represented as functional integral. The third term 
can also be represented in functional form. They read:
\begin{eqnarray}
\langle q'',x_a'' | \, \exp\left(-\frac{i}{\hbar} \hat H_{\sc tot} t\right) \,
| Q ,X_a \rangle
&=&
\int_{Q}^{q''} {\cal D} q^+(t) \int_{X_a}^{{x''_a}} 
{\cal D}{x^+_a}(t)  
\; \exp\left(\frac{i}{\hbar} S^+_{\sc tot}\right)
\label{pos}\\
\langle Q',X'_a | 
\, \exp\left(\frac{i}{\hbar} \hat H_{\sc tot} t \right) \, 
| q', {x'_a} \rangle
&=&
\int {\cal D} q^-(t) \int
{\cal D}{x^-_a}(t)  
\; \exp\left(-\frac{i}{\hbar} S^-_{\sc tot}\right)
\; .
\label{neg}
\end{eqnarray}
Interestingly enough, the evolution 
operator in Eq.~(\ref{neg})
gives rise to a path integral going backwards in time,
from $q^-(t)=q', x^-_a(t)=x'_a$ to $q^-(0)=Q', x^-_a(0)=X'_a$. 
The full time-integration can then be interpreted as being closed, going 
forwards from $t_0=0$ to $t$ and then 
backwards from $t$ to $t_0=0$. This motivates the name 
``closed time path formalism''.
A doubling of degrees of freedom $(q^+,q^-)$ 
appeared and it is 
intimately linked to the introduction of Lagrange multipliers 
in the functional 
representation of the Langevin dynamics in the classical limit.

The action $S_{\sc tot}$ has the usual four contributions,
from the system, the reservoir, the 
interaction and the counterterm. The action of the system reads
\begin{equation}
S^+_{\sc syst}[\eta^+(t)] = \int_0^t dt' \, \left[ \, \frac{M}{2} 
\left(\frac{q^+(t')}{dt'}\right)^2 - V(q^+(t')) + 
\eta^+(t') q^+(t') \, \right]
\end{equation}
where we have introduced a time-dependent source $\eta^+(t)$. Similarly,
we introduce a source $\eta^-(t)$ in the path integral going backwards in time.

As usual we are interested in the dynamics of the system under
the effect of the reservoir. Hence, we compute the reduced 
density matrix
\begin{equation}
\rho_{\sc red}(q'',q';t) = \int_{-\infty}^\infty  dx_a \; 
\langle q'' , x_a | \, \hat \rho_{\sc tot}(t) \, | q' , x_a \rangle 
\; .
\end{equation}

The initial density operator $\hat \rho(0)$ has the information 
about the initial state of the whole system. If one assumes that the 
system and the bath are set in contact at the initial time, the operator 
factorizes
\begin{equation}
\hat \rho(0) = \hat \rho_{\sc syst}(0)  \hat \rho_{\sc env}(0)
\; .
\end{equation}
(Other initial preparations, where the factorization does not hold,
 can also be considered and may be more realistic in certain cases.)
If the environment is initially in equilibrium at an inverse 
temperature $\beta$, 
\begin{equation}
\hat \rho_{\sc env}(0) = \frac{1}{Z_{\sc env}} \exp(-\beta \hat H_{\sc env})
\; ,
\end{equation}
the dependence on the bath variables is
quadratic; they can traced  away 
to yield the reduced density matrix:
\begin{eqnarray}
\rho_{\sc red}(q''; q'; t) &=& 
\int_{-\infty}^\infty dQ \int_{-\infty}^\infty
dQ' \int_{q^+(0)=Q}^{q^+(t)=q''} {\cal D} q^+(t) 
\int_{q^-(0)=Q'}^{q^-(t)=q'} {\cal D} q^-(t) 
\nonumber\\
& & 
\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; 
\exp\left(\frac{i}{\hbar} S_{\sc eff} \right) 
\langle Q | \; \hat \rho_{\sc syst}(0) \; | Q' \rangle
\label{red_quant}
\end{eqnarray}
with the effective action
\begin{equation}
S_{\sc eff} = S_{\sc syst}^+ - S_{\sc syst}^- + S_{\sc th}
\; .
\end{equation}
The last term has been 
generated by the interaction with the environment and it 
reads
\begin{eqnarray}
 S_{\sc th} &=& 
- \int_0^t dt' \int_0^t dt'' \;
\left( q^+(t') - 
q^-(t') \right) \, \eta(t'-t'') \,  
\left( q^+(t'') + q^-(t'') \right) 
\nonumber\\
& & 
+i\int_0^t dt' \int_0^{t'} dt'' \; \left( q^+(t') - 
q^-(t') \right) \, \nu(t'-t'') \,  
\left( q^+(t'') - q^-(t'') \right) 
\; .
\label{Sthermal}
\end{eqnarray}
The noise  and dissipative kernels $\nu$ and $\eta$ are given by 
\begin{eqnarray}
\nu(t) &=&  \int_0^\infty d\omega I(\omega) \, 
\coth\left( \frac12 \beta \hbar \omega \right) \; 
\cos(\omega (t))
\; ,
\label{nu}
\\
\eta(t) &=& \theta(t) \frac{d\gamma(t)}{dt} =  
-\theta(t) \, 
\int_0^\infty d\omega \; I(\omega) \, \sin(\omega (t)) 
\; .
\label{eta}
\end{eqnarray}
In these equations, $I(\omega)$ is the spectral 
density of the bath:
\begin{equation}
I(\omega) =\frac{\pi}{2} \sum_{a=1}^{N_b} \frac{c_a^2}{m_a\omega_a}
\delta(\omega-\omega_a) 
\; ,
\label{spectral}
\end{equation}
that is a smooth function of $\omega$ usually taken to be 
\begin{equation}
I(\omega) = \gamma \omega \left(\frac{\omega}{\omega_s}\right)^{s-1} 
\exp(-\omega/\Lambda)
\end{equation}
with $\gamma$ the ``friction coefficient'', $\omega_s$ a constant,
$\Lambda$ a high frequency cut-off and $s$ a parameter that characterises
different baths: $s=1$ is Ohmic (and leads to the usual white noise when
$\Lambda\to\infty$), $s>1$ is super-Ohmic and $s<1$ is sub-Ohmic.
As usual, a counterterm cancels the mass renormalization. 

Next, we have to choose an initial density matrix for the system.
One natural choice, having in mind the quenching experiments
usually performed in classical system, is the diagonal 
matrix
\begin{equation}
\langle Q | \; \hat \rho_{\sc syst}(0) \; | Q'\rangle = 
\delta (Q-Q')
\end{equation}
that corresponds to a random ``high-temperature'' situation and 
that simplifies considerably the expression in (\ref{red_quant}).

In the classical limit, $\hbar\to 0$, the effective action reduces to the one 
in the functional representation of a Langevin process with coloured noise. 
This can be easily verified by renaming the variables
\begin{eqnarray}
q \equiv \frac{1}{2} (q^+ + q^-)
\;\;\;\;\;\;\;\;\;\;
i\hat q \equiv \frac{1}{\hbar} (q^+ - q^-)
\; ,
\end{eqnarray}
expanding in powers of $\hbar$, and keeping only the leading terms.
If, moreover, the limit $\Lambda\to\infty$ is taken, the kernels $\gamma$ and 
$\nu$ become proportional to $\delta$ functions and one recovers a 
white noise. 

Any two point correlation functions that depends only on the 
system is then given by 
\begin{equation}
\langle \hat A(t) \hat B(t') \rangle 
= \frac{\mbox{Tr} \hat A(t) \hat B(t') \hat \rho_{\sc red}(0)}
{\mbox{Tr} \hat \rho_{\sc red}(0)}
\; .
\label{eq-C}
\end{equation} 
As in the classical stochastic problem, the normalization 
in given by a quantity that is trivially independent of disorder,  
${\mbox{Tr} \hat \rho_{\sc red}(0)}=1$, since we have chosen 
$\hat\rho=1$. Hence, there is no need to introduce replicas in 
such a quantum dynamic calculation (if the initial conditions are not 
correlated with disorder).

Clearly $\langle \hat A(t) \hat B(t') \neq \hat B(t') \hat A(t) \rangle$
and one can define symmetrized and anti-symmetrized correlations:
\begin{eqnarray}
C_{\{A,B\}}(t,t') &=& \frac12 \, \langle A(t) B(t') + B(t') A(t) \rangle
\; ,
\label{symm}\\
C_{[A,B]}(t,t') &=&  \frac12 \,  \langle A(t) B(t') - B(t') A(t) \rangle
\; ,
\label{antisymm}
\end{eqnarray}
respectively.

If one generalizes the above system to be one described by a field
$\phi_i$ with $i=1,N$ components, 
the symmetrized correlation function is given by
\begin{eqnarray}
C_{ij}(t,t')&\equiv&\frac{1}{2} \langle  \phi_i(t) \phi_j(t')
 + \phi_j(t') \phi_i(t) \rangle  \\
 &=&
\left.
\frac{\hbar^2}{2} 
 \left[
\frac{\delta^2}{\delta \eta_i^+(t) \delta \eta_j^-(t')}
+
\frac{\delta^2}{\delta \eta_j^+(t') \delta \eta_i^-(t) }
\right]
\rho_{\sc red}
\right|_{\eta=0}
\; 
\end{eqnarray}
($C_{ij}(t,t')=C_{ji}(t',t)$).
The linear response function $R_{ij}(t,t')$ is defined as the variation of the
averaged field $\langle \phi_i\rangle$ 
with respect to the ``strength''
$f_j$ of a perturbation that modifies the potential energy
according to $V \to V - \sum_{i=1}^N f_i \phi_i$:
\begin{eqnarray}
R_{ij}(t,t')&\equiv&
\left. \frac{\delta \langle \phi_i(t) \rangle}
{\delta f_j(t')} \right|_{f=0}
\label{respdef}
\; .
\end{eqnarray}
In linear-response theory, it can be expressed in terms of the 
averaged commutator:
\begin{eqnarray}
R_{ij}(t,t')
&=& \frac{i}{\hbar} \theta(t-t')
\langle [\hat \phi_i(t),\hat \phi_j(t')] 
\rangle  \\
&=&
\left.
\frac{\hbar}{i} 
 \left[
\frac{\delta^2}{\delta \eta_i^+(t) \delta \eta_j^+(t')}
+
\frac{\delta^2}{\delta \eta_j^+(t) \delta \eta_i^-(t')}
\right]
\rho_{\sc red}
\right|_{\eta=0}
\; .
\end{eqnarray}
It is also useful to write the correlation and response 
in terms of the fields $\phi_i^+$, $\phi_i^-$:
\begin{eqnarray}
C_{ij}(t,t') 
&=&
\frac12 \; 
\langle \phi_i^+(t) \phi_j^-(t')  + \phi_j^+(t') \phi_i^-(t)  \rangle
\; ,
\label{defcorr}
\\
R_{ij}(t,t')
&=&
\frac{i}{\hbar} \, 
\langle \phi_i^+(t) \, \left( \phi_j^+(t') - \phi_j^-(t') 
\right) \rangle
\label{defresp}
\; .
\end{eqnarray}


\section{Results}

We here summarize the results found with several analytic and numerical 
studies of the statics and dynamics of fully-connected and finite 
dimensional disordered models coupled to an environment. 

In all cases we find that the coupling to the enviornment tends to 
stabilize the ordered phase. 

\subsection{Fully-connected models}

In fully-connected models of spherical\cite{bath-spherical} 
and SU(2)\cite{bath-SU2} type
the coupling to the enviromement favors the appearance of
the spin-glass phase reducing the strength of the quantum fluctuations
that tend to destabilize it.  For $p=2$ the phase transition is always
second order. For $p \ge 3$ there exists a tricritical temperature
$T^\star$ below which quantum fluctuations drive the transition first
order. $T^\star$ decreases with the strength of the coupling to the
bath.  For $p \ge 3$ a dynamic transition precedes the equilibrium phase
transition. The coupling to the bath also stabilizes the dynamic
glassy phase.

The behaviour of the quantum system is very rich. A short summary of 
results is the following.

\begin{itemize}

\item
The model has an ordered and a disordered phase separated by a 
critical surface in the $(T,\Gamma\equiv \hbar^2/(M J),\alpha)$
parameter space, where $T$ is temperature, $\Gamma$ measures the 
strength of the quantum fluctuations in the system and $\alpha$
is the strength of the coupling to the bath. The location of the 
transition also depends on the parameter $s$ that distinguishes
different types of reservoirs (Ohmic, super-Ohmic, sub-Ohmic).

\item
Actually, there are two transition surfaces, a dynamic and a static
one. The former is obtained from the real-time dynamic analysis 
while the latter follows from the study of the partition function 
using the Matsubara formalism. (Actually, partial information about the 
nonequilibrium dynamics of the model can also be obtained from the 
replicated Matsubara formalism if a special {\it Ansatz} for the 
replica matrices, that does not correspond to equilibrium, is used.)
Below the dynamic transition  the system is out of equilibrium.
In the classical limit this double transition was already known. 
It survives when quantum fluctuations are present.

\item  
The nature of the dynamic and static transitions changes when the 
temperature decreases: close to the classical critical point 
($\Gamma\to 0$), the 
transitions are discontinuous but of second order thermodynamically.
There are no discontinuities in the thermodynamic quantities 
but there is a plateau that develops in the correlation function 
where the transition is approached from above. This is the 
expected behaviour in classical glasses. Spin-glasses have continuous 
transitions (without precursors). 
Conversely, close to the quantum critical point ($T=0,\Gamma$ finite,
$\alpha$ fixed) the transitions become of first order thermodynamically,
{\it e.g.} the energy density jumps. A change in the order of the transition
in a quantum glass has been reported experimentally
by Aeppli, Rosenbaum and collaborators\cite{aeppli,aeppli-first}. 
 
\item
In the classical case, the study of the Thouless-Anderson-Palmer
({\sc tap}) free energy landscape
has been of mcuh help to understand the behaviour of these systems.
A {\sc tap} approach can also be developed for the quantum problem\cite{Bicu}.
It helps understanding the change in nature of the transition close to the 
quantum critical point.

\item 
In the disordered phase the dynamics is fast and it occurs in equilibrium.
The auto correlation and the linear response functions are invariant 
under time-translations. They both show oscillations, as is typical of 
a quantum problem. The frequency of the oscillations depends on $\Gamma$
and the characteristics of the bath.
Correlations and responses are linked by the quantum fluctuation
dissipation theorem {\sc fdt}. 
At high temperatures and after a short transient 
there is a decoherence effect and the dynamics becomes totally classical.
For example, responses and correlations are related by the classical
{\sc fdt}. 

\item 
In the ordered phase the dynamics is very slow and the system is 
not  equilibrated with its environment.  The autocorrelation 
functions decay in two steps, with a fast decay towards a plateau and 
a slow decay below the plateau towards zero, as a function of the
time difference $t$ and for fixed waiting-time. The first decay is 
stationary while in the second decay 
the waiting-time dependences exist for all times if the thermodynamic
limit has been taken at the outset. A few quantum oscillations appear in the 
first step of the relaxation and they later disappear. It is 
important to note that these oscillations, as functions of the time difference,
will be present for any 
arbitrarily large value of the waiting time. 

\item 
The comparison of responses and correlations in the ordered phase is 
particularly interesting. The quantum {\sc fdt} is a complicated  
integral relation between the symmetrized correlation and the 
linear response. This relation holds for time-differences that are 
short as compared with the waiting time. These correspond to the
range of time-differences when the correlation decays from $1$ to the 
plateau. In the second decay the quantum {\sc fdt} is no longer 
verified, much as it happens in the classical problem. A comparison 
of the integrated responses and the symmetrized correlation in the 
parametric manner explained in \cite{kurchan} yields a surprising
result at face value: the two quantities are related by a 
{\it classical} {\sc fdt} with an effective temperature $T_{\sc eff}$ 
that depends on the parameters in the problem $T, \Gamma, \alpha$.
A non zero effective temperature appears even when the temperature of the 
bath is zero (but the coupling to the environment still exists, 
$\alpha>0$).

\item 
The generation of a non-trivial effective temperature for the slow
part of the decay gives support to the ``decoherent'' effect 
already observed in the decay of the correlations. If the second
decay is controlled by a nontrivial, rather high, effective temperature, 
it is natural that it will look classical in all respects.

\item
The similarity between the second decay in 
the classical and quantum problem can be argued as follows (it can also be 
shown explicitly by solving the dynamic equations). The response 
functions decay rather fast to zero when the time difference
increases (the response functions integrated over a time-interval 
of very long length do not vanish though, this is the ``weak long-term 
memory'' property). The explicit dependence on $\hbar$ in the 
regime of widely separated times comes from factors with higher
powers of $R$ that vanish. The effect of quantum fluctuations 
on the slow time-difference regime is simply to renormalize \cite{claudio}
certain parameters in the equations that look classical when the asymptotic
limit is taken.

\item
For both models, the classical transition temperature, $T_s^{class}$,
corresponding to $\Gamma_s\to 0$, remains unchanged by the
coupling to the quantum heat reservoir.  This value is $T_s=J$ for
when $p=2$\cite{Mepavi} and it coincides with the
one given by Gross and M\'ezard, $T_s\approx 0.67$, for the classical
problem with $p=3$\cite{Gross}.

\end{itemize}

\begin{figure}
\begin{center}
%$
%\begin{array}{c}
\resizebox{9cm}{!}{\includegraphics*[1.5in,4.7in][7in,7.7in]
{fvscampop3conex.ps}}
%\end{center}
%\begin{center}
%\\
\resizebox{9cm}{!}{\includegraphics*[1.5in,4.7in][7in,7.7in]
{svscampop3conex.ps}}
%\end{center}
%\begin{center}
\resizebox{8.5cm}{!}{\includegraphics*[1.5in,4.7in][7in,7.7in]
{chivscampop3conex.ps}}
\end{center}
%\end{array}
%$
%\vspace{-0.5cm}
%\setcaptionwidth{13cm}
\caption{ \footnotesize Free energy density, $\overline f$, entropy,
$\overline s$, and susceptibility, $\chi$, as functions of the
transverse field, $\Gamma$, for the $p=3$ model at $T=0.5>T_s^*$ for
three values of the coupling to the bath,
$\eta=0,0.5,1$. The continuous (dashed) line corresponds to the
paramagnetic (glassy) phase.
The entropy and susceptibility
%$\overline s$ and $\chi$
are continuous at the transition indicating a second order phase
transition}
\label{fvscampop3con}
\end{figure}

\begin{figure}
\begin{center}
\resizebox{9cm}{!}{\includegraphics*[1.5in,4.7in][7in,7.7in]
{fvscampop3ex.ps}}
%\end{center}
%\begin{center}
\resizebox{9cm}{!}{\includegraphics*[1.5in,4.7in][7in,7.7in]
{svscampop3ex.ps}}
%\end{center}
%\begin{center}
\resizebox{8.5cm}{!}{\includegraphics*[1.5in,4.7in][7in,7.7in]
{chivscampop3ex.ps}}
\end{center}
%\setcaptionwidth{13cm}
\caption{\footnotesize Free energy, $\overline f$, entropy, $\overline
s$, and susceptibility, $\chi$, as function of the transverse field,
$\Gamma$, for the $p=3$ model at $T=0.3 < T_s^*$ for three
values of the coupling to the bath, $\eta=0,0.5,1$. The continuous
(dashed) line corresponds to the paramagnetic (glassy) phase. The
entropy and susceptibility are discontinuous at the transition
indicating a first order phase transition}
\label{fvscampop3}
\end{figure}

\subsection{Results for $d$-dim SU(2) random spin models}

The literature on the isolated systems is very vast. We start 
by the reviewing work on this problem and, later, we briefly 
discuss our own studies for the system in contact with an Ohmic 
environment. 


\subsubsection{The role of the environment}

The effect of an environment on the static properties and, especially,
the Griffiths phase of the quantum disordered spin-$\frac12$ models in finite
$d$ is a hottly debated issue. 


*** Simplifications of the latter (with
no spin-glass coupling between the spins) are now recieving
considerable attention\cite{Antonio-etal} due to their relevance in
solid state physics (heavy fermion behaviour in rare earth materials,
manganites).  These people want to know the phase diagram and the
properties of the quantum phase transition at $T=0$ (according to
Antonio\cite{Antonio-etal}). ***


In this Section I summarize the results found in \cite{Culolo}
using MC simulations of the classical $2d$ system.

The classical equivalent system is defined on a rectangular lattice
with dimensions $N_x \times N_\tau$. For simplicity, we used a
constant tranverse magnetic field, $\Gamma_i =\Gamma$ at all sites
$i$.  We applied a MC algorithm especially adapted to study the
critical properties of the $2d$ classical system\cite{Culolo}.  We
focus on the behavior in the PM phase close to (and at) the phase
transition where a cluster algorithm that is a variation of Wolff's
algorithm\cite{Wolff} adapted to deal with long-range ferromagnetic
interactions\cite{Luijten} combined with a histogram
method\cite{Ferren} that allows to identify the phase transition with
less computational effort is rather efficient. We concentrated on the
critical behaviour at zero-temperature, that corresponds to $\tau_c
N_\tau\to\infty$, and we used a finite {\it classical} temperature
$T_{cl}$ in the MC simulations of the classical $2d$ model that is
`proportional' to the strength of the transverse field.

We identified the critical line in the $(\alpha, T_{cl})$ plane 
by studying the averaged Binder cumulant
\begin{equation}
g_{av}\equiv 
\left[3-\frac{\langle m^4 \rangle }{\langle
m^2\rangle^2} \right]
\end{equation}
where the angular brackets indicate thermal average, and the
magnetization density, $m$, is defined as $m\equiv ( N N_\tau)^{-1}
\sum_{i=1}^N \sum_{t=1}^{N_\tau} s_{i}^{t}$.
$g_{av}$ should vary from $1$ in the ordered phase to
zero in the disordered one.

Finite size scaling in the anisotropic problem\cite{Binder} implies
that dimensionless quantities should scale as functions of
$N_x/\xi_x$, with $\xi_x$ the correlation length in the spatial
direction, and $N_\tau/N_x^z$ in the case of conventional scaling or
$N_\tau/e^{N_x^{\overline z}}$ in the case of activated scaling.  At
criticality, for each system size $N$, the Binder ratio should attain
a maximum as a function of $N_\tau$, and the value, $g_{av}^{max}$, at
the maximum should be independent of $N$\cite{Guoetal}.  For each
$T_{cl}$ we looked for the value of $\alpha$ such that the
$N_x$-independent maximum is reached thus obtaining the phase diagram.
We found that the extent of the ordewred phase increases with the
value of $\alpha$. For $\alpha < 0.3$ the critical line is rather well
described by a fit using $f(x)=a x^b - a (1.42)^b$ (The critical
temperature at $\alpha=0$ is given by\cite{McCoy} $ 2 \beta_c B +
[\ln\tanh(\beta_c J_i)] =0 $ that for the uniform distribution of
exchanges and $B=1$ yields $T_{cl}(\alpha=0)=1.42$.).  Alternatively,
we taylored the dynamic critical exponents $z$ and $\overline z$ to
give the best data collapse of the Binder parameter for different
sizes. For $(T_{cl}^c=2.5, \alpha=0.311)$ we found that $z = 0.5$ and
$\overline z = 1.85$ yield equivalently good results.

Griffiths singularities are due to localized regions with spatial
bonds that are larger than on average and that are, due to the quantum
nature of the problem, highly-correlated in the imaginary-time
direction. Their effect is more clearly seen on the behaviour of local
quantities in space.  We also studied the local linear susceptibility
to a (uniform if $h_i=h$) magnetic field oriented along the Ising axis
\begin{equation}
\chi^{(loc)}
= 
\left.
\frac{\partial}{\partial h_i} \langle \sigma_i^z \rangle \right|_{h=0}
\end{equation}
that is 
related, via the fluctuation-dissipation theorem, to
the imaginary-time correlation
\begin{equation}
\chi^{(loc)} =
\frac1{N_\tau}
\sum_{\tau=1}^{N_\tau} \langle s_i(0) s_i(\tau) \rangle
\; ;
\end{equation}
and the local non-linear susceptibility
\begin{equation}
\chi^{(loc)}_{nl}
= 
\left.
\frac{\partial^3}{\partial h^3_i} \langle \sigma_i^z \rangle \right|_{h=0}
\end{equation}
that is related to 
\begin{equation}
\chi^{(loc)}_{nl}
= 
- \frac{1}{6 N_\tau}
\left( \langle m_i^4 \rangle - 3 \langle m_i^2\rangle^2 \right)
\end{equation}
with
\begin{equation}
m_i \equiv 
\sum_{\tau=1}^{N_\tau} \langle s_i(\tau) \rangle 
\end{equation}
We measured the distributions
$P(\chi^{(loc)})$
and
$P(\chi^{(loc)}_{nl})$
constructed with $N_x\times N_\tau\times xxx$ points that are 
the results of simulating $xxx$ samples (different
realizations of disorder).
The scaling
\begin{eqnarray}
\ln P(\ln \chi^{(loc)}) &=& - \frac{d}{z} \ln \chi^{(loc)} + \mbox{const}
\\
\ln P(\ln \chi_{nl}^{(loc)}) &=& 
- \frac{d}{3z} \ln \chi_{nl}^{(loc)} + \mbox{const}
\end{eqnarray}
`define' the dynamic exponent $z$. We found that $z$ varies smoothly
through the Griffiths phase and its behaviour can be described with a
divergence at values $(\alpha_c, T_{cl}^c)$ that are compatible with
the critical line found studying the Binder ratio.

\begin{figure}
\begin{center}
\resizebox{11cm}{!}{\includegraphics*[1.5in,4.7in][7in,7.7in]{phase-diagram-chain.eps}}
\caption{\footnotesize Phase diagram of the one dimensional
random spin model coupled ot an environment.}
\end{center}
\label{fvscampop3}
\end{figure}



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\bibitem{Zecchina} R. Zecchina, in this volume.

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\end{thebibliography}