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<strong>Goal:</strong> the goal of this set of problems is to derive an estimate for the transition point for the Anderson model on the Bethe lattice.  
<strong>Goal: </strong> The goal of these problems is to understand some features of glassy dynamics (power laws, aging) in a simplified description, the so called trap model.
<br>
<br>
<strong>Techniques: </strong> cavity method, Laplace transform, stability analysis.
<strong>Techniques: </strong> extreme value statistics, asymptotic analysis.
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<br>




=== A criterion for localization ===
== A dynamical dictionary:  out-of-equilibrium, aging ==
 


<ul>
<ul>
<li> <strong> Green functions and self-energies. </strong> Given a lattice with <math> N </math> sites <math>a </math>, we call <math> |a \rangle </math> the wave function completely localised in site <math> a </math>. The Anderson model has Hamiltonian
 
<center>
 
<math>
[[File:Correlation Function.png|thumb|right|x160px|Fig. 7 - Behaviour of the correlation function in a system displaying aging.]]
H= W \sum_{a} \epsilon_a |a \rangle \langle a| - \sum_{<a, b>} V_{ab} \left(|a \rangle \langle b|+ |b \rangle \langle a| \right)
<li> '''Equilibrating dynamics.''' A system evolving with thermal dynamics (e.g. Langevin dynamics) <ins> equilibrates dynamically </ins> if there is a timescale <math> \tau_{\text{eq}} </math> beyond which the dynamical trajectories sample the configurations of the system <math> \vec{\sigma} </math> with the frequency that is prescribed by the Gibbs Boltzmann measure, <math> \sim e^{-\beta E(\vec{\sigma})} </math>, where <math> \beta </math> is the inverse temperature associated to the noise. At equilibrium, one-point functions in time, like the energy of the system, reach a stationary value (the equilibrium value predicted by thermodynamics at that temperature), while two-point functions like the correlation function
<math display="block">
C(t_w+ t, t_w)= \frac{1}{N} \sum_{i=1}^N \sigma_i(t_w) \sigma_i(t_w+t)
</math>
</math>
</center>
are <ins> time-translation invariant</ins>, meaning that <math> C(t_w+ t, t_w) \sim c(t) </math> is only a function of the difference between the two times, and does not depend on <math> t_w </math>.</li><br>
where the local fields <math> \epsilon_a </math> are random variables, independent and distributed according to <math> p(\epsilon)</math>.
 
We introduce the <ins>Green functions</ins> <math> G_{ab}(z) </math> and the <ins>local self-energies</ins> <math> \sigma_a(z)</math>: these are functions of a complex variable belonging to the upper half of the complex plane, and are defined by [NOTA SU STILTJIES]
 
<center>
 
<math>
G_{ab}(z)= \langle a| \frac{1}{z-H}| b \rangle , \quad \quad G_{aa}(z)= \langle a| \frac{1}{z-H}| a\rangle  = \frac{1}{z- \epsilon_a-\sigma_a(z)}.
</math>
</center>
It is clear that when the kinetic term <math>V </math> in the Hamiltonian vanishes, the local self-energies vanish; these quantities encode how much the energy levels <math> \epsilon_a </math> are shifted by the presence of the kinetic term <math>V </math>. They are random functions, because the Hamiltonian contains randomness. They encode properties on the spectrum of the Hamiltonian; the <ins> local density of eigenvalues </ins>  <math>\rho_{a, N}(E)</math> for an Hamiltonian of size <math> N </math> is in fact given by
<center> <math>
\rho_{a,N}(E)=-\frac{1}{\pi}\lim_{\eta \to 0} \Im  G_{aa}(E+ i \eta) = \sum_{\alpha=1}^N |\langle E_\alpha| a\rangle|^2 \delta(E-E_\alpha),
</math>
</center>
where <math> E_\alpha </math> are the eigenvalues of the Hamiltonian. [NOTA SU PLEMELJI]


<li> '''Out-of-equilibrium and aging.''' In some systems the equilibration timescale <math> \tau_{\text{eq}} </math> is extremely large/diverging with some parameter of the model (like <math>N</math>), and for very large time-scales the dynamics is <ins>out-of-equilibrium </ins>.  In glassy systems, out-of-equilibrium dynamics is often characterized by <ins>aging</ins>: the relaxation timescale of a system (how slow the system evolves) depends on the age of the system itself (on how long the system has evolved so far). Aging can be seen in the behaviour of correlation function, see Fig 7: the timescale that the system needs to leave the plateau increases with the age of the system <math> t_w </math>, meaning that the system is becoming more and more slow as it gets more and more old.
</li>
</li>
<br>
<br>


== Problems ==


<li> <strong> A criterion for localization. </strong> The local self-energies encode some information on whether localization occurs. More precisely, one can claim [CITE] that localization occurs whenever the imaginary part of <math> \sigma(E+ i\eta)</math> goes to zero when <math> \eta \to 0</math>. Given the randomness, this criterion should however be formulated probabilistically. One has:
=== Problem 7: a simple model for aging ===


<center>  
[[File:Trap.png|thumb|right|x160px|Fig 6.3 - Traps in the trap model.]]
<math>
The trap model is an abstract model for the dynamics in complex landscapes. The configuration space is a collection of <math> M \gg 1 </math> traps labeled by <math> \alpha </math> (see sketch). The model is random, since each trap is associated to a random number <math> \tau_\alpha </math> that is called the mean trapping time. The dynamics is stochastic: it is a sequence of jumps between the traps, where the system spends in a trap <math> \alpha </math>  a certain amount of time with mean value <math> \tau_\alpha</math>. This means that the probability to jump out of the trap in time <math> [t, t+dt]</math> is <math> dt/\tau_\alpha </math>. When the system exits the trap, it jumps into another one randomly chosen among the <math> M</math>. The mean trapping times are distributed as
\lim_{\eta \to 0} \lim_{N \to \infty} \mathbb{P}\left(- \Im \sigma_a(E+i \eta)>0 \right)=0 \quad \Longrightarrow \quad \text{Localization}
<math display="block">
P_\mu(\tau)= \frac{\mu \tau_0^\mu}{\tau^{1+\mu}} \quad \quad \tau \geq \tau_0
</math>
</math>
</center>
where <math> \mu </math> is a parameter. In this Problem, we aim at understanding the main features of this dynamics. <br>
  </li>
Notice that in this criterion, the probability plays the role of an order parameter (like the magnetization in ferromagnets, or the overlap in spin glasses), and the <ins> imaginary part</ins> <math> \eta </math> plays the role of a symmetry breaking field (like the magnetic field in the ferromagnet, or the coupling between replicas in spin glasses). However, the localization transition has nothing to do with equilibrium, i.e., it is not related to a change of structure of the Gibbs Boltzmann measure; rather, it is a dynamical transition. Pushing the analogy with equilibrium phase transitions, one can say that the localised phase corresponds to the disordered phase (the one in which symmetry is not broken, like the paramagnetic phase). DEPINNING
<br>


=== Problem 7.1: the Bethe lattice, recursion relations and cavity  ===
The Bethe lattice is a lattice with a regular tree structure: each node has a fixed number of neighbours <math> k+1</math>, where <math> k </math> is the branching number, and there are no loops (see sketch). In these problems we consider the Anderson model on such lattice.




<ol>
<ol>
<li><em> Green functions identities. </em> Consider an Hamiltonian split into two parts, <math> H= H_0 + V </math>. Show that the following general relation holds (Hint: perturbation theory!)
<li> <em> Ergodicity breaking.</em> Compute the average trapping time (averaging between the traps) and show that there is a critical value of <math> \mu </math> below which it diverges, signalling a non-ergodic phase: the system needs infinite time to explore the whole configuration space.
<center>
</li> <br>
<math>
 
G=G^0+ G^0 V G, \quad \quad G^0 =\frac{1}{z-H_0}, \quad \quad G =\frac{1}{z-H}.
 
 
<li> <em> Condensation.</em> Consider a dynamics running from time <math>t_w</math> to some later time <math> t_w+ t</math>: compute the typical value of the maximal trapping time <math> \tau_{\text{max}}(t) </math> encountered in this time interval, assuming that the system has spent exactly a time <math> \tau_\alpha </math> in each visited trap <math> \alpha </math>. Show that in the non-ergodic phase <math> \tau_{\text{max}}(t) \sim t </math>. Why is this interpretable as a condensation phenomenon?
</li> <br>
 
 
<li> <em>Aging and weak ergodicity breaking.</em> Justify why the correlation function for this model can be written as
<math display="block">
C(t_w + t, t_w)=\Pi(t, t_w), \quad \quad   \Pi(t, t_w)= \text{probability that systems has not jumped in }[t_w, t_w+t].
</math>
In the non-ergodic regime, one finds:
<math display="block">
\Pi(t, t_w)= \frac{\sin (\pi \mu)}{\pi}\int_{\frac{t}{t+ t_w}}^1 du (1-u)^{\mu-1}u^{-\mu}.
</math>
Why is this an indication of aging? Show that
<math display="block">
\lim_{t \to \infty} C(t_w + t, t_w)=0 \quad \text{ for finite }t_w, \quad \quad \lim_{t_w \to \infty} C(t_w + t, t_w)=1 \quad \text{ for finite }t
</math>
</math>
</center> </li><br>
This behaviour is called "weak ergodicity breaking". </li>
<br>


<li><em> Cavity equations. </em>We now apply this to a specific example: we consider a Bethe lattice, and choose one site 0 as the root. We then choose <math> V </math> to be the kinetic terms connecting the root to its <math> k+1 </math> neighbours <math> a_i </math>,
<li>  
<em>Power laws.</em> Study the asymptotic behavior of the correlation function for <math> t \ll t_w </math> and <math> t \gg t_w </math> and show that the dynamics is slow, characterized by power laws (algebraic behaviour). </li>
</ol>
<br>
 
<!---->
 
<!--=== Problem 7.2: Motivating the model: from landscapes to traps ===
<!--and spherical <math>p</math>-spin model. While for the <math>p</math>-spin we think about Langevin dynamics, for the REM we consider Monte Carlo dynamics: at each time step the system in a given configuration <math> \vec{\sigma} </math> with energy <math> E_1 </math> tries to transition to another configuration that differs with respect to the previous one by a single spin flip; let the energy of this second configuration be <math> E_2 </math>. The transition occurs with probability one if <math> E_2 <E_1 </math>, and with probability <math> e^{-\beta (E_2- E_1)}</math>  otherwise.
<ol>
<li> <em> REM: distribution of depths of traps.</em> In the REM, the energy levels are independent Gaussian variables. In Lecture 1, we have shown that the Ground State <math> E_{\min} </math> has the statistics of <math> E_{\min }=E_{\min }^{\rm typ}+ \frac{1}{\sqrt{2 \log 2}}z </math>, with  <math> z </math> Gumbel. The distribution <math> P_N^{\text{extrm}}(E) </math> of the smallest energies values <math> E_\alpha </math> among the <math> M=2^N </math> can be assumed to be the same. Show that:
<center>
<center>
<math>
<math>
V= -\sum_{i=1}^{k+1} V_{0 a_i} \left( |a_i \rangle \langle 0|+ |0 \rangle \langle a_i|\right)
P_N^{\text{extrm}}(E) \approx C_N \text{exp}\left[ \sqrt{2\log 2} E  \right], \quad \quad E<0, \quad \quad C_N \text{ normalization}
</math>
</math>
</center>
For all the <math> a_i </math> with <math> i=1, \cdots, k+1 </math> we introduce the notation
<center>
<math>
G^{\text{cav}}_{a_i} \equiv G^0_{a_i a_i}, \quad \quad \sigma^{\text{cav}}_{a_i} \equiv \sigma^0_{a_i a_i},
</math>
</center>
</center>
where <math> \sigma^0 </math> is the self energy associated to <math> G^0 </math>. Show that, due to the geometry of the lattice, with this choice of <math> V </math> the Hamiltonian <math> H_0 </math> is decoupled and <math> G^{\text{cav}}_{a_i} </math> is the local Green function that one would have obtained removing the root 0 from the lattice, i.e., creating a “cavity” (hence the suffix). Moreover, using the relation above show that
(Hint: approximate the Gumbel distribution for small argument).  </li> <br>
<center>
<li> <em> REM: trapping times.</em> The Arrhenius law states that the time needed for the system to escape from a trap of energy density <math> \epsilon<0 </math> and reach a configuration of zero energy density is <math> \tau \sim e^{-\beta N \epsilon} </math>. This is a trapping time. Given the energy distribution <math> P_N^{\text{extrm}}(E) </math>, determine the distribution of trapping times <math> P_\mu(\tau) </math>: what plays the role of <math> \mu</math>? Is the non-ergodic transition in the TRAP model consistent with what we know about the REM? </li><br>
<math>  
<li> <em> Extra: p-spin and the “trap” picture.</em> In Problems 6, we have seen that the energy landscape of the spherical <math>p</math>-spin is characterized by the threshold energy, below which plenty of minima appear. Explain why the trap model corresponds to the following picture for the dynamics: the system is trapped into minima below the threshold for exponentially large times, and then jumps from minimum to minimum passing through the threshold energy.
G_{00}(z)= \frac{1}{z-\epsilon_0 - \sum_{i=1}^{k+1} t^2_{0 a_i}G^{\text{cav}}_{a_i}(z)} 
</li>
</math>  
</ol>
</center>
<br>-->
Iterating this argument, show that if <math> \partial a_i </math> denotes the collection of “descendants" of  <math> a_i</math>, i.e. sites that are nearest neighbours of <math> a_i </math> <em> except</em> the root, then
 
<center>
== Check out: key concepts and exercises ==
<math>  
 
G^{\text{cav}}_{a_i}(z)=  \frac{1}{z-\epsilon_{a_i} - \sum_{b \in \partial a_i}t^2_{a_i b}G^{\text{cav}}_{b}(z)}, \quad \quad \sigma^{\text{cav}}_{a_i}(z)=\sum_{b \in \partial a_i}t^2_{a_i b}G^{\text{cav}}_{b}(z)=\sum_{b \in \partial a_i} \frac{t^2_{a_i b}}{z- \epsilon_b - \sigma^{\text{cav}}_{b}(z)}
Key concepts: aging, activation, time-translation invariance, out-of equilibrium dynamics, power laws, decorrelation, condensation, extreme values.
</math>  
</center>
</li>


<li><em> Equations for the distribution. </em>  Justify why the cavity functions appearing in the denominators in the last equations above are independent and identically distributed random variables, and therefore the cavity equations can be interpreted as self-consistent equations for the distribution of the cavity functions.</li>
</ol>
<br>


=== Check out: key concepts of this TD ===
In <code>Exercise 13 </code>, you will see in which sense the trap model is a good effective model to describe a dynamics exploring a complicated energy landscape, focusing on the REM landscape as an example.


=== References ===
== To know more ==
* Anderson. [https://hal.science/jpa-00246652/document]
* Bouchaud. Weak ergodicity breaking and aging in disordered systems [https://hal.science/jpa-00246652/document]
* The model is solved in Abou-Chacra, Thouless, Anderson. A selfconsistent theory of localization. Journal of Physics C: Solid State Physics 6.10 (1973)
* Biroli. A crash course on aging [https://arxiv.org/abs/cond-mat/0504681]
* Kurchan. Six out-of-equilibrium lectures [https://arxiv.org/abs/0901.1271]

Latest revision as of 22:34, 16 March 2026

Goal: The goal of these problems is to understand some features of glassy dynamics (power laws, aging) in a simplified description, the so called trap model.
Techniques: extreme value statistics, asymptotic analysis.


A dynamical dictionary: out-of-equilibrium, aging

    Fig. 7 - Behaviour of the correlation function in a system displaying aging.
  • Equilibrating dynamics. A system evolving with thermal dynamics (e.g. Langevin dynamics) equilibrates dynamically if there is a timescale τeq beyond which the dynamical trajectories sample the configurations of the system σ→ with the frequency that is prescribed by the Gibbs Boltzmann measure, ∼e−βE(σ→), where β is the inverse temperature associated to the noise. At equilibrium, one-point functions in time, like the energy of the system, reach a stationary value (the equilibrium value predicted by thermodynamics at that temperature), while two-point functions like the correlation function C(tw+t,tw)=1N∑i=1Nσi(tw)σi(tw+t) are time-translation invariant, meaning that C(tw+t,tw)∼c(t) is only a function of the difference between the two times, and does not depend on tw.

  • Out-of-equilibrium and aging. In some systems the equilibration timescale τeq is extremely large/diverging with some parameter of the model (like N), and for very large time-scales the dynamics is out-of-equilibrium . In glassy systems, out-of-equilibrium dynamics is often characterized by aging: the relaxation timescale of a system (how slow the system evolves) depends on the age of the system itself (on how long the system has evolved so far). Aging can be seen in the behaviour of correlation function, see Fig 7: the timescale that the system needs to leave the plateau increases with the age of the system tw, meaning that the system is becoming more and more slow as it gets more and more old.

    • Problems

    Problem 7: a simple model for aging

    Fig 6.3 - Traps in the trap model.

    The trap model is an abstract model for the dynamics in complex landscapes. The configuration space is a collection of M≫1 traps labeled by α (see sketch). The model is random, since each trap is associated to a random number τα that is called the mean trapping time. The dynamics is stochastic: it is a sequence of jumps between the traps, where the system spends in a trap α a certain amount of time with mean value τα. This means that the probability to jump out of the trap in time [t,t+dt] is dt/τα. When the system exits the trap, it jumps into another one randomly chosen among the M. The mean trapping times are distributed as Pμ(τ)=μτ0μτ1+μτ≥τ0 where μ is a parameter. In this Problem, we aim at understanding the main features of this dynamics.


    1. Ergodicity breaking. Compute the average trapping time (averaging between the traps) and show that there is a critical value of μ below which it diverges, signalling a non-ergodic phase: the system needs infinite time to explore the whole configuration space.

    2. Condensation. Consider a dynamics running from time tw to some later time tw+t: compute the typical value of the maximal trapping time τmax(t) encountered in this time interval, assuming that the system has spent exactly a time τα in each visited trap α. Show that in the non-ergodic phase τmax(t)∼t. Why is this interpretable as a condensation phenomenon?

    3. Aging and weak ergodicity breaking. Justify why the correlation function for this model can be written as C(tw+t,tw)=Π(t,tw),Π(t,tw)=probability that systems has not jumped in [tw,tw+t]. In the non-ergodic regime, one finds: Π(t,tw)=sin⁡(πμ)π∫tt+tw1du(1−u)μ−1u−μ. Why is this an indication of aging? Show that limt→∞C(tw+t,tw)=0 for finite tw,limtw→∞C(tw+t,tw)=1 for finite t This behaviour is called "weak ergodicity breaking".

    4. Power laws. Study the asymptotic behavior of the correlation function for t≪tw and t≫tw and show that the dynamics is slow, characterized by power laws (algebraic behaviour).



    Check out: key concepts and exercises

    Key concepts: aging, activation, time-translation invariance, out-of equilibrium dynamics, power laws, decorrelation, condensation, extreme values.


    In Exercise 13 , you will see in which sense the trap model is a good effective model to describe a dynamics exploring a complicated energy landscape, focusing on the REM landscape as an example.

    To know more

    • Bouchaud. Weak ergodicity breaking and aging in disordered systems [1]
    • Biroli. A crash course on aging [2]
    • Kurchan. Six out-of-equilibrium lectures [3]
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