On strong disorder renormalization

COND-MAT, Featured — By Ariel Amir on March 18, 2011 at 8:22 pm

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Ariel Amir has recently got his PhD from the Weizmann Institute of Science, Israel. He is moving to Boston in July 2011 to join Harvard as a junior fellow.

Last time I talked about the fate of Anderson localization when the disorder fluctuates in time. This time I will talk about another localization problem, of a particular class of random matrices. The mathematical model I will describe in the following can be used to describe various physical problems: the localization of phonons in a random media, a particle diffusing in a random environment, as well as relaxations in an electron glass. The work was done together with Prof. Yuval Oreg and Prof. Yoseph Imry at the Weizmann Institute, and is published in Phys. Rev. Lett. 105, 070601 (2010).

It is well established that if we consider a tight-binding model (see my previous blog if you are not familiar with tight-binding Hamiltonians) with on-site energies which are random, independent variables, the wavefunctions can either decay exponentially (localized) or be extended over space (delocalized), depending on the system’s dimensionality and the relative strength of the disorder (the magnitude of the fluctuating on-site energy) and the tunneling matrix element.

It is also well known that if we construct the equations of motion for a system of masses and springs, we can find N independent eigenmodes (where N is the number of degrees of freedom) that oscillate with a given eigenfrequency. To find the eigenmodes and eigenfrequencies, we have to diagonalize a matrix whose off-diagonals are related to the spring constants between the different masses, and with the diagonal such that the sum of every row vanishes. If the springs are random, we have a similar problem to the previous one (the tight-binding Hamiltonian): we have to diagonalize a matrix with random entities, albeit with a different type of disorder (both “diagonal disorder” and “off-diagonal disorder” exists in the phonon problem) and a “strange” connection between the fluctuations of the diagonal and off-diagonal matrix elements, arising from the vanishing rows rule (which comes about, essentially, from momentum conservation). Can the vibrations of this disordered system be localized in space?

In the paper, we define an ensemble of random matrices which is related to the above phonon problem, but, as emphasized before, once formulated mathematically is also adequate to describe different physical problems. To define the model, let me present a prescription to generate a random matrix from the ensemble considered:

1. Choose N points randomly in a d-dimensional box, with a uniform distribution.

2. The off-diagonal matrix elements are chosen as an exponential of the Euclidean distance between the sites.

3. The diagonal of the matrix is chosen as minus the sum of the rest of the row (or column, since the matrix is symmetric).

Another simplification we make is that the system has low density, namely, that the length characterizing the exponential decay of the matrix elements, $\xi$ , is much smaller than $r_{nn}$ , the average nearest neighbor distance. We thus have a small parameter $\epsilon{}\equiv{}\xi{}/r_{nn}$ .

To connect this with the masses and springs model mentioned previously, one should think of the points as masses, and of the off-diagonal matrix elements as the springs. Then, this matrix is the one discussed before corresponding to the normal modes (“phonons”) of the mechanical system.

The question we ask in this paper, regards these random matrices. What do the eigenmodes look like? (for the masses and springs model, these are the “phonons”). What is the density-of-states of the eigenvalues? In fact, our motivation was not the phonon problem, but arose from our study of relaxations in electron glasses ,which turns out to map mathematically to the same form of matrices (I will not explain what electron glasses are or describe the relation, but you can see Phys. Rev. B 77, 165207 (2008) and Phys. Rev. Lett. 103, 1264023 (2009) for more details).

These analogies between different physical models which have the same underlying mathematical structure is very useful, and before going into the results let me mention another problem which is described by the same matrix, and which one can draw further intuition from: diffusion of a particle in a random environment. Here, we know that it is useful to consider the matrix describing the Markov process, whose off diagonals are the transition rates between sites, and the diagonal is such that the sum of columns vanishes (this property comes from conservation of the particle number this time).

In the following, I will describe the results for the density-of-states (DOS), as well as the localization properties of the model, which we obtain by using a strong disorder renormalization group approach. I should stress that the field of diffusion in a random environment is vast, as is the field of vibrations in disordered materials, and the results presented here are for a definite model, with its particular form of disorder. In the following, to have a concrete physical system in mind, I will focus on the masses and springs interpretation of the matrices.

Let me first discuss the DOS. It is possible to prove that all eigenvalues $\lambda$ are negative, except one which is exactly zero. This in fact must be the case since $\lambda=-m{}\omega^2$ in the phonon problem. This is in contrast to the “usual” Anderson localization problem. We find that the DOS is different from the spectrum of phonons in an ordered lattice: in the former, there is a broad range over which the spectrum approximately follows $P(\lambda{})\sim{}1/\lambda$ . We find a formula that also accounts for the deviations from this form, plotted in Figure 3. The results for the DOS have important implications in the glass relaxation problem I mentioned above, and lead to slow, logarithmic relaxations, measurable over the course of hours and days, see Phys. Rev. Lett. 103, 126403 (2009) for more details.

To find the DOS, we calculated its moments, which can be related to the average of certain diagrams, and then reconstructed the distribution from its moments. As far as I know, this method was used by Wigner to find the beautiful semi-circle law for the Gaussian random matrix ensembles (there, the moments turn out to be the Catalan numbers).

Let me proceed to the second main result of the work, which addresses the eigenmodes (i.e., the localization properties). Here, we employed a strong disorder renormalization group (RG), which is similar in spirit to that used by Dasgupta and Ma, Bhatt and Lee and Daniel Fisher to study certain spin systems, and recently used by Lee et al. to study a synchronization problem in 1D (see references in our paper). Here, the mechanical picture of masses and springs will be particularly useful to comprehend the RG process.

At each step of the RG process, we choose the two masses connected by the largest spring, and claim that they form an approximate eigenmode (to justify this step, we take advantage of the “small parameter” $\epsilon$ , corresponding to the disorder in the spring constants being large). We record the oscillation frequency, and now, since the masses have a strong spring between them, we combine them together to make a larger, single mass. In this RG step, we eliminated one degree of freedom of the system, and found one eigenmode.

Repeating the process over and over again would yield growing clusters, see Figure 1.

On strong disorder renormalization

So we conclude that the eigenmodes (phonons) are localized clusters. The high frequency ones correspond to pairs of points, while as we go to lower and lower frequencies their size increases. This is demonstrated numerically in Fig. 2, showing 4 eigenmodes of a particular 5000 X 5000 random matrix, found by numerical diagonalization (not using the RG). The most negative eigenvalues are close to -2, and correspond to two masses moving in anti-phase (top left eigenmode in the figure). As one goes to lower eigenvalues, the eigenmodes contain a larger number of points, corresponding to larger clusters, which are localized in space (in the figure, the x axis is an arbitrary index, so it does not reflect this fact).

On strong disorder renormalization

By combining the results of the DOS discussed earlier, with this RG procedure, we found an analytical form for the size of the cluster as a function of eigenvalue, showing that it diverges as one goes to zero frequency. This makes sense since we have an extended zero frequency mode, corresponding to the center-of-mass motion.

In a recent interesting work, http://arxiv.org/abs/1010.1627, Monthus and Garel study the same model, and correctly point out that, in general, one should choose not the two masses connected with the largest spring, but those two masses which have the largest frequency. However, in the case above in which initially all masses are equal, this is not important, since as I shall shortly explain the mass distribution stays narrow along the RG process. The following graph compares the DOS obtained by exact diagonalization with that obtained by the RG, as well as with the analytic formula mentioned above.

On strong disorder renormalization

Let me conclude with explaining why the distribution of masses stays narrow in the process. We assume that the clustering process is completely random: i.e., at every step two random clusters are chosen and glued together. So we have a simple statistical problem: initially, we have a bag full of unit masses, and at every step two of them are chosen and united to be one mass. What is the resulting distribution of masses? One can show that in the continuous limit, the flow of the mass distribution is described by the following integro-differential equation:

${}\frac{dP}{dt}=-P(m,t)+\int{}dm{}P(m')P(m-m'){}$ ,

where ${}t=-\log{}((N-k)/N)$ , and k is the number of steps.

This equation was written down almost a hundred years ago by Smoulochovskii in the context of molecules sticking to each other and clustering, and is known as the Smoluchowski coagulation equation. $e^{-m{}e^{-t}}{}e^{-t}{}$ is a solution, and it can be shown that starting with any initial conditions one would converge to this solution (see M. Aizenman and T. Bak, Commun. Math. Phys. 65, 203 (1979)).

This shows that the mass distribution will be exponential, which is narrow in comparison to the broad distribution of spring constants.

To summarize, I have discussed here the localization properties of a class of random matrices which correspond to a variety of physical models, and shown that they are characterized by a particular distribution of eigenvalues, and have localized eigenmodes whose spatial size diverges as one approaches zero eigenvalues.

There are many other important open problems related to the model I discussed here. What happens at higher densities? (a lot of work was done on this aspect of the problem by Parisi and coworkers). Is the RG process exact at low frequencies? (see Monthus and Garel’s paper). What are the elastic properties of the disordered phonon system, and its thermal conductance? (we are working on these now, in collaboration with Prof. Vincenzo Vitelli at Leiden University).

Literature

M. Mehta. Random matrices
K. Efetov. Supersymmetry in disorder and chaos
G. Anderson et al. An introduction to random matrices
S. Alexander, J. Bernasconi, W. R. Schneider and R. Orbach, Excitation dynamics in random one-dimensional systems, Rev. Mod. Phys. 53, 175?198 (1981)

15 Comments

Dmitry says:

March 18, 2011 at 8:42 pm

Dear Ariel,

Thanks again for the nice post! Before I get to the point , let me understand the formal setup a bit better.

1) I guess, we are talking systems with random Hamiltonians such that the correlation properties of their matrix elements are

$\langle{}H_{ij}H_{ij}\rangle\sim\exp{}(-r_{ij}/\xi{})$

with $i\ne{}j$ , is it correct? Or are we talking about Hamiltonians with off-diagonal matrix elements decaying as stated above?

2) did I understand correctly that diagonal matrix elements are zero? If so, is this property related to detailed balance – a particle can hope between two cites and back, the probabilities to hope forth and back are equal?

Cheers

Reply
- Ariel Amir says:
  
  March 19, 2011 at 3:46 am
  
  Dear Dmitry,
  Let me try to clarify.
  1) The points are chosen randomly in space, and then the off-diagonal matrix element is chosen as the exponential of the distance. Therefore we are talking about a Hamiltonian with (random) off-diagonals matrix elements. The average you mention above will not depend on i and j, since the points are chosen randomly and there is no difference between different indices.
  2) The diagonal matrix element is non-zero, and is chosen as minus the sum of the off-diagonal matrix elements in the same row or column. This property arise from a conservation law in the problem, and modifies the localization properties significantly. For a hopping problem, this will arise from the particle number conservation.
  
  Best,
  
  Ariel
  
  Reply
  - Dmitry says:
    
    March 19, 2011 at 7:31 pm
    
    Hi Ariel,
    
    Regarding (1) – are we talking about Gaussian disorder completely determined by the two-point function $\langle{}H_{ij}H_{ij}\rangle$ ? Later stated results regarding the distribution of eigenvalues – are they based on the assumption of Gaussianity of the disorder?
    
    Another thing regarding (1) – you say that $\xi$ is the localization length. As far as I remember, in Anderson localization $\xi$ itself is a function of the disorder strength $\Delta$ (this one is determined from the correlation properties of the random potential
    
    $\langle{}V(x)V(x')\rangle\sim\Delta\delta{}(x-x')$
    
    ). Dependence of $\xi$ on disorder strength $\Delta$ is very different for different dimensionalities (for example, we know that localization is absent for $d\ge{}3$ ). Do you now want to parametrize all physics in the problem by choosing the value of $\xi$ and keeping the exponential dependence $\exp{}(-r_{ij}/\xi)$ for any dimensionality of the problem?
    
    Cheers
    
    P.S. Regarding (2) – thanks for clarification. Indeed, diagonal matrix elements are determined from detailed balance: if you solve Fokker-Planck
    
    $\partial{}P_i/\partial{t}=\sum{}H_{ij}P_j$
    
    with random , the probability distribution settles down to equilibrium time independent value which is determined by the detailed balance eq.
    
    $\sum{}H_{ij}P_j=0$ ,
    
    so it’s indeed probability conservation (or particle number) condition.
    
    Reply
    - Ariel Amir says:
      
      March 20, 2011 at 2:51 pm
      
      In the model under study, the disorder originates from the random positions of the points in space, with a uniform distribution, and then taking matrix elements which depend on the Euclidean distances between points (exponentially). No Gaussian distribution of any variable is assumed, and therefore one does not expect the disorder to be Gaussian.
      
      One can see this, for example, by calculating the distribution of a given matrix element, which is indeed found to be different than Gaussian. But even more importantly, the way the random matrix is generated implies there are non-trivial correlations between different matrix elements (e.g.: think about the triangle inequality for the geometric distances), so even knowing the full distribution of a particular matrix element would not fully characterize the ensemble. Gaussianity assumptions, therefore, are unjustified and not used in any proof.
      
      Regarding the second point, indeed, it is interesting to ask what the localization length is in a given dimensionality and for a given disorder strength. Here, the strength of the disorder is defined by the dimensionless parameter $\epsilon$ , assumed small (which corresponds to strong disorder). In this regime, we found that the localization length diverges as one goes to zero frequency (see the second figure above), and the number of points in a localization volume is approximately given by:
      
      $N{}\sim{}\exp{(-C{}\epsilon^d{}|log[-\lambda/2]|^d)}$ ,
      
      with C a constant of order unity, and d the dimensionality.
      
      The states are localized, since we are discussing the strong disorder regime only.
      
      To obtain this formula, the result for the distribution of eigenvalues was combined with the RG approach.
      
      Reply
      - Dmitry says:
        
        March 20, 2011 at 9:00 pm
        
        > with a uniform distribution
        
        Thank you, now I (probably) got it.
        
        Let me consider a particular example among those you mention (maybe then I’ll understand better what you guys are doing) – diffusion in random environment. We have N sites and particle can hop between them. Probability to find a particle at the site i is determined by the Fokker-Planck eq.
        
        $\partial{}P_i/\partial{}t=\sum_j{}(H_{ij}P_j-H_{ji}P_i)$
        
        (the last term takes care of the probability conservation as you mentioned in the first reply).
        
        Your setup seems to correspond to the situation when $H_{ij}=exp(-r_{ij}/\xi{})$ and $r_{ij}$ is random uniformly distributed quantity. Am I correct?
        
        I a kind of have irregular lattice with different distances between adjacent sites. If adjacent sites are close, the particle hops easily, otherwise hopping is exponentially suppressed (and this is the situation you are really interested in saying that your disorder is strong).
        
        Now, at $t\to\infty$ Fokker-Planck reaches time independent asymptotics $P(\infty{})$ and you say that localization length goes to infinity in this limit, so $P(\infty{})$ is spread along the lattice.
        
        a) What is exactly the effect of disorder then – suppression of diffusion? I get to $P(\infty{})$ more slowly with disorder then without disorder?
        
        b) for a given physical setup I have a particular realization of disorder. For example, the form of $P(\infty{})$ depends on this realization. What is the meaning of averaging over disorder then?
Ariel Amir says:

March 21, 2011 at 3:05 am

Thanks for this discussion, I think it’s a nice extension of the blog.
I mentioned the connection to diffusion in random environments only very briefly in the blog (so as not to confuse, although it’s exactly the same maths as for the other applications).
Let me elaborate a little.
Indeed, as you correctly mention, we study a particular model for the random environment, in which the hops are exponential in the distance (although, I should say, diffusion was not our initial motivation for this study but the problem of relaxation in glasses). Such a diffusion problem actually occurs in reality in rather useful and interesting systems: in the 70′s, it was understood by Lax, Scher and Montroll that this is a good model for amorphous materials that were commonly used in the Zerox industry! For a nice introductory account see Physics Today, vol. 44, pp. 26-34 (1991). Their important contribution was to understand that over a large time window, the electrons in these materials will not diffuse, but move much more slowly: exactly as you correctly guessed, and showed that under certain approximations that $x^2 \sim t^{\alpha}$ , with $\alpha <1$ . This explains the experimental data they studied well.

This brings me to the other points you mention. First of all, to see that at large times the stationary distribution is uniform, one doesn't need any sophisticated tools: since the matrix is symmetric in our case, a vector of ones is a simple eigenmode with eigenvalue zero, corresponding to the uniform stationary distribution (regardless of the disorder). The more difficult question regards the exact nature of the diffusion, related to the anomalous diffusion mentioned above. It can be shown that the Laplace transform of the eigenvalue distribution which we found gives precisely the return probability, so by knowing the spectral properties one can also learn about the diffusion problem!
Finally, let me address the question of averaging over the disorder. In many problems in physics, taking a large system is equivalent to averaging over the disorder (the system is called "self-averaging" in this case), and this is also the case here. In other words, if you take a large (single) sample, and calculate its spectrum, it will be close to the one calculated by averaging over the disorder.

Reply
- Dmitry says:
  
  March 21, 2011 at 8:08 pm
  
  Hi Ariel,
  
  Thanks again for the post and explanation! Actually, as you will see in the end, I have in mind a rather nice application of your problem to cosmology hopefully, it will be interesting to you.
  
  By the way, you mention in the paper at some point that for 1D anomalous diffusion goes as
  
  $\langle{}x^2\rangle\sim\log{}t$
  
  Sinai and collaborators were studying anomalous diffusion in 1D disordered systems back in the end of 70s and found that it goes even slower:
  
  $\langle{}x^2\rangle\sim\log^{1/4}t$ .
  
  I can send you the preprint if you want. This result was known to Fisher whom you thank in the acknowledgements (he also analyzed anomalous diffusion by RG in 80s – but I think, he was doing random potential).
  
  Reply
  - Ariel Amir says:
    
    March 22, 2011 at 2:11 am
    
    Hi Dmitry,
    I am curious to hear about another possible application!
    Regarding your comment, the nature of the anomalous diffusion depends on the details of the random environment, which is the source of difference between the two forms you mention.
    In the case of exponential dependence on the distance that we study here (in any dimension), one can simplify the problem significantly in 1D in the case of strong disorder (i.e., short localization length compared to the nearest-neighbor distance). In this case, you can order the points, and neglect all couplings except the nearest neighbor. If you compute the distribution of the nearest-neighbor couplings, you get a power-law in this case! (since the distances to the nearest-neighbor are exponentially distributed). This problem (a 1D chain with a power-law distribution of springs) was solved in the beginning of the 80′s (see our paper for the references). One finds anomalous diffusion which goes as $t^ \epsilon$ , which for $\epsilon \rightarrow 0$ approaches a logarithm. In the Supplementary Materials of our paper, BTW, you can find a simple non-rigorous derivation of the power-law spectrum in 1D, as well as a derivation using the RG procedure.
    
    Reply
    - Dmitry says:
      
      March 22, 2011 at 8:28 pm
      
      Ariel,
      
      how to see that if the points are distributed uniformly, distances are distributed exponentially?
      
      Reply
      - Ariel Amir says:
        
        March 23, 2011 at 3:25 am
        
        First, let me emphasize that the exponential distance distribution is only correct in 1D. To prove it, consider the probability that a given interval is larger than some distance r. For this to happen, all other points must not lie in this interval, and therefore the probability is:
        $(1-r/L)^{N-1}$ . since , the density, one obtains that for large N the dependence is exponential. To obtain the distribution of distances, we have to differentiate, but we still have an exponential form.
        A possibly useful analogy is the following: think about the points as times of, say, radioactive decays of a bunch of atoms. Indeed, we know that in this case the intervals are exponentially distributed, and that the points are uniformly distributed.
Dmitry says:

March 27, 2011 at 9:01 pm

Hi Ariel,

thanks for the explanation!

Let me now discuss a bit the application I had in mind. It has to do with eternal inflation (if you don’t quite remember what it is about, try searching this blog, I have discussed it several times). Basically, if the underlying theory has many vacua with different values of the cosmological constant, eternal inflation is governed by the Fokker-Planck equation

$\frac{\partial{}P_i}{\partial{}t}=\sum_j\left(\Gamma_{ij}P_j-\Gamma_{ji}P_i$ ,

where

$\Gamma_{ij}\sim{}\exp\left(-S_{ij}\right)$

is the Coleman-de Luccia rate of tunneling between the vacua and . The tunneling action in the limit when cosmological constants for both vacua are small comared to the Planckian scale is of the order

$S\sim\frac{M_P^2}{\Lambda}$ ,

$\Lambda\ll{}M_P^2$

where $\Lambda$ is cosmological constant and is the Planckian mass.

Now, if there is significant disorder present in the distribution of values of $\Lambda$ among the vacua, this system seems to be reduced to the one you discuss (I am not sure though if $\Lambda$ , $1/\Lambda$ should be considered uniformly distributed to make the analogy complete).

The wild guess would be then by localization system naturally chooses vacua with small values of cosmological constant rather then with large cosmological constant, so we a kind of present a dynamical argument why the value of the cosmological constant is so small in our Hubble volume.

Another interesting thing is that since the number of vacua is extremely large and all possible tunnelings are allowed, one is effectively dealing with the infinite dimensional case $d\to\infty$ .

Generally, I think, the problem you have solved should be applicable for many different phenomena involving tunneling, since it does not really matter what we understand by $r_{ij}$ in the expression for the matrix element you use – it could be any suitable metric distance between the points satisfying appropriate inequalities – for example, classical action on the trajectory connecting points and .

Reply
- Ariel Amir says:
  
  March 29, 2011 at 1:31 am
  
  Very interesting, thanks. It indeed seems to be very close to what we discuss here. I wonder whether the distribution of modes could also play a role here? As you say, it is $1/\Lambda$ that should be uniformly distributed, but if $\Lambda$ is uniformly distributed it will only modify the form of the logarithmic corrections, and the overall picture will be similar.
  
  Reply
  - Dmitry says:
    
    April 3, 2011 at 3:53 am
    
    Yes, renormalization group should still be applicable, but crit. exponents have to be different. By the way, if we weaken the assumption of uniform distribution (say, points are distributed slightly non-uniformly), how do you expect the answer to change? is RG still applicable or will one get scaling violating corrections?
    What is the area of applicability of RG anyways?
    
    At some point I thought that it should be applicable at large times (say, large compared to the mean free time) but apparently at $t\to\infty$ RG should not be applicable since we achieve time-independent asymptotics which is not universal and depends on the details of the particular realization of disorder.
    
    Reply
James Ph. Kotsybar says:

April 2, 2011 at 12:01 pm

UNIQUE PARAMETERS
– James Ph. Kotsybar

There is only one answer to creation.
Though we don?t nearly understand it yet,
there?s but one elegant variation
emerging from initial values set
that even allows molecules to be,
much less achieve complexity of life,
or suns to burn their planets distantly
with not too much but with the needed strife.

It?s easy to view things anthropically –
nothing explained, just looked at in reverse –
or declare we see things myopically –
anomalies blind to our multiverse.

Whether we?re destined or a doubtful freak,
as far as we know so far, we?re unique.

Reply
- Dmitry says:
  
  April 3, 2011 at 3:54 am
  
  Thank you James, beautiful and inspiring as usual
  
  Reply

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On strong disorder renormalization

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