205. Multifractality and metal-insulator transition
COND-MAT, HEP-TH/PH — By Matt Foster on January 26, 2009 at 5:31 pm
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This is a guest blog post by Matthew Foster from Rutgers about his recent paper arXiv:0901.0284v1 [cond-mat.dis-nn]. Dmitry.
In most electronic materials, impurities and other defects (“quenched disorder”) play a dominant role in shaping transport phenomena. Of particular interest has been the interplay between multiple impurity scattering and quantum interference effects. The scaling theory of localization, developed nearly 30 years ago by Abrahams, Anderson, Licciardello, and Ramakrishnan, predicts that sufficiently strong impurity scattering can exponentially localize electronic wavefunctions; when all states at the Fermi energy become localized, the material is an insulator, incapable of conducting electric current at zero temperature. This effect is especially pronounced in low dimensions, where the scaling theory in fact predicts that all wavefunctions localize for arbitrarily weak disorder (in the conventional symmetry classes of disordered metals). By contrast, in three and higher dimensions, a continuous metal-insulator transition (MIT) is predicted to occur at a critical value of the sample conductance.
At the transition, electronic wavefunctions are characterized not by just a single (or a few) independent exponent(s), but by an infinite set thereof, the phenomenon known as “multifractality”. In particular, the probability density associated to a typical critical wavefunction exhibits a complex spatial structure, which is neither localized nor extended, but instead can be thought of as an intricate pattern of escalating amplitudes assigned to different, interwoven fractal measures of the sample. Multifractality is an interesting property exhibited by many complex structures in nature; see, e.g., “Fractal measures and their singularities: The characterization of strange sets,” Halsey et al., Phys. Rev. A 33, 1141 (1986). A very nice plot of a multifractal wavefunction at the Anderson transition in 3D can be found as Fig. 1 of Vasquez, Rodriguez, and Roemer’s recent preprint arXiv:0807.2217v1 [cond-mat.dis-nn]:
While the above-described picture of wave function structure at the Anderson MIT has been confirmed through extensive numerical simulations, it has proven difficult to capture some crucial aspects of the critical multifractal wavefunction statistics analytically. A major difficulty is that the conventional theoretical framework used to describe the metal-insulator transition in more than 2 dimensions, the so-called non-linear sigma model field theory, allows simple computation of average, but not typical quantities. It has been long-appreciated that the multifractal wavefunction statistics in fact characterize a broad probability distribution. For sufficiently large moments of the distribution, the typical and average wavefunction statistics are known to behave differently in certain special 2D models of Dirac fermions, subject to a particular type of quenched disorder. By contrast, up until now, it has not been possible to compute analytically the typical multifractal spectrum for the conventional Anderson localization transition in 
dimensions, despite the fact that it is the typical, rather than average spectrum obtained from a representative wavefunction in most numerical studies.
In our paper, we fuse the conventional non-linear sigma model field theory of the Anderson MIT with a functional renormalization group (FRG) technique developed by Carpentier and Le Doussal (Nucl. Phys. B 588, 565 (2000)), previously employed in the study of the aforementioned 2D Dirac fermion models. The FRG allows us to compute the typical spectrum at the MIT in a disordered metal, via the controlled epsilon-expansion approximation scheme in 
dimensions. Working at the non-trivial MIT critical point, we demonstrate that the multifractal spectrum associated to a typical wavefunction undergoes a sharp transition, controlled by epsilon. For integral powers 
of the wavefunction amplitude less than the critical value 
, we show that the typical and average spectra coincide; by contrast, for 
, we demonstrate that the typical multifractal spectrum deviates completely from the average spectrum. The transition at 
is referred to as “termination” of the typical spectrum, and has been consistently observed in numerical studies; physically, it corresponds to a crossover in which the rare amplitude extrema of a representative wavefunction begin to dominate the typical multifractal spectrum. Our paper represents the first rigorous, field-theoretic derivation of this result.
What follows is a lightning overview of the setup of our problem; more details can be found in arXiv:0901.0284v1. The multifractal spectrum 
can be defined via the inverse participation ratio (IPR) 
,
Here, 
denotes the linear system size, and 
is the probability density of a typical (representative) normalized eigenstate wavefunction 
with energy 
, evaluated at the point 
. The single particle wavefunction 
is computed, e.g., for the Hamiltonian
where 
is a random potential with sufficiently short-range correlations. For eigenenergies 
lying within a band of extended plane wave states, 
, while exponentially localized states yield 
for 
, with 
the localization length. Multifractal behavior refers to non-linear 
-dependence of the spectrum, which occurs at the MIT. Within the epsilon-expansion, for example, at the unitary class MIT studied in our paper, one finds that
In fact, this result holds only for 
, which is the termination transition we are after – more on this below.
In principle, one should compute for a typical wavefunction obtained in a given, fixed realization of the disorder potential . Of course, this is only possible numerically, and it is necessary to diagonalize relatively large matrices (in order to minimize subleading finite-size corrections). In the limit of an infinitely large system, the typical spectrum is self-averaging and universal, and thus
serves as a “fingerprint” of the spatial structure of the wavefunctions at the particular MIT being studied.
In order to make progress analytically, it is necessary to perform some kind of ensemble averaging over configurations of the random potential . Wegner showed in 1979 that an effective low-energy field theory can be derived, which takes the form of a matrix non-linear sigma model. The theory can be defined through the functional integral
where
In this equation, the “temperature” 
is inversely proportional to the dimensionless dc conductance of the disordered metal. The symbol 
denotes
a 
Hermitian matrix field satisfying
(1)
The sigma model theory can be derived from a Grassmann path integral description of the electronic system, averaged over configurations of the quenched disorder. In order to do this efficiently, it is necessary that partition function 
. There several different ways of achieving this; one is the so-called “supersymmetry” method (pioneered by K. Efetov) wherein one includes an identical, bosonic “ghost” copy of the fermion theory; the average over disorder realizations couples together the fermion and boson sectors. In that case, the target manifold for the sigma model matrix is a graded space. Considerable analytical progress has been made in understanding the physics of certain special, 2D models of disordered electrons subject to particular types of quenched randomness. For these models, the combination of the SUSY method and CFT techniques has allowed certain properties of these systems to be obtained exactly. Efetov himself (and subsequently, many other authors) have used the zero-dimensional version of the SUSY sigma model to study the properties of disordered quantum dots, for example. Using the SUSY method, it has been possible to make direct connections to random matrix theory and quantum chaos.
In our paper, we focus on the perturbative regime of the sigma model in dimensions. To do the disorder-averaging, we have employed the older replica trick, wherein one works with 
identical copies of the Grassmann theory, leading to the 
Q-matrix. At the end of the calculation, it is necessary to take the “replica limit” 
. In our case, the target space of the NL
M is the compact coset 
– this result obtains from the specific assumptions that we make about our disordered electron system. Other symmetric spaces describe the localization physics of other types of (effectively) non-interacting particles, e.g. quasiparticle excitations in a superconductor.
The inverse participation ratio can be represented, roughly speaking, as the integral over the sample of the 
moment of the local density of states operator
.
This is the same operator that appears in the action of our field theory, where it couples to the “external field parameter” 
.
One interesting aspect of the localization problem is the relatively unconventional nature of the MIT, when compared to the second order phase transitions that occur in, say, classical statistical mechanics models of clean (not disordered) condensed matter systems. Although the character of the typical wavefunction changes from localized to extended at the MIT, it turns out that the average density of states (i.e., the average of the operator 
) is not critical across the transition. The sigma model for localization is similar in structure to the conventional O(3)/O(2) model, which describes classical magnetic ordering (of SU(2) Heisenberg spins), but the Q-matrix in the former does not play the role of an order parameter across the MIT. (Formally, this result obtains only in the replica limit of the theory described above.)
Now, we are interested in the typical multifractal spectrum , as obtained from the IPR defined for a representative wavefunction, computed in a given, fixed disorder realization. We can in principle extract it from the sigma model if we can compute the scaling properties of the expectation
In other words, we compute the log of the spatial integral of the moment of the local density of states operator, averaged over disorder configurations. While it facilitates the use of the sigma model framework, the disorder-averaging is actually redundant, because the typical spectrum is both universal and self-averaging. Even so, this is not a very natural scaling object in a local field theory! By contrast, it is easy to compute the scaling properties of the composite eigenoperators
with 
a permutation of 
symbols; 
denotes the sign of the permutation. Repeated indices are summed over. One can show that the most relevant component of the LDOS moment 
(in the RG sense, at the MIT in ) can be represented by the above
operator; the analogous construction in the simpler O(3)/O(2) model is just a spherical harmonic 
of the “
” and “” field coordinates.
So the problem becomes, how do we compute the scaling properties of the log of the spatial integral of such an operator, thereby obtaining the typical at the perturbatively accessible MIT critical point, via the epsilon expansion about 
. Our answer is the functional RG technique. In deploying the FRG, we follow closely the prior work of Carpentier and Le Doussal (Phys. Rev. E 63, 026110 (2001)) and Mudry, Ryu, and Furusaki (Phys. Rev. B 67, 064202 (2003)). In order to extract the typical , one adds the entire, infinite tower of operators 
, 
to the action of the non-linear sigma model, with some coupling constants 
. We show that the logarithmic average that we need can be extracted from this “extended” sigma model. Using the epsilon-expansion, we derive the lowest order RG equations, at the MIT in , for the coupling constants , including the lowest-order nonlinear coupling between different moments (encoded by the operator product expansion at the transition).
The FRG obtains by writing a certain generating functional 
for the coupling strengths . Remarkably, it turns out that this function satisfies the Kolmogorov – Petrovsky – Piscounov (KPP) equation:
(2)
where 
, and the constant 
is a (universal) function of 
. Here, 
is the log of the system size, i.e., the RG scale, while 
is an auxiliary parameter. The KPP equation describes non-linear diffusion phenomena in 1D. Physically, the linear diffusion reflects the broad probability distribution characterizing the IPR; the diffusion constant 
becomes “large” (order one) for high moments 
of the wavefunction intensity. As with other problems of quenched disorder in condensed matter physics, a random critical point must be characterized in principle by the entire distribution functions of observables, rather than just their average values.
On the other hand, we know from other arguments that the typical multifractal spectrum is expected to be self-averaging and universal at the Anderson MIT; this is encoded in KPP by the non-linear forcing term. It turns out that the KPP equation supports traveling wave solutions, characterized by a single parameter (the front velocity) that depends only upon the diffusion constant . Since the latter is universal, as determined by the -expansion at the MIT critical point, the KPP equation predicts a universal front velocity 
for the distribution function 
. It turns out that this front velocity exhibits two different regimes of behavior, depending upon whether is greater or less than one. Translating everything back into the language of the original problem, this “velocity selection” property of KPP determines the typical multifractal spectrum both below and above termination. The final result is that
.
Thus the nonlinear behavior (multifractality) of the typical spectrum terminates at ; for larger moments, we get a simple fractal behavior, characterized by the constant 
.
We hope that our work stimulates a further re-examination of the Anderson transition, and we believe that the FRG framework will likely offer further insights into this rich and fascinating problem.
13 Comments
Thanks for the commentary. I actually don’t have very good understanding of Anderson MIT. If someone could please explain in heuristic terms why would one expect a multifractal wave-fn. at this transition, that would be very nice. I am aware of numerical work but don’t know the physics really.
Also, I was pondering over two more things (warning: just random thoughts of someone not in disorder related CMT):
1. Is there some reason to believe/not believe that the wave-fn. at the Mott MIT (no disorder) would also have multi-fractal structure? I am aware of proposal of critical spectral functions (which I guess do imply critical wave-fn?), see e.g. paper by Senthil http://arxiv.org/abs/0804.1555.
2. There is some recent work by Fradkin on entanglement entropy of 2-D conformally invariant wave-fn’s (http://arxiv.org/pdf/0812.0203). Are similar calculations already been done for the wave-fn. at Anderson MIT. What new/interesting information do one hope to get from such a calculation??
Dear tg
For the post, you mean
I’ll try to answer and Matt will correct me if need (and if he wants so
) The reason for multifractal behavior is that vicinity of the phase transition is described by conformal field theory. In the latter, w.f. cannot be described in terms of particles (Georgi likes to use the word “unparticles” in this context), and one wants to study scaling behavior of the correlation functions explicitly in the real space. The word “multifractal” essentially means that this scaling behavior is non-trivial.
If my words above are correct and if the Mott MIT is the second order phase transition, then certainly yes.
Hmm, if CFT is discussed, then probably there is a gravity dual to it. Entanglement entropy on the CFT side could bring some nice information about causal structure of the gravity dual
Cheers,
Dmitry.
Yes, but its more than that – you don’t get multifractality at the Ising critical point, for example.
See my reply to TG, but I think the answer is generally no. The equilibrium properties of a quantum phase transition in a clean system are invariably described by a nice, unitary quantum field theory (which might, however, have weird topological or other properties). This would seem to be incompatible with the way multifractality formally occurs, at least in the theories that I know about.
Best,
matt
Hi TG and Dmitry,
Not being an expert on multifractality, the best answer that I can give is that this is a property frequently exhibited by complex or chaotic phenomena, and the basic message is that Anderson localization and the MIT has more in common with quantum (and classical) chaos than with the more familiar phase transitions in classical or quantum clean statistical mechanics models. See my answer to your next point, below.
Technically, the notion of multifractality arises when you have some kind of probability distribution defined over some set. The set itself might be fractal (as in the case of, say, the Lorentz attractor), but this is not the case true in the typical Anderson localization scenario (where the background is just
). In this case, the probability measure is just the local density of states 
(at a fixed energy), and the this measure exhibits multifractal critical scaling at the MIT, according to both numerics and various (usually approximate) analytical treatments, such as the sigma model.
No – what Senthil is talking about is just the possibility of a critical state at a Mott MIT, rather than a first order transition.
There is a deep physical and formal difference between the Anderson MIT, and more generally field theories of disordered systems, vs. the Mott transition/field theories of clean systems.
The essential difference is effective non-unitarity or non-compactness. Even though the Anderson localization problem is just to compute the eigenfunctions of the Hamiltonian
with
random, and this is a perfectly Hermitian operator that would give rise to a unitary path integral, it turns out that in order to compute anything useful, you have to effectively break the unitarity of the quantum mechanics. In the field theory description, this usually occurs upon averaging disorder configurations.
The sigma model for Anderson localization is similar in structure to a conventional classical stat mech model, say the O(3)/O(2) NLsM for classical magnetism. However, there is an essential difference, and this is intimately tied to the advent of multifractality: the sigma model for the localization problem possesses operators with arbitrarily large, negative anomalous dimensions. What this means is that you can find an operator in the theory whose second moment is more relevant than its first, third more relevant than its second, and so on…
This is completely different from the usual situation in quantum field theory, where one truncates the set of operators (that are added to a Lagrangian, say) by arguing that higher powers or higher derivatives are irrelevant, and thus can be neglected in a description of the low energy physics. By contrast, in the case of Anderson localization we need to actually keep track of entire towers of relevant operators, each more relevant than the last. This is part of the motivation for our paper, and the use of the functional renormalization group. The (average) multifractal spectrum arises due to the negative anomalous dimensions of these operators.
That probably sounds quite exotic, but really it isn’t. It is the usual case in probability theory that higher moments of a random variable give larger results than lower ones. The multifractal statistics indicate an extremely broad (e.g., log-normal) distribution of wavefunction amplitudes.
There are much simpler analogs of the effective “non-unitarity” of complex (chaotic or disordered) systems in quantum chaos which avoid field theory altogether. For example, there are certain quantum chaotic iterative maps, which are unitary operators by construction, but are effectively characterized by a spectrum of eigenvalues that lie inside the unit circle of the complex plane. See, e.g., Hasegawa and Saphir, Phys. Rev. A 46, 7401 (1992).
I have no idea. My impression is that entanglement entropy is something useful for strongly-correlated systems with topological order, etc. I do not know if this is a useful resource for characterizing multifractal wavefunctions–I’ll have to do some digging to find out!
Dear Matt,
thanks! what’s
in your post and what is physical meaning of 
?
Cheers,
Dmitry.
with L the system size, while an exponentially localized state gives
for L >> the localization length.
So
is probe of the degree of localization; it can also be thought of as a measure of the degree of sensitivity of a wavefunction to its boundary conditions. Some early work by Thouless
of an eigenenergy 
corresponding to a wavefunction calculated with periodic and antiperiodic boundary conditions, respectively. A localized state has 
exponentially small, so the conductance is also very small. This argument is heuristic, and not quite correct, but it has the right ingredients. A modern version of it is actually used as a practical numerical algorithm for computing the conductance in models of disordered electrons.
and others that eventually lead to the scaling theory of localization stressed the importance boundary-dependence, arguing that the conductance of a disordered slab of material should be proportional to the variation
The most natural way to interpret the the localization transition is through the conductance, and with the advent of the non-linear sigma model field theory (where the conductance plays the role of a coupling constant, the effective inverse temperature), interest in local probes such as
waned a bit. But there was a resurgence in the 90s, when the community began working in earnest on fluctuations in mesoscopic systems, such as quantum dots, etc. In these systems, conductance fluctuations are quite large, and in fact reflect the multifractal character of the critical wavefunctions in the presence of disorder.
Hoho, nice! Is MIT second-order phase transition or not (you are saying that the average density of states is not critical across the MIT)? If not, can you comment more on the nature of MIT?
Cheers,
Dmitry.
Well, exactly!! In fact this is still a point of some contention, but it is generally agreed that the Anderson MIT is a second order transition, albeit a rather unconventional one. Indeed, the average density of states is NOT critical across the transition–in the standard universality classes of disordered metals. If you do not have access to the conductance, then you need to study fluctuations (e.g., moments) of the density of states in order to see that the transition actually occurs.
Part of the problem has been that the Anderson transition typically occurs in 3D systems, and as usual there is no rigorous field theoretic (or other framework) that works well in D > 2. What we have is the standard epsilon expansion for the sigma model, and while aspects of this work just like O(3)/O(2) model in stat mech, there are other aspects, in particular multifractality, that are hard to understand and control. I address this point in a little more detail in my response to TG.
However, there is unambigious evidence of second order delocalization transitions in 2D. The paradigmatical one is the plateau transition in the (integer) quantum Hall effect. As you pass through a Landau level the longitudinal conductance peaks, while the Hall conductivity makes its step from
to 
, say. At zero temperature, it is believed that the longitudinal conductivity narrows to a delta function spike, indicating the existence of delocalized wavefunctions at exactly one energy. Extensive numerical work (e.g., B. Huckestein, M. Janssen) has demonstrated that the plateau transition is indeed universal, exhibiting a unique localization length exponent 
, and a universal multifractal spectrum (for the IPR).
Unfortunately, there is no analytically tractable CFT for this transition (in a model of non-interacting electrons, subject to perp. B-field and quenched disorder). This is not for lack of trying – Pruisken, Zirnbauer, and many others have formulated candidate field theories, but none of these have panned out–either the theory turns out to be wrong, or we still can’t calculate anything with it. This is actually a rather deep problem having to do with CFTs that represent disordered systems. Frequently these CFTs possess so-called logarithmic operators, which, suffice to say, frell up the works.
There do exist, however, some other localization problems in 2D which are partly or completely solved by CFT or related methods. These can be thought of as cousins of the plateau transition, but in different symmetry
(universality) classes, describing the localization of, for example, non-interacting quasiparticle excitations in a superconductor.
matt
Dear Foster and Dmitry, thanks a lot for answering my questions and for the whole discussion.
One basic question: if I understood correctly then the field theory we are discussing contains infinte no. of relevant operators. Why doesn’t that make it non-renormalizable (and hence difficult to deal within perturbative RG)?
Also, what is the value of q_c estimated from numerics in 3-d?
ps: Actually it would be great if there could be a separate post someday on concept of (non)renormalizability, multifractality and phenomena involving them like disorder and turbulence (where multifractality arises with regard to various moments of velocity-velocity correlations).
Hi TG,
The issue you raise is at the heart of our paper, actually. We were neither the first to note the conceptual problem associated with an infinite tower of increasingly relevant operators (this was realized in the very early days, by Wegner and others), nor were we the first to solve this problem in the localization context (this was done by Carpentier and Le Doussal, and later by Mudry, Ryu, and Furusaki, for the special cases of certain 2D models of Dirac fermions subject to special types of quenched disorder). What we have done is to take the methodology of the latter authors, specifically the functional RG, and apply it to the standard Anderson MIT, as described by the non-linear sigma model in
dimensions.
The basic answer is that, yes, there are operators with arbitrarily negative anomalous dimensions. Without them you would not have multifractality. However, these do not destabilize the critical point describing the MIT; instead, their relevance encodes information about the probability distribution of certain observables, such as the local density of states (LDOS) or the current, which become broadly distributed at the critical point.
Remarkably, while the LDOS has a broad distribution, the multifractal spectrum associated to a typical wavefunction is in fact universal, and can be extracted from the sigma model augmented with the entire infinite tower of relevant LDOS moment operators. Through the operator product expansion between different LDOS moment operators, one can derive coupled, nonlinear RG equations for the coupling constants conjugate to the entire infinite tower. These ODEs are then traded for PDE using a generating function technique, and it turns out that the PDE is the KPP equation of nonlinear diffusion. This nonlinear diffusion equation actually supports traveling wave asymptotics, provided the initial condition meets certain requirements. And it turns out that, in this case, the wave front velocity is universal, and encodes information about the typical spectrum.
This is really quite remarkable (again–we didn’t invent it!–the FRG works for the Anderson MIT in essentially the same way as the Dirac theory/random XY model studied by Carpentier and Le Doussal). Through the asymptotics of the KPP equation, in the large system size limit the FRG replaces the infinite set of negative anomalous scaling dimensions associated to the infinite tower of LDOS moments with a single, unique typical exponent (~ the KPP wavefront velocity). We interpret this result as implying that the LDOS operator tower “fuses” under RG into a new, “typical” LDOS operator, up to less relevant perturbations, and the overall effect of this operator is quite mild–it does not destroy renormalizability (as far as we understand it), but merely tells us how the typical LDOS moment (which is essentially the inverse participation ratio, and hence, the typical multifractal spectrum) behaves.
The nontrivial result that obtains is a transition in the behavior of the IPR statistics, from multifractal to fractal.
Somewhere between 2 and 3, see e.g., Schrieber and
is q_c ~ 1.7.
expansion. The transition occurs for
, which can become large for small 
. We argue that our calculation is still controlled for small [Unparseable or potentially dangerous latex formula. Error 2 ], which is smaller than the one-loop term by a factor of
, when evaluated at q = q_c.
Grussbach, PRL 67, 607 (1991). [q_c is not usually quoted in these numerical studies]. Our result, naively extrapolated to
We are not pushing a detailed comparison, because the convergence properties of the epsilon expansion are suspect in the
Also note that in the IPR, we are free to take q non-integer or even negative. In the sigma model we derive the result for positive integer q, but can then “analytically continue” it to real q. Doing so, we fit the continuous spectrum.
Seconded! Would really like to know more about multifractality, which originated in turbulence!
Hi TG,
Do you want to actually step up and write a guest blog post about it? I can create an account for you.
Cheers,
Dmitry.
I myself hardly know anything about multifractality in turbulence (except whatever very basics I learnt in a few colloquiums I might have attended) though and I meant that if someone in the field could be invited to write it, that would be great. I was also tempted to think that you could be that person given blog subtitle
Isn’t writing a blog post (or even better – a paper
) about the subject the best way to learn it?
What I can promise you is to put it on a schedule (which is a bit overfilled right now). And of course if somebody will write anything interesting about multifractality in turbulence and push it to arxiv, I’ll definitely invite him/her to make a guest blog post.
Cheers,
Dmitry.
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