Spectral properties of quantum diffusion

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Italo Guarneri (2010), Scholarpedia, 5(11):10463. doi:10.4249/scholarpedia.10463 revision #137228 [link to/cite this article]
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Curator: Italo Guarneri

The spectral properties of quantum diffusion are a special case of the tight connections which exist in general between the long-time properties of quantum motion and the structure of spectra on small energy scales. They refine the elementary notion that bounded and unbounded motion respectively correspond to discrete and continuous energy spectra, as they also account for intermediate types of spectra, such as fractal spectra. Such properties are especially relevant to the dynamics of quantum particles in extended systems, i.e. in potentials which are spatially homogeneous, at least in a statistical sense: e.g., electrons in solids, with various degrees of order/disorder.


Coherent Quantum Diffusion.

Particles in extended systems are neither asymptotically free, nor trapped by binding potentials, so quantum diffusion may arise as a fully coherent quantum effect, with the mean displacement of a particle from its initial position increasing with some nontrivial power of time (diffusion exponent), and the probability of return decreasing with a likewise nontrivial inverse power of time. Paradigmatic examples are provided by models of electronic transport in quasi-crystals (e.g., the Fibonacci model), by the Harper model (also known as the almost-Mathieu model in the mathematical literature), by quantum maps like the Kicked Harper model, and others. In such examples quantum diffusion is typically associated with a fractal structure of the spectrum (Abe and Hiramoto 1987). The quantum diffusion is anomalous, for the following reasons: the diffusion exponent is not, typically, exactly equal to 1/2; it may even depend on the choice of an initial state; the exponent of growth of the \(q\)-th moment of the displacement does not scale proportional to \( q\) (multiscaling). It is a purely quantal effect, typically unrelated to motion in the classical limit. For instance, in many models it crucially depends on the arithmetic properties of some effective Planck's constant; notably on whether this constant is expressed by a rational number, or not. It is a long-time effect, which becomes manifest after short-time memories of the classical motion are lost. Anomalous quantum diffusion has been surmised to lie at the root of anomalous transport properties of quasi-crystals at low temperatures ( Schulz-Baldes and Bellissard 1998).

The basic spectral property underlying quantum diffusion is that the diffusion exponent is the larger, the more continuous the spectrum is: the degree of continuity of a spectrum being measured by appropriate fractal dimensions. Such dimensions range from 0 in the case of point spectra to 1 in the case of absolutely continuous spectra . The former case occurs e.g. with disordered quasi-one dimensional solids (Anderson localization), and the latter with band spectra in perfect crystals.

Heuristic Argument.

Connections between spectra and dynamics are easier to analyze for wave packets which propagate on the \(d\)-dimensional discrete lattice \(\mathbb{Z}^d\ .\) This lattice may represent a real lattice in physical space, as in the case of tight-binding models for electrons in solids, or may be one in momentum space, as in the case of certain quantum maps. In full generality, it may be defined by any complete orthonormal set of states in the Hilbert space of the system, each state in the basis corresponding to a site in the lattice. Let \(\psi\) be a state, \(\hat H\) the Hamiltonian, and let \(\psi(t)=e^{-i{\hat H}t}\psi\ ,\) where \(\hbar=1\) is assumed with no limitation of generality. If \(d=1\ ,\) the connection between fractal spectra and coherent diffusion is illustrated by the following heuristic argument. Let a wave-packet be at time \(t=0\) located at the origin in the lattice. If \({\vec{\hat X}}=({\hat X}_1,\ldots,{\hat X}_d)\) denotes the position operator, and \(|\vec{\hat X}|^2=\hat{X}_1^2+\ldots+{\hat X}_d^2\ ,\) then the mean-square displacement at time \(t>0\) is given by \(\langle\psi(t)|\;|{\vec{\hat X}}|^2|\psi(t)\rangle\ .\) Assume that it increases proportional to \(t^{\alpha}\ .\) At any given time \(t>0\ ,\) the motion will have resolved the energy spectrum on a scale \(\sim 1/t\ ;\) on the other hand, the number of energy eigenstates significantly contributing in the motion up to that time may be roughly identified with the number of site-states which have been excited at that time, i.e. \(\sim t^{\alpha/2}\ .\) So the spectrum is resolved on the scale \(\delta=1/t\) by \(\delta^{-\alpha/2}\) points, and this suggests that the spectrum is a fractal set, with a box-counting dimension not larger than \(\alpha/2\ .\)

Spectral Dimensions.

The above heuristic argument is made rigorous using the concept of spectral measure. It is the fractal properties of such measures, rather than those of the spectrum as a set of real numbers, that strongly affect the wave packet dynamics. The spectral measure of a normalized state \(\psi\) (also called local density of states) is defined on the subsets of the real line so that the measure \(\mu_{\psi}(B)\) of a given set \(B\) is equal to the probability that a measurement of energy performed when the system is in state \(\psi\) yields a value \(E\in B\ .\) Several types of fractal dimensions may be attached to a spectral measure; among these, the Renyi dimensions \( D_q(\mu_{\psi})\ ,\) where the real parameter \(q\) is the order of the dimension. The Renyi dimension \(D_2\) is called correlation dimension of the measure. The dimension \(D_1\) is called information dimension, and may be loosely identified with the minimal fractal dimension that a spectral set must have, if it is to carry the whole of the spectral measure. In the case of a pure point spectrum, which occurs whenever a complete set of normalizable energy eigenfunctions exist, the spectral measure of any state is concentrated in the countable set of points which correspond to the energy levels, so \(D_1=0\ .\) In the opposite case of an absolutely continuous spectrum (e.g., a band spectrum) no set can have zero Lebesgue measure, which has nonzero spectral measure, and then \(D_1=1\ .\) Each state has its own spectral measure, and the presence of states which have spectral measures with \(0<D_1<1\) signals that the spectrum is "of a multifractal type". The proper mathematical denotation is singular continuous spectrum. Singular continuity is not a geometric property of the spectrum as a set of real numbers; it is instead a property of the spectral measures, which convey much more information than the bare structure of spectra. Though fairly frequent in extended systems, spectra of this type are rather unstable, as they often tend to collapse into pure point spectra under tiny perturbations. A quite popular example is related to the Hofstadter Butterfly.

Exact Results.

The probability of survival in the initial state, time-averaged up to time \(T\ ,\) satisfies: \[\tag{1} \lim\limits_{T\to\infty}\frac1T\int_0^Tdt\;|\langle\psi(0)|\psi(t)\rangle|^2\;=\;\sum\limits_{E}\|{\hat P}_E\psi(0)\|^4\;, \]

where the sum is over the at most countable set of proper eigenenergies \(E\) (i.e., the energy eigenvalues which are associated with normalizable eigenfunctions) and \(\hat{P}_E\psi(0)\) is the projection of the state \(\psi(0)\) onto the eigenspace corresponding to energy \(E\ .\) If the spectrum is pure point and non-degenerate, the quantity on the rhs in (1) is the inverse participation ratio of the given state on the energy eigenbasis. In the case when the spectrum is purely continuous the survival probability decays to \(0\) as \(T\to\infty\ ,\) and the asymptotic decay law is (Ketzmerick et al. 1992): \[\tag{2} \frac1T\int_0^Tdt\;|\langle\psi(0)|\psi(t)\rangle|^2\;\sim\;C_1 T^{\;-D_2(\mu_{\psi})}\;, \]

for some constant \(C_1\) depending on the initial state. The growth with time of the time-averaged moments of the position operator \(\vec{X}\) is subject to the lower bound (Guarneri 1989, Combes 1993, Last 1996): \[\tag{3} \frac1T\int_0^Tdt\;\langle\psi(t)|\;|\vec{\hat{X}}\;|^{2r}|\psi(t)\rangle\;>\;CT^{\;2rD_1(\mu_{\psi})/d}\;, \]

which holds for all times \(T>0\ ,\) where \(r\) is an arbitrary positive real number, and \(C\) is a constant, depending on \(r\) and on the state \(\psi\ .\) A fully precise formulation of the above bound requires replacing \(D_1\) with the upper Hausdorff dimension of the measure. The bound (3) is one-sided, so, while slow diffusion implies a low spectral dimension, the converse is not true of necessity. In particular, a point spectrum may not exclude unbounded motion: a physically significant example is the random dimer model, which is a 1-dimensional solid with a special type of correlated disorder. This model has a pure point spectrum, yet motion is superdiffusive, as the exponent of growth of the 2nd moment of position is \(3/2\) (Phillips and Wu 1992, Jitomirskaya and Schulz-Baldes 2007).

Improved lower bounds on the growth of moments of the position operator can be established whenever supplementary information about the decay of generalized eigenfunctions is available (Ketzmerick et al. 1997, Kiselev and Last 2000). Other lower bounds have \(D_1\) on the right-hand side in (3) replaced by another Renyi dimension, the order \(q\) of which is related in nontrivial ways to the order \(2r\) of the moment to be bounded (Piechon 1996, Guarneri and Schulz-Baldes 1999, Barbaroux et al. 2001).

Precise estimates of the exponents of growth require that upper bounds be also established. Such bounds cannot be purely spectral ones. An elementary, fairly general upper bound which is valid for a wide class of systems of the type considered here is the ballistic bound, which says that the moment of order \(2r\) cannot grow faster than \(t^{2r}\ .\) On account of (3), for \(d=1\) this bound is always attained with absolutely continuous spectra, but this is not necessarily true when \(d>1\ .\) While (3) does not exclude that the ballistic bound may be attained even with singular continuous spectra, it can never be exactly attained with pure point spectra (Simon 1990).

More precise bounds require specific, model-dependent information. Such bounds have been proven, e.g., for the Fibonacci model (Damanik and Tcheremchantsev 2007) and for the random polymer models (Jitomirskaya and Schulz-Baldes 2007).


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See also

Fractals; Kicked Harper model

Further reading

Michael Reed and Barry Simon, Methods of Modern Mathematical Physics (vol 1), Academic Press.

An introduction to the Hofstadter butterfly can be found at Hofstadter Butterfly, which is however not updated to the most recent exact results.

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