Lifshitz tails and Griffiths singularities
In random
systems with binary disorder there occur, because of statistiscal reasons,
regions with only one of the component. For instance, in a binary mass problem,
the probability to find N light masses surrounded by M heavy masses is p^N(1-p)^M,
if p is the probability to find one light mass and 1-p the probability to
find a heavy. Now if the shape of these regions is close to spherical, they
have the highest possible eigenfrequencies and they set the density of states
near the upper frequency. This density vanishes exponentially because the
needed regions are exponentially rare.
Similar singularities occur in a whole region on the high frequency side of
the spectrum, related to finite regions made up of unit cells with more than
one particle, e.g. having one light and one heavy atom, LH, or two light ones,
LLH, or HHL, etc.
- A lot of attention was payed to this problem in one dimension [P14], [P18],
[P19]. There these exponential decays are multiplied by periodic amplitudes,
of which the structure was be analyzed.
- A natural generalization was arbitrary disorder in arbitrary dimension,
studied in [P20], [P29],[L11].
- These studies culminated in a field theoretic approach for renormalized
instantons for the description of binary disorder in three dimensions, [L10].
This had a nice implication for the number of compact random walks on ordered
lattices, a result still not "proven" by mathematicians. That prediction
was confirmed for the density of states of a binary mixture in three dimensions,
where a latice implication of the theory was compared with a numerical evaluation
of the density of states [P33].
- A short review on this matter is [P29]. It also discusses the related renormalized
instanton prediction in a one-dimensional case, which gave a very good description
of the known exact result.
- Griffiths singularites are related rare regions when considered for their
application to thermodynamics. Typically they cannot be evaluated analytically.
But for the two-dimensional Ising model we know from Onsager that the free energy is
determined by an underlying density of states. Then the Griffiths singularities
are directly related to the Lifshitz singularities of that densiy of states [L12].
- Finally, there in binary random chains there are other singularities in
the spectrum. Consider, for instance, the eigenfrequency of one light mass
in a chain of heavy masses. In the random chain the light mass will typically
be surrounded by a finite number of heavy masses. The statistics is such that
the resulting density of states has a powerlaw divergency at that point.
Now there is a dense set of such frequencies with divergencies in the density
of states, related
to more complicated insertions in heavy mass sections, such as LHL. Around
all these points the powerlaws are dressed by periodic amplitudes [P10].
[L12] Th.M. N., Griffiths singularies in two-dimensional Ising models: relation Lifshitz band tails , Phys. Rev. Lett. 63 (1989) 1760-1763
[L11] Th.M.
N. and J.M. Luck, Singularies in spectra of disordered systems: an instanton
for arbitrary dimension and randomness ,
Europhys. Lett. 9 (1989) 407-413
[L10] Th.M. N., Trapping and Lifshitz tails in random media, self-attracting polymers and the number of distinct sites visited: a renormalized instanton approach in three dimensions , Phys. Rev. Lett. 62 (1989) 357-360
[P33] M.C.W. van Rossum, Th.M. N., E. Hofstetter and M. Schreiber, Density of states of disordered systems, Phys. Rev. B 49 (1994) 13377-13782
[P29] Th.M. N., Singularies in spectra of disordered systems, Physica A 167 (1990) 43-65
[P20] J.M. Luck and Th.M. N., Lifshitz tails and long-time decay in random systems with arbitrary disorder, J. Stat. Phys. 52 (1988) 1-22
[P19] Th.M.
N. and J.M. Luck, Lifshitz singularies in the total and the wavenumber
dependent spectral density of random harmonic chains,
Physica
A 145 (1987) 161-189
[P18] Th.M.
N. and J.M. Luck, Lifshitz singularies in random harmonic chains: Periodic
amplitudes near the band edge and near special frequencies,
J. Stat. Phys. 48 (1987) 393-424
[P14] Th.M.
N., J.M. Luck, J. Canisius, J.L. van Hemmen, and W.J.Ventevogel, Special
frequencies and Lifshitz singularies in binary random chains,
J. Stat. Phys. 45 (1986) 395-417
[P10] Th.M.
N. and J.M. Luck, Singular behavior of the density of states and the Lyapunov
coefficient in binary random harmonic chains,
J. Stat. Phys.
41 (1985) 745-771