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Comment on “Localization-delocalization
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by Sun M. L. et al.
Magnus Johansson
EPL, 112 (2015) 17002
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EPL, 112 (2015) 17002
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Comment
Comment on “Localization-delocalization transition in self-dual
quasi-periodic lattices” by Sun M. L. et al.
Magnus Johansson
Department of Physics, Chemistry and Biology, Linköping University - SE-581 83 Linköping, Sweden
received 14 July 2015; accepted 29 September 2015
published online 12 October 2015
PACS
PACS
PACS
71.23.An – Theories and models; localized states
61.44.Fw – Semi-periodic solids: Incommensurate crystals
42.25.Dd – Optics: Wave propagation in random media
c EPLA, 2015
Copyright In a recent letter, Sun et al. [1] have studied a selfdual quasiperiodic tight-binding model, obtained by extending the well-known Aubry-André (AA) model with a
second-harmonic potential, as well as with next-nearest
neighbor hoppings. Unexpectedly, in spite of the selfduality the authors show, by numerical calculations for the
golden-mean irrationality, the existence of mobility edges
in the spectrum, and conclude that “generally all of the
self-dual QP models including far neighbors provide the
mobility edges”. This is particularly surprising, not only
because it is in contrast with the AA model, but even
more because it is in apparent contrast with the results
of ref. [2], whose authors, in 1991, studied exactly the
same model, and gave numerical evidence, from studies
of a bronze-mean irrationality, for an energy-independent
metal-insulator transition, and consequently no mobility
edges. The main point of this Comment is to illustrate
that the apparent contradiction between the results of
refs. [1] and [2] is rooted in a subtle dependence on the
properties of the chosen irrational number, and how it
connects to the heuristic generalization of the Thouless
formula given in ref. [2]. We also point out some apparently incorrect statements in ref. [1].
First, we must emphasize that, although the authors
of ref. [1] repeatedly in their letter propose their model
as “a new type of self-dual quasiperiodic model”, ref. [2]
(co-authored by the present author) proposed exactly the
same model 25 years ago. Equations (2), (3) of ref. [1]
are equivalent to eqs. (7), (8) of ref. [2], and the selfduality conditions of ref. [1] are the same as eq. (9) of
ref. [2]. Curiously enough, even the specific choice of
the ratio between next-nearest- and nearest-neighbor couplings (which for self-duality must be the same as the ratio
between the second-harmonic and fundamental potential
terms), σ = c2 = 1/3 with the notation of ref. [1], is the
same as in ref. [2]. It must also be mentioned that the
energy dependence of the localization length in the localized regime of the model, also presented as a “new” result
in ref. [1], was already obtained in ref. [2], too (see sect. III
and fig. 3 in ref. [2]).
Thus, if results in ref. [1] should be of general validity,
they must be consistent with results of ref. [2]. The only
essential difference between the parameter values used in
the numerical calculations of refs. [1] and [2] is the choice
of√irrationality: while ref. [1] used the golden mean α =
= 0.618034 . . . , ref. [2] had chosen the bronze
( 5 − 1)/2 √
mean α = ( 13 − 3)/2 = 0.302776 . . . . (The general definition of the “precious means” is α(µ) = [(µ2 +4)1/2 −µ]/2,
see, e.g., eqs. (10)–(12) of ref. [2].) Still, the results are
strikingly different. Figure 2 in ref. [2] shows that, with
very little doubt, all eigenstates of the infinite chain are
extended for V = 1.95 and localized for V = 2.05, and
there are no mobility edges in the spectrum (here, the
potential strength V corresponds to λ of ref. [1]). By contrast, figs. 1, 2 and 4 in ref. [1] show that, equally convincingly, there is a large interval of potential strenghts
λc1 < λ < λc3 with one or several mobility edge(s) in the
spectrum (with the exception of the fixed point of the duality transformation λ = 2 where all states are critical, as
observed also in ref. [2]). Here, by duality, λc1 = 4/λc3 ,
and judging from figs. 1 and 4 in ref. [1], λc1 ≈ 0.4 and
λc3 ≈ 10. The authors of ref. [1] also claim that they “observed qualitatively the same localization-delocalization
transition” both for the silver mean and bronze mean,
where the latter would obviously contradict the results
of ref. [2].
Apparently, the only way in which the novel results in
ref. [1] can be reconciled with the old results of ref. [2]
must be that the qualitative nature of the localizationdelocalization transition essentially depends on the choice
17002-p1
Comment
0.4
γ (µ=1)
S(µ=1)
γ( µ=2)
S(µ=2)
γ(µ=3)
S(µ=3)
γ(µ=4)
S(µ=4)
0.3
0.2
S, γ
0.1
0
-0.1
-0.2
-0.3
-3
-2
-1
0
1
2
3
4
E
Fig. 1: (Color online) Symbols: standard deviation S vs.
eigenenergy E for eigenstates of eq. (2) in ref. [1] with
λ = 2.2, c1 = 1, σ = c2 = 1/3, and α given by precious means
with different µ according to the legend. Fixed boundary conditions were used for chains of N = 4095 sites. Lines: the
evaluation of the expression (27) in ref. [2] for the characteristic exponent γ(E) for the corresponding parameter values, for
µ = 1 (red solid line), µ = 2 (green dashed line), µ = 3 (blue
dotted line), and µ = 4 (black double-dotted line), respectively.
AA expression, which could be evaluated numerically
from the iteration of a four-dimensional linear map (see
eqs. (24), (26) and (27) in ref. [2]). As noted in ref. [2],
this correction term could be either positive or negative,
and thus the localization length could, depending on the
eigenenergy, be either smaller or larger than the corresponding AA length. However, in fact the correction term
could also for some energies E be negative with a magnitude larger than ln |λ/2c1 |, in which case eq. (27) in ref. [2]
would give a negative value for the characteristic exponent. This would indicate that the assumptions of having
γ > 0 for λ > 2c1 and γ = 0 for λ < 2c1 , used to obtain
eq. (27) in ref. [2], would break down for these energies. Instead, for such energies one should apply the more general
eq. (26) in ref. [2], which then by duality gives a finite,
positive localization length at λ′ = 4c21 /λ < 2c1 , for an
eigenstate at energy E ′ = 2c1 E/λ, if the state at energy
E at λ > 2c1 is assumed to be extended with γ = 0.
We illustrate this by plotting the expression γ(E), obtained from eq. (27) in ref. [2] for λ/c1 = 2.2 by numerical
iterations of eq. (24) in ref. [2] until convergence, with different line types in fig. 1 for the four first precious means.
Note that although the numerical iteration gives a welldefined value γ(E) for any E, the interpretation as an
inverse localization length is evidently only valid for energies inside the spectrum (which for N → ∞ is expected
to have positive measure for λ/c1 = 2, like the standard
AA model). It is clear from fig. 1, that the regimes of
extended states for µ = 1, 2, 4 correspond exactly to those
energies in the spectrum where the numerical evaluation
of γ(E) becomes negative. Moreover, we see that also for
µ = 3 there is a narrow interval (−1.8 E −1.4) where
γ(E) becomes negative; however, this interval lies entirely
inside a major gap (−2.17 E −1.01), which thus explains why there are no extended states, and consequently
no mobility edges, in the spectrum for the bronze mean.
In conclusion, we have resolved the apparent contradiction between the earlier results [2] claiming an
energy-independent metal-insulator transition, and the
new results [1] claiming the general existence of mobility
edges, for the self-dual model with next-nearest-neighbor
interactions. Using a generalized version of Thouless formula from ref. [2], an energy-independent metal-insulator
transition is seen to appear only for such irrationals
where there is no overlap between the energy spectrum
and the regime where the formula gives negative values
for the characteristic exponent. We hope that pointing
out this subtle dependence on the properties of the irrational number will stimulate further, more rigorous work
on the topic.
of irrationality, where (at least for the particular coupling
ratio c2 = 1/3) the golden mean yields mobility edges
while the bronze mean yields an energy-independent
metal-insulator transition at λ = 2. That this is indeed the case is illustrated in fig. 1, where we, analogously to fig. 2 in ref. [2], have plotted the standard
deviation S for all eigenstates {an } vs. eigenenergy at
λ = 2.2, for the first four precious means with different
symbols. As
defined from the second moment,
usual, S is S 2 = N12 [ n2 |an |2 − ( n|an |2 )2 ], and, for large N , S
typically takes values between 0.18 and 0.29 for extended
states in a lattice with fixed boundary conditions, while
S ≈ 0 for localized states [2]. As can be seen, even though
λ > 2 the lowest-energy bands are indeed extended for the
golden and silver means, in agreement with ref. [1] using
the inverse participation ratio (IPR), while for the bronze
mean all states are localized in agreement
with [2]. For
√
the fourth precious mean (α = 5 − 2 = 0.236068 . . .),
there are in fact two mobility edges, with localized states
at the lowest energies as well as for all positive energies,
and a regime of extended states for −1.54 E −1.25.
Some understanding for the origin of the qualitatively
different behaviors for different incommensurabilities can
be obtained from the heuristic generalization of the Thouless formula discussed in ref. [2]. In the case of the standard AA model, the Thouless formula combined with
duality yields the celebrated expression for the energyindependent characteristic exponent (inverse localization REFERENCES
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