VOLUME 86, NUMBER 20
PHYSICAL REVIEW LETTERS
14 MAY 2001
400
Comment on “Analytic Structure of
One-Dimensional Localization Theory:
Reexamining Mott’s Law”
−2.50
300
2
Here s0 is the Drude conductivity, ´ 苷 22ivt is a small
parameter in the problem, t is the elastic scattering time,
⺓ is the Euler constant, and c 苷 O共1兲 is another constant.
In a recent Letter [4] Gogolin claimed that the expansion
in powers of ln´ inside the square brackets in Eq. (1) starts
with the ln3´ term, challenging our basic ideas about disordered systems [1], as well as previous analytical [2,5,6]
and numerical [7] work. Below we reinstate the validity of
the Mott-Berezinskii law by pointing out two calculational
errors in Gogolin’s paper. We also present numerical results for s, which fully agree with Eq. (1).
Gogolin expresses s共v兲 as a sum over certain matrix
elements fn of the eigenfunctions of Berezinskii’s secondorder differential equation:
X̀
B2n´
p 2 s0 ´
s共v兲 苷
.
(2)
fn ,
fn 苷
cosh3 pkn kn2 1 14
n苷1
The wave numbers kn are to be determined from the equation g共kn 兲 苷 2pn, where
∑ 2
∏
1
G 共1 1 ik兲G共 2 2 ik兲
g共k兲 苷 2kL 2 i ln
(3)
1
G 2 共1 2 ik兲G共 2 1 ik兲
and L 苷 ln共16兾´兲. The Poisson summation formula
brings Eq. (2) to the form
1 X̀ Z
(4)
s共v兲 苷
dk f共k兲eipg共k兲 g0 共k兲 .
4p p苷2`
The third power of ln´ was obtained in Ref. [4] because
g共k兲 苷 2kL was used instead of the full expression (3).
This is the crucial mistake in that paper. To get the correct
coefficient in front of the subleading term in Eq. (1), one
more mistake has to be rectified. Namely, one has to use
the following result for Bn´ :
1
pL
p G共 4 1
1
苷
2
2
Bn´
2kn sinhpkn
4 G共 34 1
ikn
1
2 兲G共 4
ikn
3
2 兲G共 4
2
2
ikn
2 兲
ikn
2 兲
.
(5)
The second term has to be retained because it diverges at
the point kn 苷 i兾2 where the integrand in Eq. (4) has a
pole. Once we substitute more accurate expressions (3)
and (5) into Eqs. (2) and (4), Eq. (1) is fully recovered.
As a final check, we solved Berezinskii’s recursive equations [2,4] numerically for small real ´, slightly refining the algorithm described in Ref. [7]. The solid line
in Fig. 1 is the best fit of s共´兲 to the form sB 共´兲 苷
0031-9007兾01兾86(20)兾4715(1)$15.00
2
[σ−8εζ(3)/π]/ε
s共v兲 苷 s0 兵2z 共3兲´ 2 ´ 关ln ´ 1 共2⺓ 2 3兲 ln´ 2 c兴其 ,
(1)
2
−2.70
−2.80
2
The low-frequency conductivity s共v兲 of a disordered
Fermi gas in one spatial dimension is governed by the
Mott-Berezinskii law [1–3]
(σ−σB)/ε
−2.60
−2.90
200
10
1
10
2
10
4
5
10
10
−3.00
6
1/ε
100
0 0
10
3
10
10
1
10
2
10
3
10
4
10
5
10
6
1/ε
FIG. 1.
Numerical results for s (see text).
8z 共3兲´兾p 2 b´2 共ln2 ´ 2 g ln´ 1 d兲 (s0 苷 4兾p in our
system of units), which is provided by b 苷 1.272共5兲, g 苷
1.85共1兲, and d 苷 2.11共1兲, in excellent agreement with
b 苷 4兾p 苷 1.2732 . . ., g 苷 3 2 2⺓ 苷 1.8456 . . . predicted by Eq. (1). In the inset, we plot 关s共´兲 2 sB 兴兾´2
using the latter values of b and g and d 苷 0. This quantity
tends to a constant at small ´ in accordance with Eq. (1).
Z. W. is supported by DOE Grant No. DE-FG0299ER45747.
Note added.— Similar numerical results have been independently obtained in Ref. [8].
Michael M. Fogler and Ziqiang Wang*
Department of Physics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, Massachusetts 02139
Received 2 November 2000
DOI: 10.1103/PhysRevLett.86.4715
PACS numbers: 72.15.Rn, 73.20.Fz
*On subbatical leave from Department of Physics, Boston
College, Chestnut Hill, MA 02467.
[1] N. F. Mott, Philos. Mag. 17, 1259 (1968).
[2] V. L. Berezinskii, Zh. Eksp. Teor. Fiz. 65, 1251 (1973)
[Sov. Phys. JETP 38, 620 (1974)].
[3] The ln2 ´ factor in Eq. (1) was actually pointed out to Mott
by B. I. Halperin; see Ref. [1].
[4] A. O. Gogolin, Phys. Rev. Lett. 84, 1760 (2000).
[5] A. A. Abrikosov and I. A. Ryzhkin, Adv. Phys. 27, 147
(1977); A. A. Gogolin, Phys. Rep. 86, 1 (1982).
[6] L. P. Gorkov, O. N. Dorokhov, and F. V. Prigara, Zh. Eksp.
Teor. Fiz. 85, 1470 (1983) [Sov. Phys. JETP 58, 852
(1983)].
[7] A. A. Gogolin and V. I. Melnikov, Phys. Status Solidi B 88,
377 (1978).
[8] F. Evers and B. M. Gammel, cond-mat/0010492.
© 2001 The American Physical Society
4715