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Broken time-reversal symmetry scattering at the Anderson transition.

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Ann. Phys. (Berlin) 18, No. 12, 896 – 900 (2009) / DOI 10.1002/andp.200910375
Broken time-reversal symmetry scattering
at the Anderson transition
A. Alcazar-López1,∗ , J. A. Méndez-Bermúdez1 , and Imre Varga2,3
1
2
3
Instituto de Fı́sica, Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
Elméleti Fizika Tanszék, Fizikai Intézet, Budapesti Műszaki és Gazdaságtudományi Egyetem,
1521 Budapest, Hungary
Fachbereich Physik und Wissenschaftliches Zentrum für Materialwissenschaften,
Philipps Universität Marburg, 35032 Marburg, Germany
Received 1 September 2009, accepted 17 September 2009
Published online 11 December 2009
Key words Metal-insulator transition, Anderson model, electronic transport, random matrix theory.
PACS 03.65.Nk, 71.30.+h, 73.23.-b
We study numerically the statistical properties of some scattering quantities for the Power-law Banded
Random Matrix model at criticality in the absence of time-reversal symmetry, with a small number of
single-channel leads attached to it. We focus on the average scattering matrix elements, the conductance
probability distribution, and the shot noise power as a function of the effective bandwidth b of the model.
We find a smooth transition from insulating- to metallic-like behavior in the scattering properties of the
model by increasing b. We contrast our results with existing random matrix theory predictions.
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
The study of the scattering properties of systems at the Anderson metal-insulator transition (MIT) has
a long history. In particular, the interest has been focused on the conductance probability distribution of
systems in more than two dimensions with a large number of attached leads; see for example [1]. In
the present work we study numerically the statistical properties of some scattering quantities for a onedimensional (1D) system at the MIT described by the Power-law Banded Random Matrix (PBRM) model
in the absence of time-reversal symmetry with a small number of single-channel leads attached to it.
The PBRM model [2, 3] at criticality describes 1D samples of length L represented, in its broken timereversal symmetry version, by L × L Hermitian matrices H = H S + iH A . H S and H A denote real symmetric and antisymmetric matrices whose matrix elements are statistically independent random variables
S
A
= Hij
= 0, and variance
drawn from a normal distribution with zero mean, Hij
S 2
|Hij
| =
1 + δij
1
,
2 1 + (|i − j|/b)2
A 2
|Hij
| =
1
1
,
2 1 + (|i − j|/b)2
(1)
where b is a parameter. Field-theoretical considerations [2–4] and detailed numerical investigations [3, 5,
6] verified that the PBRM model shows all the key features of the Anderson MIT at the critical point,
including multifractality of eigenfunctions and non-trivial spectral statistics. Thus model (1) possesses a
line of critical points b ∈ (0, ∞). Notice that for b → ∞, H reproduces the Gaussian Unitary Ensemble.
Using standard methods [7] we attach 2M semi-infinite single-channel leads to the 1D sample described
by the PBRM
model
at criticality. Then, we calculate the 2M × 2M scattering matrix in the standard form
r t
S(E) =
; see for example [8]. The leads are attached to the first 2M sites of the sample. That
t r
∗
Corresponding author
E-mail: aalcazar@venus.ifuap.buap.mx, Phone: +52 222 2295610, Fax: +52 222 2295611
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Ann. Phys. (Berlin) 18, No. 12 (2009)
897
0.8
1
1
0.6
2
<|S11| >
0.5
0.5
2
<|S12| >
0.4
0
0
0.01
0.1
1
10
0.02
0.1
0.6
0.2
1
D2
b
(a)
0.8
(b)
Fig. 1 (online colour at: www.ann-phys.org) |S11 |2 and |S12 |2 as a function of (a) b and (b) D2 for
M = 1. The red dashed line corresponds to |Sab |2 RMT , see the text. The black dashed lines in (a) [(b)]
are Eqs. (2) and (3) [Eqs. (4) and (5)]. The dashed-dotted line in (b) separates logarithmic and normal scales.
Error bars are not shown, they are much smaller than symbol size.
is, we attach the leads at the boundary of the system. In this way we considerably reduce finite size effects.
In fact, the quantities we analyze bellow are L-independent once L M .
There is a number of predictions made from RMT for scattering quantities for systems with and without
time-reversal invariance. Here, as we deal with a broken time-reversal invariant system, in the following,
we recall some of the corresponding RMT predictions with the additional constraint of S = 0; i.e., in the
absence of direct processes [see for example [9] for predictions (i–iii)]:
(i) The average S matrix elements are given by |Sab |2 RMT = 1/2.
(ii) The probability distribution of the dimensionless conductance T = Tr(tt† ) is given by w(T )RMT = 1
for M = 1 and w(T )RMT = 2(1 − |1 − T |)3 for M = 2.
(iii) As a function of M the prediction for the average value of T is T RMT = M/2.
(iv) The shot noise power P = Tr(tt† −tt† tt† ) as a function of M reads [10] PRMT = M 3 /2(4M 2 −1).
In the following we focus on |Sab |2 , w(T ), T , and P for the PBRM model at the MIT. In all
calculations we consider the case of perfect coupling, S ≈ 0, so that we are able to contrast our results
with the above RMT predictions which we expect to recover for b → ∞. Also notice that · above means
average over a small energy range. However, making the ergodicity assumption we will fix the energy
(E = 0) and calculate averages over disorder realizations.
First we consider the case M = 1 (two-point transmission). In Fig. 1 (a) we plot the average S-matrix
elements |S11 |2 and |S12 |2 as a function of b. We observe a strong b-dependence which we found to be
well described by
−1
,
(2)
|S12 |2 (b) = |Sab |2 RMT 1 + (γb)−2
|S11 |2 (b) = 1 − |S12 |2 (b) ,
(3)
with γ ≈ 2.58; see Fig. 1 (a). Notice that the full RMT limit (b → ∞) is already recovered for b ≥ 4.
On the other hand, it is well known that in systems at the Anderson MIT both the energy spectra and
the eigenstates exhibit multifractal characteristics [3]. The PBRM model is characterized by the effective
bandwidth b that drives the system from strong (b → 0) to weak (b → ∞) multifractality. Multifractality
can be described by the generalized dimensions Dq which describe the fluctuations of the eigenfunctions.
However among all dimensions, the correlation dimension D2 plays a prominent role. Using numerical diagonalization we extracted D2 (b) from the scaling of the typical value of the inverse participation numbers
with L. We√found good agreement between the numerically obtained D2 (b) and the analytic estimation [3]:
D2 = bπ/ 2 for b 1 and D2 = 1 − (2πb)−1 for b 1. Thus, by substituting the theoretical estimation
www.ann-phys.org
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
898
A. Alcazar et al.: Scattering at the MIT
b = 0.01
b = 0.02
b = 0.04
b = 0.1
b = 0.2
b = 0.4
w(ln T)
10
-2
10
-1
w(ln T)
-1
10
-3
-4
-10
-15
2
(b)
-2
10
10
-10
b 0.1
-3
(b)
-1
10
<ln T>
-2
10
-5
-4
-10
-4
-5
0
ln (T/Ttyp)
5
2
~ ln b
-3
10
-10
0
-5
-4
w(ln T)
w(ln T)
-4
ln T
~ ln b
0.01
(a)
10 -15
0
-5
-8
10
-3
b = 0.01
b = 0.02
b = 0.04
b = 0.1
b = 0.2
ln T
<ln T>
-1
10
-2
10
10
(a)
10 -20
10
10
-2.5
0.01
b
0.1
0
2.5
5
ln (T/Ttyp)
Fig. 2 (online colour at: www.ann-phys.org) Left [Right]: (a) w(ln T ) for several values of b < 1 in the
case M = 1 [M = 2]. (b) w(ln T ) scaled to Ttyp = expln T . Inset: ln T as a function of b (symbols).
The red dashed line is the best fit of the data to the logarithmic function A + ln b2 , with A ≈ −0.051
[A ≈ 1.006].
for D2 (b) into Eqs. (2) and (3), we obtain the following expressions for the average S-matrix elements as
a function of D2 :
⎧
−1
⎪
⎨ |Sab |2 RMT 1 + (γD2 /π)−2
,
b1
−1
|S12 |2 (D2 ) =
(4)
2
⎪
⎩ |Sab |2 RMT 1 + (2π[1 − D2 ]/γ)
, b1
|S11 |2 (D2 ) = 1 − |S12 |2 (D2 ) .
(5)
Then, in Fig. 1 (b) we show |S11 |2 and |S12 |2 as a function of the numerically obtained D2 together
with Eqs. (4) and (5). The agreement is excellent in the corresponding limits. In addition, it is interesting
to note that |Sab |2 ∝ D22 for D2 → 0, while |Sab |2 ∝ (1 − D2 )2 for D2 → 1, which may be relevant
for systems at the MIT where D2 can be tuned.
Now we turn to the conductance statistics. For b < 1, w(T ) is highly concentrated close to T = 0.
So we analyze w(ln T ) instead. In Fig. 2 (left, a) we show w(ln T ) for several values of b in the case
M = 1. Notice that the distribution functions w(ln T ) do not change their shape or width by increasing b,
thus being scale invariant. In fact, ln T clearly displays a linear behavior when plotted as a function of
ln b, see inset of Fig. 2 (left, b). Then, all distributions w(ln T ) fall one on top of the other when shifting
them along the x-axis by the typical value of T , Ttyp = expln T ∝ b2 , as shown in Fig. 2 (left, b). In
Fig. 3 (top) we show w(T ) for large b (b ≥ 0.4). In the limit b → ∞, w(T ) is expected to approach the
corresponding RMT prediction. However, once b ≥ 4, w(T ) is already well described by w(T )RMT .
In Figs. 2 (right) and 3 (bottom) we explore w(T ) in the case M = 2. As for the case M = 1, here: (i)
For small b, w(ln T ) is scale invariant with ln b2 as scaling factor; see Fig. 2 (right, b). (ii) For b ≥ 4, w(T )
is well described by the corresponding RMT prediction see Fig. 3 (bottom).
All quantities reported in Figs. 1 to 3 were obtained for L = 50 with 106 ensemble realizations. In the
following we use L = 200 and 105 ensemble realizations. We have verified that our results are invariant
for increasing L.
We now increase further M . In Figs. 4 (a) and 4 (b) we plot the average value of the conductance T and the shot noise power P for several values of b for M ∈ [1, 5]. It is clear from these plots that moving b
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.ann-phys.org
Ann. Phys. (Berlin) 18, No. 12 (2009)
b = 0.4
w(T)
4
0
b=1
4
2
899
2
0
1
0.5
0
0
0.5
b = 0.4
w(T)
0
0.5
b=1
2
1
2
0
1
0
1
T
0
b=2
2
2
0
1
0.5
1
T
0
0.5
b=4
2
0
2
1
T
b = 10
2
1
0
0
T
1
0
2
T
1
0
0
b = 10
4
2
T
1
0
1
b=4
4
2
T
2
b=2
4
1
0
T
1
2
0
0
1
T
2
T
Fig. 3 (online colour at: www.ann-phys.org) Top [Bottom]: w(T ) for several values of b ≥ 0.4 in the case
M = 1 [M = 2]. Red dashed lines are w(T )RMT , see the text.
b = 0.01
b = 0.02
b = 0.04
b = 0.2
b = 0.4
b=1
b=2
b=4
b = 10
1
0.4
P
<T>
2
b = 0.01
b = 0.02
b = 0.04
b = 0.1
b = 0.2
b = 0.4
b=1
b=2
b=4
b = 10
0.6
0.2
0
0
1
(a)
2
3
M
4
1
5
(b)
2
3
M
4
5
Fig. 4 (online colour at: www.ann-phys.org) (a) T [(b)] P as a function of M for several values of b.
The blue line in (a) and (b) corresponds to T = 0 [P = 0]. The red line in (a) [(b)] is T RMT [PRMT ],
see the text.
from small (b 1) to large (b > 1) values produces a transition from insulating- to metallic-like behavior
in the scattering properties of the PBRM model. That is, for b < 0.1, T ≈ 0 and P ≈ 0; while for b ≥ 4,
T and P are well described by the corresponding RMT predictions.
To conclude, we observed a transition from insulating- to metallic-like behavior in the scattering properties of the PBRM model by moving b from small (b 1) to large (b > 1) values. We proposed heuristic
analytical expressions for |Sab |2 (b) and |Sab |2 (D2 ). For small b, we showed that w(ln T ) is scale invariant with the typical value of T as scaling factor. Finally, we realized that the full RMT limit, expected
for b → ∞, is already recovered for relatively small values of b: b ≥ 4.
Acknowledgements This work was supported by the Hungarian-Mexican Intergovernmental S & T Cooperation
Programme under grants MX-16/2007 (NKTH) and I0110/127/08 (CONACyT) and by the Hungarian Research Fund
OTKA under contracts 73381 and 75529.
www.ann-phys.org
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
900
A. Alcazar et al.: Scattering at the MIT
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c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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