Levitation of delocalized states at weak magnetic field: critical exponents and phase diagram
We study numerically the form of the critical line in the disorder–magnetic field phase diagram of the p–q network model, constructed to study the levitation of extended states at weak magnetic fields. We use oneparameter scaling, keeping either q (related to magnetic field) or p (related to energ...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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nasplib_isofts_kiev_ua-123456789-1180972025-02-09T17:15:52Z Levitation of delocalized states at weak magnetic field: critical exponents and phase diagram Kagalovsky, V. XIX Уральская международная зимняя школа по физике полупроводников We study numerically the form of the critical line in the disorder–magnetic field phase diagram of the p–q network model, constructed to study the levitation of extended states at weak magnetic fields. We use oneparameter scaling, keeping either q (related to magnetic field) or p (related to energy) constant, to calculate two critical exponents, describing the divergence of the localization length in each case. The ratio of those two exponents defines the form of the critical line close to zero magnetic field. This work was supported by the BSF under grant No. 2010030 and by the SCE under internal grant No. 5368911113. 2013 Article Levitation of delocalized states at weak magnetic field: critical exponents and phase diagram / V. Kagalovsky // Физика низких температур. — 2013. — Т. 39, № 1. — С. 37–39. — Бібліогр.: 8 назв. — англ. 0132-6414 PACS: 72.15.Rn, 73.20.Fz, 73.43.–f https://nasplib.isofts.kiev.ua/handle/123456789/118097 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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XIX Уральская международная зимняя школа по физике полупроводников XIX Уральская международная зимняя школа по физике полупроводников |
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XIX Уральская международная зимняя школа по физике полупроводников XIX Уральская международная зимняя школа по физике полупроводников Kagalovsky, V. Levitation of delocalized states at weak magnetic field: critical exponents and phase diagram Физика низких температур |
| description |
We study numerically the form of the critical line in the disorder–magnetic field phase diagram of the p–q
network model, constructed to study the levitation of extended states at weak magnetic fields. We use oneparameter
scaling, keeping either q (related to magnetic field) or p (related to energy) constant, to calculate two
critical exponents, describing the divergence of the localization length in each case. The ratio of those two exponents
defines the form of the critical line close to zero magnetic field. |
| format |
Article |
| author |
Kagalovsky, V. |
| author_facet |
Kagalovsky, V. |
| author_sort |
Kagalovsky, V. |
| title |
Levitation of delocalized states at weak magnetic field: critical exponents and phase diagram |
| title_short |
Levitation of delocalized states at weak magnetic field: critical exponents and phase diagram |
| title_full |
Levitation of delocalized states at weak magnetic field: critical exponents and phase diagram |
| title_fullStr |
Levitation of delocalized states at weak magnetic field: critical exponents and phase diagram |
| title_full_unstemmed |
Levitation of delocalized states at weak magnetic field: critical exponents and phase diagram |
| title_sort |
levitation of delocalized states at weak magnetic field: critical exponents and phase diagram |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2013 |
| topic_facet |
XIX Уральская международная зимняя школа по физике полупроводников |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118097 |
| citation_txt |
Levitation of delocalized states at weak magnetic field:
critical exponents and phase diagram / V. Kagalovsky // Физика низких температур. — 2013. — Т. 39, № 1. — С. 37–39. — Бібліогр.: 8 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT kagalovskyv levitationofdelocalizedstatesatweakmagneticfieldcriticalexponentsandphasediagram |
| first_indexed |
2025-11-28T12:04:39Z |
| last_indexed |
2025-11-28T12:04:39Z |
| _version_ |
1850035654975553536 |
| fulltext |
© V. Kagalovsky, 2013
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 1, pp. 37–39
Levitation of delocalized states at weak magnetic field:
critical exponents and phase diagram
V. Kagalovsky
Shamoon College of Engineering, Beer-Sheva 84100, Israel
E-mail: victork@sce.ac.il
Received September 13, 2012
We study numerically the form of the critical line in the disorder–magnetic field phase diagram of the p–q
network model, constructed to study the levitation of extended states at weak magnetic fields. We use one-
parameter scaling, keeping either q (related to magnetic field) or p (related to energy) constant, to calculate two
critical exponents, describing the divergence of the localization length in each case. The ratio of those two expo-
nents defines the form of the critical line close to zero magnetic field.
PACS: 72.15.Rn Localization effects (Anderson or weak localization);
73.20.Fz Weak or Anderson localization;
73.43.–f Quantum Hall effects.
Keywords: magnetic field, phase diagram, delocalized states.
The levitation scenario, describing the divergence of
extended state energy at zero magnetic field, was proposed
by Khmelnitskii in 1984 [1]. It was introduced to reconcile
the result of the scaling theory for 2d systems [2], that
there are no extended states, and the necessity of a delocal-
ized state for a quantum Hall (QH) transition [3]. Several
approaches to prove that conjecture were performed during
last 25 years experimentally, numerically and analytically
(see [4] and references therein for more details), resulting
only in establishing the tendency of the extended states
energy to increase with the decrease of magnetic field. In
order to describe the motion of electron at really low mag-
netic field one has to allow backscattering which immedi-
ately breaks the chirality of the Chalker–Coddington (CC)
network model [5], constructed to study inter-plateaux QH
transitions in strong magnetic field. It was achieved in the
p–q network model [6] with point contacts on the links
describing the backscattering by disorder and bend-
junctions at the nodes describing the orbital action of mag-
netic field. It was demonstrated that, in restricted geome-
try, electron motion reduces to two CC networks, with op-
posite directions of propagation along the links, which are
weakly coupled by disorder. Interplay of backscattering
and bending results in the quantum Hall transition in a
non-quantizing magnetic field, which decreases with in-
creasing mobility. This is in accord with scenario of float-
ing up delocalized states.
The main goal of that model was to separate in space
the regions with phase action of magnetic field, where it
affects interference in course of multiple disorder scatter-
ing, and the regions with orbital action of magnetic field,
where it bends electron trajectories. In p–q model, the dis-
order mixes counter-propagating channels on the links (the
probability of backscattering is p ), while scattering matri-
ces at the nodes describe exclusively the bending of elec-
tron trajectories (magnetic field is proportional to
(1/ 2 )q− ). The form of the disorder–magnetic field phase
diagram was predicted (see Fig. 1) and checked numerical-
ly. This diagram contains the regions with and without
edge states, i.e., the regions with zero and quantized Hall
conductivities. Taking into account that, for a given disor-
Fig. 1. (Color online) Critical red lines (1) on the phase diagram
of the p–q model. Blue arrows (2) show two lines to approach a
critical point of infinite energy at zero magnetic field, studied in
this paper.
1 1
2
2
0.5
0.5
1.0
1.00
p
q
V. Kagalovsky
38 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 1
der, the scattering strength scales as inverse electron ener-
gy, the agreement of this phase diagram with levitation
scenario was found: energy separating the Anderson and
quantum Hall insulating phases floats up to infinity upon
decreasing magnetic field. From numerical study, based on
the analysis of quantum transmission of the network with
random phases on the links, it was concluded that the posi-
tions of the weak-field quantum Hall transitions on the
phase diagram are very close to the classical-percolation
results. It was checked that, in accord with the Pruisken
theory [7], presence or absence of time reversal symmetry
on the links has no effect on the line of delocalization tran-
sitions. It was also found that floating up of delocalized
states in energy is accompanied by doubling of the critical
exponent of the localization radius.
In this brief report we study numerically the divergence
of the localization length when the parameters approach a
tricritical point = 0, = 1/ 2p q corresponding to an infi-
nite critical energy at zero magnetic field. We compute the
normalized localization length /M Mξ in the same man-
ner as in original CC model [5] for strips of width
= 16, 32, 64, 128M with periodic boundary conditions.
We propose that near this critical point the normalized lo-
calization length /M Mξ is described by a two-parameter
scaling function
1/ 1// = (|1 / 2 | , )M M f q M pMν μξ − . (1)
When there is no backscattering, = 0p (horizontal blue
arrow in Fig. 1) the network splits into two independent
CC networks with electron propagating in the opposite
directions, producing the standard QH critical exponent
2.6ν ≈ (see [8]). Numerical results of the renormalized
localization lengths /M Mξ as function of parameter p at
zero magnetic field, = 1/ 2q (vertical blue arrow (2) in
Fig. 1) for different system widths M are presented in
Fig. 2. Note, that in a limiting case, 0p → the data
strongly fluctuate. For very small values of p the off-dia-
gonal terms in the transfer matrix are close to 0 (they are
p∼ ), leading to strong fluctuations in numerical results.
Physically, enhancement of fluctuations near = 0p is a
result of proximity to two critical points, cq and 1 cq−
(see Fig. 1), where the doubling of critical exponent takes
place [6]. Nevertheless, the data satisfies rather convincing
one-parameter scaling, presented in Fig. 3. Numerical
analysis shows that the critical exponent along the line
= 1/ 2q is 4μ ≈ . This value is in a qualitative agreement
with arguments on doubling of the critical exponent pre-
sented in [6].
On a critical line the values of renormalized localization
lengths /M Mξ are expected to be independent on width
M , and therefore the parameters 1/|1 / 2 |q M ν− and
1/pM μ serve to define a one-parameter curve
/|1 / 2 |p q M ν μ−∼ . (2)
The form of the curve presented in Fig. 1 is indeed de-
scribed by Eq. (2). To summarize, using one-parameter
scaling, we have studied numerically the critical expo-
nents, describing divergence of localization length along
= 0p and = 1/ 2q lines, and have found that the critical
line in p–q phase space obtained from these values, is in
agreement with analytical predictions and direct numerical
calculations [6].
This work was supported by the BSF under grant
No. 2010030 and by the SCE under internal grant
No. 5368911113.
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M = 16
M = 32
M = 64
M = 128
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
–0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
p
ξ M
/M
Fig. 3. A one-parameter scaling 1// = ( )M M f pM μξ of data
presented in Fig. 2. Critical exponent = 4μ .
M = 16
M = 32
M = 64
M = 128
0 0.5 1.0 1.5 2.0 2.5
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
–0.2
ξ M
/M
pM 1/4
Levitation of delocalized states at weak magnetic field: critical exponents and phase diagram
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 1 39
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