1,298 research outputs found
Density of States near the Anderson Transition in a Space of Dimensionality d=4-epsilon
Asymptotically exact results are obtained for the average Green function and
the density of states in a Gaussian random potential for the space
dimensionality d=4-epsilon over the entire energy range, including the vicinity
of the mobility edge. For N\sim 1 (N is an order of the perturbation theory)
only the parquet terms corresponding to the highest powers of 1/epsilon are
retained. For large N all powers of 1/epsilon are taken into account with their
coefficients calculated in the leading asymptotics in N. This calculation is
performed by combining the condition of renormalizability of the theory with
the Lipatov asymptotics.Comment: 11 pages, PD
Is \phi^4 theory trivial ?
The four-dimensional \phi^4 theory is usually considered to be trivial in the
continuum limit. In fact, two definitions of triviality were mixed in the
literature. The first one, introduced by Wilson, is equivalent to positiveness
of the Gell-Mann -- Low function \beta(g) for g\ne 0; it is confirmed by all
available information and can be considered as firmly established. The second
definition, introduced by mathematical community, corresponds to the true
triviality, i.e. principal impossibility to construct continuous theory with
finite interaction at large distances: it needs not only positiveness of
\beta(g) but also its sufficiently quick growth at infinity. Indications of
true triviality are not numerous and allow different interpretation. According
to the recent results, such triviality is surely absent.Comment: Latex, 14 pages, 3 figures include
A thorny path of field theory: from triviality to interaction and confinement
Summation of the perturbation series for the Gell-Mann--Low function \beta(g)
of \phi^4 theory leads to the asymptotics \beta(g)=\beta_\infty g^\alpha at
g\to\infty, where \alpha\approx 1 for space dimensions d=2,3,4. The natural
hypothesis arises, that asymptotic behavior is \beta(g) \sim g for all d.
Consideration of the "toy" zero-dimensional model confirms the hypothesis and
reveals the origin of this result: it is related with a zero of a certain
functional integral. This mechanism remains valid for arbitrary space
dimensionality d. The same result for the asymptotics is obtained for
explicitly accepted lattice regularization, while the use of high-temperature
expansions allows to calculate the whole \beta-function. As a result, the
\beta-function of four-dimensional \phi^4 theory is appeared to be
non-alternating and has a linear asymptotics at infinity. The analogous
situation is valid for QED. According to the Bogoliubov and Shirkov
classification, it means possibility to construct the continuous theory with
finite interaction at large distances. This conclusion is in visible
contradiction with the lattice results indicating triviality of \phi^4 theory.
This contradiction is resolved by a special character of renormalizability in
\phi^4 theory: to obtain the continuous renormalized theory, there is no need
to eliminate a lattice from the bare theory. In fact, such kind of
renormalizability is not accidental and can be understood in the framework of
Wilson's many-parameter renormalization group. Application of these ideas to
QCD shows that Wilson's theory of confinement is not purely illustrative, but
has a direct relation to a real situation. As a result, the problem of
analytical proof of confinement and a mass gap can be considered as solved, at
least on the physical level of rigor.Comment: Review article, 30 pages, 15 figures. arXiv admin note: substantial
text overlap with arXiv:1102.4534, arXiv:0911.114
Conductance distribution in 1D systems: dependence on the Fermi level and the ideal leads
The correct definition of the conductance of finite systems implies a
connection to the system of the massive ideal leads. Influence of the latter on
the properties of the system appears to be rather essential and is studied
below on the simplest example of the 1D case. In the log-normal regime this
influence is reduced to the change of the absolute scale of conductance, but
generally changes the whole distribution function. Under the change of the
system length L, its resistance may undergo the periodic or aperiodic
oscillations. Variation of the Fermi level induces qualitative changes in the
conductance distribution, resembling the smoothed Anderson transition.Comment: Latex, 22 pages, 11 include
Density of States of an Electron in a Gaussian Random Potential for (4-epsilon)-dimensional Space
The density of states for the Schroedinger equation with a Gaussian random
potential is calculated in a space of dimension d=4-epsilon in the entire
energy range including the vicinity of a mobility edge. Leading terms in
1/epsilon are taken into account for N \sim 1 (N is an order of perturbation
theory) while all powers of 1/epsilon are essential for N>>1 with calculation
of the expansion coefficients in the leading order in N.Comment: 11 pages, Revte
Scaling for level statistics from self-consistent theory of localization
Accepting validity of self-consistent theory of localization by Vollhardt and
Woelfle, we derive the relations of finite-size scaling for different
parameters characterizing the level statistics. The obtained results are
compared with the extensive numerical material for space dimensions d=2,3,4. On
the level of raw data, the results of numerical experiments are compatible with
the self-consistent theory, while the opposite statements of the original
papers are related with ambiguity of interpretation and existence of small
parameters of the Ginzburg number type.Comment: Latex, 18 pages, 12 figures include
Symmetry Theory of the Anderson Transition
We prove the Vollhardt and Wolfle hypothesis that the irreducible vertex
U_{kk'}(q) appearing in the Bethe--Salpeter equation contains a diffusion pole
(with the observable diffusion coefficient D(\omega,q)) in the limit k+k'\to 0.
The presence of a diffusion pole in U_{kk'}(q) makes it possible to represent
the quantum "collision operator" L as a sum of a singular operator L_{sing},
which has an infinite number of zero modes, and a regular operator L_{reg} of a
general form. Investigation of the response of the system to a change in
L_{reg} leads to a self-consistency equation, which replaces the rough
Vollhardt-Wolfle equation. Its solution shows that D(0,q) vanishes at the
transition point simultaneously for all q. The spatial dispersion of
D(\omega,q) at \omega \to 0 is found to be \sim 1 in relative units. It is
determined by the atomic scale, and it has no manifestations on the scale q
\sim \xi^{-1} associated with the correlation length \xi. The values obtained
for the critical exponent s of the conductivity and the critical exponent \nu
of the localization length in a d-dimensional space, s=1 (d>2) and \nu=1/(d-2)
(24), agree with all reliably established results. With
respect to the character of the change in the symmetry, the Anderson transition
is found to be similar to the Curie point of an isotropic ferromagnet with an
infinite number of components. For such a magnet, the critical exponents are
known exactly and they agree with the exponents indicated above. This suggests
that the symmetry of the critical point has been established correctly and that
the exponents have been determined exactly.Comment: 43 pages, 6 figures include
Conductance of Finite Systems and Scaling in Localization Theory
The conductance of finite systems plays a central role in the scaling theory
of localization (Abrahams et al, 1979). Usually it is defined by the
Landauer-type formulas, which remain open the following questions: (a)
exclusion of the contact resistance in the many-channel case; (b)
correspondence of the Landauer conductance with internal properties of the
system; (c) relation with the diffusion coefficient D(\omega,q) of an infinite
system. The answers to these questions are obtained below in the framework of
two approaches: (1) self-consistent theory of localization by Vollhardt and
Woelfle, and (2) quantum mechanical analysis based on the shell model. Both
approaches lead to the same definition for the conductance of a finite system,
closely related to the Thouless definition. In the framework of the
self-consistent theory, the relations of finite-size scaling are derived and
the Gell-Mann - Low functions \beta(g) for space dimensions d=1,2,3 are
calculated. In contrast to the previous attempt by Vollhardt and Woelfle
(1982), the metallic and localized phase are considered from the same
standpoint, and the conductance of a finite system has no singularity at the
critical point. In the 2D case, the expansion of \beta(g) in 1/g coincides with
results of the \sigma-model approach on the two-loop level and depends on the
renormalization scheme in higher loops; the use of dimensional regularization
for transition to dimension d=2+\epsilon looks incompatible with the physical
essence of the problem. The obtained results are compared with numerical and
physical experiments. A situation in higher dimensions and the conditions for
observation of the localization law \sigma\propto -i\omega for conductivity are
discussed.Comment: Latex, 24 pages, 16 figures include
Anderson Transition and Generalized Lyapunov Exponents (comment on comment by P.Markos, L.Schweitzer and M.Weyrauch, cond-mat/0402068)
The generalized Lyapunov exponents describe the growth of the second moments
for a particular solution of the quasi-1D Schroedinger equation with initial
conditions on the left end. Their possible application in the Anderson
transition theory became recently a subject for controversy in the literature.
The approach to the problem of the second moments advanced by Markos et al
(cond-mat/0402068) is shown to be trivially incorrect. The difference of
approaches by Kuzovkov et al (cond-mat/0212036, cond-mat/0501446) and the
present author (cond-mat/0504557, cond-mat/0512708) is discussed.Comment: Latex, 5 page
T_c of disordered superconductors near the Anderson transition
According to the Anderson theorem, the critical temperature T_c of a
disordered superconductor is determined by the average density of states and
does not change at the localization threshold. This statement is valid under
assumption of a self-averaging order parameter, which can be violated in the
strong localization region. Stimulating by statements on the essential increase
of T_c near the Anderson transition, we carried out the systematic
investigation of possible violations of self-averaging. Strong deviations from
the Anderson theorem are possible due to resonances at the quasi-discrete
levels, resulting in localization of the order parameter at the atomic scale.
This effect is determined by the properties of individual impurities and has no
direct relation to the Anderson transition. In particular, we see no reasons to
say on "fractal superconductivity" near the localization threshold.Comment: Latex, 19 pages, 9 figures include
- …