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

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

How to observe the localization law \sigma(\omega) (-i\omega) for conductivity?

The Berezinskii localization law \sigma(\omega) (-i\omega) for frequency-dependent conductivity was never questioned from the theoretical side, but never observed experimentally. In fact, this result is valid for closed systems, while most of actual systems are open. We discuss several possibilities for observation of this law and experimental difficulties arising at this way.Comment: Latex, 4 pages, 4 figures include

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