arXiv subjects
Publications and source records attributed to I. M. Suslov.
The paper by Mayoh and Garcia-Garcia [arXiv:1412.0029v1] is entitled "Can disorder really enhance superconductivity?". In our opinion, the answer given by the authors is not satisfactory, and we present the alternative picture. Our reply to the comment [arXiv:1502.06282] is added in the end, in order to reveal a series of untrue statements contained in it.
The Berezinskii localization law σ(ω) (-iω) 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.
Existence of the upper critical dimension d_{c2}=4 for the Anderson transition is a rigorous consequence of the Bogoliubov theorem on renormalizability of ϕ^4 theory. For dimensions d\ge 4, one-parameter scaling does not hold, and all existent numerical data should be reinterpreted. These data are exhausted by results for d=4,5 from scaling in quasi-one-dimensional systems, and results for d=4,5,6 from level statistics. All these data are compatible with the theoretical scaling dependencies obtained from self-consistent theory of localization by Vollhardt and Woelfle. The critical discussion is given for a widespread point of view that d_{c2}=\infty.
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.
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.
We present another interpretation of the data by P.Markos and give a lot of new illustrations for our conception. All existing numerical data look perfectly compatible with predictions of the self-consistent theory of localization.
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(ω,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 β(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 β(g) in 1/g coincides with results of the σ-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+εlooks 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 σ\propto -iωfor conductivity are discussed.
Accepting validity of self-consistent theory of localization by Vollhardt and Woelfle, we derive the finite-size scaling procedure used for studies of the critical behavior in d-dimensional case and based on the use of auxiliary quasi-1D systems. The obtained scaling functions for d=2 and d=3 are in good agreement with numerical results: it signifies the absence of essential contradictions with the Vollhardt and Woelfle theory on the level of raw data. The results ν=1.3-1.6, usually obtained at d=3 for the critical exponent of the correlation length, are explained by the fact that dependence L+L_0 with L_0>0 (L is the transversal size of the system) is interpreted as L^{1/ν} with ν>1. For dimensions d\ge 4, the modified scaling relations are derived; it demonstrates incorrectness of the conventional treatment of data for d=4 and d=5, but establishes the constructive procedure for such a treatment. Consequences for other variants of finite-size scaling are discussed.
According to recent results, the Gell-Mann - Low function β(g) of four-dimensional ϕ^4 theory is non-alternating and has a linear asymptotics at infinity. 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 ϕ^4 theory. This contradiction is resolved by a special character of renormalizability in ϕ^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.
It has been previously shown that calculation of renormalization group (RG) functions of the scalar ϕ^4 theory reduces to the analysis of thermodynamic properties of the Ising model. Using high-temperature expansions for the latter, RG functions of four-dimensional theory can be calculated for arbitrary coupling constant g with an accuracy 10^{-4} for the Gell-Mann - Low function β(g) and an accuracy 10^{-3} - 10^{-2} for anomalous dimensions. Expansions of RG functions up to the 13th order in g^{-1/2} are obtained.
The well-known algorithm for summing of divergent series is based on the Borel transformation in combination with the conformal mapping (Le Guillou and Zinn-Justin, 1977). Modification of this algorithm allows to determine a strong coupling asymptotics of the sum of the series through the values of the expansion coefficients. Application of the algorithm to the β-function of ϕ^4 theory leads to the asymptotics β(g)=β_\infty g^αat g\to\infty, where α\approx 1 for space dimensions d=2,3,4. The natural hypothesis arises, that asymptotic behavior is β(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. Generalization of this mechanism to the arbitrary space dimensionality leads to the linear asymptotics of β(g) for all d. The same idea can be applied for QED and gives asymptotics β(g)=g, where g is the running fine structure constant. Relation to the "zero charge" problem is discussed.
Reconstruction of the β-function for ϕ^4 theory, attempted previously by summation of perturbation series, leads to asymptotics β(g)=β_\infty g^αat g\to\infty, where α\approx 1 for space dimensions d=2,3,4. The natural hypothesis arises, that asymptotic behavior is β(g) \sim g for all d. Consideration of the zero-dimensional case confirms the hypothesis and reveals the origin of this result: it is related with a zero of a certain functional integral. Consideration can be generalized to the arbitrary space dimensionality, confirming the linear asymptotics of β(g) for all d. Asymptotical behavior for other renormalization group functions (anomalous dimensions) is found to be constant. Relation to the "zero charge" problem is discussed.
The asymptotics of the Gell-Mann - Low function in QED can be determined exactly, β(g)= g at g\to\infty, where g=e^2 is the running fine structure constant. It solves the problem of pure QED at small distances L and gives the behavior g\sim L^{-2}.
It is demonstrated, that 't Hooft's renormalization scheme (in which β-function has exactly the two-loop form) is generally in conflict with the natural physical requirements and specifies the type of the field theory in an arbitrary manner. It violates analytic properties in the coupling constant plane and provokes misleading conclusion on accumulation of singularities near the origin. It artificially creates renormalon singularities, even if they are absent in the physical scheme. The 't Hooft scheme can be used in the framework of perturbation theory but no global conclusions should be drawn from it.
The four-dimensional ϕ^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 β(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 β(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.
The possible existence of the Anderson transition in 2D systems without interaction and spin-orbit effects (such as the usual Anderson model) becomes recently a subject of controversy in the literature. Comparative analysis of approaches based on generalized Lyapunov exponents is given, in order to resolve controversy.
The 1/[-iω+ D(ω, q)q^2] diffusion pole in the localized phase transfers to the 1/ωBerezinskii-Gorkov singularity, which can be analyzed by the instanton method (M V. Sadovskii, 1982; J. L. Cardy, 1978). Straightforward use of this approach leads to contradictions, which do not disappear even if the problem is extremely simplied by taking zero-dimensional limit. On the contrary, they are extremely sharpened in this case and become paradoxes. The main paradox is specified by the following statements: (i) the 1/ωsingularity is determined by high orders of perturbation theory, (ii) the high-order behaviors for two quantities Φ^{RA} and U^{RA} are the same, and (iii) Φ^{RA} has the 1/ωsingularity, whereas U^{RA} does not have it. Solution to the paradox indicates that the instanton method makes it possible to obtain only the 1/(ω+ iγ) singularity, where the parameter γremains indefinite and must be determined from additional conditions. This conceptually confirms the necessity of the self-consistent treatment for the diffusion coefficient that is used in the Vollhardt-Wolfle type theories.
We present new evaluation of the critical exponents of O(n)- symmetric ϕ^4 theory from the field theoretical renormalization group, based on the new algorithm for summing divergent series. The central values practically coincide with those by Le Guillou and Zinn-Justin (1980) but their uncertainty is essentially smaller.