arXiv subjects
Publications and source records attributed to I. M. Suslov.
It is well-known, that for separation of variables in the eigenvalue problem, the corresponding operator should be represented as a sum of operators depending on single variables. In the case of the Fokker--Planck equations, the separation of variables is possible under essentially weaker conditions.
In the recent submission arXiv:2403.01974, Altshuler et al suggested a new approach to the Anderson transition in high dimensions. The main idea consists in the use of the branching graphs instead of high-dimensional lattices: it does not look very convincing, but we do not want to stress this point. Since the authors welcome comments, we put forward a lot of objections to their exposition of the general situation. The arising hypothesis is given in the end.
It is well-known that the dimensionless Landauer resistance ρof an 1D disordered system obeys the log-normal distribution. The average value <ρ> for such distribution is not representative, since it strongly differs from the typical value ρ_{typ} in a specific sample. In fact, this conclusion should be revised due to effects of non-ergodicity. If L is the system size, and $K$ is the number of realizations of a random potential, then a situation for L\to\infty, K\to \infty depends on the order of limiting transitions. If the limit K\to \infty is taken firstly, then the log-normal distribution is valid for all L, if the condition ρ>>1 is fulfilled. If the number of realizations K is restricted, then a situation for L\to \infty is effectively described by the delta-function distribution, and <ρ>\approxρ_{typ}$. Transformation of the log-normal distribution can be observed with the use of experimental technique developed in the context of the universal conductance fluctuations. Non-ergodicity effects are essential for understanding of the difference between the theoretical predictions for the parameters of the log-normal distribution and the results of numerical and physical experiments.
We consider the mutual distribution of two linearly independent solutions y_1(x) and y_2(x) of the 1D Schroedinger equation with a random potential. Since individual distributions of $y_1$ and $y_2$ are log-normal, it is naturally to suggest that their mutual distribution is also log-normal. Such hypothesis is confirmed in the deep of the allowed and forbidden bands, but failed near the initial band edge. The mechanism of deviations from the log-normal form is elucidated, and the first correction to it is calculated. The latter allows to demonstrate broadening of the spectral lines in the universal conductance fluctuations. A lot of new information is obtained on the phase transition in the distribution P(ψ), where ψis a combined phase entering the evolution equations. According to the previous publications, this transition is related with appearance of the imaginary part of ψat a certain energy E_0, and is not accompanied by singularities in the system resistance. The real sense of this transition consists in the change of configuration of four Lyapunov exponents, which determine the general solution: there are two pairs of complex-conjugated exponents for E>E_0, while for E<E_0 all exponents become real. Realization of two different configurations is confirmed for energies in the deep of the allowed and forbidden bands; it proves the existence of the singular point E_0 at the formal level. The phase transition can be observed in optical systems, tracing the sign of the field in a wave, when the coordinate is changed.
Localization of electrons in 1D disordered systems is usually described in the random phase approximation, when distributions of phases φand θ, entering the transfer matrix, are considered as uniform. In the general case, the random phase approximation is violated, and the evolution equations (when the system length L is increased) contain three independent variables, i.e. the Landauer resistance ρand the combined phases ψ=θ-φand χ=θ+φ. The phase χdoes not affect the evolution of ρand was not considered in previous papers. The distribution of the phase ψis found to exhibit an unusual phase transition at the point E_0 when changing the electron energy E, which manifests itself in the appearance of the imaginary part of ψ. The distribution of resistance P(ρ) has no singularity at the point E_0, and the transition looks unobservable in the electron disordered systems. However, the theory of 1D localization is immediately applicable to propagation of waves in single-mode optical waveguides. The optical methods are more efficient and provide possibility to measure phases ψand χ. On the one hand, it makes observable the phase transition in the distribution P(ψ), which can be considered as a 'trace' of the mobility edge remaining in 1D systems. On the other hand, observability of the phase χmakes actual derivation of its evolution equation, which is presented below. Relaxation of the distribution P(χ) to the limiting distribution P_\infty(χ) at L\to\infty is described by two exponents, whose exponentials have jumps of the second derivative, when the energy E is changed.
Localization of electrons in 1D disordered systems is usually described in the random phase approximation, when distributions of phases φand θ, entering the transfer matrix, are considered as uniform. In the general case, the random phase approximation is violated, and the evolution equations are written in terms of the Landauer resistance ρand the combined phases ψ=θ-φand χ=θ+φ. The distribution of the phase ψis found to exhibit an unusual phase transition at the point E_0 when changing the electron energy E, which manifests itself in the appearance of the imaginary part of ψ. The distribution of resistance P(ρ) has no singularity at the point E_0, and the transition seems unobservable in the framework of condensed matter physics. However, the theory of 1D localization is immediately applicable to the scattering of waves propagating in a single-mode optical waveguide. Modern optical methods open a way to measure phases ψand χ. As a result, the indicated phase transition becomes observable.
One-dimensional disordered systems with a random potential of a small amplitude and short-range correlations are considered near the initial band edge. The evolution equation is obtained for the mutual ditribution P(ρ,ψ) of the Landauer resistance ρand the phase variable ψ=θ-φ(θand φare phases entering the transfer matrix), when the system length L is increased. In the large L limit, the equation allows separation of variables, which provides the existence of the stationary distribution P(ψ), determinative the coefficients in the evolution equation for P(ρ). The limiting distribution P(ρ) for L\to\infty is log-normal and does not depend on boundary conditions. It is determined by the 'internal' phase distribution, whose form is established in the whole energy range including the forbidden band of the initial crystal. The random phase approximation is valid in the deep of the allowed band, but strongly violated for other energies. The phase ψappears to be a 'bad' variable, while the 'correct' vaiable is ω=-ctg (psi/2). The form of the stationary distribution P(ω) is determined by the internal properties of the system and is independent of boundary conditions. Variation of the boundary conditions leads to the scale transformation ω\to sωand translations ω\to ω+ω_0 and ψ\toψ+ψ_0, which determinates the 'external' phase distribution, entering the evolution equations. Independence of the limiting distribution P(ρ) on the external distribution P(ψ) allows to say on the hidden symmetry, whose character is revealed below.
Resistance ρof an one-dimensional disordered system of length l has the log-normal distribution in the limit of large l. Parameters of this distribution depend on the Fermi level position, but are independent on the boundary conditions. However, the boundary conditions essentially affect the distribution of phases entering the transfer matrix, and generally change the parameters of the evolution equation for the distribution P(ρ). This visible contradiction is resolved by existence of the hidden symmetry, whose nature is revealed by derivation of the equation for the stationary phase distribution and by analysis of its transformation properties.
The book by Volovik [1] contains the argument, which can be considered as the topological proof of the Luttinger theorem. The Green function of the ideal Fermi gas has a pole in the (E,|p|) plane (where E and p are energy and momentum). This pole is considered to be analogous to a vortex in liquid helium. Since a vortex is topologically stable against restricted perturbations, one can include interaction adiabatically (as a succession of small perturbations), and observe transformation of the Fermi gas pole to the Fermi liquid pole. In this argument, the topological stability arises already on the level of the ideal Fermi gas, which is in conflict with its Cooper instability. We discuss the origin of this controversy.
Universal conductance fluctuations are usually observed in the form of aperiodic oscillations in the magnetoresistance of thin wires as a function of the magnetic field B. If such oscillations are completely random at scales exceeding ξ_B, their Fourier analysis should reveal a white noise spectrum at frequencies below ξ_B^{-1}. Comparison with the results for 1D systems suggests another scenario: according to it, such oscillations are due to the superposition of incommensurate harmonics and their spectrum should contain discrete frequencies. An accurate Fourier analysis of the classical experiment by Washburn and Webb reveals a practically discrete spectrum in agreement with the latter scenario. However, this spectrum is close in shape to the discrete white noise spectrum whose properties are similar to a continuous one. More detailed analysis reveals the existence of the continuous component, whose smallness is explained theoretically. A lot of qualitative results are obtained, which confirm the presented picture. The distribution of phases, frequency differences and the growth exponents agree with theoretical predictions. Discrete frequencies depends weakly on the treatment procedure. The discovered shift oscillations confirm the analogy with 1D systems. Microscopical estimates show agreement of the obtained results with geometrical dimensions of the sample.
Universal conductance fluctuations are usually observed in the form of aperiodic oscillations in the magnetoresistance of thin wires as a function of the magnetic field B. If such oscillations are completely random at scales exceeding ξ_B, their Fourier analysis should reveal a white noise spectrum at frequencies below ξ_B^{-1}. Comparison with the results for 1D systems suggests another scenario: according to it, such oscillations are due to the superposition of incommensurate harmonics and their spectrum should contain discrete frequencies. An accurate Fourier analysis of the classical experiment by Washburn and Webb reveals a purely discrete spectrum in agreement with the latter scenario. However, this spectrum is close in shape to the discrete white noise spectrum whose properties are similar to a continuous one.
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.
Using a modification of the Shapiro scaling approach, we derive the distribution of conductance in the magnetic field applicable in the vicinity of the Anderson transition. This distribution is described by the same equations as in the absence of a field. Variation of the magnetic field does not lead to any qualitative effects in the conductance distribution and only changes its quantitative characteristics, moving a position of the system in the three-parameter space. In contrast to the original Shapiro approach, the evolution equation for quasi-1D systems is established from the generalized DMPK equation, and not by a simple analogy with one-dimensional systems; as a result, the whole approach became more rigorous and accurate.
The Dorokhov-Mello-Pereyra-Kumar (DMPK) equation, using in the analysis of quasi-one-dimensional systems and describing evolution of diagonal elements of the many-channel transfer matrix, is derived under minimal assumptions on the properties of channels. The general equation is of the diffusion type with a tensor character of the diffusion coefficient and finite values of non-diagonal components. We suggest three different forms of the diagonal approximation, one of which reproduces the usual DMPK equation and its generalization suggested by Muttalib and co-workers. Two other variants lead to equations of the same structure, but with different definitions of entering them parameters. They contain additional terms, which are absent in the first variant.
Using a modification of the Shapiro approach, we introduce the two-parameter family of conductance distributions W(g), defined by simple differential equations, which are in the one-to-one correspondence with conductance distributions for quasi-one-dimensional systems of size L^{d-1}\times L_z, characterizing by parameters L/ξand L_z/L (ξis the correlation length, d is the dimension of space). This family contains the Gaussian and log-normal distributions, typical for the metallic and localized phases. For a certain choice of parameters, we reproduce the results for the cumulants of conductance in the space dimension d=2+εobtained in the framework of the σ-model approach. The universal property of distributions is existence of two asymptotic regimes, log-normal for small g and exponential for large g. In the metallic phase they refer to remote tails, in the critical region they determine practically all distribution, in the localized phase the former asymptotics forces out the latter. A singularity at g=1, discovered in numerical experiments, is admissible in the framework of their calculational scheme, but related with a deficient definition of conductance. Apart of this singularity, the critical distribution for d=3 is well described by the present theory. One-parameter scaling for the whole distribution takes place under condition, that two independent parameters characterizing this distribution are functions of the ratio L/ξ.
Using the well-known "algebra of multifractality", we derive the functional equation for anomalous dimensions Δ_q, whose solution Δ_q=aq(q-1) corresponds to strict parabolicity of the multifractal spectrum. This result demonstrates clearly that a correspondence of σ-models with the initial disordered systems is not exact.
Summation of the perturbation series for the Gell-Mann--Low function β(g) 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. 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 β-function. As a result, the β-function of four-dimensional ϕ^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 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.
Multifractal properties of wave functions in a disordered system can be derived from self-consistent theory of localization by Vollhardt and Woelfle. A diagrammatic interpretation of results allows to obtain all scaling relations used in numerical experiments. The arguments are given that the one-loop Wegner result for a space dimension d=2+εmay appear to be exact, so the multifractal spectrum is strictly parabolical. The σ-models are shown to be deficient at the four-loop level and the possible reasons of that are discussed. The extremely slow convergence to the thermodynamic limit is demonstrated. The open question on the relation between multifractality and a spatial dispersion of the diffusion coefficient D(ω,q) is resolved in the compromise manner due to ambiguity of the D(ω,q) definition. Comparison is made with the extensive numerical material.