Dorokhov-Mello-Pereyra-Kumar Equation Research Articles

Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder

New aspects of electron transport in quantum wires with Lévy-type disorder are described. We study the weak scattering and the incoherent sequential tunneling in one-dimensional quantum systems characterized by a tempered Lévy stable distribution of spacing between scatterers or tunneling barriers. The generalized Dorokhov–Mello–Pereyra–Kumar equation contains the tempered fractional derivative on wire length. The solution describes the evolution from the anomalous conductance distribution to the Dorokhov function for a long wire. For sequential tunneling, average values and relative fluctuations of conductance and resistance are calculated for different parameters of spatial distributions. A tempered Lévy stable distribution of spacing between barriers leads to a transition in conductance scaling.

General form of DMPK Equation

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.

Strong and weak localization in a ballistic quantum chain

We study coherent electron transport in a quantum network or superlattice formed by connecting chaotic ballistic quantum dots via barriers of arbitrary transparencies in a chain topology. We show that the emergence of localization effects, both weak and strong, depend on the way the infinite dot scaling limit is taken. In the localized regime, we found evidence of single parameter scaling and deviations from uniformity in the distribution of the unitary matrices in the polar decomposition of the system’s transfer-matrix. In the metallic regime, we show that recent predictions of anomalous properties in a double scaling limit, in which the barriers’ transparencies are changed along with the number of dots, can be interpreted as a two-parameter scaling theory, which cannot be derived from the Dorokhov–Mello–Pereyra–Kumar equation.

Anderson localization of one-dimensional hybrid particles

We solve the Anderson localization problem on a two-leg ladder by the Fokker-Planck equation approach. The solution is exact in the weak disorder limit at a fixed interchain coupling. The study is motivated by progress in investigating the hybrid particles such as cavity polaritons. This application corresponds to parametrically different intrachain hopping integrals (a ``fast'' chain coupled to a ``slow'' chain). We show that the canonical Dorokhov-Mello-Pereyra-Kumar (DMPK) equation is insufficient for this problem. Indeed, the angular variables describing the eigenvectors of the transmission matrix enter into an extended DMPK equation in a nontrivial way, being entangled with the two transmission eigenvalues. This extended DMPK equation is solved analytically and the two Lyapunov exponents are obtained as functions of the parameters of the disordered ladder. The main result of the paper is that near the resonance energy, where the dispersion curves of the two decoupled and disorder-free chains intersect, the localization properties of the ladder are dominated by those of the slow chain. Away from the resonance they are dominated by the fast chain: a local excitation on the slow chain may travel a distance of the order of the localization length of the fast chain.

The superspin approach to a disordered quantum wire in the chiral-unitary symmetry class with an arbitrary number of channels

We use a superspin Hamiltonian defined on an infinite-dimensional Fock space with positive definite scalar product to study localization and delocalization of noninteracting spinless quasiparticles in quasi-one-dimensional quantum wires perturbed by weak quenched disorder. Past works using this approach have considered a single chain. Here, we extend the formalism to treat a quasi-one-dimensional system: a quantum wire with an arbitrary number of channels coupled by random hopping amplitudes. The computations are carried out explicitly for the case of a chiral quasi-one-dimensional wire with broken time-reversal symmetry (chiral-unitary symmetry class). By treating the space direction along the chains as imaginary time, the effects of the disorder are encoded in the time evolution induced by a single site superspin (non-Hermitian) Hamiltonian. We obtain the density of states near the band center of an infinitely long quantum wire. Our results agree with those based on the Dorokhov–Mello–Pereyra–Kumar equation for the chiral-unitary symmetry class.

Asymptotic Behavior of the Conductance in Disordered Wires with Perfectly Conducting Channels

We study the conductance of disordered wires with unitary symmetry focusing on the case in which m perfectly conducting channels are present due to the channel-number imbalance between two-propagating directions. Using the exact solution of the Dorokhov–Mello–Pereyra–Kumar (DMPK) equation for transmission eigenvalues, we obtain the average and second moment of the conductance in the long-wire regime. For comparison, we employ the three-edge Chalker–Coddington model as the simplest example of channel-number-imbalanced systems with m = 1, and obtain the average and second moment of the conductance by using a supersymmetry approach. We show that the result for the Chalker–Coddington model is identical to that obtained from the DMPK equation.

Symmetric space description of carbon nanotubes

Using an innovative technique arising from the theory of symmetric spaces, we obtain anapproximate analytic solution of the Dorokhov–Mello–Pereyra–Kumar (DMPK) equationin the insulating regime of a metallic carbon nanotube with symplectic symmetry and anodd number of conducting channels. This symmetry class is characterized by the presenceof a perfectly conducting channel in the limit of infinite length of the nanotube. Thederivation of the DMPK equation for this system has recently been performedby Takane, who also obtained the average conductance both analytically andnumerically. Using the Jacobian corresponding to the transformation to radialcoordinates and the parametrization of the transfer matrix given by Takane, we identifythe ensemble of transfer matrices as the symmetric space of negative curvatureSO*(4m + 2)/[SU(2m + 1) × U(1)] belonging to the DIII-odd Cartan class. We rederive the leading-order correction to theconductance of the perfectly conducting channel and its variance Var(ln δg). Our results are in complete agreement with Takane’s. In addition, our approach based onthe mapping to a symmetric space enables us to obtain new universal quantities: auniversal group theoretical expression for the ratio , and as a by-product a novel expression for the localization length for the most general case of a symmetricspace with BCm root system, in which all three types of roots are present.

DMPK Equation for Transmission Eigenvalues in Metallic Carbon Nanotubes

The Dorokhov-Mello-Pereyra-Kumar (DMPK) equation for transmission eigenvalues is derived for metallic carbon nanotubes with several conducting channels when the potential range of scatterers is larger than the lattice constant. With increasing system length L, the system approaches a fixed point, where only one channel is perfectly conducting and other channels are completely closed. The asymptotic behavior of the conductance in the long-L regime is investigated on the basis of the DMPK equation. It is shown that the length scale for the exponential decay of the typical conductance is reduced due to the presence of the perfectly conducting channel. If a magnetic field is applied, the system falls into the unitary class. It is pointed out that this transition is characterized by the disappearance of the perfectly conducting channel and the increase in decay length for the typical conductance.

A new class of solutions of the Dorokhov–Mello–Pereyra–Kumar equation

We introduce and discuss a new class of solutions of the Dorokhov–Mello–Pereyra–Kumar(DMPK) equation in which some of the eigenvalues are grouped into clusters which areconserved in the asymptotic large distance limit (i.e. as the length of the wireincreases). We give an explicit expression for the asymptotic expansion of thesesolutions and suggest some possible applications. In particular, these new solutionscould be useful for avoiding the quasi-one-dimensional constraint in the DMPKequation and for studying the crossover between the metallic and insulating phases.

Generalization of the DMPK equation beyond quasi one dimension

Electronic transport properties in a disordered quantum wire are very well described by the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation, which describes the evolution of the transmission eigenvalues as a function of the length of a multichannel conductor. However, the DMPK equation is restricted to quasi one dimensional systems only. We derive a generalized DMPK equation for higher dimensions, containing dependence on the dimensionality through the properties of the transmission eigenvectors, by making certain statistical assumptions about the transfer matrix. An earlier phenomenological generalization is obtained as a special case.

Generalized Fokker-Planck Equation for Multichannel Disordered Quantum Conductors

The Dorokhov-Mello-Pereyra-Kumar (DMPK) equation, which describes the distribution of transmission eigenvalues of multichannel disordered conductors, has been enormously successful in describing a variety of detailed transport properties of mesoscopic wires. However, it is limited to the quasi-one-dimensional regime only. We derive a one parameter generalization of the DMPK equation, which should broaden the scope of the equation.

Path-integral approach to the scattering theory of quantum transport

The scattering theory of quantum transport relates transport properties of disordered mesoscopic conductors to their transfer matrix $\bbox{T}$. We introduce a novel approach to the statistics of transport quantities which expresses the probability distribution of $\bbox{T}$ as a path integral. The path integal is derived for a model of conductors with broken time reversal invariance in arbitrary dimensions. It is applied to the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation which describes quasi-one-dimensional wires. We use the equivalent channel model whose probability distribution for the eigenvalues of $\bbox{TT}^{\dagger}$ is equivalent to the DMPK equation independent of the values of the forward scattering mean free paths. We find that infinitely strong forward scattering corresponds to diffusion on the coset space of the transfer matrix group. It is shown that the saddle point of the path integral corresponds to ballistic conductors with large conductances. We solve the saddle point equation and recover random matrix theory from the saddle point approximation to the path integral.

EXACTLY SOLVABLE SCALING THEORY OF CONDUCTION IN DISORDERED WIRES

Recent developments in the scaling theory of phase-coherent conduction through a disordered wire are reviewed. The Dorokhov–Mello–Pereyra–Kumar equation for the distribution of transmission eigenvalues has been solved exactly, in the absence of time-reversal symmetry. Comparison with the previous prediction of random-matrix theory shows that this prediction was highly accurate but not exact: the repulsion of the smallest eigenvalues was overestimated by a factor of two. This factor of two resolves several disturbing discrepancies between random-matrix theory and microscopic calculations, notably in the magnitude of the universal conductance fluctuations in the metallic regime, and in the width of the log-normal conductance distribution in the insulating regime.