*Quantum interference phenomena in transport*

We study electron transport in mesoscopic systems focusing on different quantum interference phenomena. This includes Anderson localization, effects of Coulomb interaction, superconductivity and proximity effect, mesoscopic fluctuations. Our main subjects are recently discovered graphene and topological insulators. We employ a variety of tools, both analytical and numerical, including non-linear sigma model (supersymmetric and replica version), renormalization group, unfolded scattering matrix formalism, Keldysh diagrammatic techniques, quantum kinetic equation, transfer matrix method for one-dimensional systems, symmetry and topology analysis.

*Electron transport in disordered graphene*

A hallmark of graphene is its unconventional electronic spectrum (see Fig. 1). Specifically, low-energy excitations in graphene are «relativistic» Dirac fermions, with an effective «velocity of light» 10^{6} cm/s. This leads to remarkable electronic properties of this material. in particular, graphene shows anomalous half-integer quantum Hall effect, which is observed up to room temperature. This gives a clear signature of extremely robust, as compared to usual 2D systems, electron coherence in graphene. Recent progress in purification of graphene and the novel technology of suspending graphene samples have lead to the experimental observation of fractional quantum Hall effect demonstrating the significant role of electron-electron interaction in graphene. Another remarkable discovery is that the conductivity of the undoped (zero gate voltage) graphene in a broad temperature range (from 300 K down to 30 mK) is essentially independent of temperature and has a value close to the conductance quantum 4*e*^{2}/*h*. Already first experiments on small graphene samples have revealed very good ballistic electron transport with the mean free path of the order of micron and more.

**Figure 1:**

*(Left)* Honeycomb graphene lattice consists of two sublattices: A and B (red and blue dots).

*(Right)* Band structure of graphene. Two non-equivalent Dirac points (valleys) are referred to as *K* and *K'*.

One of our recent achievments is the theory of electron transport in graphene with resonant scatterers. We have developed a novel general approach to the calculation of various transport properties of the graphene sample with strong impurities. This approach is based on the specifically designed unfolded representation of the scattering matrix of the system. It allows extremely efficient numerical simulations of the electron transport far overperforming the standard method of recursive Green's functions. While complexity of the conductance calculation within the latter approach is polynomial in the system size (e.g., in the number of atoms), the algorithm based on the unfolded scattering matrix representation is polynomial in just the number of impurities. Using this method, we have studied electron transport of graphene with vacancies and revealed rich phase diagram of various critical transport regimes. One example of critical scaling of conductance in graphene with vacancies is shown in Fig. 2. These results are not attainable with any other known computational tool. Potential further applications of the developed technique are not limited to graphene and include transport characteristic of any disordered systems. This new approach is particularly beneficial for studying systems with strong impurities and, in particular, close to metal-insulator transition, where all other known methods become inefficient.

**Figure 2:** *(Left)* Conductivity of graphene as a function of the concentration of vacancies *n* = *n _{A}* +

*n*, where

_{B}*n*are the concentrations in the two sublattices. Conductivity strongly depends on the sublattice imbalance parameter δ = (

_{A,B}*n*-

_{A}*n*)/

_{B}*n*.

*(Right)*The same plot on a logarithmic scale. All curves collapse on a single universal crossover flow between an unstable fixed point with

*n*=

_{A}*n*and the stable fixed point

_{B}*n*≠

_{A}*n*. Inset demonstrates the power-law scaling of the crossover length ξ.

_{B}We study Coulomb drag in double-layer graphene near the Dirac point. We have considered the limit of relatively strong disorder taking into account Coulomb interaction as a perturbation. The non-monotonous dependence of the drag resistivity on the two electron concentrations (top and bottom) was found. This non-monotonicity is the result of the competition between screening effects and electron-hole symmetry of the Dirac spectrum. A more general theory applicable both to relatively clean systems and close to the Dirac point was also developed. Using the quantum kinetic equation framework, we have shown that the drag becomes universal (independent of temperature) in the clean limit. For disordered samples, the kinetic equation agrees with the leading-order perturbative result. At low enough temperatures, a diffusive regime sets in yielding a peak of drag signal centered at the Dirac point.

**Figure 3:** Drag coefficient in a double-layer graphene with 9 nm layer separation in the ballistic regime as a function of carrier densities (in units of 10^{11} cm^{-2}). The left panel shows ρ_{D} at 250 K in the ultra-clean limit *τ*^{-1} = 0.5 K. The right panels shows the same quantity in a disordered sample with *τ*^{-1} = 50 K.

*Topological insulators*

Quantum interference can completely suppress the diffusion of a particle in random potential, a phenomenon known as Anderson localization. For a given energy and disorder strength the quantum states are either all localized or all delocalized. This implies the existence of Anderson transitions between insulating and metallic phases in disordered electronic systems. Anderson localization is a very rich phenomenon. The properties of Anderson transitions are sensitive to the symmetry and dimensionality of the disordered system. Moreover, subtle topological effects may significantly alter the critical behavior.

One of the most recent arenas where novel peculiar localization phenomena have been studied is physics of topological insulators. Topological insulators are bulk insulators with delocalized (topologically protected) states on their surface. As discussed above, the critical behavior of a system depends on the underlying topology. This is particularly relevant for topological insulators. The famous example of a topological insulator is a two-dimensional (2D) system on one of quantum Hall plateaus in the integer quantum Hall effect. Such a system is characterized by an integer (Chern number) *n* = …, -2, -1, 0, 1, 2,… which counts the edge states (here the sign determines the direction of chiral edge modes). The integer quantum Hall edge is thus a topologically protected one-dimensional (1D) conductor realizing the group **Z**.

Another (**Z**_{2}) class of topological insulators can be realized in systems with strong spin-orbit interaction and without magnetic field (symplectic symmetry class) – and was discovered in 2D HgTe/HgCdTe structures. A 3D **Z**_{2} topological insulator has been found and investigated for the first time in Bi_{1âˆ’x}Sb_{x} crystals. Both in 2D and 3D, **Z**_{2} topological insulators are band insulators with the following properties: (i) time reversal invariance is preserved (unlike ordinary quantum Hall systems); (ii) there exists a topological invariant, which is similar to the Chern number in quantum Hall effect; (iii) this invariant belongs to the group **Z**_{2} and reflects the presence or absence of delocalized edge modes (Kramers pairs). Topological insulators exist in all ten symmetry classes in different dimensions. Very generally, the classification of topological insulators in *d* dimensions can be constructed by studying the Anderson localization problem in a *(d - 1)*-dimensional disordered system. Indeed, absence of localization of surface states due to the topological protection implies the topological character of the insulator.

**Figure 4:** Symmetry classes and «Periodic Table» of topological insulators and superconductors. The first column contains the symmetry classes of disordered systems. Next three columns list the symmetries (time-reversal, chiral, and their combination — particle-hole symmetry). The last columns show the possibility of existence of **Z** and **Z**_{2} topological insulators in each symmetry class in dimensions d = 1, 2, 3,….

We have studied the combined effect of disorder and interaction on the transport properties of both 2D and 3D topological insulators. Without interaction, the Dirac fermions on the surface of a 3D topological insulator exhibit antilocalization and hence perfect conductivity. They are protected from localization by topological properties of the Dirac Hamiltonian. Coulomb interaction overpowers antilocalization effects and suppresses the conductivity by the mechanism of Altshuler-Aronov corrections. However, topological effects are robust in the presence of interaction and do not allow complete localization. As a result, the surface Dirac fermions exhibit a critical state with the conductivity of order *e*^{2}/*h*. This interaction-induced criticality happens automatically without any fine tuning of parameters. In the 2D case, three possible phases exist in the absence of interaction: normal insulator, topological insulator (with delocalized edge modes), and normal metal. Which phase occurs in reality depends on the strength of spin-orbit interaction in the system. The two insulating phases do not touch each other; they are always separated by the metallic phase. When the interaction is taken into account, the metallic phase is destroyed by the Altshuler-Aronov corrections while the two insulating phases survive. This results in the direct transition (quantum spin-Hall transition) between the two distinct insulating phases via the novel critical state. The latter is a remnant of the metallic phase and is analogous to the quantum Hall critical state. This critical state can be described within the symplectic class Finkelstein sigma model including Coulomb interaction and **Z**_{2} vortices.

**Figure 5:** Phase diagrams of a disordered 2D topological insulator — a system demonstrating the quantum spin-Hall effect. *(a)* Noninteracting case. The two insulating phases are separated by the metallic phase for any strength of disorder. *(b)* Coulomb interaction included. Interaction «kills» metallic phase: the two insulators are separated by the critical line analogous to the quantum Hall critical state.

*Localization of Majorana fermions*

We consider low-lying electron levels in an ``antidot'' capturing a vortex on the surface of a three-dimensional topological insulator in the presence of disorder. The surface is covered with a superconductor film with a hole which induces superconductivity via proximity effect. We assume a hole in the superconducting layer capturing a superconducting vortex. The spectrum of electron states inside the hole is sensitive to disorder, however, topological properties of the system give rise to a robust Majorana bound state at zero energy. We calculate the subgap density of states with both energy and spatial resolution using the supersymmetric sigma model method. Tunneling into the hole region was also considered. We have found that the tunneling conductance is sensitive to the Majorana level and exhibits resonant Andreev reflection at zero energy.

**Figure 6:** Global density of states inside a vortex core as a function of energy. The solid blue oscillating curve represents the low-energy result for tunneling conductance 0.1 *e*^{2}/*h*, peak at zero energy is the smeared Majorana state. The dashed curve shows the high-energy asymptotics, and the red curve presents the numerical solution interpolating between the two limits.

Breaking time-reversal symmetry destroys proximity effect in the superconductor/normal metal junctions on the semiclassical level. We consider proximity effect in the quasi-1D normal wire without time-reversal symmetry attached to a superconductor fully taking into account localization phenomena. This yields suppression of the density of states. The effect is much stronger in the case of preserved spin symmetry (class C) than in the system without spin-rotation invariance (class D). These results can be straightforwardly generalized to the case of topological superconductors and yield the localization properties of the Majorana state.

**Figure 7:** Density of states in a normal wire with broken time-reversal symmetry attached to a superconductor. Energy *(left)* and spatial *(right)* profile of the density of states depends on the spin symmetry of the system. If spin symmetry is preserved (class C), density of states is suppressed up to the distances ≈ *t _{M}*/2 = ln(Δ

_{ξ}/

*E*). If the spin symmetry is broken (class D), proximity effect enhances the density of states on the scale of localization length.

*Local experts:*

*External collaborators:*

- Alexander Mirlin (Karlsruhe)
- Igor Gornyi (Karlsruhe)
- Dmitry Ivanov (ETH Zürich)
- Mikhail Titov (Nijmegen, Netherlands)
- Björn Trauzettel (Würzburg)

*Key publications:*

- M. A. Skvortsov, P. M. Ostrovsky, D. A. Ivanov, and Ya. V. Fominov,

**Superconducting proximity effect in quantum wires without time-reversal symmetry**,

arXiv:1211.0202. - P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin,

**Symmetries and weak localization and antilocalization of Dirac fermions in HgTe quantum wells**,

Phys. Rev. B**86**, 125323 (2012) arXiv:1207.4102 - M. Schütt, P. M. Ostrovsky, M. Titov, I. V. Gornyi, B. N. Narozhny, and A. D. Mirlin,

**Coulomb drag in graphene near the Dirac point**,

arXiv:1205.5018. -
P. A. Ioselevich, P. M. Ostrovsky, and M. V. Feigel'man,

**Majorana state on the surface of a disordered 3D topological insulator**,

Phys. Rev. B**86**, 035401 (2012) arXiv:1205.4193

Last modified: 4 November 2012