We propose and investigate numerically a one-dimensional model which exhibits a non-Anderson disorder-driven transition. Such transitions have recently been attracting a great deal of attention in the context of Weyl semimetals, one-dimensional systems with long-range hopping and high-dimensional semiconductors. Our model hosts quasiparticles with the dispersion $\pm |k|^α\mathrm{sign} k$ with $α<1/2$ near two points (nodes) in momentum space and includes short-range-correlated random potential which allows for scattering between the nodes and near each node. In contrast with the previously studied models in dimensions $d<3$, the model considered here exhibits a critical scaling of the Thouless conductance which allows for {an accurate} determination of the critical properties of the non-Anderson transition, with a precision significantly exceeding the results obtained from the critical scaling of the density of states, usually simulated at such transitions. We find that in the limit of the vanishing parameter $\varepsilon=2α-1$ the correlation-length exponent $ν=2/(3|\varepsilon|)$ at the transition is inconsistent with the prediction $ν_{RG}=1/|\varepsilon|$ of the perturbative renormalisation-group analysis. Our results allow for a numerical verification of the convergence of $\varepsilon$-expansions for non-Anderson disorder-driven transitions and, in general, interacting field theories near critical dimensions.
We numerically analyze the transmission through a thin disordered wire of finite length attached to perfect leads, by making use of banded random Hamiltonian matrices. We compare the Landauer and the Thouless conductances, and find that they are proportional to each other in the diffusive regime, while in the localized regime the Landauer conductance is approximately proportional to the square of the Thouless one. Fluctuations of the Landauer conductance were also numerically computed; they are shown to slowly approach the theoretically predicted value.
We investigate the longitudinal conductance of a disordered three-dimensional (3D) quantum Hall system within a tight-binding lattice model using numerical Thouless conductance calculations. For the bulk, we confirm that the mobility edges are independent of the propagating directions in this anisotropic system. As disorder increases, the conductance peak of the lowest subband in the horizontal direction floats to the central subband as in the two-dimensional (2D) case, while there is no clear evidence of floating in the vertical direction. We thus conclude that for extended states, the longit…
The single parameter scaling hypothesis is the foundation of our understanding of the Anderson transition. However, the conductance of a disordered system is a fluctuating quantity which does not obey a one parameter scaling law. It is essential to investigate the scaling of the full conductance distribution to establish the scaling hypothesis. We present a clear cut numerical demonstration that the conductance distribution indeed obeys one parameter scaling near the Anderson transition.
Paper reviews recent numerical data for the conductance distribution of disordered systems in the critical regime and in the localized regime. Of particular interest is the non-analytical form of the critical conductance distribution in the 3D and 4D systems, non-Gaussian form of the distribution of P(ln g) in localized 3D systems.
We show that a spin-up electron from a normal metal entering a superconductor propagates as a composite object consisting of a spin-down hole and a pair in the condensate. This leads to a factorization of the non local conductance as two local Andreev reflections at both interfaces and one propagation in the superconductor, which is tested numerically within a one dimensional toy model of reflectionless tunneling. Small area junctions are characterized by non local conductance fluctuations. A treatment ignoring weak localization leads to a Thouless energy inverse proportional to the sample siz…
An analytical realization is suggested for the finite-size scaling algorithm based on the consideration of auxiliary quasi-1D systems. Comparison of the obtained analytical results with the results of numerical calculations indicates that the Anderson transition point is splitted into the band of critical states. This conclusion is supported by direct numerical evidence (Edwards and Thouless, 1972; Last and Thouless, 1974; Schreiber, 1985; 1990). The possibility of restoring the conventional picture still exists but requires a radical reinterpretetion of the raw numerical data.
We analyze the conductance distribution function in the one-dimensional Anderson model of localization, for arbitrary energy. For energy at the band center the distribution function deviates from the universal form assumed in single-parameter scaling theory. A direct link to the break-down of the random-phase approximation is established. Our findings are confirmed by a parameter-free comparison to the results of numerical simulations.
We obtain an analytic expression for the full distribution of conductance for a strongly disordered three dimensional conductor within a perturbative approach based on the transfer-matrix formulation. Our results confirm numerical evidence that the log-normal limit of the distribution is not reached even in the deeply insulating regime. We show that the variance of the logarithm of the conductance scales as a fractional power of the mean, while the skewness changes sign as one approaches the Anderson metal-insulator transition from the deeply insulating limit, all described as a function of a …
The disorder-driven metal-insulator transition in the quantum spin Hall systems is studied by scaling analysis of the Thouless conductance $g$. Below a critical disorder strength, the conductance is independent of the sample size $M$, an indication of critically delocalized electron states. The calculated beta function $β=d\ln g/d\ln M$ indicates that the metal-insulator transition is Kosterlitz-Thouless (KT) type, which is characterized by bounding and unbounding of vortex-antivortex pairs of the local currents. The KT like metal-insulator transition is a basic characteristic of the quantum s…
The Thouless conjecture states that the average conductance of a disordered metallic sample in the diffusive regime can be related to the sensitivity of the sample's spectrum to a change in the boundary conditions. Here we present results of a direct numerical study of the conjecture for the Anderson model. They were obtained by calculating the Landauer-Büttiker conductance $g_L$ for a sample connected to perfect leads and the distribution of level curvatures for the same sample in an isolated ring geometry, when the ring is pierced by an Aharonov-Bohm flux. In the diffusive regime ($L\gg l_e$…
We measure the field transmission matrix t for microwave radiation propagating through random waveguides in the crossover to Anderson localization. From these measurements, we determine the dimensionless conductance, g, and the individual eigenvalues $τ_n$ of the transmission matrix $tt^\dagger$ whose sum equals g. In diffusive samples, the highest eigenvalue, $τ_1$, is close to unity corresponding to a transmission of nearly 100%, while for localized waves, the average of $τ_1$, is nearly equal to g. We find that the spacing between average values of $\lnτ_n$ is constant and demonstrate that …
We present a numerical study of the quasi-particle density of states (DoS) of two-dimensional d-wave superconductors in the presence of disorder, focusing on the influence of the range of the disorder. We find qualitatively different behavior for smooth and short-ranged disorder. In the former case, we find power law scaling of the DoS with an exponent depending on the strength of the disorder and the superconducting order parameter in quantitative agreement with the theory of Nersesyan {\em et al.}. For strong disorder, a qualitative change to an energy independent DoS occurs. In contrast, fo…
We demonstrate the presence of energy dependent fluctuations in the localization length, which depend on the disorder distribution. These fluctuations lead to Ensemble Averaged Conductance Fluctuations (EACF) and are enhanced by large disorder. For the binary distribution the fluctuations are strongly enhanced in comparison to the Gaussian and uniform distributions. These results have important implications on ensemble averaged quantities, such as the transmission through quantum wires, where fluctuations can subsist to very high temperatures. For the non-fluctuating part of the localization l…
We calculate conductance through a quantum dot weakly coupled to metallic contacts by means of Keldysh out of equilibrium formalism. We model the quantum dot with the SU(2) Anderson model and consider the limit of infinite Coulomb repulsion. We solve the interacting system with the numerical diagrammatic Non-Crossing Approximation (NCA). We calculate the conductance as a function of temperature and gate voltage, from differential conductance (dI/dV) curves. We discuss these results in comparison with those from the linear response approach which can be performed directly in equilibrium conditi…
An exact relation between the conductance maximum $G_0$ at zero temperature and a ratio of lead densities is derived within the framework of the single impurity Anderson model: $G_0={\mathfrak R}[n] \frac{2e^2}{h}$, where ${\mathfrak R}[n]=4ΔN_{{\cal L},x} ΔN_{{\cal R},x}/(ΔN_{{\cal L},x}+ΔN_{{\cal R},x})^2$ and $ΔN_{{\cal L},x}$, $ΔN_{{\cal R},x}$ denote the excess density in the left/right lead at distance $x$ due to the presence of the impurity at the origin, $x=0$. The relation constitutes a parameter-free expression of the conductance of the model in terms of the ground state density that…
The nature of transport of electrons and classical waves in disordered systems depends upon the proximity to the Anderson localization transition between freely diffusing and localized waves. The suppression of average transport and the enhancement of relative fluctuations in conductance in one-dimensional samples with lengths greatly exceeding the localization length, $L\gg ξ$, are related in the single parameter scaling (SPS) theory of localization. However, the difficulty of producing an ensemble of statistically equivalent samples in which the electron wavefunction is temporally coherent h…
We show how a recent proposal to obtain the distribution of conductances in three dimensions (3D) from a generalized Fokker-Planck equation for the joint probability distribution of the transmission eigenvalues can be implemented for all strengths of disorder by numerically evaluating certain correlations of transfer matrices. We then use this method to obtain analytically, for the first time, the 3D conductance distribution in the insulating regime and provide a simple understanding of why it differs qualitatively from the log-normal distribution of a quasi one-dimensional wire.
We show that an Anderson Hamiltonian describing a quantum dot connected to multiple leads is integrable. A general expression for the non-linear conductance is obtained by combining the Bethe ansatz exact solution with Landauer-Büttiker theory. In the Kondo regime, a closed form expression is given for the matrix conductance at zero temperature and when all the leads are close to the symmetric point. A bias-induced splitting of the Kondo resonance is possible for three or more leads. Specifically, for $N$ leads, with each at a different chemical potential, there can be $N-1$ Kondo peaks in the…
In models of hopping disorder in the absence of external fields and at the band center, the electrons are less localized in space than the standard exponential Anderson localization. A signature of this anomalous localization is the square root dependence of the logarithmic average of the conductance on the system length, in contrast to the linear length dependence for Anderson localized systems. We study the effect of a time-periodic external field in the scaling and distribution of the conductance of a quantum wire with hopping disorder. In the low-frequency regime, we show a transition betw…
The notion of Thouless energy plays a central role in the theory of Anderson localization. We investigate the scaling of Thouless energy across the many-body localization (MBL) transition in a Floquet model. We use a combination of methods that are reliable on the ergodic side of the transition (e.g., spectral form factor) and methods that work on the MBL side (e.g. typical matrix elements of local operators) to obtain a complete picture of the Thouless energy behavior across the transition. On the ergodic side, the Thouless energy tends to a value independent of system size, while at the tran…
Anomalies near the conductance threshold of nearly perfect semiconductor quantum wires are explained in terms of singlet and triplet resonances of conduction electrons with a single weakly-bound electron in the wire. This is shown to be a universal effect for a wide range of situations in which the effective single-electron confinement is weak. The robustness of this generic behavior is investigated numerically for a wide range of shapes and sizes of cylindrical wires with a bulge. The dependence on gate voltage, source-drain voltage and magnetic field is discussed within the framework of an e…
In this work we exploit the integrability of the two-lead Anderson model to compute transport properties of a quantum dot, in and out of equilibrium. Our method combines the properties of integrable scattering together with a Landauer-Buttiker formalism. Although we use integrability, the nature of the problem is such that our results are not generically exact, but must only be considered as excellent approximations which nonetheless are valid all the way through crossover regimes. The key to our approach is to identify the excitations that correspond to scattering states and then to compute t…
We investigate quantum transport through strongly disordered barriers, made of a material with exceptionally high resistivity that behaves as an Anderson insulator or a ``bad metal'' in the bulk, by analyzing the distribution of Landauer transmission eigenvalues for a junction where such barrier is attached to two clean metallic leads. We find that scaling of the transmission eigenvalue distribution with the junction thickness (starting from the single interface limit) always predicts a non-zero probability to find high transmission channels even in relatively thick barriers. Using this distri…
We study the steady-state dc transport characteristics of a system comprised of an interacting quantum dot, modelled as an Anderson impurity, coupled to two metallic non-interacting leads with Rashba SOC, using an interpolative perturbative approach (IPA). The single-particle spectra, current and differential conductance are obtained in weak and strong coupling regimes over a wide range of SOC and bias values. Extensive benchmarking of the IPA validates the method in the linear as well as non-linear response regime. The universal, zero bias ($V_{sd}=0$) peak with a width proportional to the Ko…
We study the disordered topological Anderson insulator in a 2-D (square not strip) geometry. We first report the phase diagram of finite systems and then study the evolution of phase boundaries when the system size is increased to a very large $1120 \times 1120$ area. We establish that conductance quantization can occur without a bulk band gap, and that there are two distinct scaling regions with quantized conductance: TAI-I with a bulk band gap, and TAI-II with localized bulk states. We show that there is no intervening insulating phase between the bulk conduction phase and the TAI-I and TAI-…
We analyze the scaling behavior of the higher Lyapunov exponents at the Anderson transition. We estimate the critical exponent and verify its universality and that of the critical conductance distribution for box, Gaussian and Lorentzian distributions of the random potential.
The magneto-conductance in YBCO grain boundary Josephson junctions, displays fluctuations at low temperatures of mesoscopic origin. The morphology of the junction suggests that transport occurs in narrow channels across the grain boundary line, with a large Thouless energy. Nevertheless the measured fluctuation amplitude decreases quite slowly when increasing the voltage up to values about twenty times the Thouless energy, of the order of the nominal superconducting gap. Our findings show the coexistence of supercurrent and quasiparticle current in the junction conduction even at high nonequil…
We investigate the possibility of an Anderson transition below two dimensions in disordered systems of non-interacting electrons with symplectic symmetry. Numerical analysis of energy level statistics and conductance statistics on Sierpinski carpets with spin-orbit coupling indicates the occurrence of an Anderson transition below two dimensions.
Occurrence of topological Anderson insulator (TAI) in HgTe quantum well suggests that when time-reversal symmetry (TRS) is maintained, the pertinent topological phase transition, marked by re-entrant $2e^2/h$ quantized conductance contributed by helical edge states, is driven by disorder. Here we show that when TRS is broken, the physics of TAI becomes even richer. The pattern of longitudinal conductance and nonequilibrium local current %Unlike for conventional topological insulators that, in the %absence of an external magnetic field, support only a single quantized %conductance in the quantu…
The boundary condition dependence of the critical behavior for the three dimensional Anderson transition is investigated. A strong dependence of the scaling function and the critical conductance distribution on the boundary conditions is found, while the critical disorder and critical exponent are found to be independent of the boundary conditions.
The mean and the variance of the logarithm of the conductance ($ln g$) in the localized regime in the one-dimensional Anderson model are calculated analytically for weak disorder, starting from the recursion relations for the complex reflection- and transmission amplitudes. The exact recursion relation for the reflection amplitudes is approximated by improved Born approximation forms which ensure that averaged reflection coefficients tend asymptotically to unity in the localized regime, for chain lengths $L=Na\to\infty$. In contrast the familiar Born approximation of perturbation theory would …
We study the size effects on the transport properties in topological Anderson insulators by means of the Landauer-Büttiker formalism combined with the nonequilibrium Green function method. Conductances calculated for serval different widths of the nanoribbons reveal that there is no longer quantized plateaus for narrow nanoribbons. The local spin-resolved current distribution demonstrates that the edge states on the two sides can be coupled, leading to enhancement of backscattering as the width of the nanoribbon decreases, thus destroying the perfect quantization phenomena in the topological A…
We examine the power spectrum of the energy level fluctuations of a family of critical power-law random banded matrices with properties similar to those of a disordered conductor at the Anderson transition. It is shown both analytically and numerically that the Anderson transition is characterized by a power spectrum which presents $1/f^2$ noise for small frequencies but $1/f$ noise for larger frequencies. The analysis of the transition region between these two power-law limits gives an accurate estimation of the Thouless energy of the system. Finally we discuss under what circumstances these …
The Floquet topological superconducting state is a nonequilibrium time-periodic state hosting Majorana fermions. We study its transport properties by using the Kitaev model with time-periodic incommensurate potentials, which experiences phase transition from the Floquet topological superconducting phase to the Anderson localized phase with increasing driving strength. We study both the real time dynamics of the current and the non-analytic behavior of the tunneling conductance at the transition. Especially, we find that the tunneling conductance changes continuously at the transition, being a …
We revisited the scaling behavior of the transport properties of a quantum dot system described by the spin-1/2 Anderson model using analytical methods. In the low temperature limit we show that the conductance has a universal behavior with universality between temperature and bias. We compare this result with the empirical formula used to fit the experimental data for conductance in the case of the equilibrium transport through a single channel quantum dot. In the high temperature limit the conductance obtained from the Anderson model is compared with previous results obtained from the Kondo …
We study the effect of the edge disorder on the conductance of the graphene nanoribbons (GNRs). We find that only very modest edge disorder is sufficient to induce the conduction energy gap in the otherwise metallic GNRs and to lift any difference in the conductance between nanoribbons of different edge geometry. We relate the formation of the conduction gap to the pronounced edge disorder induced Anderson-type localization which leads to the strongly enhanced density of states at the edges, formation of surface-like states and to blocking of conductive paths through the ribbons.
Anderson localization of electron states on graphene lattice with diagonal and off-diagonal (OD) disorder in the absence of magnetic field is investigated by using the standard finite-size scaling analysis. In the presence of diagonal disorder all states are localized as predicted by the scaling theory for two-dimensional systems. In the case of OD disorder, the states at the Dirac point (E=0) are shown to be delocalized due to the specific chiral symmetry, although other states ($E \neq 0$) are still localized. In OD disorder the conductance at E=0 in an $M\times L$ rectangular system at the …
We study a one-dimensional (1d) XXZ spin-chain in a random field on the metallic side of the many-body localization transition by level statistics. For a fixed interaction, and intermediate disorder below the many-body localization transition, we find that, asymptotically, the number variance grows faster than linear with a disorder dependent exponent. This is consistent with the existence of an anomlaous Thouless energy in the spectrum. In non-interacting disordered metals this is an energy scale related to the typical time for a particle to diffuse across the sample. In the interacting case …
In the late seventies an increasing interest in the scaling theory of Anderson localization led to new efforts to understand the conductance of systems which scatter electrons elastically. The conductance and its relation to the scattering matrix emerged as an important subject. This, coupled with the desire to find explicit manifestations of single electron interference led to the emergence of mesoscopic physics. We review electron transport phenomena which can be expressed elegantly in terms of the scattering matrix. Of particular interest are phenomena which depend not only on transmission …
Nonlinear tunneling current through a quantum dot (an Anderson impurity system) subject to both constant and alternating electric fields is studied in the Kondo regime. A systematic diagram technique is developed for perturbation study of the current in physical systems out of equilibrium governed by time - dependent Hamiltonians of the Anderson and the Kondo models. The ensuing calculations prove to be too complicated for the Anderson model, and hence, a mapping on an effective Kondo problem is called for. This is achieved by constructing a time - dependent version of the Schrieffer - Wolff t…
We report a finite size scaling study of the Anderson transition. Different scaling functions and different values for the critical exponent have been found, consistent with the existence of the orthogonal and unitary universality classes which occur in the field theory description of the transition. The critical conductance distribution at the Anderson transition has also been investigated and different distributions for the orthogonal and unitary classes obtained.
The M-channel Anderson impurity model (M=1,2) is studied in the Kondo limit with a finite voltage bias applied to the conduction electron reservoirs. Using the Non-Crossing Approximation (NCA), we calculate the local spectral functions, the differential conductance, and susceptibility at non-zero bias for symmetric as well as asymmetric coupling of the impurity to the leads. We describe an effective procedure to solve the NCA integral equations which enables us to reach temperatures far below the Kondo scale. This allows us to study the scaling regime where the conductance depends on the bias …
We study non-equilibrium electron transport through a quantum impurity coupled to metallic leads using the equation of motion technique at finite temperature T. Assuming that the interactions are taking place solely in the impurity and focusing in the infinite Hubbard limit, we compute the out of equilibrium density of states and the differential conductance G_2(T,V) to test several scaling laws. We find that G_2(T,V)/G_2(T,0) is a universal function of both eV/T_K and T/T_K, being T_K the Kondo temperature. The effect of an in plane magnetic field on the splitting of the zero bias anomaly in …
We study the Anderson model in which a configuration with a doublet is hybridized with another with a singlet and a triplet. We calculate the conductance through the system as a function of temperature and bias voltage, near the quantum critical line for which the system is exactly solvable. The results explain recent transport measurements in a single-molecule quantum dot.
We describe linear and nonlinear transport across a single impurity Anderson model quantum dot with intermediate coupling to the leads, i.e., with tunnel coupling of the order of the thermal energy k_B T. The coupling is large enough that sequential tunneling processes alone do not suffice to properly describe the transport characteristics. Upon applying a density matrix approach, the current is expressed in terms of rates obtained by considering a very small class of diagrams which dress the sequential tunneling processes by charge fluctuations. We call this the "dressed second order" (DSO) a…
We study the disorder effect on the transport properties in the HgTe/CdTe semiconductor quantum wells. We confirm that at a moderate disorder strength, the initially un-quantized two terminal conductance becomes quantized, and the system makes a transition to the novel topological Anderson insulator (TAI). Conductances calculated for the stripe and cylinder samples reveal the topological feature of TAI and supports the idea that the helical edge states may cause the anomalous quantized plateaus. The influence of disorder is studied by calculating the distributions of local currents. Base on th…
We study the singlet-triplet Anderson model (STAM) in which a configuration with a doublet is hybridized with another with a singlet and a triplet, as a minimal model to describe two-level quantum dots coupled to two metallic leads in effectively a one-channel fashion. The model has a quantum phase transition which separates regions of a doublet and a singlet ground state. The limits of integer valence of the STAM (which include a model similar to the underscreened spin-1 Kondo model) are derived and used to predict the behavior of the conductance through the system at both sides of the transi…
We study the transition between the two-dimensional topological insulator (TI) featuring quantized edge conductance and the trivial Anderson insulator (AI) induced by strong disorder. We discover a distinct scaling behavior of TI near the phase transition where the longitudinal conductance approaches the quantized value by a power law with system size, instead of an exponential law in clean TI. This region is thus called the weak quantization topological insulator (WQTI). By using the self-consistent Born approximation, we associate the emergence of the weak quantization with the imaginary par…
We study the disorder-induced localisation transition in a three-dimensional network model that belongs to symmetry class C. The model represents quasiparticle dynamics in a gapless spin-singlet superconductor without time-reversal invariance. It is a special feature of network models with this symmetry that the conductance and density of states can be expressed as averages in a classical system of dense, interacting random walks. Using this mapping, we present a more precise numerical study of critical behaviour at an Anderson transition than has been possible previously in any context.
The conductance of two Anderson impurity models, one with two-fold and another with four-fold degeneracy, representing two types of quantum dots, is calculated using a world-line quantum Monte Carlo (QMC) method. Extrapolation of the imaginary time QMC data to zero frequency yields the linear conductance, which is then compared to numerical renormalization group results in order to assess its accuracy. We find that the method gives excellent results at low temperature (T<Tk) throughout the mixed valence and Kondo regimes, but it is unreliable for higher temperature.
We show how the density-matrix numerical renormalization group (DM-NRG) method can be used in combination with non-Abelian symmetries such as SU(N), where the decomposition of the direct product of two irreducible representations requires the use of a so-called outer multiplicity label. We apply this scheme to the SU(3) symmetrical Anderson model, for which we analyze the finite size spectrum, determine local fermionic, spin, superconducting, and trion spectral functions, and also compute the temperature dependence of the conductance. Our calculations reveal a rich Fermi liquid structure.
Numerical studies of the Anderson transition are based on the finite-size scaling analysis of the smallest positive Lyapunov exponent. We prove numerically that the same scaling holds also for higher Lyapunov exponents. This scaling supports the hypothesis of the one-parameter scaling of the conductance distribution. From the collected numerical data for quasi one dimensional systems up to the system size 24 x 24 x infinity we found the critical disorder 16.50 < Wc < 16.53 and the critical exponent 1.50 < ν< 1.54. Finite-size effects and the role of irrelevant scaling parameters are discussed.
Roughly half of numerical investigations of the Anderson transition are based on consideration of an associated quasi-1D system and postulation of one-parameter scaling for the minimal Lyapunov exponent. If this algorithm is taken seriously, it leads to unumbiguous prediction of the 2D phase transition. The transition is of the Kosterlitz-Thouless type and occurs between exponential and power law localization (Pichard and Sarma, 1981). This conclusion does not contradict numerical results if the raw data are considered. As for interpretation of these data in terms of one-parameter scaling, suc…
We study the influence of boundary conditions transverse to the transport direction for disordered mesoscopic conductors both at the Anderson metal-insulator transition and in the metallic regime. We show that the boundary conditions strongly influence the conductance distribution exactly at the metal-insulator transition and we discuss implications for the standard picture of one-parameter scaling. We show in particular that the scaling function that describes the change of conductance with system size depends on the boundary conditions from the metallic regime up to the metal-insulator trans…
Recent experiments have probed quantum dots through transport measurements in the regime where they are described by a two lead Anderson model. In this paper we develop a new method to analytically compute for the first time the corresponding transport properties. This is done by using the exact solvability of the Anderson Hamiltonian, together with a generalization of the Landauer-Buttiker approach to integrable systems. The latter requires proper identification of scattering states, a complex and crucial step in our approach. In the Kondo regime, our results include the zero-field, finite te…
A field theory of the Anderson transition in two dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which the proliferation of vortex-like topological defects is essential for localization. The sign of vortex fugacity determines the $Z_2$ topological class of the localized phase. There are two distinct, but equivalent transitions between the metallic phase and the two insulating phases. The critical conductivity and correlation length exponent of these transitions are computed in a $N=1-ε$ expansion in the number of replicas, where for small…
