Three-dimensional topological insulators feature Dirac-like surface states which are topologically protected against the influence of weak quenched disorder. Here we investigate the effect of surface disorder beyond the weak-disorder limit using large-scale numerical simulations. We find two qualitatively distinct regimes: Moderate disorder destroys the Dirac cone and induces diffusive metallic behavior at the surface. Even more remarkably, for strong surface disorder a Dirac cone reappears, as new weakly disordered "surface" states emerge in the sample beneath the disordered surface layer, which can be understood in terms of an interface between a topological and an Anderson insulator. Together, this demonstrates the drastic effect of disorder on topological surface states, which cannot be captured within effective two-dimensional models for the surface states alone.Topological insulators (TIs) are a novel form of insulators, where topological properties of the band structure lead to the formation of metallic states at the surface [1,2]. Two-dimensional (2d) TIs were predicted [3,4] and then realized in quantum-well heterostructures [5]. Subsequently, three-dimensional (3d) generalizations were discussed theoretically [6][7][8] and realized afterwards, primarily in Bi-based compounds [9,10]. 3d TIs are characterized by a set of topological quantum numbers, which lead to a classification into strong and weak TIs.Strong TIs possess low-energy surface states which can be described in terms of massless 2d Dirac electrons, with an odd number of Dirac cones per surface. Remarkably, these surface states are topologically protected against the influence of disorder due to their helical nature: The spin-momentum locking of the Dirac theory implies that the states k and − k have opposite spin. This in turn suppresses the lowest-order matrix element for backscattering, k → − k, leading to the absence of weak localization for weak potential disorder. What about strong disorder? Theoretical results obtained within 2d continuum Dirac models suggest that such states remain metallic even for arbitrarily large disorder: The beta-function describing the renormalization-group flow of the conductivity is always positive [11][12][13]. However, by construction the applicability of these effective theories to TI surface states is restricted to energies much smaller than the bulk gap, ∆, and the effect of surface disorder with a strength γ ∆ can therefore not be captured. Intuitively, this is because strong impurities will influence both surface and bulk states, causing a non-trivial coupling between the two which cannot be described by a theory for surface states alone.To the best of our knowledge, the effect of strong surface disorder has only been discussed locally for isolated impurities [14], but the global fate of TI surface states under moderate or strong disorder is not known. This issue is not only of fundamental interest in order to understand the range and limitations of "topological protection", but is also of enormous pra...

Numerical approaches to Anderson localization face the problem of having to treat large localization lengths while being restricted to finite system sizes. We show that by finite-size scaling of the probability distribution of the local density of states (LDOS) this long-standing problem can be overcome. To this end we reexamine the approach, propose numerical refinements, and apply it to study the dependence of the distribution of the LDOS on the dimensionality and coordination number of the lattice. Particular attention is given to the graphene lattice. We show that the system-size dependence of the LDOS distribution is indeed an unambiguous sign of Anderson localization, irrespective of the dimension and lattice structure. The numerically exact LDOS data obtained by us agree with a log-normal distribution over up to ten orders of magnitude and thereby fulfill a nontrivial symmetry relation previously derived for the non-linear σ-model.

ObjectivesTo assess the safety and feasibility of MRI-guided high-intensity focused ultrasound (MR-HIFU) ablation in breast cancer patients using a dedicated breast platform.MethodsPatients with early-stage invasive breast cancer underwent partial tumour ablation prior to surgical resection. MR-HIFU ablation was performed using proton resonance frequency shift MR thermometry and an MR-HIFU system specifically designed for breast tumour ablation. The presence and extent of tumour necrosis was assessed by histopathological analysis of the surgical specimen. Pearson correlation coefficients were calculated to assess the relationship between sonication parameters, temperature increase and size of tumour necrosis at histopathology.ResultsTen female patients underwent MR-HIFU treatment. No skin redness or burns were observed in any of the patients. No correlation was found between the applied energy and the temperature increase. In six patients, tumour necrosis was observed with a maximum diameter of 3–11 mm. In these patients, the number of targeted locations was equal to the number of areas with tumour necrosis. A good correlation was found between the applied energy and the size of tumour necrosis at histopathology (Pearson = 0.76, p = 0.002).ConclusionsOur results show that MR-HIFU ablation with the dedicated breast system is safe and results in histopathologically proven tumour necrosis.Key Points• MR-HIFU ablation with the dedicated breast system is safe and feasible • In none of the patients was skin redness or burns observed • No correlation was found between the applied energy and the temperature increase • The correlation between applied energy and size of tumour necrosis was good

We examine several numerical techniques for the calculation of the dynamics of quantum systems. In particular, we single out an iterative method which is based on expanding the time evolution operator into a finite series of Chebyshev polynomials. The Chebyshev approach benefits from two advantages over the standard time-integration Crank-Nicholson scheme: speedup and efficiency. Potential competitors are semiclassical methods such as the Wigner-Moyal or quantum tomographic approaches. We outline the basic concepts of these techniques and benchmark their performance against the Chebyshev approach by monitoring the time evolution of a Gaussian wave packet in restricted one-dimensional (1D) geometries. Thereby the focus is on tunnelling processes and the motion in anharmonic potentials. Finally we apply the prominent Chebyshev technique to two highly non-trivial problems of current interest: (i) the injection of a particle in a disordered 2D graphene nanoribbon and (ii) the spatiotemporal evolution of polaron states in finite quantum systems. Here, depending on the disorder/electron-phonon coupling strength and the device dimensions, we observe transmission or localisation of the matter wave.Comment: 8 pages, 3 figure