Entanglement Properties of Localized States in 1D Topological Quantum Walks

The symmetries associated with discrete-time quantum walks (DTQWs) and the flexibilities in controlling their dynamical parameters allow to create a large number of topological phases. An interface in position space, which separates two regions with different topological numbers, can, for example, be effectively modelled using different coin parameters for the walk on either side of the interface. Depending on the neighbouring numbers, this can lead to localized states in one-dimensional configurations and here we carry out a detailed study into the strength of such localized states. We show that it can be related to the amount of entanglement created by the walks, with minima appearing for strong localizations. This feature also persists in the presence of small amounts of $sigma_x$ (bit flip) noise.

Conductance distributions in disordered quantum spin-Hall systems

We study numerically the charge conductance distributions of disordered quantum spin-Hall (QSH) systems using a quantum network model. We have found that the conductance distribution at the metal-QSH insulator transition is clearly different from that at the metal-ordinary insulator transition. Thus the critical conductance distribution is sensitive not only to the boundary condition but also to the presence of edge states in the adjacent insulating phase. We have also calculated the point-contact conductance. Even when the two-terminal conductance is approximately quantized, we find large fluctuations in the point-contact conductance. Furthermore, we have found a semi-circular relation between the average of the point-contact conductance and its fluctuation.

Boundary multifractality at the integer quantum Hall plateau transition: implications for the critical theory

We study multifractal spectra of critical wave functions at the integer quantum Hall plateau transition using the Chalker-Coddington network model. Our numerical results provide important new constraints which any critical theory for the transition will have to satisfy. We find a non-parabolic multifractal spectrum and we further determine the ratio of boundary to bulk multifractal exponents. Our results rule out an exactly parabolic spectrum that has been the centerpiece in a number of proposals for critical field theories of the transition. In addition, we demonstrate analytically exact parabolicity of related boundary spectra in the 2D chiral orthogonal `Gade-Wegner symmetry class.

Multifractality and Conformal Invariance at 2D Metal-Insulator Transition in the Spin-Orbit Symmetry Class

We study the multifractality (MF) of critical wave functions at boundaries and corners at the metal-insulator transition (MIT) for noninteracting electrons in the two-dimensional (2D) spin-orbit (symplectic) universality class. We find that the MF exponents near a boundary are different from those in the bulk. The exponents at a corner are found to be directly related to those at a straight boundary through a relation arising from conformal invariance. This provides direct numerical evidence for conformal invariance at the 2D spin-orbit MIT. The presence of boundaries modifies the MF of the whole sample even in the thermodynamic limit.

Spatial Structures of Anomalously Localized States in Tail Regions at the Anderson Transition

We study spatial structures of anomalously localized states (ALS) in tail regions at the critical point of the Anderson transition in the two-dimensional symplectic class. In order to examine tail structures of ALS, we apply the multifractal analysis only for the tail region of ALS and compare with the whole structure. It is found that the amplitude distribution in the tail region of ALS is multifractal and values of exponents characterizing multifractality are the same with those for typical multifractal wavefunctions in this universality class.

Anomalously localized states and multifractal correlations of critical wavefunctions in two-dimensional electron systems with spin-orbital interactions

Anomalously localized states (ALS) at the critical point of the Anderson transition are studied for the SU(2) model belonging to the two-dimensional symplectic class. Giving a quantitative definition of ALS to clarify statistical properties of them, the system-size dependence of a probability to find ALS at criticality is presented. It is found that the probability increases with the system size and ALS exist with a finite probability even in an infinite critical system, though the typical critical states are kept to be multifractal. This fact implies that ALS should be eliminated from an ensemble of critical states when studying critical properties from distributions of critical quantities. As a demonstration of the effect of ALS to critical properties, we show that the distribution function of the correlation dimension of critical wavefunctions becomes a delta function in the thermodynamic limit only if ALS are eliminated.

Critical level statistics and anomalously localized states at the Anderson transition

We study the level-spacing distribution function $P(s)$ at the Anderson transition by paying attention to anomalously localized states (ALS) which contribute to statistical properties at the critical point. It is found that the distribution $P(s)$ for level pairs of ALS coincides with that for pairs of typical multifractal states. This implies that ALS do not affect the shape of the critical level-spacing distribution function. We also show that the insensitivity of $P(s)$ to ALS is a consequence of multifractality in tail structures of ALS.

Correlation Exponent and Anomalously Localized States at the Critical Point of the Anderson Transition

We study the box-measure correlation function of quantum states at the Anderson transition point with taking care of anomalously localized states (ALS). By eliminating ALS from the ensemble of critical wavefunctions, we confirm, for the first time, the scaling relation z(q)=d+2tau(q)-tau(2q) for a wide range of q, where q is the order of box-measure moments and z(q) and tau(q) are the correlation and the mass exponents, respectively. The influence of ALS to the calculation of z(q) is also discussed.