The quantum phase transition between the three dimensional Dirac semimetal and the diffusive metal can be induced by increasing disorder. Taking the system of disordered Z2 topological insulator as an important example, we compute the single particle density of states by the kernel polynomial method. We focus on three regions: the Dirac semimetal at the phase boundary between two topologically distinct phases, the tricritical point of the two topological insulator phases and the diffusive metal, and the diffusive metal lying at strong disorder. The density of states obeys a novel single parameter scaling, collapsing onto two branches of a universal scaling function, which correspond to the Dirac semimetal and the diffusive metal. The diverging length scale critical exponent ν and the dynamical critical exponent z are estimated, and found to differ significantly from those for the conventional Anderson transition. Critical behavior of experimentally observable quantities near and at the tricritical point is also discussed. Topological classification of different insulating phases [1,2] is an emerging new paradigm in condensed matter physics. Unlike in the Landau theory of phase transitions that is rooted in the idea of spontaneous breaking of symmetry [3], it is less clear how to describe different universality classes of the transitions between topologically different phases. This is because the usual notion of the local order parameter characterizing the different phases is often lacking. At the transition between topologically distinct phases, on the other hand, the gap closes, and the system becomes a semimetal. In three dimensions (3D) such a critical phase is stable in presence of weak disorder [4], but as disorder is increased it gives way to a diffusive metallic state [5]. This transition belongs to a distinct universality class that exhibits nontrivial dynamical and diverging length scale exponents z and ν, for example [5,6]. The 3D Dirac Hamiltonian in presence of disorder is ubiquitous: it applies to certain phases of superfluid 3 He [7], degenerate semiconductors [5], and to the Weyl semimetals [8][9][10][11]. Related theories of disordered critical points for two-dimensional interacting Dirac fermions and bosons were also advanced in the past [12,13].In this paper we discuss how this disorder-induced fermionic criticality is reflected in the scaling behavior of a readily available physical quantity, the single particle density of states (DOS), which can be understood as a proper order parameter that characterizes such a transition. We then express the critical behavior of Dirac electron velocity, diffusion coefficient, conductivity and anomalous diffusion exponent in terms of z and ν. Such a surprisingly simple description is contrasted with the conventional Anderson transition [14][15][16], where the DOS remains smooth through the transition. In order to produce and control the semimetallic phase, we focus on a 3D time-reversal symmetric topological insulator under disorder. The Z 2 topological insu...

The electronic spectrum on the spherical surface of a topological insulator reflects an active property of the helical surface state that stems from a constraint on its spin on a curved surface. The induced effective vector potential (spin connection) can be interpreted as an effective vector potential associated with a fictitious magnetic monopole induced at the center of the sphere. The strength of the induced magnetic monopole is found to be g=2pi, -2pi, being the smallest finite (absolute) value compatible with the Dirac quantization condition. We have established an explicit correspondence between the bulk Hamiltonian and the effective Dirac operator on the curved spherical surface. An explicit construction of the surface spinor wave functions implies a rich spin texture possibly realized on the surface of topological insulator nanoparticles. The electronic spectrum inferred by the obtained effective surface Dirac theory, confirmed also by the bulk tight-binding calculation, suggests a specific photo absorption/emission spectrum of such nanoparticles.Comment: 13 pages, 2 figure

A global phase diagram of disordered weak and strong topological insulators is established numerically. As expected, the location of the phase boundaries is renormalized by disorder, a feature recognized in the study of the so-called topological Anderson insulator. Here, we report unexpected quantization, i.e., robustness against disorder of the conductance peaks on these phase boundaries. Another highlight of the work is on the emergence of two subregions in the weak topological insulator phase under disorder. According to the size dependence of the conductance, the surface states are either robust or "defeated" in the two subregions. The nature of the two distinct types of behavior is further revealed by studying the Lyapunov exponents.PACS numbers: 73.20.Fz, 71.30.+h Robustness against disorder is a defining property of the topological quantum phenomena. Depending on the degree of this robustness, three-dimensional (3D) Z 2 topological insulators (TIs) [1][2][3] are classified into strong and weak (STI and WTI). Bulk-surface correspondence implies that an STI exhibits a single helical Dirac cone that is protected, while a WTI manifests generally an even number (possibly null) of Dirac cones depending on the orientation of the surface [4].Unusual robustness of Dirac electrons (especially in the case of a single Dirac cone) against disorder has been widely recognized in the study of graphene [5,6]. As a consequence of the absence of backward scattering [7], the Dirac electrons do not localize. However, in the presence of valleys (even number of Dirac cones) they do localize mediated by intervalley scatterings [8]. Does this mean that an STI continues to be an STI in the presence of arbitrarily strong disorder, while a WTI simply collapses on the switching on of the short-ranged potential disorder that induces intervalley scattering?Recent studies on the disordered WTI [9,10] seem to suggest that the reality is much different. Our global phase diagram depicted in Fig. 1 finds its way also in this direction. This phase diagram is established by a combination of the study of the averaged two-terminal conductance and of the quasi-1D decay length in the transfer matrix approach. In the actual computation the 3D disordered Z 2 topological insulator is modeled as an WilsonDirac-type tight-binding Hamiltonian with an effective (k-dependent) mass term m(k) = m 0 + m 2 µ=x,y,z (1 − cos k µ ) [11], implemented on a cubic lattice. The topological nature of the model is controlled by the ratio of two mass parameters m 0 and m 2 such that an STI phase with Z 2 (one strong and three weak) indices [4] (ν 0 , ν 1 ν 2 ν 3 ) = (1, 000) appears when −2 < m 0 /m 2 < 0, while the regime of parameters −4 < m 0 /m 2 < −2 falls on a WTI phase with (ν 0 , ν 1 ν 2 ν 3 ) = (0, 111) [12].The obtained "global" phase diagram depicted in Fig. 1 highlights the main results of the Letter. This phase diagram shows how disorder modifies the above topological classification in the clean limit (naturally as a function of the strength of disorde...

Three-dimensional topological insulators are characterized by the presence of protected gapless spin helical surface states. In realistic samples these surface states are extended from one surface to another, covering the entire sample. Generally, on a curved surface of a topological insulator an electron in a surface state acquires a spin Berry phase as an expression of the constraint that the effective surface spin must follow the tangential surface of real space geometry. Such a Berry phase adds up to pi when the electron encircles, e.g., once around a cylinder. Realistic topological insulators compounds are also often layered, i.e., are anisotropic. We demonstrate explicitly the existence of such a pi Berry phase in the presence and absence (due to crystal anisotropy) of cylindrical symmetry, that is, regardless of fulfilling the spin-to-surface locking condition. The robustness of the spin Berry phase pi against cylindrical symmetry breaking is confirmed numerically using a tight-binding model implementation of a topological insulator nanowire penetrated by a pi-flux tube.Comment: 9 pages, 4 figures (6 panels

Motivated by a recent proposal on the possibility of observing a monopole in the band structure, and by an increasing interest on the role of Berry phase in spintronics, we studied the adiabatic motion of a wave packet of Bloch functions, under a perturbation varying slowly and incommensurately to the lattice structure. We show, using only the fundamental principles of quantum mechanics, that the effective wave-packet dynamics is conveniently described by a set of equations of motion (EOM) for a semiclassical particle coupled to a nonabelian gauge field associated with a geometric Berry phase.Our EOM can be viewed as a generalization of the standard Ehrenfest's theorem, and their derivation was asymptotically exact in the framework of linear response theory. Our analysis is entirely based on the concept of local Bloch bands, a good starting point for describing the adiabatic motion of a wave packet. One of the advantages of our approach is that the various types of gauge fields were classified into two categories by their different physical origin: (i) projection onto specific bands, (ii) timedependent local Bloch basis. Using those gauge fields, we write our EOM in a covariant form, whereas the gauge-invariant field strength stems from the noncommutativity of covariant derivatives along different axes of the reciprocal parameter space. On the other hand, the degeneracy of Bloch bands makes the gauge fields nonabelian.For the purpose of applying our wave-packet dynamics to the analyses on transport phenomena in the context of Berry phase engineering, we focused on the Hall-type and polarization currents. Our formulation turned out to be useful for investigating and classifying various types of topological current on the same footing. We highlighted their symmetries, in particular, their behavior under time reversal (T ) and space inversion (I).

The non-trivialness of a topological insulator (TI) is characterized either by a bulk topological invariant or by the existence of a protected metallic surface state. Yet, in realistic samples of finite size this non-trivialness does not necessarily guarantee the gaplessness of the surface state. Depending on the geometry and on the topological indices, a finite-size energy gap of different nature can appear, and correspondingly, exhibits various scaling behaviors of the gap. The spin-to-surface locking provides one of such gap-opening mechanisms, resulting in a power-law scaling of the energy gap. Weak and strong TI's show different degrees of sensitivity to the geometry of the sample. As a noteworthy example, a strong TI nanowire of a rectangular prism shape is shown to be more gapped than that of a weak TI of precisely the same geometry.Comment: 16 pages, 18 figure

Effects of disorder on two-dimensional Z 2 topological insulator are studied numerically by the transfer matrix method. Based on the scaling analysis, the phase diagram is derived for a model of HgTe quantum well as a function of disorder strength and magnitude of the energy gap. In the presence of s z non-conserving spin-orbit coupling, a finite metallic region is found that partitions the two topologically distinct insulating phases. As disorder increases, a narrowgap topologically trivial insulator undergoes a series of transitions; first to metal, second to topological insulator, third to metal, and finally back to trivial insulator. We show that this multiple transition is a consequence of two disorder effects; renormalization of the band gap, and Anderson localization. The metallic region found in the scaling analysis corresponds roughly to the region of finite density of states at the Fermi level evaluated in the self-consistent Born approximation.

A modified periodic boundary condition adequate for non-hermitian topological systems is proposed. Under this boundary condition a topological number characterizing the system is defined in the same way as in the corresponding hermitian system and hence, at the cost of introducing an additional parameter that characterizes the non-hermitian skin effect, the idea of bulk-edge correspondence in the hermitian limit can be applied almost as it is. We develop this framework through the analysis of a non-hermitian SSH model with chiral symmetry, and prove the bulk-edge correspondence in a generalized parameter space. A finite region in this parameter space with a nontrivial pair of chiral winding numbers is identified as topologically nontrivial, indicating the existence of a topologically protected edge state under open boundary. arXiv:1908.09438v2 [cond-mat.mes-hall]