Density of States Scaling at the Semimetal to Metal Transition in Three Dimensional Topological Insulators

Abstract

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...

Disorder Effect on Chiral Edge Modes and Anomalous Hall Conductance in Weyl Semimetals

“…To find a clear difference between them, we need to consider the density of bulk states at Weyl nodes, which jumps to a finite value from zero at the transition to a diffusive metal phase. [33][34][35][36][37][38][39]46) Note that, in previous studies on this subject, the role of Fermi arc surface states is not explicitly considered, except in Refs. 46, 48, and 49.…”

Rare-Region-Induced Avoided Quantum Criticality in Disordered Three-Dimensional Dirac and Weyl Semimetals

“…A main focus has been the direct calculation of the low-energy DOS ρðEÞ since the DOS is expected to be singular at E ¼ 0 at the transition. Moreover, the DOS can be related to the critical exponents via the scaling hypothesis [24]. Following this, the dynamic exponent z and the correlation length exponent ν have now been numerically estimated for several models using the directly numerically calculated DOS [24,[33][34][35][36].…”

“…Interestingly, the one-loop perturbative renormalization group (RG) calculations of the critical exponents for the proposed SM to DM QCP are consistent with the CCFS inequality (since ν ¼ 1, Refs. [22,23]) as, in fact, are the two-loop RG calculations [26,39] and all numerical estimates in the literature [24,25,32,33,35,36]; therefore, it is not a priori obvious that rare region effects should change the universality of this transition. Given the field theoretic RG analyses and the large body of direct numerical studies of the disorder-driven SM-DM QCP, finding the various critical exponents and identifying the critical coupling, as well as the apparent consistency between the theoretical (and numerical) correlation exponent with the CCFS inequality, it seems reasonable to assume that the rare regions arising out of nonperturbative disorder effects do not change the nature of the QCP in any substantial manner.…”

“…The goal of the current work is to settle this question definitively. Although the disordered Dirac-Weyl systems have been theoretically studied very extensively in the literature [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39], essentially all of this work, except for a very recent one (Ref. [27]), study the properties of the disorder-driven SM-DM quantum phase transition, taking it for granted that such a disorder-induced QCP indeed exists in three dimensions following the predictions of the perturbative field theory.…”

“…Because of the invariable presence of disorder in all solid-state materials, there has been a substantial amount of theoretical activity studying the effect of disorder on noninteracting Dirac and Weyl fermions [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Focusing on the undoped (i.e., Fermi energy at E ¼ 0) Dirac point (i.e., the band touching point), the quadratically vanishing density of states at zero energy [ρðEÞ ∼ E 2 ] associated with the linear three-dimensional energy band dispersion places these problems in a different class than that of a conventional metal with a parabolic energy dispersion and a nonzero Fermi energy.…”