Dirty higher-order Dirac semimetal: Quantum criticality and bulk-boundary correspondence

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Dirty higher-order Dirac semimetal: Quantum criticality and bulk-boundary correspondence

András L. Szabó and Bitan Roy

Phys. Rev. Research 2, 043197 – Published 6 November 2020

Abstract

We analyze the stability of time-reversal ( $T$ ) and lattice fourfold ( $C_{4}$ ) symmetry breaking three-dimensional higher-order topological (HOT) Dirac semimetals (DSMs) and the associated one-dimensional hinge modes in the presence of random pointlike charge impurities. Complementary real space numerical and momentum space renormalization group (RG) analyses suggest that a HOTDSM, while being a stable phase of matter for sufficiently weak disorder, undergoes a continuous quantum phase transition into a trivial metal at finite disorder. However, the corresponding critical exponents (numerically obtained from the scaling of the density of states) are extremely close to the ones found in a dirty, but first-order DSM that on the other hand preserves $T$ and $C_{4}$ symmetries, and support two Fermi arc surface states. This observation suggests an emergent superuniversality (insensitive to symmetries) in the entire family of dirty DSMs, as also predicted by a leading-order RG analysis. As a direct consequence of the bulk-boundary correspondence, the hinge modes in a system with open boundaries gradually fade away with increasing randomness, and completely dissolve in the trivial metallic phase at strong disorder.

Received 18 March 2020
Accepted 12 October 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.043197

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Density of states Topological phase transition Topological phases of matter

Dirac semimetal Weyl semimetal

Lattice models in condensed matter Quantum field theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

András L. Szabó ¹ and Bitan Roy ²

¹Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany
²Department of Physics, Lehigh University, Bethlehem, Pennsylvania, 18015, USA

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Vol. 2, Iss. 4 — November - December 2020

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Images

Figure 1
Phase diagrams of a dirty HOTDSM obtained from (a) the scaling of DOS at zero energy (numerically) in a lattice model [Eq. (1)] and (b) a leading-order RG analysis of the continuum model [Eq. (7)], which are in qualitative agreement. A first-order DSM is realized along the red dashed lines (a) $t_{1} = 0$ or (b) $x = 0$ , where $x = b Λ / v$ , with $Λ (v)$ as the ultraviolet cutoff (Fermi velocity), and $v \sim t a$ , $b \sim t_{1} a^{2}$ , and $Λ \sim a^{- 1}$ ( $a$ is the lattice spacing). When disorder $W$ or $g$ (dimensionless disorder coupling in the continuum limit) is weak HOT and first-order DSMs are stable. However, at stronger disorder they undergo a QPT into a metallic phase, where DOS at zero energy is finite. The blue dots in (a) are numerically obtained transition points between DSM and metal, across which we find single-parameter scaling of DOS (Fig. 3).
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Figure 2
(a) Melting of topological hinge states of a HOTDSM with increasing disorder ( $W$ ) in an open cubic system with $L = 18$ , obtained by averaging the local DOS over 300 independent disorder realizations. To recover the particle-hole symmetry (on average), we switch off the $Γ_{5}$ term in the lattice model [see Eq. (1)], which is not responsible for the hinge states when the crystal is cleaved such that its four corners are at $(\pm \frac{L}{2}, \pm \frac{L}{2})$ for any $z$ . The critical disorder $W_{c} = 3.7 \pm 0.1$ is obtained from the scaling of average DOS in a periodic system of $L = 200$ . The dissolution of the hinges into the bulk sets in for $W < W_{c}$ due to the finite size effects. (b) Scaling of the hinge localization ( $W_{h}$ ) of the topological hinge modes with increasing disorder ( $W$ ), confirming its gradual melting.
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Figure 3
(Top) Data collapses for $ρ (E)$ in a system of $L = 200$ . For clarity, we shift the data sets for different values of $t$ according to ${log}_{10} {[ρ (E) | δ |}^{- ν (d - z)}] + Δ y$ (quoted in each panel). All data sets collapse on three branches. The semimetallic (lower) and metallic (upper left) branches meet inside the quantum critical regime (upper right). (Bottom) Finite size data collapses for $ρ (0) (\equiv ρ_{0})$ inside the metallic phase. We vertically shift the data sets for different values of $t$ according to $ρ_{0} L^{d - z} + Δ y$ .
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Figure 4
Scaling of DOS $ρ (E)$ with (a) the $C_{4}$ symmetry breaking Wilson mass ( $t_{1}$ ), yielding a HOTDSM, for fixed $t$ or Fermi velocity and (b) $t$ for fixed $t_{1}$ in a cubic system of $L = 200$ with periodic boundaries for $W = 0.05$ . With increasing $t_{1}$ or $t$ the DOS decreases, without altering the $ρ (E) \sim {| E |}^{2}$ scaling, leading to the enhancement of critical disorder strength ( $W_{c}$ ) for metallicity (Fig. 1 and Table 1). Feynman diagrams contributing to the leading-order RG analysis are shown in (c) and (d). Solid (dashed) lines represent fermion (disorder) fields.
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Figure 5
Scaling analysis of DOS in a first-order DSM ( $t_{1} = 0$ ). With increasing $t$ , the Fermi velocity increases and DOS decreases, without altering $ρ (E) \sim {| E |}^{2}$ scaling. So, $W_{c}$ increases with increasing $t$ (see second column). Details of the numerical analysis are discussed in Appendix pp2. For additional numerical results see Ref. [29].
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