Topological diffusive metal in amorphous transition metal monosilicides

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Letter

Topological diffusive metal in amorphous transition metal monosilicides

Selma Franca and Adolfo G. Grushin

Phys. Rev. Materials 8, L021201 – Published 14 February 2024

Abstract

In chiral crystals crystalline symmetries can protect multifold fermions, pseudorelativistic masless quasiparticles that have no high-energy counterparts. Their realization in transition metal monosilicides has exemplified their intriguing physical properties, such as long Fermi arc surface states and unusual optical responses. Recent experimental studies on amorphous transition metal monosilicides suggest that topological properties may survive beyond crystals, even though theoretical evidence is lacking. Motivated by these findings, we theoretically study a tight-binding model of amorphous transition metal monosilicides. We find that topological properties of multifold fermions survive in the presence of structural disorder that converts the semimetal into a diffusive metal. We characterize this topological diffusive metal phase with the spectral localizer, a real-space topological indicator that we show can signal multifold fermions. Our findings showcase how topological properties can survive in disordered metals, and how they can be uncovered using the spectral localizer.

Received 4 July 2023
Revised 12 December 2023
Accepted 19 January 2024

DOI:https://doi.org/10.1103/PhysRevMaterials.8.L021201

Physics Subject Headings (PhySH)

Geometric & topological phases Topological materials

Amorphous materials Semimetals

Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Selma Franca ^* and Adolfo G. Grushin ^†

Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France

^*selma.franca@neel.cnrs.fr
^†adolfo.grushin@neel.cnrs.fr

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Vol. 8, Iss. 2 — February 2024

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Images

Figure 1
(a) Schematic phase diagram found in this work, including the multifold semimetal (MSM), the topological diffusive metal (TDM), the diffusive metal (DM), and the Anderson insulator (AI) phases, as a function of the disorder variance $σ$ . The TDM persists until $σ_{TDM}$ that is defined in Fig. 4 and is signaled by in-gap states of the localizer as well as a finite DOS at $E_{F}$ typical of a diffusive metal. The trivial DM phase is delimited by $σ_{AI}$ , defined in Fig. 3, from the AI phase. (b) Orbitals of the crystalline unit cell. (c) The nearest-neighbor interorbital hoppings $\frac{v_{1} \pm v_{p}}{4}$ represented by solid and dashed lines. The color denotes hoppings between different sets of orbitals: red for A-B/C-D, blue for A-D/B-C, and green for A-C/B-D. (d) Band structure for parameter regime (i) with $v_{1} = v_{2} = 0$ and $v_{p} = - 0.762$ . (e) Band structure for parameter regime (ii), a-RhSi, with $v_{1} = 0.55, v_{2} = 0.16$ , and $v_{p} = - 0.762$ . Solid and dashed curves correspond to maximum hopping radii $d_{c} = 1.01$ and 1.5, respectively. In panel (d), the spectrum is doubly degenerate with two double-Weyl fermions occurring at $Γ$ and $R$ points. In panel (e), a-RhSi regime, we see a threefold fermion at the $Γ$ point and a double-Weyl fermion at the $R$ point.
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Figure 2
(a) $ρ (E)$ vs $E$ as a function of $σ$ . Dashed and dotted lines represent fit functions $α E^{2}$ and $α + β | E |$ , respectively. (b) The averaged zero-energy DOS ${\bar{ρ}}_{0}$ as a function of disorder strength $σ$ ; $ρ_{0}$ becomes nonzero at $σ \approx 0.1$ . (c) ${\bar{ρ}}^{(2)} (0)$ vs $σ$ peaks around $σ_{c} = 0.08$ independent of system size ( $L = 12$ and 20) and KPM order $N_{C}$ . We define ${\bar{ρ}}^{(2)} (0) = \sum_{λ = 1}^{N_{dis}} {(ρ^{λ})}^{(2)} (0)$ , where ${(ρ^{λ})}^{(2)} (0)$ is estimated from a fit $ρ^{λ} (E) = ρ_{0}^{λ} + {(ρ^{λ})}^{(2)} (0) E^{2}$ in the energy range $(- 0.2, 0.2)$ for independent disorder realizations $λ$ .
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Figure 3
Panels (a) and (b) show disorder-averaged adjacent level spacing ratios (solid lines) and inverse participation ratios (dashed lines) at $E_{F} = 0$ as a function of disorder strength $σ$ for a system in parameter regime (i) and the a-RhSi regime, respectively. Panels (c) and (d) show the disorder-averaged typical DOS for different orders of the KPM expansion $N_{C}$ as a function of $σ$ for a system in parameter regime (i) and the a-RhSi regime, respectively. Here, $σ_{AI}$ is the critical disorder strength at which the system transitions from a DM phase to an AI phase.
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Figure 4
Panels (a) and (c) show ${\bar{ρ}}_{L_{0}}$ —the disorder-averaged DOS of the operator $L_{0}$ as a function of disorder strength $σ$ for regime (i) and the a-RhSi regime, respectively. In panels (b) and (d) are plotted disorder-averaged energies of the midgap and first-excited states ${\bar{ε}}_{0}$ and ${\bar{ε}}_{1}$ and their difference ${\bar{ε}}_{1} - {\bar{ε}}_{0}$ as a function of $σ$ for the two parameter regimes. Here, dashed green lines represent the topological phase transition point $σ_{TDM}$ at which ${\bar{ε}}_{1} \approx 2 {\bar{ε}}_{0}$ . The dashed black line represents a fit ${\bar{ε}}_{0} = a \sqrt{σ}$ , where $a = 0.0692$ in regime (i) and $a = 0.0696$ in regime (ii). The insets of panels (b) and (d) show how well ${\bar{ε}}_{0}$ matches with the predicted form $κ^{0.75} σ$ [68] (gray line) in the case of small disorder strengths. For panels (a)–(d), we consider $κ = 0.1$ and $N_{dis} = 25$ . In panel (e) are plotted disorder-averaged ( $N_{dis} = 10$ ) momentum-resolved spectral functions $\bar{A} (k, E = - 0.3)$ for different disorder strengths $σ$ .
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