Geometry-Enforced Topological Chiral Fermions in Amorphous Chiral Metals

Abstract

Since the prediction and observation of topological Weyl semimetals (chiral TSMs), there have been enormous efforts to characterize further condensed matter realizations of chiral fermions. These efforts were dramatically accelerated by the subsequent discovery of a profound link between low-energy topological and lattice chirality in structurally chiral crystals. Though TSMs are well understood in the limit of perfect translation symmetry, real solid-state materials host defects and disorder, and may even be rendered amorphous down to all but the smallest system length scales. Previous theoretical studies have concluded that chiral TSMs transition into trivial diffusive metals at moderate disorder scales, raising concerns that chiral TSM states may only be accessible in highly crystalline samples. In this work, we in contrast identify large families of chiral TSMs that persist under strong structural disorder - even into the amorphous regime. We show that amorphous chiral TSM phases can in particular be stabilized by the presence of long-range order in the local structural chirality. We present extensive analytic and numerical calculations demonstrating the existence of both Weyl and higher-charge chiral fermions in amorphous metals whose topology and spin and orbital angular momentum textures are tunable via the interplay of average symmetry and geometry. To distinguish and generate new realizations of strongly disordered chiral fermions, we introduce an analytic approach grounded in symmetry group theory. We then introduce an amorphous Wilson loop numerical method to characterize chiral fermions with quantized Berry curvature fluxes in metals with 3D structural disorder. Our findings bridge the crystalline and strongly disordered regimes of chiral TSMs, and indicate a clear route towards engineering geometry-enforced topology in non-crystalline materials and metamaterials.

Figures (46)

• Figure 1: Schematic numerical workflow. (a) To construct a non-crystalline chiral topological semimetal [TSM] state, we begin by introducing a tight-binding Hamiltonian $\mathcal{H}$ [Eq. (\ref{['eq:Genrealspacemodel']})] that is structurally chiral when placed on a crystalline lattice and can be realized in well-defined right-handed [$R$, blue helix and lattice] and left-handed [$L$, yellow helix and lattice] enantiomers [see Supplementary Appendix (SA) \ref{['app:symDefs']}]. The lattice-scale structural chirality enforces that chiral [e.g. Kramers-Weyl] fermions at time-reversal-invariant crystal momenta ${\bf k}$ carry oppositely signed chiral charges $C$ [Eq. (\ref{['eq:MainChiralCharge']}), blue and red $\pm$ signs at $\Gamma$ and $X$ in (a), see Refs. KramersWeylAlPtObservePdGaObserve and SA \ref{['app:models']}]. (b) Each position-space $\mathcal{H}$ in this work is defined such that $\mathcal{H}$ may equally straightforwardly be implemented on both crystalline and structurally disordered lattices. As detailed in SA \ref{['app:lattices']} and the text preceding Eq. (\ref{['eq:deltaComponentsMain']}), we employ three distinct forms of structural disorder -- Gaussian disorder, random lattices, and Mikado lattices -- to approximate the amorphous regime. The models in this work also contain internal spin and orbital degrees of freedom on each lattice site, such that the hopping reference frame orientation relative to the global coordinate axes may also be disordered independently from the lattice regularity [see SA \ref{['app:DiffTypesDisorder']} and the text following Eq. (\ref{['eq:TmatrixRotFrameMain']})]. The orientational disorder in each model specifically admits a decomposition into local chirality disorder characterized by the $\mathbb{Z}_{2}$ scalar order parameter $\chi({\bf r})=\pm 1$ [blue and yellow lattice sites in (b)], and local frame orientation disorder characterized by the SO(3) matrix order parameter $R({\bf r})$ [green and purple axes and regions in (b)]. To identify a TSM state, it is necessary to establish a momentum-space system description in which nodal degeneracies can be assigned topological invariants Armitage2018Wieder22. Though structurally disordered systems cannot be characterized in terms of an exact crystal momentum ${\bf k}$, they may still be characterized in terms of an approximate plane-wave pseudo-momentum ${\bf p}$ [Eq. (\ref{['eq:planewavesMain']})]. (c) Using ${\bf p}$, we Fourier-transform the real-space system Green's function [$\mathcal{G}(E)$ in Eq. (\ref{['eq:RealSpaceGreenMain']})] to obtain the momentum-resolved matrix Green's function $\mathcal{G}(E,{\bf p},{\bf p}')$ [Eq. (\ref{['eq:MomGreenFuncMain']}) and SA \ref{['app:PhysicalObservables']}]. Though $\mathcal{G}(E,{\bf p},{\bf p}')$ is an exact quantity, it is generically a function of two pseudo-momenta ${\bf p}$ and ${\bf p}'$, obscuring a direct connection to crystalline quantities computed from Green's functions that only depend on a single ${\bf k}$. However, we find that after averaging over an ensemble of replicas with independently generated structural and internal degree-of-freedom disorder, $\mathcal{G}(E,{\bf p},{\bf p}')$ in practice reduces on the average to a one-momentum [average] matrix Green's function $\bar{\mathcal{G}}(E,{\bf p})$ [Eq. (\ref{['eq:averageOneMomentumGreenMain']}) and SA \ref{['app:PhysicalObservables']}]. (d) Using $\bar{\mathcal{G}}(E,{\bf p})$, we then compute the spectral function $\bar{A}(E,\mathbf{p})$ [Eq. (\ref{['eq:SpecFuncMain']}) and SA \ref{['app:PhysicalObservables']}] and spin and orbital angular momentum textures [Eqs. (\ref{['eq:SpinDOSMain']}) and (\ref{['eq:OAMDOSMain']}) and SA \ref{['app:amorphousKramers']} and \ref{['app:amorphousMultifold']}]. We also use $\bar{\mathcal{G}}(E,{\bf p})$ to construct a single-particle effective Hamiltonian $\mathcal{H}_\text{Eff}({\bf p})$, which we find to exhibit numerically stable dispersion relations and Berry phases near ${\bf p}={\bf 0}$ over a wide range of reference energy cuts $E_{C}$ in Eq. (\ref{['eq:AvgHEffMain']}) [SA \ref{['app:EffectiveHamiltonian']}]. (e) Motivated by this observation, we lastly introduce an amorphous Wilson loop method to quantitatively evaluate the topology of ${\bf p}={\bf 0}$ nodal degeneracies in strongly disordered systems [Eq. (\ref{['eq:WilsonElementsMain']}) and SA \ref{['sec:WilsonBerry']}]. Across all three structural disorder implementation methods in (b), we find that each of the models in this work exhibits the same spectral and topologically nontrivial features in the presence of long-range local chirality order in $\chi({\bf r})$, providing strong evidence that each model realizes an amorphous chiral TSM state.
• Figure 2: Kramers-Weyl and multifold fermions in the crystalline limit. (a) The Brillouin zone [BZ] of chiral space group 16 ($P222$), the orthorhombic space group of the Kramers-Weyl model in the crystalline limit [Eq. (\ref{['eq:HKWMaink']}), see SA \ref{['app:symDefs']} and \ref{['app:PristineKramers']} and Ref. KramersWeyl]. (b) Bulk band structure of the right-handed [$R$] enantiomer of the crystalline Kramers-Weyl model. The spectrum at each time-reversal-invariant momentum [TRIM] point in (b) consists of a twofold nodal degeneracy with linear dispersion [Eq. (\ref{['eq:mainHKP']})], and corresponds to a Kramers-Weyl chiral fermion with chiral charge $|C|=1$ [Eq. (\ref{['eq:MainChiralCharge']})]. (c) The spectral function $A(E,\mathbf{p})$ [SA \ref{['app:PhysicalObservables']}] of the Kramers-Weyl fermion at the $\Gamma$ point [${\bf k}={\bf 0}$] in (b). (d) The Abelian Wilson loop spectrum [Eq. (\ref{['eq:WilsonElementsMain']}) and SA \ref{['sec:WilsonBerry']}] of the lower band, computed on a sphere surrounding the nodal degeneracy at the $\Gamma$ point in (b,c). The Wilson loop spectrum in (d) winds once in the positive direction, indicating that the Kramers-Weyl fermion at $\Gamma$ carries a chiral charge of $C=1$ for the $R$ model enantiomer. In the $L$ model enantiomer, the chiral charge of the Kramers-Weyl fermion at $\Gamma$ is instead $C=-1$, and is therefore directly determined by the lattice-scale structural chirality $C^{\mathrm{KW}}_{\mathcal{H}}$ [Eq. (\ref{['eq:mainChiralCharge']}), see SA \ref{['app:PristineKramers']} and Ref. KramersWeyl]. (e) The BZ of chiral space group 195 ($P23$), the cubic space group of the symmorphic multifold fermion model introduced in this work tuned to the crystalline limit [Eq. (\ref{['eq:HamBloch3FMain']}), see SA \ref{['app:symDefs']} and \ref{['app:PristineMulifold']}]. (f) Bulk band structure of the $R$ enantiomer of the multifold fermion model. In (f), the lower three bands meet in symmetry-enforced threefold nodal degeneracies at the TRIM points $\Gamma$ and $R$ that correspond to spin-1 chiral multifold fermions ManesNewFermionNewFermionschang2017largetang2017CoSiKramersWeyl. (g) The spectral function of the multifold fermion at the $\Gamma$ point in (f), plotted along $k_{111} = (1/\sqrt{3})(k_{x}+k_{y}+k_{z})$. (h) The non-Abelian Wilson loop spectrum of the lower two bands, computed on a sphere surrounding the threefold degeneracy at the $\Gamma$ point. In (h), the Wilson loop eigenvalues as a set wind twice in the positive direction, indicating that the lower two bands of the multifold fermion at $\Gamma$ together carry a chiral charge of $C=2$. In the $L$ model enantiomer, the lower two bands of the multifold fermion at $\Gamma$ instead carry a chiral charge of $C=-2$ [SA \ref{['app:PristineMulifold']}]. Like the Kramers-Weyl fermions in (b,c) and the chiral multifold fermions experimentally studied in Refs. AlPtObservePdGaObserve, the spin-1 chiral fermion at $\Gamma$ in (f,g) therefore similarly carries a low-energy topological chirality that is inherited from and controlled by the lattice-scale structural chirality $C^{\mathrm{3F}}_{\mathcal{H}}$ [Eq. (\ref{['eq:multifoldStructuralChiMain']})]. Numerical details for panels (b-d) and (f-h) are provided in SA \ref{['app:PristineKramers']} and \ref{['app:PristineMulifold']}, respectively.
• Figure 3: Energy spectrum and spin texture of the disordered Kramers-Weyl model. (a-f) The spectral function $\bar{A}(E,{\bf p})$ [Eq. (\ref{['eq:SpecFuncMain']})] averaged over 50 replicas of the Kramers-Weyl model [Eqs. (\ref{['eq:amorphousKWTmatrixFinalChiralityMain']}), (\ref{['eq:KWHeavisideMain']}), and (\ref{['eq:KWmainMats']}) and Ref. KramersWeyl] with increasing Gaussian structural disorder and local frame disorder implemented with the same standard deviation $\eta$ [Eq. (\ref{['eq:GaussianDisorderMain']})], and with local chirality domains of unequal volume [70% right-handed, 30% left-handed] within each disorder replica [see Fig. \ref{['fig:Fig1flow']}(b) and SA \ref{['app:DiffTypesDisorder']}]. Panels (a-c) show $\bar{A}(E,{\bf p})$ as a function of $E$ and $p_{x}$ [$k_{x}$ in the $\eta=0$ crystalline limit in (a)] and panels (d-f) show $\bar{A}(E,{\bf p})$ as a function of $p_{x,y}$ [$k_{x,y}$ in (d)] at $p_{z}=0.1$ and fixed $E$ [the dashed lines in (a-c), respectively]. In both the moderate [$\eta=0.2$ in (b)] and strong [$\eta=0.5$ in (c)] structural disorder regimes [see SA \ref{['app:PhysicalObservables']}], $\bar{A}(E,{\bf p})$ exhibits a linearly dispersing feature centered around ${\bf p}={\bf 0}$ that becomes increasingly diffuse as $\eta$ is increased and upwardly shifted in energy by a disorder-renormalized chemical potential, but nevertheless continues to strongly resemble the crystalline Kramers-Weyl fermion in (a). In (b,c) we also plot in light blue circles the band structure of the mean-field effective Hamiltonian $\mathcal{H}_{\mathrm{Eff}}({\bf p})$ [Eq. (\ref{['eq:AvgHEffMain']}) and SA \ref{['app:EffectiveHamiltonian']}], which for all $\eta$ exhibits a linear nodal degeneracy at ${\bf p}={\bf 0}$ that precisely corresponds to a $|C|=1$ Kramers-Weyl fermion [see Figs. \ref{['fig:Fig3randomKW3F']}(d) and \ref{['fig:Fig4_KW_main_chiralities']}(a-e)]. In addition to the linear spectral feature at ${\bf p}={\bf 0}$, the disordered systems in (e,f) exhibit broadened, ring- [sphere-] like spectral features in the vicinity of $|{\bf p}|=\pi/\bar{a}$ and $|{\bf p}|=\pi\sqrt{2}/\bar{a}$ where $\bar{a}=1$ is the average nearest-neighbor spacing, similar to the ring-like, higher-Brillouin-zone Dirac surface-state repetitions recently observed in amorphous Bi$_{2}$Se$_{3}$corbae_evidence_2020Ciocys2023. (g-i) The spin texture [Eq. (\ref{['eq:SpinDOSMain']})] of the non-crystalline Kramers-Weyl systems in (d-f), respectively. In panels (g-i), the in-plane components of the spin texture $\langle S^{x,y}(E,\mathbf{p})\rangle$ are represented as arrows with log-scale lengths, while the out-of-plane component $\langle S^{z}(E,\mathbf{p})\rangle$ is represented through a log-scale color map in which orange is positive and teal is negative. (g) In the crystalline limit, the Kramers-Weyl fermions at each TRIM point exhibit perfect monopole-like spin textures that are locked to their chiral charges KramersWeyl. (h,i) As the disorder scale $\eta$ is increased, the Kramers-Weyl fermions away from ${\bf p}={\bf 0}$ become merged into ring-like spectral features with largely isotropic spin textures that are inherited from their crystalline-limit [$\eta=0$] spin textures, and hence chiral charges. This suggests the possibility that the ring-like feature in (e,f,h,i) at $|{\bf p}|=\pi$ [$|{\bf p}|=\pi\sqrt{2}$] is a many-particle disordered Kramers-Weyl fermion with the opposite [same] chiral charge as the linear nodal degeneracy at ${\bf p}={\bf 0}$. Complete calculation details for all panels are provided in SA \ref{['app:amorphousKramers']}.
• Figure 4: Random-lattice Kramers-Weyl and chiral multifold fermions. (a-c) Energy spectrum $\bar{A}(E,\mathbf{p})$ [Eq. (\ref{['eq:SpecFuncMain']})] and spin texture $\langle\mathbf{S}(E,\mathbf{p})\rangle$ [Eq. (\ref{['eq:SpinDOSMain']})] averaged over 50 replicas of the non-crystalline Kramers-Weyl model [Eqs. (\ref{['eq:amorphousKWTmatrixFinalChiralityMain']}), (\ref{['eq:KWHeavisideMain']}), and (\ref{['eq:KWmainMats']}) and Ref. KramersWeyl] on 3D random lattices with strong random frame disorder parameterized by $\eta=0.5$ and contiguous chirality domains of unequal volume such that $2/3$ of sites are right-handed [$n_R=2/3$] and $1/3$ of sites are left-handed [$n_L=1-n_{R}=1/3$] within each disorder replica [see Fig. \ref{['fig:Fig1flow']}(b) and SA \ref{['app:DiffTypesDisorder']}]. Panel (a) shows $\bar{A}(E,{\bf p})$ as a function of $E$ and $p_{x}$, and panels (b,c) respectively show $\bar{A}(E,{\bf p})$ and $\langle\mathbf{S}(E,\mathbf{p})\rangle$ as functions of $p_{x,y}$ at $p_{z}=0.1$ and fixed $E$ [the dashed line in (a)]. In (c), $\langle S^{x,y}(E,\mathbf{p})\rangle$ are represented as arrows with log-scale lengths and $\langle S^{z}(E,\mathbf{p})\rangle$ is represented using a log-scale color map in which orange is positive and teal is negative. Like in the Kramers-Weyl model with strong Gaussian disorder [Fig. \ref{['fig:Fig2KWmainspec']}(c,f)], the random-lattice Kramers-Weyl model exhibits (a) a linear spectral feature at ${\bf p}={\bf 0}$ and (b) broadened, ring- [sphere-] like features at $|{\bf p}|=\pi/\bar{a}$ and $|{\bf p}|=\pi\sqrt{2}/\bar{a}$, where $\bar{a}=1$ is the average nearest-neighbor spacing. Also like in the Gaussian-disordered Kramers-Weyl model [Fig. \ref{['fig:Fig2KWmainspec']}(i)], the ${\bf p}={\bf 0}$ feature exhibits (c) a nearly perfect outward-pointing, monopole-like spin texture, and the ring-like features at $|{\bf p}|=\pi$ and $|{\bf p}|=\pi\sqrt{2}$ respectively exhibit inward- and outward-pointing spin textures. (d) The Abelian amorphous Wilson loop spectrum [Eq. (\ref{['eq:WilsonElementsMain']}) and SA \ref{['sec:WilsonBerry']}] of the linear spectral feature at ${\bf p}={\bf 0}$ in (a), computed over the lower band of the effective Hamiltonian $\mathcal{H}_{\text{Eff}}(\mathbf{p})$ [light blue circles in (a), see Eq. (\ref{['eq:AvgHEffMain']}) and SA \ref{['app:EffectiveHamiltonian']}]. The Wilson spectrum in (d) winds once in the positive direction as a function of the sphere polar angle $\theta$, quantitatively identifying the nodal spectral feature at ${\bf p}={\bf 0}$ in (a) as an amorphous Kramers-Weyl fermion with a chiral charge of $C=1$ [Eq. (\ref{['eq:MainChiralCharge']})]. (e-g) $\bar{A}(E,{\bf p})$ and orbital angular momentum [OAM] texture $\langle\mathbf{L}(E,\mathbf{p})\rangle$ [Eq. (\ref{['eq:OAMDOSMain']})] averaged over 50 replicas of the symmorphic multifold fermion model [see Eq. (\ref{['eq:HamBloch3FMain']}) and SA \ref{['sec:Multifold']}] on 3D random lattices with $\eta=0.5$ random frame disorder and contiguous chirality domains with $n_R=2/3$ and $n_L=1/3$. Panel (e) shows $\bar{A}(E,{\bf p})$ as a function of $E$ and $p_{111} = (1/\sqrt{3})(p_{x}+p_{y}+p_{z})$, and panels (f,g) respectively show $\bar{A}(E,{\bf p})$ and $\langle\mathbf{L}(E,\mathbf{p})\rangle$ as functions of $p_{x,y}$ at $p_{z}=0.1$ and fixed $E$ [the dashed line in (e)]. In (g), $\langle L^{x,y}(E,\mathbf{p})\rangle$ are represented as arrows with log-scale lengths and $\langle L^{z}(E,\mathbf{p})\rangle$ is represented using a log-scale color map in which red is positive and blue is negative. The random-lattice multifold model in (e) exhibits a disorder-broadened threefold nodal degeneracy at ${\bf p}={\bf 0}$ that to leading order consists of two linearly dispersing bands and a central nondispersing band, closely resembling the crystalline $\Gamma$-point chiral multifold fermion in Fig. \ref{['fig:MainBands']}(g). $\bar{A}(E,{\bf p})$ in (f) also exhibits ring-like spectral features at larger $|{\bf p}|$ whose origin and properties are further detailed in SA \ref{['app:amorphousMultifold']}. The multifold spectral feature at ${\bf p}={\bf 0}$ in (e,f) exhibits (g) a nearly perfect outward-pointing, monopole-like OAM texture like that observed in recent experimental studies of crystalline chiral multifold fermions OAMmultifold2OAMmultifold3. (h) The non-Abelian amorphous Wilson loop spectrum of the threefold spectral feature at ${\bf p}={\bf 0}$ in (e), computed over the lower two bands of $\mathcal{H}_{\text{Eff}}({\bf p})$ [light purple circles in (e)]. The two Wilson loop eigenvalues in (h) wind twice as a set in the positive direction, confirming that the threefold spectral feature in (e) is an amorphous chiral multifold fermion whose lower two bands together carry $C=2$. Complete calculation details for panels (a-d) and (e-h) are provided in SA \ref{['app:amorphousKramers']} and \ref{['app:amorphousMultifold']}, respectively.
• Figure 5: Controlling topology with average structural chirality in disordered chiral semimetals. To generate this figure, we first construct five ensembles of (a-e) the non-crystalline Kramers-Weyl model [Eqs. (\ref{['eq:amorphousKWTmatrixFinalChiralityMain']}), (\ref{['eq:KWHeavisideMain']}), and (\ref{['eq:KWmainMats']}) and Ref. KramersWeyl] and (f-j) five ensembles of the symmorphic multifold model [Eq. (\ref{['eq:HamBloch3FMain']}) and SA \ref{['sec:Multifold']}] with 50 replicas each, where each replica is subject to Gaussian lattice and local frame disorder with $\eta=0.2$ in Eq. (\ref{['eq:GaussianDisorderMain']}) and contains contiguous right- and left-handed chirality domains with varying concentrations respectively given by $n_R=N_R/N_{\mathrm{sites}}$ and $n_{L}=1-n_{R}$ [respectively schematically depicted with blue $R$ and yellow $L$ regions in the top row, see Fig. \ref{['fig:Fig1flow']}(a,b) and SA \ref{['app:DiffTypesDisorder']}]. For each ensemble, we construct an effective Hamiltonian $\mathcal{H}_{\mathrm{Eff}}({\bf p})$ using the replica-averaged Green's function $\bar{\mathcal{G}}(E,\mathbf{p})$ [Eqs. (\ref{['eq:averageOneMomentumGreenMain']}) and (\ref{['eq:AvgHEffMain']})], and then use the eigenstates of $\mathcal{H}_{\mathrm{Eff}}({\bf p})$ to compute the amorphous [disordered] Wilson loop spectrum on a sphere surrounding the nodal degeneracy at ${\bf p}={\bf 0}$ [Eqs. (\ref{['eq:MainChiralCharge']}) and (\ref{['eq:WilsonElementsMain']}) and SA \ref{['app:EffectiveHamiltonian']} and \ref{['sec:WilsonBerry']}]. For the Abelian [scalar] Wilson loop of the lower band of $\mathcal{H}_{\mathrm{Eff}}({\bf p})$ in the Kramers-Weyl model, we observe a quantized winding number as a function of the sphere polar angle $\theta$ of (a,b) $C=1$ for $n_{L}<0.5$, (d,e) $C=-1$ for $n_{L}>0.5$, and (c) a region in the vicinity of $n_{L}\approx 0.5$ with a non-smooth Wilson spectrum. Similarly, for the non-Abelian Wilson loop of the lowest two bands of $\mathcal{H}_{\mathrm{Eff}}({\bf p})$ in the multifold model, the two Wilson loop eigenvalues as a set exhibit quantized winding numbers of (f,g) $C=2$ for $n_{L}<0.5$ and (i,j) $C=-2$ for $n_{L}>0.5$, outside of again (h) a region in the vicinity of $n_{L}\approx 0.5$ with a non-smooth Wilson spectrum. The Wilson spectra in (a-j) therefore indicate that both the linear spectral feature at ${\bf p}={\bf 0}$ in Figs. \ref{['fig:Fig2KWmainspec']}(b,c) and \ref{['fig:Fig3randomKW3F']}(a) and the threefold spectral feature at ${\bf p}={\bf 0}$ in Fig. \ref{['fig:Fig3randomKW3F']}(e) are disordered chiral fermions with quantized topological chirality that is tunable via the average system structural chirality, generalizing the results of Refs. KramersWeylAlPtObservePdGaObserve to the structurally disordered regime. Complete calculation details for panels (a-e) and (f-j) are provided in SA \ref{['app:amorphousKramers']} and \ref{['app:amorphousMultifold']}, respectively.