Quasi-symmetry protected topology in a semi-metal

Abstract

The crystal symmetry of a material dictates the type of topological band structures it may host, and therefore symmetry is the guiding principle to find topological materials. Here we introduce an alternative guiding principle, which we call ‘quasi-symmetry’. This is the situation where a Hamiltonian has an exact symmetry at lower-order that is broken by higher-order perturbation terms. This enforces finite but parametrically small gaps at some low-symmetry points in momentum space. Untethered from the restraints of symmetry, quasi-symmetries eliminate the need for fine-tuning as they enforce that sources of large Berry curvature will occur at arbitrary chemical potentials. We demonstrate that a quasi-symmetry in the semi-metal CoSi stabilizes gaps below 2 meV over a large near-degenerate plane that can be measured in the quantum oscillation spectrum. The application of in-plane strain breaks the crystal symmetry and gaps the degenerate point, observable by new magnetic breakdown orbits. The quasi-symmetry, however, does not depend on spatial symmetries and hence transmission remains fully coherent. These results demonstrate a class of topological materials with increased resilience to perturbations such as strain-induced crystalline symmetry breaking, which may lead to robust topological applications as well as unexpected topology beyond the usual space group classifications.

The introduction of concepts of topology in the past years into the field of electronic dispersions has captured the imagination of condensed matter physics and sparked a flurry of new research directions. In topologically non-trivial metals, the wavefunctions are inherently nonlocal, and novel electronic states are expected and observed at the crystal surface terminating them (1–10). Beyond these surface properties, interesting anomalies in the bulk dispersion lead to novel physical phenomena, such as the emergence of quasiparticles mimicking ultra-relativistic Weyl- and Dirac-Fermions in 3D; a solid-state analogon of the Adler-Bell-Jackiw anomaly; a planar Hall effect; topological piezoelectric effect; second harmonic generation in Weyl semimetals; or strong Berry curvature that impacts the semi-classical and quantum dynamics of quasiparticles (11–19).

Given the wealth of new phenomena and potential applications, it is important to identify such “ideal” Weyl-, Dirac- or nodal-line semi-metals in which the a priori unrelated chemical potential and the topological anomaly coincide. That such a coincidence is rather rare can be understood from the Wigner-Von Neumann theorem, which states that a two-level crossing generally requires three tuning parameters (20, 21). To stabilize crossings of more than two bands (nodal points) or achieve higher dimensional degeneracies (nodal lines, planes), additional crystalline symmetries must provide further constraints on the form of the Hamiltonian, such as rotational symmetries for the fourfold Dirac points in the 3D Dirac semi-metals Cd₃As₂ and Na₃Bi (6, 7) or two-fold screw axis for the nodal plane in MnSi (22).

Hence crystalline symmetry and band topology are inseparably intertwined, and it is natural that symmetry should guide our search for materials. Here we refine this picture, and argue for the importance of approximate symmetries as an additional concept (Fig. 1) with clearly observable experimental consequences. A crystalline symmetry operation, M, commutes with the Hamiltonian of the system, [H(k), M] = 0. The well-known representation theory dictates that low-symmetry points usually support only one-dimensional representations, and hence bands anti-cross at these points. It may be possible, however, that at these low symmetry points approximate symmetries emerge. They are not exact symmetries of the crystal yet still commute with a dominant part of the Hamiltonian. Such approximate symmetries would stabilize almost linear band crossings by enforcing finite yet perturbatively small energy gaps. A trivial example is a weak lattice distortion that removes a spatial symmetry that otherwise stabilizes a band crossing. Graphene could be considered as an alternative example in the spin channel, as its Dirac point is weakly gapped when spin-orbit coupling is considered, turning spin-rotation into an approximate symmetry (23, 24). However, as approximate symmetries leave the bounds set by crystalline point-groups, more exciting approximate symmetries may emerge that have no counterpart in the crystal symmetry.

Figure 1.

(a) Illustration of a mirror symmetry operation, which acts at the whole object consistently. (b) In contrast to the mirror symmetry, quasi-symmetry operation acts differently on different parts of the system. (c) If a crystalline symmetry operation commutes with the Hamiltonian system, the band crossing point is therefore symmetry-protected. (d) In this case the Hamiltonian itself does not commute with the symmetry operation yet its first order perturbation does. This lead to the situation where though the band crossing is numerically avoided but the resulted gap size is negligible for the physical properties of the system. This type of symmetry is called quasi-symmetry. (e) If the symmetry operation does not commute with the Hamiltonian to any order of perturbation of the system, the crossing is not protected by any symmetry. This will result in a sizeable gap and lost of topological character.

We adopt the term “quasi-symmetry” to denote a special type of approximate symmetry of electronic band structures, that emerges from the hierarchy of a k · p type of perturbation expansion around a high-symmetry point. Quasi-symmetries are operators M_eff that are the exact symmetries of the lower-order expansion but not of higher-order perturbation terms. The main goal of this paper is to show such a quasi-symmetry exists in the semi-metal CoSi as well as a large set of similar materials, and that it enforces a low-symmetry plane of near-degeneracies spanning the Brillouin zone which pin a source of Berry curvature to the Fermi level (Fig. 2). Excitingly, it falls into the latter category of quasi-symmetries that are unrelated to approximate point-group symmetries, but can be treated as a symmetry of internal degrees of freedom, akin to spin in graphene. Unlike graphene, however, the form of quasi-symmetry is non-trivial as the corresponding symmetry operator is generally k-dependent and forms a part of the Hamiltonian. The key distinction is that while a proper crystal symmetry acts on all atomic coordinates uniformly, the quasi-symmetry operation exchanges 3d orbitals on subsets of inequivalent Co-sites selectively. A simplified example may be an incomplete mirror image [Fig. 1(a) and (b)], in which the mirror operation acts only on parts of the object but not the whole consistently.

Figure 2.

(a) Crystal structure of CoSi. Co and Si atoms are presented as red and blue spheres respectively. (b) Ab-initio-calculated band structure of CoSi around R-point. Here 1/2 denotes orbital character while +/- stands for the spin character of the band. (c) 3D view of all Fermi surfaces, which are centered around either R- or Γ- point of the Brilloiun zone. (d) 3D view of Fermi surfaces centered around R-point with a quadrant cut. (e) Quasi-symmetry and crystalline-symmetry protected degenerate planes. (f) The Berry curvature distribution of the 1^–-band defined in (b) at different Fermi energy calculated from the model Hamiltonian Eq. (1).

CoSi crystallizes in a chiral cubic structure of space group P2₁3 without an inversion center (Fig. 2). Recent studies revealed the existence of exceptionally long Fermi arcs which connect the sixfold and fourfold band degenerate points (9, 25–27). These two degenerate points correspond to two branches of Fermi surfaces located at at the Γ- and R-point, respectively, yet only the R-point pockets are observed in quantum oscillation experiments. Band structure calculations reveal four split bands due to spin-orbit coupling which correspond to four Fermi surfaces at R, which we label as 1⁺, 1⁻, 2⁺, 2⁻, representing the orbital and spin degree of freedom respectively. States at the boundary of the Brillouin zone are doubly degenerate due to crystalline symmetry, hence the non-degenerate Fermi surfaces are enforced to touch there (5). The spin-orbit coupling and the orbital gap are both small yet similar in magnitude, which explains how four slightly degenerate, interpenetrating spheroids make up a surprisingly rich internal structure of the Fermi surfaces.

CoSi single crystals resemble octahedra, indicating a dominant growth along the [111] direction (see supplement Sec.I.A. for growth details). These crystals were fabricated into microstructures using focused-ion-beam machining (28) (see supplement Sec.I.B.) to increase the current path homogeneity and signal of Shubnikov-de Haas oscillations. This microfabrication will later be the key to study the effects of unidirectional strain. The resistivity as well as the large residual resistivity ratio in the samples agrees well with previous bulk measurements (29), indicating an unchanged material quality during fabrication (see supplement Fig. S1). Hence not surprisingly, large quantum oscillations of the magnetoresistance are readily observed at low fields (Fig. 3). Subtracting a polynomial background uncovers strong oscillations that resemble a beating pattern of two frequencies around F₁ ~ 550 T and F₂ ~ 660 T, in good agreement with existing literature (29–32). The temperature dependence of the oscillations follows well the Lifshitz-Kosevich form, leading to a cyclotron effective mass of m_c ~ 0.84 m_e (see supplement Fig. S14).

Figure 3.

(a) Temperature-dependent SdH oscillations with field and current applied along [100] axis. Here ρ_osc = Δρ/ρ_BG, with Δρ the oscillatory part of the magnetoresistivity, and ρ_BG the background obtained from a 3^rd-order polynomial fit to the magnetoresistivity. (b) Fast-Fourier-transformation spectrum of the SdH oscillations presented in (a) with the field window of 3 to 14 T. Two main peaks, as well as their higher-harmonic components, can be clearly observed. The suppression of peak amplitude with increasing temperature is due to the thermal damping effect. (c) Fast-fourier-transformation (FFT) spectrum of angle-dependent quantum oscillations measured at T = 2 K. Here the magnetic field is rotated within (100) plane and the angle is defined between the field direction and [001] axis. (d) Summary of angular dependence of oscillation frequencies. Here the fitting is genereated by calculating the orbital area based on band structure calculations with taking extrem magnetic breakdown due to quasi-symmetry into account (see supplement). The near-perfect fitting clearly demonstrates that quasi-symmetry is the only option to explain the experimental results of nearly angle-independent oscillation frequencies.

In light of the complexity of self-intersecting Fermi surfaces it appears at first surprising that the quantum oscillation spectrum is tantalizingly simple, with only two frequences that, most importantly, only weakly depend on the direction of the applied magnetic field (Fig. 3). When magnetic quantum oscillations arise from such highly degenerate Fermi surfaces, the main question is which, if any, possible trajectories become quantum coherent. As the Fermi-surface is centered around the R-point, any orbit necessarily crosses the symmetry-enforced degeneracies at the Brillouin zone boundary multiple times. This has been argued to stabilize band degeneracies pinned to the Fermi level in this crystal structure (22, 32) on the planes of Brilloiun zone boundary [marked purple in Fig. 4(f)].

Figure 4.

(a) Scanning electron microscope image of CoSi microdevice. A long bar roughly along [110] direction with a 2.5 by 2.4 μm² cross section is fabricated by FIB. (b) Illustration of tensile strain approximately along [110] which breaks the C₂ rotational symmetry of the crystal structure. (c) SdH oscillations with both field and current applied along the fabricated bar direction at T = 50 mK. (d) Logarithmic-scaled FFT spectrum of SdH oscillations displayed in (c). The main peaks are always accompanied with satellite peaks up to the third harmonics. (e) Enlarged view of satellite peaks correspond to the 1^st to 3^rd harmonic oscillations. The red, purple and blue vertical lines correspond to the FFT spectrum produced by the fully symmetric, crystalline-symmetry-preserved and quasi-symmetry-preserved scenarios respectively. (f) Corresponding Landau orbits for three different scenarios. Here the colored area illustrates the orbital area difference compared to the fully-symmetric case, and the black crosses represent the degeneracies that are lifted in different scenarios. Only the quasi-symmetry-preserved scenario reproduces FFT peaks that match perfectly well with the experimental data.

Yet the orbits further intersect at low-symmetry points at which a gap must open [marked blue in Fig. 4(f)]. Neither the frequencies nor their angle dependence obtained from DFT calculations match the data if this gap is considered to separate the orbits via significant avoided-crossing. As the gap is small, the quasiparticle can tunnel across it continuing on the original trajectory, a phenomenon known as magnetic breakdown (33–36). Based on estimations of the gap from DFT and further analytical calculations, vanishingly small breakdown fields fall below 0.11 T for any field orientation, and hence the experimental data is always obtained in an extreme breakdown regime (see supplement Sec. V. for details). Thus the wavefunctions propagate through these points at near-perfect transmission, and indeed an excellent match of frequencies and their angle dispersion between theory and experiment is found (Fig. 3).

These observations raise a key question about the electronic structure, namely why in a metal with eV bandwidth two bands anti-cross with a gap no larger than 2 meV in extended regions of the Brillouin zone. To obtain such a parametrically small gap accidentally, an unreasonable degree of fine-tuning is required that acts simultaneously at many k-points, as well as at a wide range of energies as we will show. Instead, a quasi-symmetry enforces the uniform smallness of this gap, and thus explains why magnetic breakdown at full transparency occurs at any arbitrary angle.

To understand this quasi-symmetry, we next consider an effective model around the R-point. Without spin-orbit coupling (SOC), all the bands are eight-fold degenerate at R due to the combination of two-fold screw axis symmetries along the x and y direction and time reversal symmetry. Including spin doubles the degeneracy and the SOC splits the eight-fold degenerate states into six-fold and two-fold degenerate states at the R-point. An effective model for these eight bands around the R-point can be constructed as

$${\mathscr{H}}_R={\mathscr{H}}_0\left(\mathbf{\text{k}}\right)+{\mathscr{H}}_{soc}+{\mathscr{H}}_{k^2}\left(\mathbf{\text{k}}\right),$$ (1)

where 𝓗₀(k) = C₀ + 2A₁ (k · L) is the lower-order expansion of the spin-independent Hamiltonian, and three 4-by-4 matrices L form an emergent angular momentum algebra [L_i, L_j] = iε_ijkL_k with Levi-Civita symbol ε_ijk and i, j, k = x, y, z. The SOC term is given by 𝓗_soc = 2λ₀(s · L), where s is the spin operator. 𝓗_k²(k) is the higher-order spin-independent term with its form given in the supplement. Here 𝓗₀(k) + 𝓗_k² (k) is the SOC-free expanded Hamiltonian to second order. By choosing appropriate parameters such as C₀, A₁, λ₀, the energy dispersion of 𝓗_R well reproduces that from the DFT calculations (see supplement Sec. VII.).

A striking feature in the Hamiltonian 𝓗_R is that the SOC term 𝓗_soc takes a similar form as the linear-momentum term, just by replacing the momentum k by spin s. This is because spin, as a pseudo-vector, behaves exactly the same as a vector due to the lack of inversion, mirror or other roto-inversion symmetries for a chiral crystal. Due to this similarity, up to the first-order perturbation, we find that the spin of the eigen-states is parallel or anti-parallel to the momentum k and thus we can label the spin states of these bands by ± in Fig. 2(b). We derive an effective Hamiltonian for the near degenerate bands. This can be done by first projecting the full Hamiltonian 𝓗_R into the 4 closest bands (1^± and 2^± bands) to get a 4-band model and then constructing a 2-band model for the 1⁻ and 2⁺ bands, as discussed in supplement. Up to the first-order perturbation, the 2-band Hamiltonian takes the form H_eff = ϵ₀ + d_z(k)σ_z, where $d_z\left(\mathbf{\text{k}}\right)={\lambda }_0-\sqrt{3}\tilde{C}\frac{\left|k_x{k}_y{k}_z\right|}{k}$ and the form of the function ϵ₀ is given in supplement. Here σ_z is the Pauli matrix on the basis of 1⁻ and 2⁺ bands, λ₀ term comes from the SOC Hamiltonian 𝓗_soc and the C̃ term from 𝓗_k²(k). It is striking to see that the operator 𝓜_eff = σ_z commutes with H_eff and thus serves as an exact symmetry at the first-order expansion. The existence of 𝓜_eff forbids any terms that are coupled to σ_x,y in the effective Hamiltonian, thus reducing the codimension of a two-level crossing from 3 to 1 and stabilizing a nodal plane, which is defined by the equation ${\lambda }_0=\sqrt{3}\tilde{C}\frac{\left|k_x{k}_y{k}_z\right|}{k}$ in this two-band model. A more accurate and complete description around the Fermi energy is the 4-band model, which shares a similar k-dependent hidden quasi-symmetry and is essential in deriving the 2-band model (see supplement Sec. VII.), and the whole perturbation theory reveals a striking hierarchy structure. Higher-order perturbation expansion breaks 𝓜_eff through inducing additional terms coupled to σ_x,y in the effective model and thus leads to a small gap opening of this nodal plane (see supplement). Therefore, 𝓜_eff is not a symmetry of the system, but an emergent quasi-symmetry in the sense of low-energy effective theory. Importantly, this curved nodal plane spans low-symmetry regions of the Brillouin zone, in contrast to symmetry-enforced exact degeneracies. It is this combined structure of symmetry and quasi-symmetry induced degeneracies that forms the basis of the complex structure of the Fermi surface centered at R, and reduces the complexity of the quantum orbits in magnetic fields to two, angle-independent frequencies.

Physically, quasi-symmetry is linked to the orbital structure of the wavefunctions, and its operation only interchanges the orbitals at different atomic sites (internal degrees). Thus, unlike crystalline symmetries, they remain robust against symmetry-breaking lattice distortions (Fig. 4). This is probed experimentally by the application of tensile strain to a bar cut approximately along the [110] direction. The low-symmetry distortion removes the critical screw symmetries at the zone boundaries, and gaps the otherwise symmetry protected crossing points. Experimentally, such strain can be easily applied by thinning the transport bar by FIB-milling, thus reducing its effective spring constant and increasing the total distortion under equal force. The microstructures are coupled to a sapphire substrate and their differential thermal contraction generates the desired tensile strain along the bar [see supplement and (37)]. As such strain lifts the screw symmetry protecting the degeneracy, one naturally expects the sample to gradually leave the magnetic breakdown regime as the strain in increased and a zoo of sum frequencies to emerge that correspond to all possible area differences between the orbits (34–36). Indeed, the strained bar exhibits pronounced satellite peaks that are symmetrically offset from the main peaks by ΔF ≈ 32 T for all harmonics. This matches exactly with the calculated area difference between the crystalline symmetry protected points, fully explaining the observed spectrum. However, those areas associated with a quasiparticle tunneling at a quasi-symmetry-protected point are not observed, implying that the magnetic breakdown field associated with that gap remains much smaller than the lowest field at which quantum oscillations are observed. This is a direct experimental proof that the quasi-symmetry protection is insensitive to breaking of crystalline symmetries, a notion that can be well corroborated by calculations (Fig. 4). This resilience clearly distinguishes symmetry and quasi-symmetry enforced topological band anomalies.

Their unique implications in the Berry curvature distribution in momentum space are another key characteristic of the topological character of near-degeneracies enforced by quasi-symmetry, compared to the exact degeneracies due to symmetry [Fig. 2(f)]. The Berry curvature almost vanishes in the vicinity of the exact degenerate plane, yet it is concentrated to a ring on the near nodal plane at any fixed energy. This large Berry curvature around the near nodal lines, which always occurs at the Fermi energy, may strongly affect physical phenomena that are related to local Berry curvature, including intrinsic spin Hall effect (38, 39) and quantum non-linear Hall effect (40).

The quasi-symmetry is the mathematical and physical framework behind the unexpectedly simple and strain-resistant quantum oscillation spectrum of CoSi. It is, however, just one example of a large class of materials in which quasi-symmetries can exist. Clearly, this new class of near degenerate manifolds at low-symmetry k-points is not found by current search programs for topological materials based on crystalline symmetry without further guideline. To search for materials with quasi-symmetries, a systematic expansion of the k · p-type effective Hamiltonian order by order is required for all space groups, which is in principle feasible (41). Typically, lower-order terms with higher symmetries than the crystalline symmetry itself possess larger prefactors than the higher-order terms with lower symmetries (42), and a symmetry hierarchy is expected for the terms of different orders in the k · p-type Hamiltonian. Further interesting flavors of quasi-symmetry may exist which emerge when higher-order terms with additional symmetries become dominant. Experimentally, we propose that quantum oscillations may provide guidance by pointing to materials with similarly extreme magnetic breakdown on extended regions on the Fermi surface, which a quasi-symmetry would naturally explain. While the here described search methods need to be implemented in the future, it is natural to anticipate quasi-symmetries in related compounds crystallizing in the same structure, such as e.g. PtGa, PtAl, RhSi and the magnetic MnSi where the influence of magnetism can be explored. (22, 43–51). Our DFT calculations indeed reveal similar energy dispersions and Fermi surface shapes with quasi-symmetry-induced near degeneracies in PtAl, PtGa and RhSi (see supplement Sec. VI.). The concept of the quasi-symmetry raises the interesting prospect that even those materials in which crystalline symmetries do not allow stable band crossings may still host exotic quasi-particles and strong Berry curvature at the Fermi level. If such features at the Fermi level are desired, the current approach of automated symmetry classifications may not prove to lead to the most practical materials. Without the necessity of a crystalline symmetry protecting higher-order band crossing points, quasi-symmetries hold an advantage for topological applications. The topological anomalies in real materials often are far away from the Fermi level, and in thin-film form they are exposed to mismatch strains inherent to heterostructures. These key bottlenecks of topological materials do not restrict quasi-symmetric materials. A consensus of high-throughput efforts emerges that 20-30% of materials are topological (52–55). Yet it appears that crystal symmetry may not hold up as the strict barrier it was thought to be, and via quasi-symmetries concepts, topology may reach even further than these estimates predict.

Methods

Data Availability

Data that support the findings of this study is deposited to Zendo with the access link: https://doi.org/10.5281/zenodo.6336000.

Code Availability

Matlab code used for this study is deposited to Zendo with the access link: https://doi.org/10.5281/zenodo.6336013.