Hidden quasi-symmetries stabilize non-trivial quantum oscillations in CoSi


 Unlocking the exotic properties promised to occur in topologically non-trivial semi-metals currently requires significant fine-tuning. Crystalline symmetry restricts the location of topological defects to isolated points (0D) or lines (1D), as formalized by the Wigner-Von Neumann theorem. The scarcity of materials in which these anomalies occur at the chemical potential is a major obstacle towards their applications. Here we show how non-crystalline quasi-symmetries stabilize near-degeneracies of bands over extended regions in energy and in the Brillouin zone. Specifically, a quasi-symmetry is an exact symmetry of a k∙p Hamiltonian to lower-order that is broken by higher-order terms. Hence quasi-symmetric points are gapped, yet the gap is parametrically small and therefore does not influence the physical properties of the system. We demonstrate that in the eV-bandwidth semi-metal CoSi an internal quasi-symmetry stabilizes gaps in the 1-2 meV range over a large near-degenerate plane (2D). This quasi-symmetry is key to explaining the surprising simplicity of the experimentally observed quantum oscillations of four interpenetrating Fermi surfaces around the R-point. Untethered from the limitations of crystalline symmetry, quasi-symmetries eliminate the need for fine-tuning as they enforce sources of large Berry curvature to occur at the chemical potential, and thereby lead to new Wigner-Von Neumann classifications of solids. Quasi-symmetries arise from a comparable splitting of degenerate states by spin-orbit coupling and by orbital dispersion - suggesting a hidden classification framework for symmetry groups and materials in which quasi-symmetries are critical to understand the low-energy physics.

symmetries arise from a comparable splitting of degenerate states by spin-23 orbit coupling and by orbital dispersion -suggesting a hidden classification 24 framework for symmetry groups and materials in which quasi-symmetries are 25 critical to understand the low-energy physics. 26 The introduction of concepts of topology in the past years into the field of electronic dis-27 persions has captured the imagination of condensed matter physics and sparked a flurry of new 28 research directions in the past years. At its core is the realization that the electronic wave-29 forms constituting a material cannot always be derived from an entirely local description. In 30 these topologically non-trivial metals, the wavefunctions are inherently non-local, and as such 31 novel electronic states are expected and observed at the crystal surface terminating them (1-10). 32 Beyond these surface properties, interesting anomalies in the bulk dispersion lead to novel phys- 33 ical phenomena, such as the emergence of quasiparticles mimicking ultra-relativistic Weyl-and 34 Dirac-Fermions in 3D; a solid-state analogon of the Adler-Bell-Jackiw anomaly; a planar Hall 35 2 effect; topological piezoelectric effect; second harmonic generation in Weyl semimetals; or 36 strong Berry curvature that impacts the semi-classical and quantum dynamics of quasiparti-37 cles (11)(12)(13)(14)(15)(16)(17)(18)(19). 38 Given the wealth of new phenomena and potential applications, it is important to identify 39 such "ideal" Weyl-, Dirac-or nodal-line semi-metals in which a priori unrelated chemical po-40 tential and the topological anomaly coincide. That such a coincidence is rather rare can be 41 understood from the Wigner-Von Neumann theorem, which states that a two-level crossing 42 generally requires three tuning parameters (24, 25). Any two-level model can be expanded as Dirac points in the 3D Dirac semi-metals Cd 3 As 2 and Na 3 Bi (6, 7). As symmetry and topology 51 are inseparably intertwined, it is natural that symmetry should guide our search for interesting 52 materials.

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Here we refine this picture, and argue for the importance of approximate symmetries as 54 an additional concept [ Fig. 1 ple of a hidden symmetry is the Laplace-Runge-Lenz vector in classical gravitation. It is the 70 conserved quantity stabilized by the hidden SO(4) symmetry of the r −1 potential of the Kepler 71 problem that enforces strictly closed classical gravitational orbits. Weak perturbations, such as 72 r −1+ǫ , mathematically reduce the symmetry to SO(3), yet the almost closed orbits for small ǫ 73 evidence that the system senses the proximity to that hidden symmetry. 74 We adopt the term "quasi-symmetry" to denote a special type of approximate symmetry 75 of electronic band structures, that emerges from the hierarchy of a k · p type of perturbation

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In light of the complexity of the self-intersecting Fermi surfaces it appears at first surprising 114 that the quantum oscillation spectrum is tantalizingly simple, with only two frequences that, 115 most importantly, only weakly depend on the direction of the applied magnetic field (Fig. 4). their angle dispersion between theory and experiment is found (Fig. 4).

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The oscillation spectrum thus can be completely rationalized a priori without quasi-symmetries.

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However one overlooks a key question about the structure of the wavefunctions, namely why 134 in a metal with eV bandwidth two bands anti-cross with a gap no larger than 2 meV over the 135 6 entire Fermi surface. To obtain such a parametrically small gap accidentally, an unreasonable 136 degree of fine-tuning is required that acts simultaneously at many k-points, as well as at a wide 137 range of energies as we will show. Instead, a hidden quasi-symmetry enforces the smallness of 138 the gap at these points, and thus explains why magnetic breakdown at full transparency occurs 139 at any arbitrary angle.

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To understand this quasi-symmetry, we next consider an effective model around the R-point.
where H 0 (k) = C 0 + 2A 1 (k · L) is the lower-order expansion of the spin-independent Hamil-147 tonian, and three 4-by-4 matrices L form an emergent angular momentum algebra [L i , L j ] = 148 iε ijk L k with Levi-Civita symbol ε ijk and i, j, k = x, y, z. The SOC term is given by H soc = 149 2λ 0 (s · L), where s is the spin operator. H k 2 (k) is the higher-order spin-independent term with 150 its form given in the supplement. perturbation, we find that the spin of the eigen-states is parallel or anti-parallel to the momen-158 tum k and thus we can label the spin states of these bands by ± in Fig. 1(e). We derive an effective Hamiltonian for the near degenerate bands. This can be done by first projecting the    Here the magnetic field is rotated within (100) plane and the angle is defined between the field direction and [001] axis. (c) Landau orbits and corresponding FFT spectrum for Quasi-symmetry and avoided-crossing scenarios. It is clear that only the quasi-symmetry scenario reproduces FFT peaks that match perfectly well with the experimental data. For the avoided crossing case the predicted frequency is far off. (d) Summary of angular dependence of oscillation frequencies. The contrast between two different scenario again clearly demonstrates that quasi-symmetry is the only option to explain the experimental results of nearly angle-independent oscillation frequencies.