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Register a new userSpectral properties of Dirac billiards at the van Hove singularities
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We study distributions of the ratios of level spacings of a rectangular and an Africa-shaped superconducting microwave resonator containing circular scatterers on a triangular grid, so-called Dirac billiards (DBs). The high-precision measurements allowed the determination of all 1651 and 1823 eigenfrequencies in the first two bands, respectively. The resonance densities are similar to that of graphene. They exhibit two sharp peaks at the van Hove singularities, that separate the band structure into regions with a linear and a quadratic dispersion relation, respectively. In their vicinity we observe rapid changes, e.g., in the wavefunction structure. Accordingly, the question arose, whether there the spectral properties are still determined by the shapes of the DBs. The commonly used statistical measures, however, are no longer applicable whereas, as demonstrated in this Letter, the ratio distributions provide most suitable ones.
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Wave kinetic theory has been developed to describe the statistical dynamics of weakly nonlinear, dispersive waves. However, we show that systems which are generally dispersive can have resonant sets of wave modes with identical group velocities, leading to a local breakdown of dispersivity. This shows up as a geometric singularity of the resonant manifold and possibly as an infinite phase measure in the collision integral. Such singularities occur widely for classical wave systems, including acoustical waves, Rossby waves, helical waves in rotating fluids, light waves in nonlinear optics and also in quantum transport, e.g. kinetics of electron-hole excitations (matter waves) in graphene. These singularities are the exact analogue of the critical points found by Van Hove in 1953 for phonon dispersion relations in crystals. The importance of these singularities in wave kinetics depends on the dimension of phase space $D=(N-2)d$ ($d$ physical space dimension, $N$ the number of waves in resonance) and the degree of degeneracy $delta$ of the critical points. Following Van Hove, we show that non-degenerate singularities lead to finite phase measures for $D>2$ but produce divergences when $Dleq 2$ and possible breakdown of wave kinetics if the collision integral itself becomes too large (or even infinite). Similar divergences and possible breakdown can occur for degenerate singularities, when $D-deltaleq 2,$ as we find for several physical examples, including electron-hole kinetics in graphene. When the standard kinetic equation breaks down, then one must develop a new singular wave kinetics. We discuss approaches from pioneering 1971 work of Newell & Aucoin on multi-scale perturbation theory for acoustic waves and field-theoretic methods based on exact Schwinger-Dyson integral equations for the wave dynamics.
The Hofstadter butterfly is a quantum fractal with a highly complex nested set of gaps, where each gap represents a quantum Hall state whose quantized conductivity is characterized by topological invariants known as the Chern numbers. Here we obtain simple rules to determine the Chern numbers at all scales in the butterfly fractal and lay out a very detailed topological map of the butterfly. Our study reveals the existence of a set of critical points, each corresponding to a macroscopic annihilation of orderly patterns of both the positive and the negative Cherns that appears as a fine structure in the butterfly. Such topological collapses are identified with the Van Hove singularities that exists at every band center in the butterfly landscape. We thus associate a topological character to the Van Hove anomalies. Finally, we show that this fine structure is amplified under perturbation, inducing quantum phase transitions to higher Chern states in the system.
We discuss solutions of an algebraic model of the hexagonal lattice vibrations, which point out interesting localization properties of the eigenstates at van Hove singularities (vHs), whose energies correspond to Excited-State Quantum Phase Transitions (ESQPT). We show that these states form stripes oriented parallel to the zig-zag direction of the lattice, similar to the well-known edge states found at the Dirac point, however the vHs-stripes appear in the bulk. We interpret the states as lines of cell-tilting vibrations, and inspect their stability in the large lattice-size limit. The model can be experimentally realized by superconducting 2D microwave resonators containing triangular lattices of metallic cylinders, which simulate finite-sized graphene flakes. Thus we can assume that the effects discussed here could be experimentally observed.
The possibility of triggering correlated phenomena by placing a singularity of the density of states near the Fermi energy remains an intriguing avenue towards engineering the properties of quantum materials. Twisted bilayer graphene is a key material in this regard because the superlattice produced by the rotated graphene layers introduces a van Hove singularity and flat bands near the Fermi energy that cause the emergence of numerous correlated phases, including superconductivity. While the twist angle-dependence of these properties has been explored, direct demonstration of electrostatic control of the superlattice bands over a wide energy range has, so far, been critically missing. This work examines a functional twisted bilayer graphene device using in-operando angle-resolved photoemission with a nano-focused light spot. A twist angle of 12.2$^{circ}$ is selected such that the superlattice Brillouin zone is sufficiently large to enable identification of van Hove singularities and flat band segments in momentum space. The doping dependence of these features is extracted over an energy range of 0.4 eV, expanding the combinations of twist angle and doping where they can be placed at the Fermi energy and thereby induce new correlated electronic phases in twisted bilayer graphene.
We determine with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics of the constant-width billiards is unidirectional, a change of the direction of motion is possible in the corresponding quantum system via dynamical tunneling. This becomes manifest in a splitting of the vast majority of resonances into doublets of nearly degenerate ones. The fluctuation properties of the two respective spectra are demonstrated to coincide with those of a random-matrix model for systems with violated time-reversal invariance and a mixed dynamics. Furthermore, we investigate tunneling in terms of the splittings of the doublet partners. On the basis of the random-matrix model we derive an analytical expression for the splitting distribution which is generally applicable to systems exhibiting dynamical tunneling between two regions with (predominantly) chaotic dynamics.
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