Dirac branch-cut modes

Abstract

Bound states arising in Dirac fields are usually attributed to two kinds of features: domain walls where a real Dirac mass field changes sign, which host Jackiw-Rebbi states, and phase singularities in a complex Dirac mass field, which host Jackiw-Rossi zero modes. We show that phase discontinuities, such as branch-cuts of complex branch functions, supply a third distinct binding mechanism. We derive the existence of guided modes that propagate along the cut, called Dirac branch-cut (DBC) modes, which obey an effective one-dimensional relativistic Dirac equation with a reduced mass determined by the phase difference across the cut. When the mass field has fixed magnitude, the DBC modes' transverse confinement lengths are energy-independent, unlike Jackiw-Rebbi and Jackiw-Rossi states or conventional boundary modes. Using acoustic metamaterials, we realize DBC modes experimentally, and verify their relativistic dispersion, robust transverse confinement length, and ability to propagate along freeform paths. These results show that phase discontinuities in a complex Dirac mass field constitute a versatile design principle for guided modes, with interesting application possibilities for photonic and acoustic metamaterials.

Figures (9)

• Figure 1: Properties of Dirac branch-cut (DBC) modes.a, Upper panel: complex plane for the Jackiw--Rossi mass parameter $m$, indicating two points with the same $|m|$ and phase difference $\Delta \theta$. Lower panel: a schematic of a piecewise-constant $m(\mathbf{r})$ distribution in 2D space, constructed from those two $m$ values with phase dislocation along $y = 0$. b, Phase dislocation mode (DBC mode) energies versus wavenumber $k_x$ for $\Delta\theta=\pi$ (left panel) and $\Delta\theta=2\pi/3$ (middle panel), and versus $\Delta\theta$ at $k_x = 0$ (right panel). Cyan dashes show the results obtained analytically using Eqs. \ref{['dispersion']}--\ref{['kappa_fixed']}. Color maps show the minimum eigenvalue magnitude of the scattering matrix, computed directly from the Hamiltonian \ref{['eq:hamiltonian']}; a zero value corresponds to a boundary state (see Supplementary Information). The bulk bands are shaded in gray. c, Phase profile of a mass distribution made from a branch function $m = (z-3)^{1/2}(z+3)^{1/2}$, where $z = x + iy$. d, Numerically-obtained DBC mode energies versus mode order with the phase of mass distribution in c, and the amplitude extracted from the branch function $|m|$. e, Spatial profiles for two of the modes in d: the Jackiw--Rossi state at $E = 0$ (left panel), and the order-4 mode (right panel). f, Phase profile of a "phase-only" mass distribution $m = m_0 \mathrm{arg}[(z-3)^{1/2}(z+3)^{-1/2}]$, with $m_0 = 5$. g, DBC mode energies for the mass distribution in f (blue markers), and for another mass distribution with $\Delta\theta = 2\pi/3$ using the same cut (yellow markers). The theoretically-predicted minigap for the latter is indicated by horizontal dashes. h Representative spatial profiles of two of the DBC modes for $\Delta\theta=\pi$. For the DBC mode profiles for $\Delta\theta=2\pi/3$, see Supplementary Information Figs. S1 and S2. All the DBC modes have very similar confinement lengths, up to high orders approaching the bulk bands.
• Figure 2: Implementation of DBC modes in an acoustic crystal.a, Schematic of the acoustic structure, consisting of solid pillars (pink circles) surrounded by air. The pillar positions are fixed, and their radii are modulated according to Eq. \ref{['eq:Kekule_acoustic']}; in this plot, we take $\theta = -0.35\pi$. Dashed and solid circles indicate the radii before and after modulation, respectively. b, A phase map $\theta(\mathbf{r})$ with a $\pi$ phase dislocation along $y = 0$ (left panel), and the corresponding acoustic structure (right panel). c, DBC mode dispersion for $\Delta\theta = \pi$ (left panel) and $\Delta\theta = 2\pi/3$ (right panel). Color maps show the Fourier transformed excitation spectrum obtained from experiments, while cyan dashes show the dispersion relations from FEM simulations of the acoustic structure (see Methods). The bulk bands are shaded in gray. Markers indicate four frequencies between 10.7 kHz--11.3 kHz (with 0.2 kHz spacing) used in subsequent subplots. d, Experimental setup for measuring spatially-resolved excitation spectra. The location of the acoustic source is indicated by the loudspeaker symbol. e, Measured profiles for DBC modes at 10.9 kHz (upper panel) and 11.3 kHz (lower panel). f, Semilogarithmic plot of the transverse acoustic pressure profiles, measured along a line of constant $x$ with the four frequencies marked in c. These experimental results are normalized to the value at $y = 0$, and agree with the frequency-independent theoretical prediction derived from the Jackiw--Rossi model (solid green line).
• Figure 3: Coupled waveguide-cavity system based on DBC modes.a, Upper panel: diagram $\theta (\textbf{r})$ created from branch function $f(z) = (z-150)^{1/2}(z+150)^{-1/2}$, with $z=x+i4y$. Lower panel: the acoustic crystal modulated with the diagram $\theta (\textbf{r})$. Blue dashed lines indicate the positions of cuts I and II, which are separated by a distance of $70$ mm. Scale bar: 100 mm. b, Upper panel: acoustic power transmittance to the right side of cut II. Middle panel: transmittance spectrum from FEM simulations. Lower panel: quality (Q) factors of the DBC modes on cut I. The bulk bands are shaded in gray. c, The profiles of DBC modes from experiment (upper panel) and simulations (lower panel) at the frequency marked by the star in b.
• Figure 4: DBC modes on a spiral.a, A modulation phase profile $\theta (\textbf{r})$ with a spiral-shaped cut. b, Photograph of the acoustic lattice generated from this phase profile, with the cut indicated by the cyan curve. The location of the acoustic source in the experiment is indicated by the loudspeaker symbol. c, The acoustic pressure distribution measured at 10.95 kHz. d, Spatial profiles measured along a line transverse to the spiral (dashes in c) at two different frequencies (blue dots and red circles). The solid green line is the theoretical prediction.
• Figure S1: DBC modes on a bounded cut. a, Spatial profiles of DBC modes on a curved cut with $\Delta\theta = \pi$. All the other parameters are the same as in Fig. 1h of the main text. b, Profiles along the line $x=0$ for the two DBC modes in a, as well as the two modes plotted in Fig. 1h of the main text. All four modes have very similar confinement lengths. c--d, Mode profiles for $\Delta\theta = 2\pi/3$.