Exceptional topology in ordinary soft matter

Hydrodynamics is shown to induce non-Hermitian topological phenomena in ordinary, passive soft matter. This is demonstrated by subjecting a two-dimensional elastic lattice to a low-Reynolds viscous flow. The interplay of hydrodynamics and elasticity splits Dirac cones into bulk Fermi arcs, pairing exceptional points with opposite half-integer topological charges. The bulk Fermi arc is a generic hallmark of the system exhibited in all lattice and flow symmetries. An analytic model and simulations explain how the emergent singularities shape the spectral bands and give rise to a web of van Hove singularity lines in the density of states. The present findings suggest that non-Hermitian physics can be explored in a broad class of ordinary soft matter, living and artificial alike, opening avenues for topology-based technology in this regime.

Figure 1

(a) A triangular lattice made of spheres connected by elastic struts is immersed in a thin layer of viscous fluid between two walls in the $z$ direction (perpendicular to the page). The lattice is driven at a velocity $u$ relative to the fluid. (b) A pair of moving particles induce dipolar flow fields (gray streamlines), thereby exerting on each other equal hydrodynamic forces, ${\mathbf{f}}_{ij}^{\mathrm{hyd}}={\mathbf{f}}_{ji}^{\mathrm{hyd}}$ (blue arrows), in the direction of the induced flow. The elastic forces along the elastic strut are opposite, ${\mathbf{f}}_{ij}^{\mathrm{elas}}=-{\mathbf{f}}_{ji}^{\mathrm{elas}}$ (red arrows), thus conserving linear momentum.

Figure 2

(a) A purely hydrodynamic system, $\epsilon =0$. Left: The operator $\mathcal{H}={\mathcal{H}}_{\mathrm{hyd}}$ is Hermitian with real frequency bands ${\omega }_{+}=-{\omega }_{-}=\sqrt{{{\mathrm{\Omega }}_x^2+{\mathrm{\Omega }}_y^2}^{\phantom{\int }}}$. The spectrum exhibits six Dirac points (green, one denoted as $D$) on the boundary of the Brillouin zone (black hexagon). Middle: At each Dirac point, the bands merge, forming a graphenelike double-cone, “diabolo” shape. Right: The 3D shape of the frequency bands in the first Brillouin zone (gray hexagon), showing the double-cones, which are halved by the zone's boundary. (b) At $\epsilon \ne 0$, the symmetry is broken when $\mathcal{H}$ includes a skew-Hermitian component, $\epsilon {\mathcal{H}}_{\mathrm{elas}}$. Left: The real part of the bands, $\text{Re}\left({\omega }_{+}\right)=-\text{Re}\left({\omega }_{-}\right)$, drawn for $\epsilon =\pi$, exhibits six bulk Fermi arcs (white lines) in the first Brillouin zone (the whole spectrum is shown in Fig. 7). Each arc splits from a Dirac point (green) and joins two exceptional points (ExPs) with topological charges $\pm \frac{1}{2}$ (orange and light blue, $E_{+}$ and $E_{-}$ are two ExPs that split from $D$). Middle: The real parts of the bands merge along the Fermi arc, forming a double-wedge shape. Right: 3D shape of the frequency bands in the first Brillouin zone (gray hexagon), showing the double-wedges (halved by the zone's boundary).

Figure 3

The double branching transition in the real (blue) and imaginary (red) parts of the bands at the exceptional points (10). Zoom on the upper right bulk Fermi arc in Fig. 2.

Figure 4

The vorticity of the band gap, $\mathrm{\Delta }\omega ={\omega }_{+}-{\omega }_{-}$ (11), for $\epsilon =\pi$. The argument of the band gap, $arg\mathrm{\Delta }\omega$, is color-coded, and arrows denote its gradient field, ${\nabla }_{\mathbf{k}}(arg\mathrm{\Delta }\omega )$. The hexagonal black line shows the Brillouin zone boundary, with Dirac points (green) and exceptional points with $\pm \frac{1}{2}$ charges (orange and blue).

Figure 5

Left: The density of states $g\left(\omega \right)$ plotted in log-scale, showing the Dirac points (green circles) and the exceptional points (blue/orange circles), all connected by bulk Fermi arcs (white line) at $\text{Re}\omega =0$. Right: 3D representation of $g\left(\omega \right)$.

Figure 6

The physical limitations I–IV in the $\ell \text{−}\kappa$ plane, drawn for typical values of a microfluidic system [31]: Velocity $u=500\mu \mathrm{m}/\mathrm{s}$, viscosity $\eta =30\mathrm{cP}$, geometric factor ${(\ell /a)}^3=0.1$, and lattice size ${(L/a)}^2={10}^4$. The regions excluded by bounds II–VI are shaded, and condition I ($\epsilon \sim 1$) is the dashed line. The accessible region is in the remaining white domain.

Figure 7

The spectrum of the triangular lattice. The real (top) and imaginary (bottom) components of the eigenfrequencies ${\omega }_{+}$ (left) and ${\omega }_{-}$ (right) of a triangular lattice at $\epsilon =\pi$ [Fig. 2]. The black hexagon is the boundary of the Brillouin zone. Shown are the Dirac points (green) at the midpoints of the Brillouin zone edges, and the ExPs with their $+\frac{1}{2}$ (red) and $-\frac{1}{2}$ (blue) charges. Fermi arcs are gray curves.

Figure 8

Hydrodynamic and elastic interactions and frequency bands. The black hexagon is the boundary of the Brillouin zone. The eigenfrequencies of the purely hydrodynamic systems with negligible elasticity, ${\omega }_{\pm }^{\mathrm{hyd}}$ (A16), are real because the effective Hamiltonian is Hermitian, $\mathcal{H}={\mathcal{H}}_{\mathrm{hyd}}$ (top-right and center-right). In the absence of flow, the elastic system is purely damping. $\mathcal{H}={\mathcal{H}}_{\mathrm{elas}}$ is skew-Hermitian, and therefore its spectrum, ${\omega }_{\pm }^{\mathrm{elas}}$ (A17), is purely imaginary (top-center and center). The elastic interactions, ${\mathrm{\Omega }}_x$ and ${\mathrm{\Omega }}_y$, vanish at the Dirac points, the middle of Brillouin zone edges (bottom left and center), while the elastic interactions, ${\omega }_x$, ${\omega }_y$, and ${\omega }_1$, exhibit extrema (left column).

Figure 9

The spectrum of a square lattice. The real (top) and imaginary (bottom) components of the eigenfrequencies, ${\omega }_{+}$ (left) and ${\omega }_{-}$ (right), of a square lattice at $\epsilon =\pi$. The black square is the boundary of the Brillouin zone. Shown are the Dirac points (green) and the ExPs with their $+\frac{1}{2}$ (red) and $-\frac{1}{2}$ (blue) charges. Bulk Fermi arcs are gray curves.

Figure 10

Density of states $g\left(\omega \right)$: Analytic solution vs simulation. (a), (b) $g\left(\omega \right)$ for a triangular lattice, analytic (a) and simulation (b), for $\epsilon =\pi$. The simulated system is a $121\times 121$ triangular lattice with periodic boundary conditions (a torus). The spectrum is obtained by Fourier analysis of the motion computed from the dynamical equation (4). The simulated $g\left(\omega \right)$ exhibits general similarity to the analytic solution. The incommensurately of the fourfold symmetry of the torus and the sixfold symmetry of the lattice results. (c), (d) $g\left(\omega \right)$ for a square lattice, analytic (a) and simulation of a $121\times 121$ lattice (b), for $\epsilon =\pi$. The correspondence between the simulation and the analytic solution is excellent, due to the commensurable fourfold symmetry of the lattice and the periodic boundary conditions, with some coarseness due to the finite size of the simulated system.