Hydrodynamics · topology · rotation space
Quantum structures in classical matter
Quasiparticles, flat bands, exceptional points, and SU(2) do not all require quantum matter.
The mathematics of condensed matter often outlives the system in which it was discovered. A periodic classical medium has bands. Degeneracies can form Dirac cones. Dissipation can make the spectrum non-Hermitian. Ordinary rotations live in SO(3) and lift to SU(2). The useful question is then precise: which structures require quantization, and which require only the right spectrum, symmetry, or geometry?
Hydrodynamic matter
Dirac cones, flat bands, and long-lived particle pairs
Exceptional topology
Flow and elasticity split a Dirac cone into exceptional points
Rotation space
A walk in SO(3), lifted to SU(2), can return when doubled and scaled
The spectrum is not a quantum property
Band structure belongs to linear dynamics, not to electrons. A periodic array of classical objects also has normal modes, and those modes can cross, flatten, and become singular.
In a two-dimensional crystal of hydrodynamically coupled particles, the measured disordered phase contains long-lived particle pairs. The ordered-phase spectrum identifies these pairs with excitations emerging near Dirac cones. Their formation can propagate as fast pairing avalanches and melt the crystal.
Hexagonal order gives a different limit. When the symmetry of the lattice matches the three-fold structure of the hydrodynamic interaction, an almost dispersionless band appears. The resulting accumulation of ultraslow modes changes the character of melting.
A quasiparticle is useful when a many-body disturbance behaves as a persistent particle-like degree of freedom. Quantization is one way to obtain it, not its definition.
Exceptional points without exotic matter
Dissipative dynamics need not have a Hermitian mode operator. In a two-dimensional elastic lattice exposed to low-Reynolds-number flow, hydrodynamics and elasticity combine to split a Dirac cone. The split ends are exceptional points: eigenvalues and eigenvectors coalesce there.
The two exceptional points carry opposite half-integer topological charge and are joined by a bulk Fermi arc. No gain medium or quantum material is required. The non-Hermiticity comes from ordinary viscous dynamics.
SO(3), SU(2), and a return home
A sequence of three-dimensional rotations is a walk in SO(3). The same walk can be lifted to SU(2), the double cover familiar from spin. This geometry entered through trajectoids: bodies whose shape is designed so that rolling traces a prescribed path.
With Jean-Pierre Eckmann, we later found a general return property. For almost every walk in SO(3) or SU(2), traversing the walk twice and uniformly scaling all rotation angles produces a scale at which the walk returns to the identity. A single traversal almost never has this property.
The result is geometric. It applies equally to ordinary rotations and to systems—spins, qubits, polarization—in which the same group acts on a quantum or optical state.
What remains genuinely quantum?
None of this makes a classical suspension a quantum system. Superposition amplitudes, quantum statistics, entanglement, and quantization are absent.
The narrower point is more useful. Dirac cones follow from spectral degeneracy. Flat bands follow from dispersion. Exceptional topology follows from non-Hermitian mode structure. SO(3) and SU(2) follow from rotation geometry. These structures can therefore survive when the microscopic physics is entirely classical.
Papers
- 2021 Tlusty T — Exceptional topology in ordinary soft matter · Phys. Rev. E 104, 025002
- 2023 Saeed I, Pak HK & Tlusty T — Quasiparticles, flat bands and the melting of hydrodynamic matter · Nature Physics 19, 536–544
- 2023 Sobolev YI, Dong R, Tlusty T, Eckmann J-P et al. — Solid-body trajectoids shaped to roll along desired pathways · Nature 620, 310–315
- 2025 Eckmann J-P & Tlusty T — Walks in Rotation Spaces Return Home when Doubled and Scaled · Phys. Rev. Lett. 135, 147201