Non-dissipative Hall viscosity in 2D quantum fluids — a topological transport coefficient tied to the Chern number
Hall Viscosity
In 2D parity-violating fluids, the viscosity tensor has an antisymmetric (non-dissipative) part:
σᵢⱼ = ηˢ(∂ᵢvⱼ + ∂ⱼvᵢ) + η_H ε^{ik}(∂_k v_j + ∂_j v_k)
The Hall viscosity η_H is related to the mean orbital angular momentum:
η_H = ℏ n̄ s̄ / 2
where s̄ is the mean orbital spin per particle. For the ν = p/q fractional quantum Hall state, s̄ = p(2p−q)/(2q), and η_H quantizes in units of ℏn̄/4.
Key property: η_H is non-dissipative (stress ⊥ strain rate), unlike shear viscosity η_s.
Topological Origin & Chern Number
Hall viscosity is a topological transport coefficient — it cannot change continuously under smooth deformations of the Hamiltonian that preserve the gap.
The Chern number counts the topological winding of the Berry curvature over the Brillouin zone:
C = (1/2π) ∫_BZ Ω_k d²k
For the integer QHE at filling ν=C, η_H = ℏnC/4. For FQHE:
η_H = ℏn̄ · s̄(ν)/2
The guiding center metric g_ab controls η_H — it describes the shape of the lowest Landau level guiding center wavefunction. Adiabatic deformation of g_ab generates momentum currents proportional to η_H.