Hydrodynamic effective field theories with discrete rotational symmetry

Abstract

We develop a hydrodynamic effective field theory on the Schwinger-Keldysh contour for fluids with charge, energy, and momentum conservation, but only discrete rotational symmetry. The consequences of anisotropy on thermodynamics and first-order dissipative hydrodynamics are detailed in some simple examples in two spatial dimensions, but our construction extends to any spatial dimension and any rotation group (discrete or continuous). We find many possible terms in the equations of motion which are compatible with the existence of an entropy current, but not with the ability to couple the fluid to background gauge fields and vielbein.

Goldstone bosons and fluctuating hydrodynamics with dipole and momentum conservation

“…Unlike the dipole symmetry however, [K i , H] = 0 and this causes the η i "dipole Goldstone" to also show up in E µ a,0 as well as its quadratic form in C a,0 , which causes η i to relax. 12 In the literature it is remarked that the Goldstone of broken boosts is the conventional sound wave [47], but we remark that even without any boost or rotational symmetries, such sound waves still exist [59]. So the sound wave is not crucially reliant on the broken continuous boost symmetry, while the "sound mode" of the dipole-momentum theory cannot be disentangled from symmetry breaking, as far as we can tell.…”

“…We now proceed to writing down the invariant blocks that will be used to write the effective action. We assume rotational invariance, but a generalization to discrete rotational symmetry is straightforward [59]. We recall that the energy conservation is not assumed, so we take β 0 as a constant inverse temperature.…”

Odd Viscosity and Odd Elasticity

“…Anisotropic 2D fluids can have more odd viscosities (7,8): this has been discussed in the context of nematic systems (9,10,8) as well as in solid-state physics, where the viscosity tensor can be constrained by the crystallographic symmetries of the underlying lattice (7,11,12,13,14). In the language of rheology, odd viscosity can be expressed in terms of the so-called normal stress difference (more precisely, the part that is odd in shear rate), see Ref.…”

Anomalous hydrodynamics with triangular point group in 2 + 1 dimensions

“…On the other hand, there are a variety of exotic fluids, arising in (or at least inspired by) quantum matter. Anomalies lead to clear signatures even within classical hydrodynamics [4][5][6], while electron liquids may have reduced spatial symmetries which lead to unconventional transport coefficients [7][8][9][10][11][12][13][14]. Most recently, kinetically constrained "fracton hydrodynamics" have been intensely studied [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”

“…With triangular symmetry, there is a naive possibility of finding a ballistic contribution to this viscous mode. Yet recent work has found that such a ballistic contribution does not exist, either because it violated the KMS-invariance of the geometric action (in the case where the vector conserved charge is momentum) [9], or because it is not compatible with kinetic theory of liquids with anisotropic kinetic energy [8]. This raised the intriguing possibility that there may truly be constraints on hydrodynamics, arising from fundamental statistical mechanics, that are wholly invisible within the canonical Landau paradigm.…”