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arXiv:cond-mat/9607164 · Modern Physics Letters B

EFFECTIVE FIELD THEORY OF IDEAL-FLUID HYDRODYNAMICS

Adriaan M. J. Schakel

Published Sep 10, 1996• 18 citations
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Abstract

Starting from a standard description of an ideal, isentropic fluid, we derive the effective theory governing a gapless non-relativistic mode — the sound mode. The theory, which is dictated by the requirement of Galilei invariance, entails the entire set of hydrodynamic equations. The gaplessness of the sound mode is explained by identifying it as the Goldstone mode associated with the spontaneous breakdown of Galilei invariance. Differences with a superfluid are pointed out.

DOI
10.1142/s0217984996001139

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The first paper in this series developed a field-theoretical approach to hydrodynamics. The objective of this second paper is to extend that formalism to superfluid hydrodynamics. Given the viewpoint adopted here, the fundamental objectives of the theory are the superfluid order-parameter function, Ψ(x), the collective and the individual motions, in terms of which one can define Galilei invariants. Basic to the present theory is that the Schrödinger field Ψ(x) can by expressed in terms of the Galilei invariants (ϕn(x), ϕs(x)) and the phase ϕ(x) as φ(x) = exp {(im/h) ϕ(x)} F[x: Ψ(x), ϕn(x), ϕs(x), φ(x)] where the complex field φ(x) describes the individual motions (or is the asymptotic field). A plausible Lagrangian density, which in invariant, under the Galilei boost with a constant velocity, then permits the derivation of the equation of motion for the order-parameter.

Fortschritte der Physik/Progress of Physics•Jan 1, 1991•0 citations
Starting from a standard description of an ideal, isentropic fluid, we derive the effective theory governing a gapless non-relativistic mode — the sound mode. The theory, which is dictated by the requirement of Galilei invariance, entails the entire set of hydrodynamic equations. The gaplessness of the sound mode is explained by identifying it as the Goldstone mode associated with the spontaneous breakdown of Galilei invariance. Differences with a superfluid are pointed out.
We discuss the effective theory of a bosonic superfluid whose microscopic behavior is described by a nonrelativistic, weak-coupling Φ4 theory in the phase with broken particle number symmetry, both at zero temperature and in the vicinity of the phase transition. In the zero-temperature regime, the theory is governed by the gapless Goldstone mode resulting from the broken .symmetry. Although this mode is gapless, the effective theory turns out to be Gallilei invariant. The regime just below the critical temperature is approached in a high-temperature expansion which is shown to be consistent with the weak-coupling assumption of the theory. We calculate the critical temperature, the co-efficients of the Landau theory, and the finite-temperature sound velocity. A comparison with BCS theory is given.
We develop an effective field theory for dissipative fluids which governs the dynamics of long-lived gapless modes associated with conserved quantities. The resulting theory gives a path integral formulation of fluctuating hydrodynamics which systematically incorporates nonlinear interactions of noises. The dynamical variables are mappings between a "fluid spacetime" and the physical spacetime and an essential aspect of our formulation is to identify the appropriate symmetries in the fluid spacetime. The theory applies to nonlinear disturbances around a general density matrix. For a thermal density matrix, we require an additional Z2Z_2Z2​ symmetry, to which we refer as the local KMS condition. This leads to the standard constraints of hydrodynamics, as well as a nonlinear generalization of the Onsager relations. It also leads to an emergent supersymmetry in the classical statistical regime, and a higher derivative deformation of supersymmetry in the full quantum regime.
Abstract We construct a theory of hydrodynamic transport for systems with conserved dipole moment, U(1) charge, energy, and momentum. These models have been considered in the context of fractons, since their elementary and isolated charges are immobile by symmetry, and have two known translation-invariant gapless phases: a “p-wave dipole superfluid” phase where the dipole symmetry is spontaneously broken and a “s-wave dipole superfluid” phase where both the U(1) and dipole symmetries are spontaneously broken. We argue on grounds of symmetry and thermodynamics that there is no transitionally-invariant gapless fluid with unbroken dipole symmetry. In this work, we primarily focus on the hydrodynamic description of p-wave dipole superfluids, including leading dissipative corrections. That theory has, in a sense, a dynamical scaling exponent z = 2, and its spectrum of fluctuations includes novel subdiffusive modes ω ∼ −ik4 in the shear sector and magnon-like sound mode ω ∼ ±k2 − ik2. By coupling the fluid to background fields, we find response functions of the various symmetry currents. We also present a preliminary generalization of our work to s-wave dipole superfluids, which resemble z = 1 fluids and feature sound waves and diffusive shear modes, as in an ordinary fluid. However, the spectrum also contains a magnon-like second-sound mode ω ∼ ±k2 ± k4 − ik4 with subdiffusive attenuation.
We present a dissipative hydrodynamic theory of "s-wave dipole superfluids" that arise in phases of translation-invariant and dipole-symmetric models in which the U(1) symmetry is spontaneously broken. The hydrodynamic description is subtle on account of an analogue of dangerously irrelevant operators, which requires us to formalize an entirely new derivative counting scheme suitable for these fluids. We use our hydrodynamic model to investigate the linearized response of such a fluid, characterized by sound modes ω∼±k−ik2ω\sim \pm k - ik^2ω∼±k−ik2, shear modes ω∼−ik2ω\sim-ik^2ω∼−ik2, and magnon-like propagating modes ω∼±k2−ik4ω\sim \pm k^2 - ik^4ω∼±k2−ik4 that are the dipole-invariant version of superfluid "second sound" modes. We find that these fluids can also admit equilibrium states with "dipole superflow" that resemble a polarized medium. Finally, we couple our theory to slowly varying background fields, which allows us to compute response functions of hydrodynamic operators and Kubo formulas for hydrodynamic transport coefficients.
A bstract We present a dissipative hydrodynamic theory of “s-wave dipole superfluids” that arise in phases of translation-invariant and dipole-symmetric models in which the U(1) symmetry is spontaneously broken. The hydrodynamic description is subtle on account of an analogue of dangerously irrelevant operators, which requires us to formalize an entirely new derivative counting scheme suitable for these fluids. We use our hydrodynamic model to investigate the linearized response of such a fluid, characterized by sound modes ω ~ ± k – ik 2 , shear modes ω ~ – ik 2 , and magnon-like propagating modes ω ~ ± k 2 – ik 4 that are the dipole-invariant version of superfluid “second sound” modes. We find that these fluids can also admit equilibrium states with “dipole superflow” that resemble a polarized medium. Finally, we couple our theory to slowly varying background fields, which allows us to compute response functions of hydrodynamic operators and Kubo formulas for hydrodynamic transport coefficients.
The first paper in this series developed a field-theoretical approach to hydrodynamics. The objective of this second paper is to extend that formalism to superfluid hydrodynamics. Given the viewpoint adopted here, the fundamental objectives of the theory are the superfluid order-parameter function, Ψ(x), the collective and the individual motions, in terms of which one can define Galilei invariants. Basic to the present theory is that the Schrödinger field Ψ(x) can by expressed in terms of the Galilei invariants (ϕn(x), ϕs(x)) and the phase ϕ(x) as φ(x) = exp {(im/h) ϕ(x)} F[x: Ψ(x), ϕn(x), ϕs(x), φ(x)] where the complex field φ(x) describes the individual motions (or is the asymptotic field). A plausible Lagrangian density, which in invariant, under the Galilei boost with a constant velocity, then permits the derivation of the equation of motion for the order-parameter.