Theory of Diffusive Fluctuations

Abstract

The recently developed effective field theory of fluctuations around thermal equilibrium is used to compute late-time correlation functions of conserved densities. Specializing to systems with a single conservation law, we find that the diffusive pole is shifted in the presence of non-linear hydrodynamic self-interactions, and that the density-density Green's function acquires a branch point half way to the diffusive pole, at frequency ω = − i 2 Dk 2 . We discuss the relevance of diffusive fluctuations for strongly correlated transport in condensed matter and cold atomic systems.

Heavy operators and hydrodynamic tails

“…which should be contrasted to the pole in the tree-level part of the correlator (2.13a), at ω = ±c s k − i 2 Γ s k 2 . This is the sound analog of the two-diffuson branch cut at ω = − i 2 Dk 2 found in [42]. The analytic structure of hydrodynamic correlators is shown in Fig.…”

“…In this sense we are justified in approximating the action as Gaussian in evaluating (2.9) and in the following. Systematically accounting for corrections to our results would require knowing the structure of interactions in the effective field theory -this was done for simple diffusion in [42].…”

“…The exponent corresponds to the energy threshold for production of two diffusive fluctuations, which is half that of a single diffusive mode [42]. More generally, the operator O(k, t) can decay into n diffusive modes, distributing its momentum such that each mode carries k = k/n so that the exponential factor becomes e −Dk 2 t n = e − 1 n Dk 2 t .…”

“…(A.18) The quantity in square brackets is formally divergent -the divergence can be treated in dimensional regularization by expanding around integer d and throwing away the divergent piece (this UV divergence can be absorbed in the bare transport parameters [42]). This gives…”

The complex life of hydrodynamic modes

“…has branch points located at p = ±im, and for which the failure of the convergence of the gradient expansion corresponds to the breakdown of the non-relativistic approximation. We hope our results may be of interest for studies of higher-order hydrodynamics necessary for improving the precision of hydrodynamic predictions and also for justifying the construction of the effective field theory of hydrodynamics formulated as a gradient expansion [85][86][87][88][89][90][91][92][93][94][95]. As already mentioned in ref.…”