^{3}.

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 nonlinear hydrodynamic self-interactions, and that the density-density Green’s function acquires a branch point halfway to the diffusive pole, at frequency

Diffusion was invented by Fourier to describe the dynamics of heat

The traditional approach to address this class of problems is to couple the degrees of freedom (d.o.f.) of interest to stochastic noise fields and solve perturbatively a nonlinear Langevin equation

Another motivation for a systematic study of hydrodynamic fluctuations is thermalization. The local thermalization (or equilibration) time

In the following, we use the general formalism of Ref.

Our objective is to understand the structure of energy density correlation functions in nonintegrable quantum systems at nonzero temperature

The traditional stochastic approach to hydrodynamic fluctuations with Gaussian noise

In the remainder we will show precisely how the interactions in

The one-loop diagrams contributing to

The diagrams shown in Fig.

The retarded Green’s function is analytic in the upper-half frequency plane, as required by causality. The interactions have induced a branch point at

On-shell condition for the two internal legs (left), and analytic structure of the retarded Green’s function

In previous treatments, similar physics to what we have just described was found in the coupled diffusion of two modes

Furthermore, we have also found nonanalytic corrections to the Green’s function even with a single diffusing mode. These are not as strong as those arising with two modes, as we now explain. The functions

Strong renormalization of the transport parameters due to hydrodynamic fluctuations occurs if the ratios in

If microscopic interactions are weak then

Fluctuation effects can also become important for transport close to a thermal phase transition. The thermalization length diverges as

Transport length scales approaching or exceeding the lattice spacing are also seen in “bad metals”

Condensed matter systems—including most bad metals—are typically at degenerate temperatures

Finally, diffusion with a short thermalization length has also been seen in spin transport in strongly interacting ultracold atoms in a trap

In summary, long wavelength fluctuations about diffusive dynamics may be relevant in condensed matter and cold atom systems of widespread interest. We have seen that a systematic derivation of these effects leads to different results than previous, more phenomenological, approaches. For this reason, it will be important to revisit the computation of fluctuations in relativistic hydrodynamics

We would like to thank Erez Berg, Debanjan Chowdhury, Paolo Glorioso, and Andrew Lucas for illuminating discussions. S. A. H. and L. V. D. are partially supported by the U.S. Department of Energy Office of Science under Award No. DE-SC0018134. X. C. L. is supported by the Knut and Alice Wallenberg Foundation.

That these interactions should arise is already clear from the stochatistic approach, where the fluctuation dissipation theorem imposes

Additional poles in

Using Eqs.

If thermoelectric effects are strong, one should instead work with coupled heat and charge diffusion. This is done in the Supplemental Material