Theory of diffusive fluctuations

TL;DR

The paper develops a systematic effective field theory of hydrodynamic fluctuations for systems with a single conserved density and computes the one-loop corrections to the density-density retarded Green's function. It shows that non-linear self-interactions renormalize the diffusion constant and conductivity and introduce a non-analytic branch point at $\omega = -\frac{i}{2} D k^2$, organizing these effects in terms of a thermalization length $\ell_{\rm th}$ and thermodynamic derivatives. It further analyzes the analytic vs non-analytic structure of $G^R_{\varepsilon\varepsilon}$ and extends to two interacting diffusive charges, where stronger non-analytic conductivities arise. The results clarify discrepancies with traditional stochastic treatments and have implications for thermal transport in strongly correlated materials and ultracold atomic systems, especially in regimes with short thermalization lengths or near phase transitions.

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 $ω= -\frac{i}{2}Dk^2$. We discuss the relevance of diffusive fluctuations for strongly correlated transport in condensed matter and cold atomic systems.

Figures (4)

• Figure 1: The one-loop diagrams contributing to $G_{\varepsilon \varepsilon}$. Solid lines denote the energy density field $\varepsilon$ and squiggly lines denote the auxiliary field $\varphi_a$.
• Figure 2: On-shell condition for the two internal legs (left), and analytic structure of the retarded Green's function $G^R_{\varepsilon\varepsilon}(\omega,k)$ at one loop (right). In imposing the on-shell condition, it is important to consider two poles in opposite halves of the complex $\omega'$ plane, otherwise the loop contribution vanishes. The pole in the upper half plane arises from an advanced Green's function in the loop: $G^A = \left(G^R\right)^*$.
• Figure 3: Closed time path.
• Figure 4: On-shell condition when two different diffusive poles are picked up in the loop (left), and analytic structure of $G^R_{nn}(\omega,k)$ in the presence of two coupled diffusive charges. The branch cuts have been rotated for clarity, and the splitting of the diffusive poles (see Fig. \ref{['fig_non_ana']}) is not shown.