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arXiv:2607.22527

Effective field theories of nonlinear fluctuating hydrodynamics in one dimension

Matija Koterle, Enej Ilievski

Published Jul 24, 2026• 0 citations
arXivHTMLPDF

Abstract

Describing emergent macroscopic phenomena in low spatial dimensions is known to be notoriously challenging, primarily due to strong interactions that render perturbative approaches inapplicable. On the other hand, low-dimensional systems host a wealth of unorthodox phenomena. A prominent example is the emergence of superdiffusive transport in one-dimensional interacting systems, traditionally studied in the framework of nonlinear fluctuating hydrodynamics. After identifying and discussing internal inconsistencies in the previous formulations, in this work we develop a general and systematic approach for constructing effective field theories of one-dimensional hydrodynamic systems in the form of coupled stochastic Langevin-type equations compatible with the physical requirements of local equilibrium states such as the thermodynamic Maxwell relation and fluctuation-dissipation symmetry. We implemented a general numerical integration scheme and exemplified our construction on a simple model of two interacting hydrodynamic modes with a non-Gaussian stationary equilibrium measure.

DOI
10.48550/arxiv.2607.22527

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Describing emergent macroscopic phenomena in low spatial dimensions is known to be notoriously challenging, primarily due to strong interactions that render perturbative approaches inapplicable. On the other hand, low-dimensional systems host a wealth of unorthodox phenomena. A prominent example is the emergence of superdiffusive transport in one-dimensional interacting systems, traditionally studied in the framework of nonlinear fluctuating hydrodynamics. After identifying and discussing internal inconsistencies in the previous formulations, in this work we develop a general and systematic approach for constructing effective field theories of one-dimensional hydrodynamic systems in the form of coupled stochastic Langevin-type equations compatible with the physical requirements of local equilibrium states such as the thermodynamic Maxwell relation and fluctuation-dissipation symmetry. We implemented a general numerical integration scheme and exemplified our construction on a simple model of two interacting hydrodynamic modes with a non-Gaussian stationary equilibrium measure.
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.
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We present a symmetry-based formulation of nonlinear fluctuating hydrodynamics (NFH) for one-dimensional many-particle systems with generic homogeneous nearest-neighbor interactions. We derive the hydrodynamic equations solely from symmetry and conservation principles, ensuring full consistency with thermalization. Using the dynamic renormalization group, we identify a Kardar-Parisi-Zhang (KPZ)-type fixed point, characterized by the dynamical exponent z=3/2 for both the sound and heat modes. Extensive numerical simulations of the derived NFH equations confirm this exponent and further reveal that both modes are close to the universal KPZ scaling function-the Prähofer-Spohn function. These findings establish a unified, symmetry-based framework for understanding universal transport and fluctuation phenomena in one-dimensional nonequilibrium systems, independent of microscopic details.
Recent ground-breaking experiments have realized strongly interacting quantum degenerate Fermi gas in a cold atomic system with tunable interactions. This has provided a table-top system which is extremely hydrodynamic in nature. This experimental realization helps us to investigate several aspects such as the interplay between nonlinearity, dissipation and dispersion. We find, for instance, that the dynamics in such a system shows near perfect agreement with a hydrodynamic theory. In collaboration with the group of John Thomas at Duke we interpreted studies of collision of two strongly interacting Fermi gases that led to shock waves which are a hallmark of nonlinear physics. Due to reasons such as the nature of interactions, higher dimensionality, these cold atomic systems are non-integrable and moreover the underlying field theory construction is mostly phenomenological in nature. On the other hand there are certain one-dimensional systems which are not only integrable but also facilitate more formal and rigorous ways of deriving the corresponding integrable field theories. One such family of models is the family of Calogero models (and their generalizations). They provide an extraordinary insight into the field of strongly correlated systems and hydrodynamics. We study the collective field theory of such models and address aspects of nonlinear physics such as Spin-Charge Interaction, Emptiness Formation Probability, Solitons etc; We derive a two-component nonlinear, nonlocal, integrable field theory. We also show that the Calogero family which is integrable even in an external harmonic trap (usually unavoidable in cold atom setups) is relatively short ranged thereby qualifying as a toy model for cold atom experiments. Transport in certain strongly correlated systems (impurity models) was studied using few low-dimensional techniques such as a 1/N diagrammatic expansion, Slave Boson Mean Field Theory and the Bethe Ansatz. A mesoscopic setup such as parallel quantum dots forms an ideal platform for such an investigation and comparison between different low-dimensional techniques. We studied transport, correlations and nature of the ground state of double quantum dots. We probed several non-perturbative aspects of this double-impurity model. For example, we showed that the RKKY interaction in closely spaced dots can be non-ferromagnetic due to its non-pertubative nature. This study helped us to point some discrepancies between different methods (such as the Numerical Renormalization Group). We give possible reasons for these discrepancies.
Fluctuating hydrodynamics emerges in essentially any local many-body system at nonzero temperature. Effective field theory (EFT) approaches enable the quantitative study of this emergence, providing a controlled framework to capture late-time observables. These lectures introduce the organizing principles behind equilibrium and out-of-equilibrium dynamics in these thermalizing systems. A central focus is the modern construction of these EFTs, which frames fluctuating hydrodynamics through the lens of "strong-to-weak" spontaneous symmetry breaking. Drawing examples from both high-energy and condensed matter physics, we show how this paradigm adapts to systems ranging from spin chains to relativistic quantum field theories, including models with generalized symmetries or symmetries with 't Hooft anomalies. Finally, we discuss UV/IR constraints on transport parameters -- viewed as the Wilson coefficients of hydrodynamic EFTs -- both in continuum and on the lattice.
We study the dynamics of a locally conserved energy in ergodic, local many-body quantum systems on a lattice with no additional symmetry. The resulting dynamics is well approximated by a coarse grained, classical linear functional diffusion equation for the probability of all spatial configurations of energy. This is equivalent to nonlinear stochastic hydrodynamics, describing the diffusion of energy in physical spacetime. We find the absence of nonhydrodynamic slow degrees of freedom, a nonlinear fluctuation-dissipation theorem, and the emergence of a (weakly interacting) kinetic theory for hydrodynamic modes near thermal equilibrium. The observable part of the microscopic entropy obeys the local second law of thermodynamics, and quantitatively agrees with the phenomenological predictions of hydrodynamics. Our approach naturally generalizes to ergodic systems with additional symmetries, may lead to numerical algorithms to calculate diffusion constants for lattice models, and implies sufficiency conditions for a rigorous derivation of hydrodynamics in quantum systems.