Daniel A Schubring

Wave turbulence is a quintessential cross-disciplinary field in physics, with applications to oceanography, condensed matter, non-linear optics, and cosmology, to name just a few. At its core it involves solving a kinetic equation for non-equilibrium solutions. Traditionally the kinetic equation is determined only to low order in perturbation theory, but with more sophisticated methods we can get a window on strong-turbulence phenomena.

Together with Vladimir Rosenhaus and collaborators, I worked on a path integral approach to wave turbulence which allows for calculation of higher order corrections to the kinetic equation. Simplified diagrammatic rules for these corrections were derived in [2308.00740], and a non-perturbative understanding of the framework was presented in [2309.08484].

In [2406.18475] we used a large N approximation to construct a differential approximation model valid across weak and strong turbulence regimes. In the strong regime, stationary solutions exhibiting Phillips (critical-balance) scaling were identified. Subsequent work with Simon Thalabard [2509.23199] extended the analysis to the time-dependent approach to stationarity, revealing a regime with oscillatory evolution of the wave-action spectrum. In forthcoming work we study a melonic large-N model and show how a large-q approximation developed in the context of the SYK model leads to a simple calculation of nonlinear resonance broadening.

The S-matrix should, in principle, encode how a system’s thermodynamics departs from ideal-gas behavior. While the thermodynamic Bethe ansatz captures this connection in integrable models, the Dashen–Ma–Bernstein (DMB) approach extends it to non-integrable systems. I revisited and clarified the DMB formalism, showing how it applies to multiparticle scattering using a set of exactly solvable two-dimensional examples [2408.00729]. More recently, I have been exploring how a version of the DMB formula can be used to derive higher-order corrections to the kinetic equation relevant for wave turbulence.

This topic is directly relevant for the study of confining flux tubes in Yang-Mills. The free energy of an infinitely long flux tube is equivalent to the ground state energy of a finite-length tube, and that may be calculated on the lattice. The ground state energy (and indeed higher levels) may then be connected to the S-matrix in a model of effective string theory. Shortly after my work appeared, DMB was successfully applied to the simplest effective string theory (the D=3 GGRT model) [2408.06729], but much remains to be done for realistic, non-integrable theories.

There is an old idea due to Lord Kelvin in the 19th century that distinct atoms might be related to distinct topologies of knotted vortices. Of course today we know nuclei may be described in terms of protons and neutrons, but in [2111.06385] my collaborators and I show that there may be some validity to the idea after all.

Nuclei may be represented as topological defects ("Skyrmions") in an effective field theory for QCD known as the Skyrme model. If we turn instead to magnetic systems, a Skyrmion usually refers to a 2D topological defect, but recent experimental work has produced 3D loops of magnetic Skyrmion strings known as "Hopfions". Our work ties the two notions together. We studied a 3D magnetic system which at a fixed value of control parameter is close to the Skyrme model of QCD. As the parameter is varied, the nuclei-like Skyrmions deform into knotted Hopfions. We further showed that even in the conventional Skyrme model topological charge can be interpreted from this Hopfion viewpoint, and that a Skyrmion can be 'untied' into a long string carrying baryon charge per length.

I began my theoretical physics career by looking at relativistic fluids and uncovered a connection between a coarse-grained description of cosmic strings and magnetohydrodynamics [1412.3135]. This and my related papers with Vitaly Vanchurin (e.g. [1410.5843]) were formulated in terms of conserved higher-form currents at a time just before the explosion of interest in generalized global symmetries. Subsequent work by other groups re-examined this construction in the modern language of higher-form symmetries [1610.07392] and holography [1707.04182, 1707.08577].

Since then, there have been exciting developments (e.g. [1510.02494, 1511.03646]) that formulate hydrodynamics in the same Schwinger–Keldysh framework I am familiar with from my work on non-equilibrium field theory. I have recently returned to this area, working with Sriram Ganeshan on odd-viscosity fluids.

Interdisciplinary theoretical physics

My work draws together methods and ideas from across theoretical physics, from high-energy theory to condensed-matter and fluid dynamics. This breadth makes it easy to communicate with researchers in different areas and to design concrete, accessible projects for students that connect formal theory with observable phenomena.

Low-dimensional QFT as a laboratory

Low-dimensional field theories such as Chern–Simons, sine-Gordon, and SYK form a shared toolkit for theoretical physicists across high-energy, condensed-matter, and statistical physics. These systems provide a controlled setting in which to study non-perturbative effects. In particular, I have worked extensively with non-linear sigma models [2207.10549] and more recently I have become especially interested in the ’t Hooft model as a minimal framework for confinement and string-like behavior.

Bridging perturbative and non-perturbative physics

A recurring theme in my research is understanding how perturbative expansions encode non-perturbative physics. In [2107.11017, 2207.10549], using various non-linear sigma models, I showed how a finite truncation of the large-N expansion already contains information about resurgent trans-series in the ordinary coupling. I continue to apply large-N and saddle point methods in much of my work.