The Euler-scale power-law asymptotics of space-time correlation functions in many-body systems, quantum and classical, can be obtained by projecting the observables onto the hydrodynamic modes admitted by the model and state. This is the Boltzmann-Gibbs principle; it works for integrable and non-integrable models alike. However, certain observables, such as some order parameters in thermal of generalised Gibbs ensembles, do not couple to any hydrodynamic mode: the Boltzmann-Gibbs principle gives zero. I will explain how hydrodynamics can still give the leading exponential decay of order parameter correlation functions. With the examples of the quantum XX chain and the sine-Gordon model, I will explain how large deviations of the spin and U(1) current fluctuations are related to such exponential decay. Exact predictions are given by the ballistic fluctuation theory based on generalised hydrodynamics. In the XX model, this is in agreement with results obtained previously by a more involved Fredholm determinant analysis and other techniques, and even gives a new formula for a parameter regime not hitherto studied. In the sine-Gordon model, these are new results, inaccessible by other techniques. Works in collaboration with Giuseppe Del Vecchio Del Vecchio, and Márton Kormos. ---- Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed on Tuesday.

Join here (you need Microsoft Teams). Typical systems of many particles in strong interaction have extremely complex behaviours which are hard to study in detail. But when the system is very large, simplicity resurfaces: typically just a few degrees of freedom are relevant, which follow new, simple laws. Understanding what the emergent behaviours are from the underlying microscopic interactions is one of the foremost problems in modern science. A very powerful set of ideas and tools at our disposal is hydrodynamics. Although the Navier-Stokes and related equations have been studied for a very long time, we are now starting to uncover the full potential of the fundamental principles of hydrodynamics. In particular, in a recent breakthrough it was understood how to apply these principles to quantum and classical integrable models, where infinitely many conserved currents exist, giving ``generalised hydrodynamics”. I will overview the fundamental principles of hydrodynamics and their adaptation to integrable systems, with simple examples such as the quantum Lieb-Liniger model, the classical Toda model, and the soliton gases. I will discuss a recent cold-atom experiment that confirmed generalised hydrodynamics, and, if time permits, show some of the exact results that can be obtained with this formalism, such as exact nonequilibrium steady states and exact asymptotic of correlation functions at large space-time separations.

This will be presenting mainly some of my recent work, and also aspects of recent work of Herbert Spohn, both developed in parallel. The classical Toda system is a one-dimensional integrable many-body system, which can be seen either as a gas of particles or as a chain of degrees of freedom. Herbert has shown how the generalised Gibbs ensembles of the Toda chain can be obtained from a certain limit of the beta-ensemble in random matrix theory. Analysing and connecting the gas and chain viewpoints, I have obtained both the generalised Gibbs ensembles and generalised hydrodynamics from a quasiparticle scattering description. Thus we make a connection between quasiparticle scattering and random matrix theory.

The hydrodynamic approximation is an extremely powerful tool to describe the behavior of many-body systems such as gases. At the Euler scale (that is, when variations of densities and currents occur only on large space-time scales), the approximation is based on the idea of local thermodynamic equilibrium: locally, within fluid cells, the system is in a Galilean or relativistic boost of a Gibbs equilibrium state. This is expected to arise in conventional gases thanks to ergodicity and Gibbs thermalization, which in the quantum case is embodied by the eigenstate thermalization hypothesis. However, integrable systems are well known not to thermalize in the standard fashion. The presence of infinitely-many conservation laws preclude Gibbs thermalization, and instead generalized Gibbs ensembles emerge. In this talk I will introduce the associated theory of generalized hydrodynamics (GHD), which applies the hydrodynamic ideas to systems with infinitely-many conservation laws. It describes the dynamics from inhomogeneous states and in inhomogeneous force fields, and is valid both for quantum systems such as experimentally realized one-dimensional interacting Bose gases and quantum Heisenberg chains, and classical ones such as soliton gases and classical field theory. I will give an overview of what GHD is, how its main equations are derived and its relation to quantum and classical integrable systems. If time permits I will touch on the geometry that lies at its core, how it reproduces the effects seen in the famous quantum Newton cradle experiment, and how it leads to exact results in transport problems such as Drude weights and non-equilibrium currents. This is based on various collaborations with Alvise Bastianello, Olalla Castro Alvaredo, Jean-Sébastien Caux, Jérôme Dubail, Robert Konik, Herbert Spohn, Gerard Watts and my student Takato Yoshimura, and strongly inspired by previous collaborations with Denis Bernard, M. Joe Bhaseen, Andrew Lucas and Koenraad Schalm.

The talk is based on a very recent work of the speaker on generalising Zamolodchikov's c-theorem to non-unitary theories.

Vertex operator algebra (VOA) is the algebraic setup formalising conformal field theory. It develops in a mathematically complete way the idea of constructing quantum field theory using the algebra of symmetry currents and their modules. On the other hand, conformal loop ensembles (CLE) are measures on random loop configurations that are known, in certain cases, to describe the continuous limit of statistical models at critical points. There is a one-parameter family of such measures, supposed to correspond to all central charges between 0 and 1. These two constructions enjoy complete mathematical rigour, and give the opportunity to understand with more precision the relation between the statistical interpretation of QFT, and its algebraic description. I will describe some of my recent works in this direction: I will explain how to construct the Virasoro VOA (the stress-energy tensor and its descendents) in terms of random objects in CLE. No prior knowledge of either VOA or CLE is needed as I will review both subjects.

TBA