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arXiv:2111.08420 · Journal of Statistical Mechanics Theory and Experiment

The hydrodynamic theory of dynamical correlation functions in the XX chain

Giuseppe Del Vecchio Del Vecchio, Benjamin Doyon

Published May 1, 2022• 19 citations
arXivHTMLPDF

Abstract

Abstract By the hydrodynamic linear response theory, dynamical correlation functions decay as power laws along certain velocities, determined by the flux Jacobian. Such correlations are obtained by hydrodynamic projections, and physically, they are due to propagating ‘sound waves’ or generalisation thereof, transporting conserved quantities between the observables. However, some observables do not emit sound waves, such as order parameters associated to symmetry breaking. In these cases correlation functions decay exponentially everywhere, a behaviour not captured by the hydrodynamic linear response theory. Focussing on spin–spin correlation functions in the XX quantum chain, we first review how hydrodynamic linear response works, emphasising that the necessary fluid cell averaging washes out oscillatory effects. We then show how, beyond linear response, Euler hydrodynamics can still predict the exponential decay of correlation functions of order parameters. This is done by accounting for the large-scale fluctuations of domain walls, via the recently developed ballistic fluctuation theory. We use the framework of generalised hydrodynamics, which is particularly simple in this model due to its free fermion description. In particular, this reproduces, by elementary calculations, the exponential decay in the celebrated formulae by Its et al (1993) and by Jie (1998), which were originally obtained by intricate Fredholm determinant analysis; and gives a new formula in a parameter domain where no result was obtained before. We confirm the results by numerical simulations.

DOI
10.1088/1742-5468/ac6667

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Abstract By the hydrodynamic linear response theory, dynamical correlation functions decay as power laws along certain velocities, determined by the flux Jacobian. Such correlations are obtained by hydrodynamic projections, and physically, they are due to propagating ‘sound waves’ or generalisation thereof, transporting conserved quantities between the observables. However, some observables do not emit sound waves, such as order parameters associated to symmetry breaking. In these cases correlation functions decay exponentially everywhere, a behaviour not captured by the hydrodynamic linear response theory. Focussing on spin–spin correlation functions in the XX quantum chain, we first review how hydrodynamic linear response works, emphasising that the necessary fluid cell averaging washes out oscillatory effects. We then show how, beyond linear response, Euler hydrodynamics can still predict the exponential decay of correlation functions of order parameters. This is done by accounting for the large-scale fluctuations of domain walls, via the recently developed ballistic fluctuation theory. We use the framework of generalised hydrodynamics, which is particularly simple in this model due to its free fermion description. In particular, this reproduces, by elementary calculations, the exponential decay in the celebrated formulae by Its et al (1993) and by Jie (1998), which were originally obtained by intricate Fredholm determinant analysis; and gives a new formula in a parameter domain where no result was obtained before. We confirm the results by numerical simulations.
Abstract One of the most profound questions of mathematical physics is that of establishing from first principles the hydrodynamic equations in large, isolated, strongly interacting many-body systems. This involves understanding relaxation at long times under reversible dynamics, determining the space of emergent collective degrees of freedom (the ballistic waves), showing that projection occurs onto them, and establishing their dynamics (the hydrodynamic equations). We make progress in these directions, focussing for simplicity on one-dimensional systems. Under a model-independent definition of the complete space of extensive conserved charges, we show that hydrodynamic projection occurs in Euler-scale two-point correlation functions. A fundamental ingredient is a property of relaxation: we establish ergodicity of correlation functions along almost every direction in space and time. We further show that to every extensive conserved charge with a local density is associated a local current and a continuity equation; and that Euler-scale two-point correlation functions of local conserved densities satisfy a hydrodynamic equation. The results are established rigorously within a general framework based on Hilbert spaces of observables. These spaces occur naturally in the C∗C^*C∗ C ∗ algebra description of statistical mechanics by the Gelfand–Naimark–Segal construction. Using Araki’s exponential clustering and the Lieb–Robinson bound, we show that the results hold, for instance, in every nonzero-temperature Gibbs state of short-range quantum spin chains. Many techniques we introduce are generalisable to higher dimensions. This provides a precise and universal theory for the emergence of ballistic waves at the Euler scale and how they propagate within homogeneous, stationary states.
Abstract We propose a general formalism, within large-deviation theory, giving access to the exact statistics of fluctuations of ballistically transported conserved quantities in homogeneous, stationary states. The formalism is expected to apply to any system with an Euler hydrodynamic description, classical or quantum, integrable or not, in or out of equilibrium. We express the exact scaled cumulant generating function (or full counting statistics) for any (quasi-)local conserved quantity in terms of the flux Jacobian. We show that the “extended fluctuation relations” of Bernard and Doyon follow from the linearity of the hydrodynamic equations, forming a marker of “freeness” much like the absence of hydrodynamic diffusion does. We show how an extension of the formalism gives exact exponential behaviours of spatio-temporal two-point functions of twist fields, with applications to order-parameter dynamical correlations in arbitrary homogeneous, stationary state. We explain in what situations the large-deviation principle at the basis of the results fail, and discuss how this connects with nonlinear fluctuating hydrodynamics. Applying the formalism to conformal hydrodynamics, we evaluate the exact cumulants of energy transport in quantum critical systems of arbitrary dimension at low but nonzero temperatures, observing a phase transition for Lorentz boosts at the sound velocity.
In the hydrodynamic theory, the nonequilibrium dynamics of a many-body system is approximated, at large scales of space and time, by irreversible relaxation to local entropy maximization. This results in a convective equation corrected by viscous or diffusive terms in a gradient expansion, such as the Navier-Stokes equations. Diffusive terms are evaluated using the Kubo formula, and possibly arising from an emergent noise due to discarded microscopic degrees of freedom. In one dimension of space, diffusive scaling is often broken as noise leads to superdiffusion. But in linearly degenerate hydrodynamics, such as that of integrable models, diffusive behaviors are observed, and it has long been thought that the standard diffusive picture remains valid. In this Letter, we show that in such systems, the Navier-Stokes equation breaks down beyond linear response. We demonstrate that diffusive-order corrections do not take the form of a gradient expansion. Instead, they are completely determined by ballistic transport of initial-state fluctuations, and obtained from the nonlocal two-point correlations recently predicted by the ballistic macroscopic fluctuation theory; the resulting hydrodynamic equations are reversible. To do so, we establish a regularized fluctuation theory, putting on a firm basis the recent idea that ballistic transport of initial-state fluctuations determines fluctuations and correlations beyond the Euler scale. This extends the idea of "diffusion from convection" previously developed to explain the Kubo formula in integrable systems to generic nonequilibrium settings.
Abstract We review the recent advances on exact results for dynamical correlation functions at large scales and related transport coefficients in interacting integrable models. We discuss Drude weights, conductivity and diffusion constants, as well as linear and nonlinear response on top of equilibrium and non-equilibrium states. We consider the problems from the complementary perspectives of the general hydrodynamic theory of many-body systems, including hydrodynamic projections, and form-factor expansions in integrable models, and show how they provide a comprehensive and consistent set of exact methods to extract large scale behaviours. Finally, we overview various applications in integrable spin chains and field theories.
We study the dynamics of charge fluctuations after homogeneous quantum quenches in one-dimensional systems with ballistic transport. For short but macroscopic times where the non-trivial dynamics is largely dominated by long-range correlations, a simple expression for the associated full counting statistics can be obtained by hydrodynamic arguments. This formula links the non-equilibrium charge fluctuation after the quench to the fluctuations of the associated current after a charge-biased inhomogeneous modification of the original quench which corresponds to the paradigmatic partitioning protocol. Under certain assumptions, the fluctuations in the latter case can be expressed by explicit closed form formulas in terms of thermodynamic and hydrodynamic quantities via the Ballistic Fluctuation Theory. In this work, we identify precise physical conditions for the applicability of a fully hydrodynamic theory, and provide a detailed analysis explicitly demonstrating how such conditions are met and how this leads to such hydrodynamic treatment. We discuss these conditions at length in non-relativistic free fermions, where calculations become feasible and allow for cross-checks against exact results. In physically relevant cases, strong long-range correlations can complicate the hydrodynamic picture, but our formula still correctly reproduces the first cumulants.
Hydrodynamic noise is the Gaussian process that emerges at larges scales of space and time in many-body systems. It is justified by the central limit theorem, and represents degrees of freedom forgotten when projecting coarse-grained observables onto conserved quantities. It is the basis for fluctuating hydrodynamics, where it appears along with ‘’bare’’ diffusion terms via the Einstein relation. In one dimension of space, nonlinearities may modify the corrections to ballistic behaviours by superdiffusive effects. But in systems where no shocks appear, such as linearly degenerate and integrable systems, it turns out that the diffusive scaling stays intact. Nevertheless, anomalies remain. We show that in such systems, the noise covariance is given by a modification of the Kubo formula, where effects of ballistic long-range correlations – quadratic charges – have been projected out. We further show that nonlinearities are tamed by a point-splitting regularisation. We then obtain a well-defined hydrodynamic fluctuation theory in the ballistic scaling of space-time, as a stochastic PDE. It describes the asymptotic expansion in the inverse variation scale of connected correlation functions, self-consistently organised via a cumulant expansion. The resulting anomalous hydrodynamic equation takes into account both long-range correlations and bare diffusion, generalising recent works. Despite these anomalies, two-point functions satisfy an ordinary diffusion equation, with a normal Kubo formula. In integrable systems, we show that hydrodynamic noise, hence bare diffusion, must vanish, as was conjectured recently, and argue that under an appropriate gauge, this is true at all orders. Thus initial-state fluctuations do not affect coarse-grained currents, and the Ballistic Macroscopic Fluctuation Theory give the all-order hydrodynamic theory for integrable models.