The hydrodynamic theory of dynamical correlation functions in the XX chain

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 A.R. Its, A.G. Izergin, V.E. Korepin, N.A. Slavnov (1993) and by X. 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.

v1: 40 pages, 5 figures. v2: 43 pages, one appendix added, typos corrected, references added. v3: 44 pages, moved appendix A in main text, typos corrected, few comments removed/added for clarification, references added

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• Long-time dynamics in quantum spin lattices: ergodicity and hydrodynamic projections at all frequencies and wavelengths
• Entanglement Rényi Entropies from Ballistic Fluctuation Theory: the free fermionic case
• More on the Operator Space Entanglement (OSE): Rényi OSE, revivals, and integrability breaking
• Quantum fluctuating theory for one-dimensional shock waves
• Dynamical correlation functions in the Ising field theory
• Macroscopic Quantum States and Universal Correlations in a Disorder-Order Interface Propagating over a 1D Ground State