Fluctuations in Ballistic Transport from Euler Hydrodynamics

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. AppendicesA Derivation of the main result (3.6) with (3.2)

Cumulants of conserved charges in GGE and cumulants of total transport in GHD: exact summation of matrix elements?

“…We confirm our conjecture by a non-trivial matching with the result of [23] up to the fourth cumulant of the same current. We first remind how this quantity was derived in [14] and [3,23].…”

“…We conjecture that they are given by the same diagrams presented in the previous section, with only two modifications: the operator at the root is the energy derivative E ′ (instead of the momentum derivative) and each internal vertex θ of odd degree carries an extra sign of the effective velocity sign[v eff (θ)]. We confirm our conjecture by a non-trivial matching with the result of [23] up to the fourth cumulant of the same current. We first remind how this quantity was derived in [14] and [3,23].…”

Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains

“…We believe that our approach provides useful insights and essential prerequisites for the solution of the interacting case, and at the very least can serve for benchmarking and comparison. We remark that the Levitov-Lesovik type of asymptotic was already implemented in the generalized hydrodynamic approach to interacting systems [24]. Another interesting question our research can be related to is the exploration of dynamical defects (quantum impurities) and moving defects [84][85][86][87].…”

“…We see that only the last terms contributes to the current, resulting in Eqs. (24), (25). For zero temperature J hydro and J bound from Eq.…”

“…Remarkably, it was demonstrated that the leading contribution of the long-time behavior of FCS is universal and can be described by the Levitov-Lesovik formula [21][22][23]. From the generalized hydrodynamics point of view, this formula corresponds to the ballistic transport and can be obtained from the large deviation theory for the statistics of non-equilibrium currents [24].…”