Full counting statistics in the free Dirac theory

Abstract

We study charge transport and fluctuations of the (3+1)-dimensional massive free Dirac theory. In particular, we focus on the steady state that emerges following a local quench whereby two independently thermalized halves of the system are connected and let to evolve unitarily for a long time. Based on the two-time von Neumann measurement statistics and exact computations, the scaled cumulant generating function associated with the charge transport is derived. We find that it can be written as a generalization of Levitov-Lesovik formula to the case in three spatial dimensions. In the massless case, we note that only the first four scaled cumulants are nonzero. Our results provide also a direct confirmation for the validity of the extended fluctuation relation in higher dimensions. An application of our approach to Lifshitz fermions is also briefly discussed. arXiv:1802.01154v2 [cond-mat.stat-mech]

Fluctuations in Ballistic Transport from Euler Hydrodynamics

“…This appears to be physically sensible, as without interactions, initial-state fluctuations are not affected during transport. The statement of the extended fluctuation relations [21] was obtained by extracting principles found in [23,24] in the context of energy and charge transport in 1+1dimensional CFT, and was argued to hold also in free particle models, later confirmed by various explicit calculations [16,17,19]. This shows that the present formalism fully agrees with these results, and that, effectively, it generalises the method to nonlinear Euler hydrodynamics.…”

Transport fluctuations in integrable models out of equilibrium

“…The results have been obtained by combining recent developments in integrable systems in the context of generalised hydrodynamics [21,22], in particular the flux Jacobian obtained in [32], with a new ballistic fluctuation formalism developed in [92]. They significantly generalise previous expressions and studies in free particle models [50][51][52][53][54][55] and in one-dimensional conformal field theory [57][58][59]. To our knowledge, these are the first exact results for transport SCGFs in interacting homogeneous integrable systems, and provide an entirely new application of the hydrodynamic theory of integrable systems.…”

“…In an ensemble formulation, fluctuations originate from those in the initial state. Despite many efforts, only a few results exist: free-fermions with the celebrated Lesovik-Levitov formula [50][51][52], harmonic chains [53] and free field theory [54,55], particular integrable impurity models [56], and one-dimensional critical systems [57,58]; see the review [59]. Some results also exist for fluctuation statistics of other quantities, not related to transport, in certain integrable models, see for instance [60][61][62][63].…”

Euler-scale dynamical fluctuations in non-equilibrium interacting integrable systems

“…The second cumulant is related to the Gaussianity of the distribution close to the mean value and, from Eqs. (43) and (63), its expression reads…”

“…The large deviation function can be computed from the associated scaled cumulant generating function (SCGF) [49], which is the non-equilibrium analogue of the free energy. In the context of isolated systems, until recently only few results for transport fluctuations were present, mostly regarding one-dimensional critical systems [52][53][54] and non-interacting models, such as the Levitov-Lesovik formula for free fermions [55][56][57][58][59][60][61][62][63][64], the free Klein-Gordon field theory [65] and harmonic chains [66]. Only recently, results became available in interacting integrable systems for the statistics of various observables not related to transport, see e.g., Refs.…”