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arXiv:2601.05079 · SciPost Physics

Full counting statistics in the sine-Gordon model

Botond C. Nagy, Márton Kormos, G. Takács

Published Sep 10, 2026• 0 citations
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Abstract

Full counting statistics (FCS) is a dynamical generalisation of the free energy, encapsulating detailed information about the distribution and large-scale correlation functions of conserved charges and their associated currents. In this work, we present a comprehensive numerical study of the FCS and the cumulants of the three lowest charges across the full parameter space of the sine–Gordon field theory. To this end, we extend the thermodynamic Bethe Ansatz (TBA) formulation of the FCS to the sine–Gordon model, emphasise the methodological subtleties for a reliable numerical implementation, and compare numerical results with analytical predictions in certain limits.

DOI
10.21468/scipostphys.21.3.061

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Unless constrained by symmetry, measurement of an observable on an ensemble of identical quantum systems returns a distribution of values which are encoded in the full counting statistics. While the mean value of this distribution is important for determining certain properties of a system, the full distribution can also exhibit universal behavior. In this paper we study the full counting statistics of particle number in one-dimensional interacting Bose and Fermi gases which have been quenched far from equilibrium. In particular, we consider the time evolution of the Lieb-Liniger and Gaudin-Yang models quenched from a Bose-Einstein condensate initial state and calculate the full counting statistics of the particle number within a subsystem. We show that the scaled cumulants of the charge in the initial state and at long times are simply related and, in particular, the latter are independent of the model parameters. Using the quasiparticle picture we obtain the full time evolution of the cumulants and find that although their end points are fixed, the finite-time dynamics depends strongly on the model parameters. We go on to construct the scaled cumulant generating functions and from this determine the limiting charge probability distributions at long time which are shown to exhibit distinct, nontrivial, and non-Gaussian fluctuations and large deviations. Published by the American Physical Society 2024

arXiv:2312.02929•Physical Review A•Apr 1, 2024•16 citations
Full counting statistics (FCS) is a dynamical generalisation of the free energy, encapsulating detailed information about the distribution and large-scale correlation functions of conserved charges and their associated currents. In this work, we present a comprehensive numerical study of the FCS and the cumulants of the three lowest charges across the full parameter space of the sine–Gordon field theory. To this end, we extend the thermodynamic Bethe Ansatz (TBA) formulation of the FCS to the sine–Gordon model, emphasise the methodological subtleties for a reliable numerical implementation, and compare numerical results with analytical predictions in certain limits.
Abstract We study the counting statistics of charge transport in the Anderson impurity model (AIM) employing both Keldysh perturbation theory in a Fermi liquid picture and the Bethe ansatz. In the Fermi liquid approach, the object of our principal interest is the generating function for the cumulants of the charge current distribution. We derive an exact analytic formula relating the full counting statistic (FCS) generating function to the self‐energy of the system in the presence of a measuring field. We first check that our approach reproduces correctly known results in simple limits, like the FCS of the resonant level system (AIM without Coulomb interaction). We then proceed to study the FCS for the AIM perturbatively in the Coulomb interaction. By comparing this perturbative analysis with a strong coupling expansion, we arrive at a conjecture for an expression for the FCS generating function at 𝒪( V 3 ) ( V is the voltage across the impurity) valid at all orders in the interaction. In the second part of the article, we examine a Bethe ansatz analysis of the current noise for the AIM. Unlike the Fermi liquid approach, here the goal is to obtain qualitative, not quantitative, results for a wider range of voltages both in and out of a magnetic field. Particularly notable are finite field results showing a double peaked structure in the current noise for voltages satisfying eV ∼ μ B H . This double peaked structure is the “smoking gun” of Kondo physics in the current noise and is directly analogous to the single peak structure predicted in the differential conductance of the AIM.
We analyze the full counting statistics (FCS) of the charge transport through the Anderson impurity model (AIM) and similar systems with a single conducting channel. The object of principal interest is the generating function for the cumulants of charge current distribution. We derive an exact analytic formula relating the FCS generating function to the self energy of the system in the presence of the measuring field. We first check that our approach reproduces correctly known results in simple limits, such as the FCS of the resonant level system (AIM without Coulomb interaction). We then proceed to study the FCS for the AIM both perturbatively in the Coulomb interaction and in the Kondo regime at the Toulouse point (we also study a related model of a spinless single-site quantum dot coupled to two half-infinite metallic leads in the Luttinger liquid phase at a special interaction strength). At zero temperature the FCS turns out to be binomial for small voltages. For the generic case of arbitrary energy scales the FCS is shown to be captured very well by generalizations of the Levitov-Lesovik type formula. Surprisingly, the FCS for the AIM indicates a presence of coherent electron pair tunnelling in addition to conventional single-particle processes. By means of perturbative expansions around the Toulouse point we succeeded in showing the universality of the binomial FCS at zero temperature in linear response. Based on our general formula for the FCS we then argue for a more general binomial theorem stating that the linear response zero-temperature FCS for any interacting single-channel setup is always binomial.
Full-counting statistics (FCS) provides a systematic framework for characterising current fluctuations in quantum transport, quantum optics, and open quantum systems. We introduce QuantumFCSjl, an open-source Julia package for efficient and flexible numerical FCS calculations. The package defines currents through monitored jump operators and weights, allowing particle, electric, and heat currents to be treated within the same workflow. It implements a recursive cumulant algorithm with dense, sparse, and iterative solver backends, making it suitable for models that are challenging for direct dense methods. We demonstrate the package on photon counting in a driven-dissipative Jaynes-Cummings system near the blockade-breakdown phase transition and on heat-current fluctuations in a non-linear circuit-QED heat engine. These examples show how higher cumulants reveal intermittency between bright and dim emission, antibunching, and provide probes of thermodynamic uncertainty relations. Benchmarks against an existing FCS implementation show substantial speed-ups.
We set up a hydrodynamic description of the non-equilibrium dynamics of sine-Gordon quantum field theory for generic coupling. It is built upon an explicit form of the Bethe Ansatz description of general thermodynamic states, with the structure of the resulting coupled integral equations encoded in terms of graphical diagrams. The resulting framework is applied to derive results for the Drude weights of charge and energy. Quantities associated with the charge universally have fractal dependence on the coupling, which is notably absent from those associated with energy, a feature explained by the different roles played by reflective scattering in transporting these quantities. We then present far-from-equilibrium results, including explicit time evolution starting from bipartite initial states and dynamical correlation functions. Our framework can be applied to explore numerous other aspects of non-equilibrium dynamics, which opens the way to a wide array of theoretical studies and potential novel experimental predictions.
This thesis investigates the full-counting statistics (FCS) of charge and spin transport using nonequilibrium Green's function (NEGF) in non-interacting systems. \nGenerating function (GF) encodes all the distribution information and is the key of FCS. For the charge statistics, GF is expressed with respect to the modified evolution operator on the complex contour using the two time measurement scheme, the modified evolution operator is expressed in terms of the modified Hamiltonian. Using Keldysh NEGF technique, GF has the form of Fredholm determinant in terms of Greens functions and self-energies in the time domain. In the long time limit, the celebrated Levitov-Lesovik's formula and the fluctuation relation are derived from this formalism. GF of charge statistics for the quantum point contact system in the transient regime is obtained as well. The transient dynamics in lead-dot-lead transport system, including the cumulants of transferred charges and waiting time distribution defined under the transient regime for different temperatures is investigated numerically. A detailed description on how to calculate the GF in the time domain is presented. Generalizations of the formalism to the ferromagnet-normal-ferromagnet system to investigate the FCS of charge current, spin current and spin transfer torque (STT) are presented. GFs in time domain are Fredholm determinants expressed by NEGF as well. As an application of FCS, a formalism using FCS of STT in the long time limit to calculate the switching probability of a nanomagnet system is proposed. The formalism enables us to calculate the switching probability for nonequilibrium systems which are non-Gaussian. From the stochastic Landau-Lifshitz-Gilbert (LLG) equation, the contributions to the change of the anisotropic energy, one is from the power gain due to Gilbert damping and the other is from power dissipation due to the spin transfer torque, were derived. Optimal path approximation which requires the nanomagnet a modestly large volume is presented and the approximation greatly reduces the numerical complexities. This formalism could be used to predict the switching probability numerically for real systems in combination with the first principle calculation.
Unless constrained by symmetry, measurement of an observable on an ensemble of identical quantum systems returns a distribution of values which are encoded in the full counting statistics. While the mean value of this distribution is important for determining certain properties of a system, the full distribution can also exhibit universal behavior. In this paper we study the full counting statistics of particle number in one-dimensional interacting Bose and Fermi gases which have been quenched far from equilibrium. In particular, we consider the time evolution of the Lieb-Liniger and Gaudin-Yang models quenched from a Bose-Einstein condensate initial state and calculate the full counting statistics of the particle number within a subsystem. We show that the scaled cumulants of the charge in the initial state and at long times are simply related and, in particular, the latter are independent of the model parameters. Using the quasiparticle picture we obtain the full time evolution of the cumulants and find that although their end points are fixed, the finite-time dynamics depends strongly on the model parameters. We go on to construct the scaled cumulant generating functions and from this determine the limiting charge probability distributions at long time which are shown to exhibit distinct, nontrivial, and non-Gaussian fluctuations and large deviations. Published by the American Physical Society 2024