Full counting statistics in the sine--Gordon model

TL;DR

The paper develops and applies a full thermodynamic Bethe Ansatz framework for full counting statistics in the sine--Gordon model, enabling the calculation of cumulants for energy, momentum, and topological charge across temperatures, couplings, and space-time directions. By combining the Bethe Ansatz with Ballistic Fluctuation Theory, it derives flow equations for pseudo-energies and explicit expressions for the cumulants, validating them in analytically tractable limits and via extensive numerics. A key finding is the fractal-like dependence of topological-charge cumulants on the coupling at low temperatures and small ray angles, contrasting with the smooth behavior of energy and momentum cumulants, which align with high-temperature conformal predictions. The results pave the way for experimental tests in cold-atom setups and provide a detailed benchmark for integrable transport in non-quadratic quantum field theories.

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.

Figures (14)

• Figure 1: Variance $c_2^q$ as a function of the coupling at $T=0.1$, $\alpha=0$ (left) and as a function of $\alpha$ at $T=0.2$, $\xi=1/3$ (right). The black dashed lines are the second cumulant derived from Eq.(\ref{['eq:lowT']}). In the left panel, the colours refer to the number of magnonic levels.
• Figure 2: Variance $c_2^q$ for $\alpha=0$, $\xi=3$ as a function of temperature. Note that the $y$ scale is logarithmic and rather wide, so Eq.(\ref{['eq:low_T_repulsive_approx']}) is only a good approximation at low and medium temperatures. See also Table \ref{['table:low_T_repulsive_approx']}.
• Figure 3: Charge cumulants as a function of $\alpha$, probing Eqs.(\ref{['eq:c2q_low_T_nonzero_alpha']},\ref{['eq:c4q_low_T_nonzero_alpha']},\ref{['eq:c2q_low_T_big_alpha']}).
• Figure 4: High temperature limit of $c_2^q$ as a function of the coupling, $\alpha$," and the temperature.
• Figure 5: Variance $c_2^q$ for $T=0.5$ at different ray angles $\alpha$. The colours refer to the number of magnonic levels. For small angles, this quantity exhibits fractal behaviour, as shown in subfigures (a) and (b).