Thermodynamics and fractal Drude weights in the sine-Gordon model

The sine-Gordon model is a paradigmatic quantum field theory that provides the low-energy effective description of many gapped one-dimensional systems. Despite this fact, its complete thermodynamic description in all its regimes has been lacking. Here, we fill this gap and derive the framework that captures its thermodynamics and serves as the basis of its hydrodynamic description. As a first application, we compute the Drude weight characterizing the ballistic transport of topological charge and demonstrate that its dependence on the value of the coupling shows a fractal structure, similar to the gapless phase of the XXZ spin chain. The thermodynamic framework can be applied to study other features of nonequilibrium dynamics in the sine-Gordon model using generalized hydrodynamics, opening the way to a wide array of theoretical studies and potential novel experimental predictions.

Figure 1

Drude weight in the sine-Gordon model calculated from Eq. (7) as a function of the coupling strength ${\beta }^2/8\pi$ for (a) high, (b) moderate, and (c) low temperatures, at values of $\xi$ with at most two magnonic levels in the TBA system (3). The Drude weight was computed at discrete points which are joined by a red line in the plot to emphasize their discontinuous nature. At high temperatures, the fractal structure is washed out, while it gets more pronounced as the temperature is lower. We verified that the results from the bipartition protocol Eq. (8) agree with the TBA results within $0.1\%$ relative difference. The dashed line in (a) is the high-$T$ limit Eq. (9), while in (c) the dotted blue line is the low-$T$ prediction for the reflectionless points (11), and the black dashed line shows (10). The vertical grid lines in (b) and (c) indicate the reflectionless points. Note also the logarithmic scale in (c).

Figure 2

Diagram encoding the TBA kernels $K_{ij}\left(\theta \right)$ in terms of their Fourier transforms ${\tilde{K}}_{ab}\left(t\right)=\int \mathrm{d}\theta /2\pi K_{ab}\left(\theta \right)e^{-i\theta t}$ for the generic case with two magnonic levels (for the exceptions cf. Ref. [48]). The diagram is similar to the one presented in Ref. [50] for the boundary sine-Gordon model. The kernel parameters are $p_0={\nu }_1+1/{\nu }_2, p_1=1, p_2=1/{\nu }_2$.

Figure 3

A close-up view of the fractal structure of Drude weights from Fig. 1 (b) ($T=M/2$) with separate markings for different numbers of magnon levels. Green diamonds: reflectionless couplings ($N=0$, no magnons), blue squares: $N=1$ magnonic level, red triangles: $N=2$ magnonic levels.