Thermodynamics of the Double-Layer Quantum Hall Systems

In this paper we apply the exact solution of the sine-Gordon model to describe thermodynamic properties of the soliton liquid in the incommensurate phase of the double-layer quantum Hall systems. In this way we include thermal fluctuations and extend to finite temperatures the results obtained by C.B. Hanna, A.H. MacDonald and S.M. Girvin [Phys. Rev. B {\bf 63}, 125305 (2001)]. In addition we calculate the specific heat of the system. While the results obtained for the sine-Gordon model are available in a temperature interval $(0,T_c)$, where $T_c=8\pi \rho_s$, $\rho_s$ the pseudospin stiffness, they can be applied in the bilayer system up to temperatures $3T_{\rm BKT}$, where $T_{\rm BKT}=\pi\rho_s/2$ is the vortex mediated Berezinskii-Kosterlitz-Thouless transition temperature. Above this temperature the operators $\cos\beta \phi$ and $\cos(2\pi \vartheta/\beta)$ are both relevant and the system is in a phase with coexisting order parameters. $\vartheta$ is the dual field of $\phi$ and $\beta$ is the sine-Gordon coupling constant. We provide numerical estimates for thermodynamic quantities for the range of parameters relevant for GaAs.