Clausius-Clapeyron relations for first-order phase transitions in bilayer quantum Hall systems

A bilayer system of two-dimensional electron gases in a perpendicular magnetic field exhibits rich phenomena. At total filling factor ${\nu }_{tot}=1$, as one increases the layer separation, the bilayer system goes from an interlayer-coherent exciton condensed state to an incoherent phase of, most likely, two decoupled composite-fermion Fermi liquids. Many questions still remain as to the nature of the transition between these two phases. Recent experiments have demonstrated that spin plays an important role in this transition. Assuming that there is a direct first-order transition between the spin-polarized interlayer-coherent quantum Hall state and spin partially polarized composite Fermi-liquid state, we calculate the phase boundary ${\left(d/l\right)}_c$ as a function of parallel magnetic field, NMR/heat pulse, temperature, and density imbalance, and compare with experimental results. Remarkably good agreement is found between theory and various experiments.

Figure 1

Total electron density deduced from the critical $d/l={\delta }_c$ vs the total magnetic field for the parallel magnetic field experiments. Open and solid circles are experimental results of Giudici et al. (Ref. 30) [cf. Fig. 4a there]. Solid line is our theoretical calculation with the fitting parameter $\eta =0.8\times {10}^{-3}$. The boundary condition in our calculation is chosen as $n_{tot}=11\times {10}^{10}{\text{cm}}^{-2}$ when $B=10\text{T}$.

Figure 2

The phase boundary $d/l$ vs the temperature $T$ (in kelvin) for the finite-temperature experiments. Circles are experimental results of Champagne et al. (Ref. 31) [cf. Fig. 2c there]. Solid line is our theoretical calculation with the fitting parameter $\eta =0.7\times {10}^{-3}$. The boundary condition in our calculation is chosen as ${\delta }_c=1.83$ when $T=50\text{mK}$.

Figure 3

Summary of the value of $\eta$ extracted from various experiments. Experiment 1: parallel field experiment of Ref. 30. Experiment 2: NMR/heat-pulse experiment of Ref. 26. Experiment 3: finite-temperature transition experiment of Ref. 31. Experiments 4–7: density imbalance experiments of Ref. 32 with $T=55$, 85, 125, and 200 mK. Experiments 8 and 9: density imbalance experiments of Ref. 24 with phase boundary determined by Hall drag and tunneling. To obtain this result we used the numerical result of Ref. 68 for unpolarized composite Fermi-liquid ground-state energy to estimate ${\kappa }^{-1}$. The horizontal line is the average value of $\eta$ weighted by inverse of error square, which is $\sim \left(1\pm 0.1\right)\times {10}^{-3}$.

Figure 4

(Color online) Standard deviation of $\eta$ among various experiments divided by their average value (which measures the goodness of agreement between $\eta$’s) within the Chern-Simons framework as a function of the composite-fermion mass and the exchange contribution to ${\kappa }^{-1}$ [see Eq. (52)]. Horizontal axis: composite-fermion mass in units of $m_e\sqrt{B_{\perp }}$, $m_e$ being the vacuum electron mass, $B_{\perp }$ is in units of tesla. Vertical axis: exchange contribution to ${\kappa }^{-1}$, which is $F_0^s\pi /m_{\ast }$ (in units of $e^{2}l/ϵ$, $l$ being the magnetic length). Gray color denotes the region where at least one of the $\eta$’s becomes negative, thus unphysical. The horizontal line denotes the Hubbard approximation to the exchange effect $\left(-\sqrt{2}\pi \right)$. The vertical line denotes the experimental value of the activation mass $m_{\ast }\approx 0.2m_e\sqrt{B_{\perp }}$, which is the value of composite-fermion mass we used in calculations for Figs. 2, 3.